Distributed decomposition of string-automated reasoning using predicates

Abstract

A computer-implemented procedure that includes the following: Receiving a request by a policy analysis service (146) of a cloud provider network (100) to determine whether a formula is satisfyable, wherein the formula relates to an analysis of one or more policies (126) applicable to one or more compute resources (314A-314N) connected to a user of the cloud provider network (100); Identifying (604) a set of predicates (306) in the formula on the basis of string variables contained in the formula, wherein each predicate of the set of predicates is a binary-valued function of at least one string variable contained in the formula; Partitioning (606) a search space associated with the formula into a plurality of subformulas, each subformula of the plurality of subformulas containing the formula under the assumption that a corresponding predicate of the set of predicates (306) is either true or false; Sending (608) a respective subformula from the multitude of subformulas to a Satisfiability Modulo-Theories (SMT) solver (152) running on each of a multitude of computing resources (314A-314N); Receiving (610) a notification from the SMT solver (152) running on a computing resource of the multitude of computing resources (314A-314N) that the respective subformula being analyzed by the SMT solver (152) running on the computing resource is satisfiable; Causing the display of information indicating that the formula is satiable, wherein the satiability of the formula determines a property of the one or more policies (126) that define permissions applicable to the one or more compute resources (314A-314N) connected to the user of the cloud provider network (100); Controlling access to the one or more computing resources (314A-314N) according to the one or more policies (126) that define permissions applicable to the one or more computing resources assigned to the user of the cloud provider network (100), and Grouping the set of predicates (306) into a plurality of predicate groups based on an estimated computational complexity associated with each predicate of the set of predicates (306), grouping predicates of lower complexity of the set of predicates together and predicates of higher complexity of the set of predicates separately, and partitioning the search space based on the plurality of predicate groups.

Description

STATE OF THE ART

[0001] Cloud provider networks enable users to use a variety of compute-related resources, such as compute resources, storage resources, network resources, and the like. When a user or application interacts with a cloud provider network (e.g., using an application programming interface (API) or a command-line interface (CLI) provided by the cloud provider network), the user or application typically needs to set security credentials to indicate who the user or application is and whether the user or application is authorized to access the requested resources. A cloud provider network, in turn, uses the security credentials to authenticate the user or application and authorize them to perform various actions.Access to resources and actions within a cloud provider network can be further managed through policies. A policy is a data object that, when associated with a user or resource, defines their permissions. For example, resource-based policies can be attached to a storage resource, a compute instance, encryption keys, etc., and specify who has access to the resource and what actions those identities can perform on the resource. US 10 922 423 B1 describes a security policy analysis service from a computing resource service provider that performs assessments of the security policies provided by the service provider's users to determine whether the security policies are valid, satisfiable, accurate, and / or sufficiently secure.US 2020 / 0 073 739 A1 describes a constraint solver service from a computing resource service provider that performs evaluations of logic problems provided by the service provider's users and / or services by using a plurality of constraint solvers to evaluate the logic problem simultaneously. The document by HYVARINEN, Antti EJ; MARESCOTTI, Matteo; SHARYGINA, Natasha, Search-space partitioning for parallelizing SMT solvers, in: Theory and Applications of Satisfiability Testing-SAT 2015: 18th International Conference, Austin, TX, USA, September 24-27, 2015, Proceedings 18. Springer International Publishing, 2015. pp. 369-386 describes how parallel computing can be used to reduce the time required to solve instances of the Satisfiability Modulo Theories Problem (SMT).It is an object of the invention to propose computer-implemented methods and a system for distributed decomposition of string-automated reasoning using predicates. This object is achieved by an inventive method according to claim 1, an inventive method according to claim 4, and a system according to claim 16. BRIEF DESCRIPTION OF THE DRAWINGS

[0002] Various embodiments according to the present disclosure will be described with reference to the drawings, in which: Fig. 1 is a diagram illustrating an environment including a policy analysis service used to analyze policies associated with user accounts and organizations of user accounts of a cloud provider network, according to some embodiments. Fig. 2 is a diagram illustrating a process for encoding policy-defined permissions into propositional logic statements including string variables, according to some embodiments. Fig. 3 is a diagram illustrating a process for partitioning a search space associated with a formula expressed in propositional logic and involving string variables into a set of subformulas whose execution can be distributed across multiple computational resources, according to some embodiments. Fig. 4 is a diagram illustrating the use of a satisfiability solver executing across multiple computational resources to analyze a set of subformulas derived from a formula expressed in propositional logic and involving string variables, according to some embodiments. Fig. 5 is a diagram illustrating a graphical user interface (GUI) that displays policy results derived from an analysis of one or more policies, according to some embodiments. Fig. 6 is a flow diagram illustrating operations of a method for using reasoning techniques to analyze formulas expressed in propositional logic and including string variables, according to some embodiments. Fig. 7 illustrates an example provider network environment according to some embodiments. Fig. 8 is a block diagram of an example provider network that provides a storage service and a hardware virtualization service to customers, according to some embodiments. Fig. 9 is a block diagram illustrating an example computer system that may be used in some embodiments. DETAILED DESCRIPTION

[0003] The present disclosure relates to methods, apparatus, systems, and non-transitory computer-readable storage media for efficiently distributing the processing of satisfiability modulo theories (SMT) queries expressed in first-order logic and involving theory variables (e.g., variables related to the theory of strings, the theory of integers, the theories of data structures, etc.) across multiple compute resources. According to some embodiments, many cloud providers also provide, as part of the compute-related services provided by a cloud provider network, identity and access management services that generally assist users in controlling access to and permissions for the services and resources (e.g., compute instances, storage resources, etc.) acquired by users across a cloud provider network.For example, using identity-based and resource-based policies, users can precisely control which identities can access specific resources associated with user accounts and how those identities can use those resources. However, configuring such policies can often become very complex, and it can quickly become challenging for users to understand all the security implications of such policies and their interrelationships.

[0004] To address some of these concerns, a cloud provider network may provide various analysis tools to help users analyze the security-related properties of the resources and associated policies in their accounts. An example of such a security tool is an access analyzer, which can be used to help users understand which identities can access specific resources associated with their account and, for example, to help users determine whether their current policy configurations might provide unintended access to users outside their organization. In this example, an access analyzer service can perform such analyses in part by translating a user's or organization's stored policies into equivalent logical statements (e.g.,Propositional logic (statements expressed in first-order logic) and a number of general and specialized logical solvers (e.g., SMT solvers) are used to verify whether certain security-related behaviors are possible or not. In this context, the logical solvers can make inferences about propositional logic statements involving various string variables, e.g., corresponding to aspects of policies such as account identifiers, resource identifiers, and the like. In addition to identity and access management services, some cloud provider networks also provide source code review and optimization services, program or computer network verification services, and other types of services that use automated inference to assist in analyzing the correctness of various types of computing systems.

[0005] The SMT solvers described above generally attempt to confirm or falsify formulas expressed in first-order logic using combinations of theories such as Presburger arithmetic, uninterpreted functions, or strings. Existing SMT solvers are generally monolithic, single-process applications, and there is no successful method for efficiently distributing the search for confirmations or refutations of such formulas across multiple computational resources. As the size and complexity of user and organizational policies and computational resources increase, the resources required to reason about these resources and provide timely information using existing solvers can quickly exceed the resources available on individual computational resources.

[0006] These challenges are addressed, among other things, by the techniques described herein for efficiently distributing the analysis of SMT queries expressed in first-order logic and involving theory variables across any number of separate computational resources (e.g., between separate processes, compute instances, containers, etc.). For example, according to embodiments described herein, a cloud provider network service receives a request to determine whether a formula is satisfiable (e.g., to verify an expected behavior of a user's or organization's policy set, or other such automated reasoning-based analysis). The service identifies a set of predicates in the formula based on a type of theory associated with the formula, where each predicate is a binary-valued function of at least one theory variable included in the formula.In some embodiments, a search space associated with the formula is then partitioned into a set of subformulas, each subformula defined by a unit of the formula with an assumption that a respective predicate of the set of predicates is either true or false. In some embodiments, a respective subformula of the set of subformulas is sent to an SMT solver executing on each of a plurality of separate computational resources. Upon receiving an indication from the SMT solver executing one of the computational resources that its respective subformula is satisfiable, the policy analysis service may cause information indicating that the formula is satisfiable to be displayed; otherwise, the policy analysis service may cause information indicating that the formula is unsatisfiable to be displayed or otherwise communicated.Among other advantages, the described analysis techniques enable efficient computation of SMT queries expressed in first-order logic and involving theory variables, thereby also contributing to improving the security posture of organizations' computing resources provided by cloud provider networks and other operating environments.

[0007] Fig. 1 is a diagram illustrating an environment in which, according to some embodiments, security policy analysis is performed with respect to user accounts, organizations (e.g., including sets of users, roles, and policies), and policies defined by an identity and access management service 102 of a cloud provider network 100. A cloud provider network 100 (sometimes simply referred to as a "cloud") refers to a pool of network-accessible computing resources (such as compute, storage, and network resources, applications, and services), which may be virtualized or a bare-metal system. The cloud may provide convenient, on-demand network access to a shared pool of configurable computing resources that may be programmatically provisioned and released in response to customer commands.These resources can be dynamically provisioned and reconfigured to adapt to changing loads. Cloud computing can therefore be understood as both the applications delivered as services over a publicly accessible network (e.g., the internet, a mobile network) and the hardware and software in the data centers of cloud providers that provide these services.

[0008] A provider network 100 (or "cloud" provider network) provides users with the ability to utilize one or more of a variety of computation-related resources 160, such as compute resources (e.g., running virtual machine (VM) instances or containers, running batch jobs, running code without requiring servers), data / storage resources (e.g., object storage, block-level storage, data archiving storage, databases and database tables, etc.), network-related resources (e.g., configuring virtual networks including groups of compute resources, content delivery networks (CDNs), domain name service (DNS)), application resources (e.g., databases, application build / deployment services), access policies or roles, identity policies or roles, machine images, routers, and other computing resources, etc.These and other computing resources may be provided as services 106, such as a hardware virtualization service 118 that can run compute instances, a storage service 110 that can store data objects, etc. Users (or "customers") of provider networks 100 may utilize one or more user accounts associated with a customer account, although these terms may be used interchangeably to some extent depending on the context of use. Users may interact with a provider network 100 through one or more intermediate networks 104 (e.g., the Internet) via one or more interfaces, such as through the use of application programming interface (API) calls, through a console implemented as a website or application, etc.An API refers to an interface and / or communication protocol between a client and a server such that, when the client submits a request in a predefined format, the client should receive a response in a specific format or initiate a defined action. In the context of the cloud provider network, APIs provide a gateway for customers to access the cloud infrastructure by allowing customers to obtain data from the cloud provider network or initiate actions within the cloud provider network, enabling the development of applications that interact with resources and services hosted in the cloud provider network. APIs can also enable different services of the cloud provider network to exchange data with each other.The interface(s) may be part of, or serve as a frontend to, a control plane of the provider network 100, which includes “backend” services that support and enable the services that can be offered more directly to customers.

[0009] A cloud provider network 100 may be formed from a number of regions, where a region is a geographic area in which the cloud provider groups data centers. Each region includes multiple (e.g., two or more) availability zones (AZs) connected by a private high-speed network, such as a fiber optic communications link. An AZ (also known as an availability domain or simply a "zone") provides an isolated failure domain that includes one or more data center facilities with separate power, networking, and cooling from those in another AZ. A data center refers to a physical building or enclosure that houses and supplies power and cooling to servers of the cloud provider network.Preferably, AZs within a region are far enough apart that a natural disaster (or other outage-causing event) should not affect or take offline more than one AZ at a time.

[0010] Customers can connect to AZs of the cloud provider network 100 via a publicly accessible network (e.g., the internet, a cellular network), e.g., via a transit center (TC). TCs are the primary backbone locations that link customers to the cloud provider network and can be hosted at other network providers (e.g., internet service providers (ISPs), telecommunications providers) and securely connected (e.g., via a VPN or a direct connection) to the AZs. Each region can operate two or more TCs for redundancy. Regions are connected to a global network that includes private network infrastructure (e.g., fiber optic links controlled by the cloud provider) that connects each region to at least one other region.The cloud provider network can deliver content from the nearest points of presence (POPs) outside these regions, but interconnected to them, via edge locations and regional edge cache servers. This compartmentalization and geographical distribution of computing hardware enables the cloud provider network to provide customers with low-latency resource access on a global scale with a high degree of fault tolerance and stability.

[0011] In general, the traffic and operations of a provider network can be roughly divided into two categories: control plane operations transmitted over a logical control plane, and data plane operations transmitted over a logical data plane. While the data plane represents the movement of user data through the distributed computing system, the control plane represents the movement of control signals through the distributed computing system. The control plane generally includes one or more control plane components distributed across and implemented by one or more control servers. Control plane traffic generally includes administrative operations such as system configuration and management (e.g., resource placement, hardware capacity management, diagnostic monitoring, system status information). The data plane includes customer resources implemented in the provider network (e.g.,Data plane traffic generally involves non-administrative operations, such as transferring customer data to and from customer resources. The control plane components are typically implemented on a separate set of servers from the data plane servers, and control plane and data plane traffic may be sent over separate / different networks.

[0012] To provide these and other computing resource services, provider networks 100 often rely on virtualization techniques. For example, virtualization technologies can be used to provide users with the ability to control or utilize computing resources (e.g., a "compute instance" such as a VM using a guest operating system (O / S) operating using a hypervisor that may run on top of an underlying host operating system, a container that may run in a VM, a compute instance that may run on "bare metal" hardware without an underlying hypervisor), where one or more computing resources may be implemented using a single electronic device. Thus, a user can directly access a computing resource hosted by the provider network (e.g.,provided by a hardware virtualization service) to perform a variety of computing tasks. Additionally or alternatively, a user may indirectly utilize a computing resource by submitting code to be executed by the provider's network (e.g., via an on-demand code execution service), which in turn utilizes one or more computing resources to execute the code—typically without the user having control or knowledge of the underlying computing instances involved.

[0013] For example, in various embodiments, a "serverless" function may include code provided by a user or other entity—such as the provider network itself—that can be executed on demand. Serverless functions may be maintained within the provider network 100 through an on-demand code execution service and may be associated with a specific user or account or be generally accessible to multiple users / accounts. A serverless function may be associated with a Uniform Resource Locator (URL), a Uniform Resource Identifier (URI), or other reference that can be used to invoke the serverless function. A serverless function may be executed by a compute resource, such as a virtual machine, a container, etc., when triggered or invoked.In some embodiments, a serverless function may be invoked via an application programming interface (API) call or a specially formatted HyperText Transport Protocol (HTTP) request message. Accordingly, users can define serverless functions that can be executed on demand, without requiring the user to maintain dedicated infrastructure to run the serverless function. Instead, the serverless functions can be executed on demand using resources maintained by the provider network 100. In some embodiments, these resources can be maintained in a "ready" state (e.g., with a pre-initialized runtime environment configured to execute the serverless functions), allowing the serverless functions to execute in near real time.

[0014] The hardware virtualization service 118 (referred to in various implementations as an elastic compute service, virtual machine service, computing cloud service, compute engine, or cloud compute service) may enable users of the provider network 100 to provision and manage computing resources, such as virtual machine instances. The virtual machine technology may use a physical server to run the equivalent of many servers (each of which is referred to as a virtual machine), for example, using a hypervisor that may run on at least one offload card of the server (e.g., a card connected to the physical CPUs via PCI or PCIe), and other components of the virtualization host may be used for some virtualization management components.Such a host offload card may include one or more CPUs that are not available to customer instances, but are instead specialized for instance management tasks, such as virtual machine management (e.g., hypervisor), input / output virtualization for network-attached storage volumes, local migration management tasks, instance health monitoring, and the like. Virtual machines are commonly referred to as compute instances or simply "instances." As used herein, providing a virtual compute instance generally includes reserving resources (e.g., compute and memory resources) of an underlying physical compute instance for the client (e.g., from a pool of available physical compute instances and other resources), installing or starting required software (e.g.,operating system) and making the virtual computing instance available to the client to perform tasks specified by the client.

[0015] In some embodiments, the provider network 100 includes a container service. The container service may be a container orchestration and management service (referred to in various implementations as a container service, cloud container service, container engine, or container cloud service) that enables users of the cloud provider network to instantiate and manage containers. In some embodiments, the container service may be a Kubernetes-based container orchestration and management service (referred to in various implementations as a container service for Kubernetes, Azure Kubernetes Service, IBM Cloud Kubernetes Service, Kubernetes Engine, or Container Engine for Kubernetes).A container, as referred to herein, bundles code and all its dependencies so that an application (also called a task, pod, or cluster in various container services) can be moved quickly and reliably from one computing environment to another. A container image is a self-contained, executable software package that includes everything required to run an application process: code, runtime, system tools, system libraries, and settings. Container images become containers at runtime. Containers are thus an abstraction of the application layer (that is, each container simulates a different software application process). Although each container runs isolated processes, multiple containers can share a common operating system, for example, by being launched within the same virtual machine.In contrast, virtual machines are an abstraction of the hardware layer (meaning that each virtual machine simulates a physical machine capable of running software). While multiple virtual machines can run on a single physical machine, each virtual machine typically has its own copy of an operating system, applications, and associated files, libraries, and dependencies. Some containers can run on instances running a container agent, and some containers can run on bare-metal servers or on a server's offload card.

[0016] In some embodiments, an identity and access management service 102 is a service that enables users to securely control access to resources of the cloud provider network (e.g., resources 160 associated with various provider network services 106, such as storage objects 108 associated with a storage service 110, databases 112 associated with a database service 114, compute instances 116 associated with a hardware virtualization service 118, and the like). The identity and access management service 102 is used to control who is allowed to authenticate (e.g., log in) to the cloud provider network 100 and who is authorized (e.g., has permissions) to use the resources provided by the cloud provider network.In general, a resource is a concept used to capture the domain of items that can be created, read, modified, or deleted by customers in a cloud provider network 100. Examples of resources also include principals (e.g., principals 120, including example users 122A-122N and roles 124A-124N) and policies 126 (e.g., including identity-based policies 128, resource-based policies 130, and other policies 132). Fig. 1 further illustrates the concept of an organization 134, which may include any number of linked accounts 136A-136N, which in turn may include any number of users and roles (e.g., role(s) 138 linked to account 136B and role(s) 140 linked to account 136N).

[0017] In some embodiments, when a person first creates an account with cloud provider network 100, the person begins with a single login identity that has full access to all cloud provider network services and resources associated with the account (e.g., a root user of principals 120). The root user identity may be accessed, for example, by logging in with a username (e.g., an email address) and password used to create the account. Cloud provider networks 100 often advise users not to use a root user for most tasks and instead to create additional user accounts with defined permissions (e.g., including one or more of user accounts 122A-122N). In some embodiments, a user may grant different permissions to different user accounts for different resources.For example, a user account could be configured to allow some users full access to a hardware virtualization service 118, a storage service 110, and other resources of the cloud provider network 100. For other users, a user account might only allow read access to some storage buckets or permission to manage some instances 116, etc.

[0018] In some embodiments, an account includes identity-related objects stored as part of the identity and access management service 102, including, for example, users 122A-122N, groups (not illustrated), roles 124A-124N, policies 126, and the like. These resources can be added, edited, and removed by users of the cloud provider network 100 with sufficient permissions, e.g., using a web-based console, API, CLI, or other interface provided by the identity and access management service 102.

[0019] In some embodiments, a principal 120 represents a person or application that can make a request for an action or operation on a resource of the cloud provider network 100 (e.g., a resource 160 or a resource of the identity and access management service 102). The set of principals 120 associated with an account 136A may include any number of users 122A-122N and roles 124A-124N. A cloud provider network request occurs when a principal (e.g., a user or role) sends a request for an action or operation on a resource. A request may include some or all of the following information: the action or operations the principal wishes to perform, the resource object on which the actions or operations are performed, the person or application that used an entity (e.g., a user or role) to send the request, environmental data (e.g.,Information about the IP address, user agent, SSL activation status, time of day, etc.) and resource data (e.g., data related to the requested resource, such as a resource identifier or tag name). The identity and access management service 102 collects the information contained in a request into a request context, which is used to evaluate and authorize the request.

[0020] In some embodiments, in order for a request to be completed, the identity and access management service 102 determines whether the requesting principal is authorized (e.g., entitled) to complete the request. During authorization, the identity and access management service 102 uses values ​​included in the request context to check for policies that apply to the request (e.g., one or more of policies 126). The identity and access management service 102 uses the policies 126 to determine whether to allow or deny the request. In some embodiments, the policies 126 are stored in the identity and access management service 102 as JavaScript Object Notation (JSON) documents (or using another data format) and specify the authorization statements that apply to principal entities, resources, or combinations thereof.In some embodiments, there are multiple types of policies 126 that can affect whether a given request is authorized, including, for example, identity-based policies 128, resource-based policies 130, and other possible types of policies 132. For example, to provide users with permissions to access resources within their own account, identity-based policies can be configured, while resource-based policies can be used to grant cross-account access to resources. In some embodiments, the identity and access management service 102 reviews each policy that applies to the context of a request. If an individual permission policy includes a denied action, the identity and access management service 102 can deny the entire request.In some embodiments, an identity and access management service 102 rejects requests by default, so that a request is only authorized if each part of a request is permitted by applicable authorization policies.

[0021] In some embodiments, once a request is authenticated and authorized, the identity and access management service 102 validates the actions or operations in the request. Operations are defined by a service and include actions that can be performed on or relative to a resource, such as viewing, creating, editing, and deleting that resource. For example, the identity and access management service 102 may support actions such as creating a user, deleting a user, creating a role, and assuming a role, among many other possible actions. For a principal to perform an operation, the action is included in a policy applicable to the principal or the affected resource.

[0022] In some embodiments, identity-based policies 128 are authorization policies attached to an identity, such as a user, group, or role in an account. In some embodiments, resource-based policies are authorization policies attached to a resource, such as a storage object 108 or a role trust policy. A resource-based policy controls what actions a specific principal can perform on that resource and under what conditions. In some embodiments, the identity and access management service 102 further supports trust policies that can be attached to a role (e.g., one or more of roles 124A-124N).Because a role is both an identity and a resource that supports resource-based policies, in some embodiments, both a trust policy and an identity-based policy are added to a role. Trust policies define which principal entities (accounts, users, roles, and federated users) can assume the role.

[0023] In some embodiments, a role is an identity that a user creates in an account that has specific permissions. A role is similar to a user in that it is an identity with permission policies that determine what the identity can and cannot do. However, a role is not uniquely associated with an individual but is intended to be assumed by anyone who needs it. Also, a role may not have standard long-term credentials associated with it, such as a password or access keys. Instead, when an entity assumes a role, it is provided with temporary security credentials for a role session. Roles can be used to delegate access to users, applications, or services that would not normally have access to the resource.For example, a person may want to give users in one account access to resources that those users do not normally have access to, or give users in one account access to resources in another account.

[0024] As mentioned above, users may often want to be assured that their configured policies are configured to help protect their data and resources. In some embodiments, a policy analysis service 146 uses various types of automated reasoning to perform such analyses and present policy results to users based on the analyses. At a higher level, automated reasoning is a method of formal verification that automatically generates and verifies mathematical proofs that help confirm the correctness of systems (e.g., to analyze policies and the future consequences of policies).As stated above, policies dictate who can (or cannot) perform specific actions with respect to specific resources, and a policy analysis service 146 may use automated reasoning to verify the properties of the policies. Although some of the examples in this document relate to the analysis of policies managed by an identity and access management service 102, similar automated reasoning techniques may be used to analyze the correctness of source code, analyze network configurations, or generally perform any type of analysis related to various types of computing resources or computing systems.

[0025] In some embodiments, to perform such analyses, a policy analysis service 146 translates policies into equivalent logical statements and runs a series of general and specialized logical solvers (e.g., SMT solvers) on the problem. Generally, an SMT solver uses a mixture of numbers, strings, regular expressions, dates, IP addresses, etc., to confirm and refute logical formulas. For example, a policy analysis service 146 may not examine access logs to determine whether an external entity has accessed a resource within a trust zone. Rather, it may generate a result if a resource-based policy allows access to a resource, even if the resource has not yet been accessed by an external entity.In addition, when conducting such analyses, the Service may not take into account the state of external accounts when making its decision.

[0026] In Fig. 1, the numbered circles labeled "1" through "6" illustrate a process by which one or more users configure accounts, principals, policies, etc., via an identity and access management service 102 and a policy analysis service 146 that performs an analysis of one or more of the users' policies, as described above. In particular, the illustrated process includes distributing the processing of one or more SMT queries expressed in propositional logic with string variables across multiple computing resources (e.g., across multiple computing devices 150), as described in more detail below.

[0027] In some embodiments, use circle “1” in Fig. 1 one or more users associated with an organization 134, (an) electronic device(s) 144 to generate (an) account and policy configuration request(s) 142 to configure a set of accounts 136A-136N, principals 120 associated with an organization (e.g., an organization represented by organization 134), etc., and to further configure policies 126 associated with some or all of these principals and resources 160 associated with an organization. These principals may be created, for example, to provide authentication for users and processes within accounts (e.g., account 136A-136N) of cloud provider network 100.As mentioned above, identities represent a user and can be authenticated and then authorized to perform actions on the cloud provider network 100. Each identity can be associated with one or more policies 126 to determine which actions a user or role can perform on which cloud provider network resources and under what conditions. The collection of accounts, principals, and policies may be created, for example, by an organization that intends to use various services 106 of the cloud provider network 100 for different purposes. Furthermore, the collection of accounts, principals, and policies comprising an organization may change over time depending on the organization's desires.

[0028] In some embodiments, at circle "2," in response to the account and policy configuration request(s) 142, the identity and access management service 102 creates and stores data representing the accounts, principals, and policies. As further mentioned above, these principals and policies may be added, edited, and removed by external users of the cloud provider network 100 with sufficient permissions, e.g., using a web-based console, API, CLI, or other interface provided by the identity and access management service 102, and data representing the principals and policies may be stored using various types of storage resources managed by the identity and access management service 102.

[0029] Once a user or organization has created one or more policies, users may wish to analyze the policies to gain assurance that the configured policies are configured to help protect their data and resources (e.g., to help ensure that resources are inaccessible to unwanted entities, to help ensure that users are not inadvertently permitted to perform unwanted actions, etc.). In some embodiments, at circle "3," a user optionally requests at 158 ​​to perform one or more analyses of their policies, e.g., using a web-based console or other interface. In other embodiments, a policy analysis service 146 automatically performs one or more analyses, e.g.,in response to requests to display more general information about various types of resources 160 associated with one or more user accounts.

[0030] At circle "4," the policy analysis service 146 obtains one or more policies 148 relevant to the requested analysis. For example, if the requested analysis involves determining whether an external entity may access one or more resources associated with a user account, the policy analysis service 146 may obtain one or more resource-based policies associated with the applicable resources 160. In other examples, identity-based policies or other policies 132 may be obtained depending on the type of analysis to be performed.

[0031] At circle "5," the policy analysis service 146 generates an encoded version of the one or more policies 148 and uses one or more SMT solvers 152 to verify one or more properties of the policies. In some embodiments, the encoded policy 154 is generated by translating the authorization statements contained in one or more policies 126 (e.g., expressed in a JSON-based format or other syntax) into dependencies expressed using first-order logic (e.g., expressed using the SMT LIB format or other formal syntax). At a high level, the encoded policy includes a set of dependencies that, when analyzed by an SMT solver 152, generates an output indicating that the associated formula (e.g.,the formulation of the properties to be tested) is satisfiable if there is an assignment of values ​​to the variables of the dependencies for which the formula is satisfied; otherwise, the formula is not satisfiable if there is no such assignment of values ​​to the variables.

[0032] Fig. 2 is a diagram illustrating a process for encoding policy-defined permissions into propositional logic statements including string variables, according to some embodiments. Fig. 2, a resource policy (e.g., including a resource policy fragment 200) is provided as input to a first-order language encoder 202 to generate an encoded policy (e.g., including an example encoded policy fragment 204). In some embodiments, the first-order language encoder 202 implements the Satisfiability-Modulo-Theory Library and Standard (SMT-LIB) or other syntax for formally specifying formulas associated with policies of a cloud provider network 100. As shown, the encoded policy includes a number of variables (e.g., including a string variable 206) and Boolean operators (e.g., the Boolean operator 208, which corresponds to the logical AND operator), which together may form various dependencies (e.g., including a dependency 210).In some embodiments, a collection of one or more dependencies defines a formula that can be passed to a solver, where the solver checks the satisfiability of the formula by determining whether a satisfactory assignment exists for the variables. Although the formula in this example includes string variables, in other examples, depending on the type of information to be analyzed, other types of theory variables may be included in the encoded representation. For example, in other embodiments, an encoded representation of information to be analyzed may include variables associated with a theory of integers, a theory of real arithmetic, a theory of bit vectors, a theory of arrays, a theory of list structures, etc.

[0033] In some embodiments, the policy analysis service 146 partitions the formula defined by an encoded policy into a plurality of subformulas to distribute the processing of SMT queries expressed in first-order logic as described above. Each of the subformulas can be processed by an SMT solver executing on an independent computational resource (e.g., as a separate process, independently executable thread, VM instance, container, on-demand function, etc.). In this way, the plurality of subformulas divides the entire search space for a satisfactory allocation for the formula among the set of subformulas, the processing of which can be parallelized as described above, thereby significantly reducing the time required to solve the formula in most cases.Some of the examples provided herein illustrate the partitioning of a formula according to the string variables identified in the formula; more generally, the techniques described can be used to partition formulas in a theory-based manner, e.g., depending on a type or types of theory variables included in the SMT query.

[0034] Fig. 3 is a diagram illustrating, according to some embodiments, a process for partitioning a search space associated with a formula expressed in first-order logic and involving string variables into a set of subformulas whose execution can be distributed across multiple computational resources 314A-314N. In Fig. 3, a process of a coordinator 300 of a policy analysis service 146 takes one or more coded policies (e.g., illustrated by coded policy fragments 204) as input and causes a formula partitioner 302 to identify a set of predicates 306 used to partition the formula represented by the coded policies into a set of subformulas 308 (e.g., including subformulas 310A-310N). An illustrative algorithm for partitioning a formula into a set of subformulas 308 is as follows:

[0035] In the above example, the routine pulp takes a formula f (e.g., defined by the coded policy) as input and returns a set of subformulas, represented by the set pulped. The routine "Predicates(f)" generates predicates over strings in formula f that name the most frequently mentioned string variable. For example, in coded policy fragment 204, the string variable csp:crn is the most frequently mentioned string variable (e.g., it is mentioned more often than the string variables csp:crn prefix and csp:crn_region ). In this example, the routine Predicates(f) splits the formula containing the string variable into a set of predicates that contain an instance of the most frequently mentioned string variable csp:crn (e.g., "(= "arn:aws:sts::111 :AAA" |aws:arn|)", "(str.Prefixof "arn:aws:sts::111:BBB / " |aws:arn|)", "(= "arn:aws:sts::111: CCC" |aws:arn|)", "(= "arn:aws:sts::111:DDD" |aws:arn|)", etc.).In some embodiments, each predicate is thus a binary-valued function of at least one string variable included in the formula. In other embodiments, other predicate generation techniques may be used, for example, by splitting the formula into predicates containing the top N most frequently occurring string variables, predicates with approximately equal computational complexity, etc. In some embodiments, predicates may be categorized, for example, based on an estimated computational complexity of inference about the theory variable(s) included in each predicate. A formula may then be partitioned by grouping predicates into a plurality of predicate groups based on their estimated computational complexity (e.g.,multiple predicates of lower complexity can be grouped into a single partition, while predicates of higher complexity can be partitioned separately, etc.).

[0036] The routine further iterates through the obtained list of predicates(f) and, for each predicate, adds a subformula to the set pulped, where the subformula contains the formula assuming that the predicate is either true or false. In this way, each subformula restricts the search space associated with the original formula to a partition of the search space where one of the predicates in predicates(f) is either true or false. Each of these subformulas represents an independent instance that can be analyzed by an SMT solver 152 and can be distributed across a set of independent computational resources (e.g., separate threads of a multithreaded execution environment, separate compute instances, containers, on-demand functions provided by various services of a cloud provider network, etc.), each executing an SMT solver.In some embodiments, if one of the SMT solvers returns an indication that its subformula is satisfiable, the formula may be determined to be satisfiable. Otherwise, if all SMT solvers return an indication that the respective subformulas are unsatisfiable, the formula may be determined to be unsatisfiable.

[0037] Fig. 4 is a diagram illustrating, according to some embodiments, the use of an SMT solver executing across multiple compute resources to analyze a set of subformulas derived from a formula expressed in first-order logic and including string variables. As shown, each of the plurality of compute resources 314A-314N executes an SMT solver 152 and is assigned by the coordinator 300 to a respective subformula of a set of subformulas 308 (e.g., subformula 310A is assigned to compute resource 314A, subformula 310B is assigned to compute resource 314B, and subformula 310N is assigned to compute resource 314N). In general, the subformulas of the set of subformulas 308 may be distributed among the multiple compute resources in any manner, for example, depending on a number of subformulas and a number of available compute resources.

[0038] In some embodiments, an exemplary algorithm executed by a process of coordinator 300 to determine the satisfiability of a formula f using a set of computational resources is illustrated below:

[0039] In the above example, a worker routine is defined that takes as input a solver s and a formula g and returns an indication that the formula f is either satisfiable (SAT) (e.g., if there is a satisfactory assignment of values ​​to the subformula variables) or unsatisfiable (UNSAT) (e.g., if it is determined that there is no satisfactory assignment of values ​​to the subformula variables). In some embodiments, each of the computational resources 314A-314N includes a process that implements a routine similar to the worker routine illustrated above using an SMT solver 152.

[0040] In some embodiments, the Distribution_Solver routine takes as input a formula f and a solver s and returns an indication of whether the formula is satisfiable or unsatisfiable. In particular, the Distribution_Solver routine generates the pulped set of subformulas, which are preceded by Fig. 3, and distributes the subformulas to a plurality of separate workers (e.g., a plurality of separate compute resources 314A, which may include separate processes, threads, compute instances, containers, on-demand functions, etc.). In some embodiments, the Dispatch Solver routine returns an indication that the formula is satisfiable whenever any of the workers returns an indication that a respective subformula is satisfiable. Otherwise, if none of the workers returns an indication that the formula is satisfiable (e.g., all workers return an indication that their respective subformula is unsatisfiable), the routine returns an indication that the formula is unsatisfiable. In some embodiments, the routine may return an indication that the formula is unsatisfiable (e.g.,before all workers return an indication that their respective subformula is unsatisfiable), if an indication is received from respective workers that a subformula is unsatisfiable both under the assumption that its predicate is true and under the assumption that its predicate is false.

[0041] In Fig. 4, for example, the compute resource 314A-314N each processes a respective subformula from the subformulas 310A-310N. At time 406A throughout the worker execution time 410, the compute resource 314N returns an indication that its subformula 310N is unsatisfiable as an unsatisfiable result 400. At time 406B, the compute resource 314A returns a satisfiable result 402 indicating that its subformula 310A is satisfiable. As mentioned above, the process of the coordinator 300 may determine that the overall formula f is satisfiable because the subformula 310A is satisfiable. In some embodiments, the coordinator 300 process may optionally terminate the processing of other subformulas by other worker compute resources (e.g., as illustrated by the analysis termination request 404 sent to compute resource 314B).Thus, while solving the formula as a whole may take at least until time 406N to determine whether the formula is satisfiable, distributed processing of the formula only takes until time 406B, at which it is determined that one of the subformulas is satisfiable.

[0042] Returning to Fig. 1, in some embodiments, the policy analysis service 146 generates policy results 156 based on the analysis performed by the SMT solvers 152 at circle "5." The policy results 156 may generally include any information obtained based on the inferences performed in connection with the one or more policies 148. For example, the policy results may include an indication that one or more resources 160 are accessible to one or more entities outside of a defined trust zone, that a policy permits one or more unintended operations to be performed on one or more resources 160, that a user may assume a role that the user should not be able to assume, and the like.In some embodiments, coordinator 300 may more generally transmit a message indicating whether the formula is satisfiable or unsatisfiable based on the analysis performed by SMT solvers 152. The transmitted message may result in the display of information associated with the result, may be sent to one or more downstream SMT solvers or other automated reasoning tools for further analysis, or may be used by other processes.

[0043] Fig. 5 is a diagram illustrating a graphical user interface (GUI) displaying policy results derived from an analysis of one or more policies, according to some embodiments. For example, GUI 500 illustrates a console interface displaying a list of storage resources (e.g., "Example 1 Resource," "Example ab Resource," etc.). In some embodiments, the interface further includes at least one policy result 502 indicating information about a resource derived from an automated inference-based analysis, as described above.In this example, result 502 indicates that one of the storage resources is accessible to users outside a defined trust zone, which may prompt a user to further analyze and modify the policies associated with the resource to mitigate inadvertent access to the resource. In general, the representation of such policy results may be presented in other types of interfaces (e.g., CLIs, standalone application interfaces, etc.) and may relate to other types of policy analysis as described herein.

[0044] Fig. 6 is a flowchart illustrating acts 600 of a method for using reasoning techniques to analyze formulas expressed in propositional logic and including string variables, according to some embodiments. Some or all of acts 600 (or other processes described herein, or variations and / or combinations thereof) may be performed under the control of one or more computer systems configured with executable instructions, and are implemented as code (e.g., executable instructions, one or more computer programs, or one or more applications) executing together on one or more processors, by hardware, or combinations thereof. The code is stored on a computer-readable storage medium, for example, in the form of a computer program comprising instructions executable by one or more processors.The computer-readable storage medium is non-transitory. In some embodiments, one or more (or all) of the operations 600 are performed by a policy analysis service 146 of the other figures.

[0045] Operations 600 include, at block 602, receiving a request by a policy analysis service of a cloud provider network to determine whether a formula is satisfiable, where the formula relates to an analysis of policies attached to one or more compute resources associated with a user of the cloud provider network.

[0046] The operations 600 further include, at block 604, identifying a set of predicates in the formula based on a type of theory associated with the set of predicates, wherein each predicate of the set of predicates is a binary-valued function of at least one theory variable included in the formula. For example, the type of theory may be at least one of the following: a theory of strings, a theory of integers, a theory of real arithmetic, a theory of bit vectors, a theory of arrays, a theory of list structures, etc. In some embodiments, the type of theory is a theory of strings and the theory variable is a string variable, and the operations further include identifying a most frequently occurring theory variable in the formula, and wherein each predicate of the set of predicates includes an instance of the most frequently occurring theory variable.

[0047] The operations 600 further include, at block 606, partitioning a search space associated with the formula into a plurality of subformulas, each subformula including the formula under an assumption that a respective predicate of the set of predicates is either true or false.

[0048] Operations 600 further include, at block 608, sending a respective subformula of the plurality of subformulas to an SMT solver executing on each of a plurality of compute resources. In some embodiments, each compute resource of the plurality of compute resources is an independently executable thread of a plurality of threads, and wherein the plurality of threads execute on one or more computing devices. In some embodiments, each compute resource of the plurality of compute resources is one of the following: a compute instance provided by a hardware virtualization service of a cloud provider network, a container provided by a container service of the cloud provider network, or an on-demand function provided by an on-demand code service of the cloud provider network.

[0049] The operations 600 further include, at block 610, receiving an indication from the SMT solver executing on a compute resource of the plurality of compute resources that the respective subformula analyzed by the SMT solver executing on the compute resource is satisfiable.

[0050] Operations 600 further include, at block 612, transmitting a message indicating that the formula is satisfiable. For example, the message may be used to prompt the display of information indicating that the formula is satisfiable, may be sent to one or more downstream SMT solvers or other automated reasoning tools for further analysis, or may be used by other processes.

[0051] In some embodiments, the formula relates to at least one of the following: an analysis of policies applicable to one or more computing resources associated with a user of a cloud provider network, an analysis of the correctness of a computer program, or an analysis of the correctness of a computer network configuration. In some embodiments, the satisfiability of the formula determines whether a computing resource associated with an account or organization defined by a cloud provider network is accessible to an entity outside the account or organization, and wherein the information indicating that the formula is satisfiable indicates that the computing resource is accessible to an entity outside the account or organization.In some embodiments, the request identifies a policy managed by an identity and access management service of a cloud provider network, and wherein the method further comprises generating the formula by encoding the policy in a first-order logic format.

[0052] In some embodiments, the compute resource of the plurality of compute resources is a first compute resource and the indication is a first indication, and wherein the first indication that the formula is satisfiable is received from the SMT solver executing on the first compute resource before receiving a second indication from a second compute resource of the plurality of compute resources.

[0053] In some embodiments, the operations further include: receiving a second query to determine whether a second formula expressed in first-order logic is satisfiable; generating a second set of predicates based on the second formula; partitioning a search space associated with the second formula into a second plurality of subformulas; sending a respective subformula of the second plurality of subformulas to the SMT solver executing on each of the respective plurality of second computational resources; receiving an indication that the respective subformula is unsatisfiable from each SMT solver executing on a computational resource of the plurality of computational resources; and transmitting a message indicating that the second formula is unsatisfiable.

[0054] In some embodiments, the operations further include: receiving a second query to determine whether a second formula expressed in first-order logic is satisfiable; generating a second set of predicates based on the second formula; partitioning a search space associated with the second formula into a second plurality of subformulas; sending a respective subformula of the second plurality of subformulas to the SMT solver executing on each of the respective plurality of second computational resources; receiving a first indication that a particular subformula of the second plurality of subformulas is unsatisfiable under an assumption that its respective predicate is true, and a second indication that the particular subformula of the second plurality of subformulas is unsatisfiable under an assumption that the respective predicate is false;and transmitting a message indicating that the second formula is not satisfiable.;

[0055] In some embodiments, the operations further include grouping the set of predicates into a plurality of predicate groups based on an estimated computational complexity associated with each predicate of the set of predicates, wherein the search space is partitioned based on the plurality of predicate groups.

[0056] In some embodiments, the operations further include causing a graphical user interface (GUI) to be displayed, including information indicating that the formula is satisfiable, the information indicating a value for the at least one string variable that causes the formula to be satisfiable. In some embodiments, the message indicating that the formula is satisfiable is used as input to another SMT solver.

[0057] Fig. 7 illustrates an example provider network environment (or "service provider system") according to some embodiments. A provider network 700 may provide resource virtualization to customers through one or more virtualization services 710 that enable customers to purchase, rent, or otherwise obtain instances 712 of virtualized resources, including, but not limited to, compute and storage resources implemented on devices within the provider network or networks in one or more data centers. Local Internet Protocol (IP) addresses 716 may be associated with the resource instances 712; the local IP addresses are the internal network addresses of the resource instances 712 in the provider network 700. In some embodiments, the provider network 700 may also include public IP addresses 714 and / or public IP address ranges (e.g.,Internet Protocol Version 4 (IPv4) or Internet Protocol Version 6 (IPv6) addresses) that customers can obtain from the provider 700.

[0058] Conventionally, via the virtualization services 710, the provider network 700 may enable a customer of the service provider (e.g., a customer operating one or more client networks 750A-750C including one or more customer devices 752) to associate at least some public IP addresses 714 assigned or allocated to the customer with specific resource instances 712 assigned to the customer. The provider network 700 may also enable the customer to reassign a public IP address 714 previously assigned to a virtualized computing resource instance 712 assigned to the customer to another virtualized computing resource instance 712 also assigned to the customer. Using the virtualized computing resource instances 712 and public IP addresses 714 provided by the service provider, a customer of the service provider, such as the operator of the customer network orthe customer networks 750A-750C, for example, implement customer-specific applications and present the customer applications on an intermediate network 740, such as the Internet. Other network entities 720 in the intermediate network 740 may then generate traffic to a public destination IP address 714 published by the customer network(s) 750A-750C; the traffic is routed to the service provider's data center and, in the data center, routed across a network substrate to the local IP address 716 of the virtualized computing resource instance 712 currently associated with the public destination IP address 714. Similarly, response traffic from the virtualized computing resource instance 712 may be routed across the network substrate back onto the intermediate network 740 to the source entity 720.

[0059] Local IP addresses, as used herein, refer to the internal or "private" network addresses, for example, of resource instances within a provider's network. Local IP addresses may be located within address blocks reserved by Internet Engineering Task Force (IETF) Request for Comments (RFC) 1918 and / or an address format specified by IETF RFC 4193, and may be changeable within the provider's network. Network traffic from outside the provider's network is not routed directly to local IP addresses; instead, the traffic uses public IP addresses mapped to the local IP addresses of the resource instances. The provider's network may include network devices or equipment that provide network address translation (NAT) or similar functionality to perform the mapping of public IP addresses to local IP addresses and vice versa.

[0060] Public IP addresses are variable network addresses on the Internet that are assigned to resource instances either by the service provider or by the customer. Traffic routed to a public IP address, for example, is translated using 1:1 NAT and forwarded to the respective local IP address of a resource instance.

[0061] Some public IP addresses may be assigned to specific resource instances by the provider's network infrastructure; these public IP addresses may be referred to as default public IP addresses or simply default IP addresses. In some embodiments, mapping a default IP address to a resource instance's local IP address is the default launch configuration for all types of resource instances.

[0062] At least some public IP addresses may be assigned to or acquired by customers of the provider network 700; a customer may then assign their assigned public IP addresses to specific resource instances allocated to the customer. These public IP addresses may be referred to as customer public IP addresses or simply customer IP addresses. Instead of being assigned to resource instances by the provider network 700, as is the case with standard IP addresses, customer IP addresses may be assigned to resource instances by the customer, for example, via an API provided by the service provider. Unlike standard IP addresses, customer IP addresses are assigned to customer accounts and may be reassigned by the respective customer to other resource instances as needed or desired.A customer IP address is associated with a customer account, not with a specific resource instance, and the customer controls that IP address until they release it. Unlike traditional static IP addresses, customer IP addresses allow the customer to mask resource instance or availability zone outages by reassigning the customer's public IP addresses to each resource instance associated with the customer's account. For example, customer IP addresses allow a customer to work around issues with the customer's resource instances or software by reassigning customer IP addresses to backup resource instances.

[0063] Fig. 8 is a block diagram of an example provider network that provides a storage service and a hardware virtualization service to customers, according to some embodiments. The hardware virtualization service 820 provides customers with a plurality of computing resources 824 (e.g., computing instances 825 such as VMs). The computing resources 824 may be rented or leased, for example, by customers of the provider network 800 (e.g., by a customer implementing the customer network 850). Each computing resource 824 may be provided with one or more local IP addresses. The provider network 800 may be configured to route packets from the local IP addresses of the computing resources 824 to public internet destinations and from public internet sources to the local IP addresses of the computing resources 824.

[0064] The provider network 800 may provide a customer network 850, coupled to an intermediate network 840, for example, via a local area network 856, with the ability to implement virtual computing systems 892 via the hardware virtualization service 820 coupled to the intermediate network 840 and the provider network 800. In some embodiments, the hardware virtualization service 820 may provide one or more APIs 802, such as a web service interface, through which a customer network 850 may access the functionality provided by the hardware virtualization service 820, for example, via a console 894 (e.g., a web-based application, standalone application, mobile application, etc.).In some embodiments, in the provider network 800, each virtual computing system 892 in the customer network 850 may correspond to a computing resource 824 that is leased, rented, or otherwise provided to the customer network 850.

[0065] From an instance of a virtual computing system 892 and / or another customer device 890 (e.g., via console 894), the customer may access the functionality of the storage service 810, for example, via one or more APIs 802, to access and store data in storage resources 818A-818N of a virtual data store 816 (e.g., a folder or "bucket," a virtualized volume, a database, etc.) provided by the provider network 800.In some embodiments, a virtualized data storage gateway (not shown) may be deployed in the customer network 850, which may locally cache at least some data, such as frequently accessed or critical data, and may communicate with the storage service 810 over one or more communication channels to upload new or modified data from a local cache so that the primary data store (virtualized data store 816) is maintained. In some embodiments, a user may load and access volumes of the virtual data store 816 via a virtual computing system 892 and / or on another customer device 890 via the storage service 810 acting as a storage virtualization service, and these volumes may appear to the user as local (virtualized) storage 898.

[0066] Although in Fig. 8, the virtualization service(s) may also be accessed from resource instances within the provider network 800 via the API(s) 802. For example, a customer, a device service provider, or another entity may access a virtualization service via an API 802 from within a respective virtual network in the provider network 800 to request allocation of one or more resource instances within the virtual network or within another virtual network.

[0067] In some embodiments, a system implementing some or all of the techniques described herein may include a general-purpose computer system that includes or is configured to access one or more computer-accessible media, such as the one described in Fig. 9 illustrates the computer system 900. In the illustrated embodiment, the computer system 900 includes one or more processors 910 coupled to a system memory 920 via an input / output (I / O) interface 930. The computer system 900 further includes a network interface 940 coupled to the I / O interface 930. While Fig. 9 shows the computer system 900 as a single computing device, in various embodiments, a computer system 900 may include one or any number of computing devices configured to cooperate as a single computer system 900.

[0068] In various embodiments, computer system 900 may be a single-processor system having one processor 910 or a multiprocessor system having multiple processors 910 (e.g., two, four, eight, or any other suitable number). Processors 910 may be any suitable processors capable of executing instructions. For example, in various embodiments, processors 910 may be general-purpose or embedded processors implementing any of various instruction set architectures (ISAs), such as x86, ARM, PowerPC, SPARC, or MIPS ISAs, or any other suitable ISA. In multiprocessor systems, each of processors 910 may typically, but not necessarily, implement the same ISA.

[0069] System memory 920 may store instructions and data accessible by processor(s) 910. In various embodiments, system memory 920 may be implemented using any suitable memory technology, such as random-access memory (RAM), static RAM (SRAM), synchronous dynamic RAM (SDRAM), non-volatile memory / flash memory, or any other type of memory. In the illustrated embodiment, program instructions and data implementing one or more desired functions, such as the methods, techniques, and data described above, are shown stored in system memory 920 as policy analysis service code 925 (e.g., executable to implement all or part of policy analysis service 146) and data 926.

[0070] In one embodiment, I / O interface 930 may be configured to coordinate I / O traffic between processor 910, system memory 920, and any peripherals in the device, including network interface 940 or other peripheral interfaces. In some embodiments, I / O interface 930 may perform any necessary protocol, timing, or other data transformations to convert data signals from one component (e.g., system memory 920) into a format suitable for use by another component (e.g., processor 910). In some embodiments, I / O interface 930 may include support for devices connected via various types of peripheral buses, such as a variant of the Peripheral Component Interconnect (PCI) bus standard or the Universal Serial Bus (USB) standard.In some embodiments, the functionality of I / O interface 930 may be split into two or more separate components, such as a northbridge and a southbridge. Furthermore, in some embodiments, some or all of the functionality of I / O interface 930, such as an interface to system memory 920, may be integrated directly into processor 910.

[0071] The network interface 940 may be configured to enable data exchange between the computer system 900 and other devices 960 connected to a network or networks 950, such as other computer systems or devices, as in Fig.1. In various embodiments, network interface 940 may support communication over any suitable wired or wireless general data networks, such as Ethernet networks. Furthermore, network interface 940 may support communication over telecommunications / telephone networks, such as analog voice networks or digital fiber optic communication networks, over storage area networks (SANs), such as Fibre Channel SANs, or over I / O of any other suitable type of network and / or protocol.

[0072] In some embodiments, a computer system 900 includes one or more offload cards 970A or 970B (including one or more processors 975 and possibly including one or more network interfaces 940) connected using an I / O interface 930 (e.g., via a bus implementing a version of the Peripheral Component Interconnect Express (PCI-E) standard or other interconnect, such as QuickPath Interconnect (QPI) or UltraPath Interconnect (UPI)). For example, in some embodiments, the computer system 900 may function as an electronic host device (e.g., operating as part of a hardware virtualization service) that hosts computing resources such as compute instances, and the one or more offload cards 970A or 970B execute a virtualization manager that may manage compute instances executing on the electronic host device.For example, in some embodiments, the offload card(s) 970A or 970B may perform compute instance management operations, such as stopping and / or restarting compute instances, starting and / or terminating compute instances, performing memory transfer / copy operations, etc. These management operations may, in some embodiments, be performed by the offload card(s) 970A or 970B in coordination with (e.g., upon request from) a hypervisor executed by the other processors 910A-910N of the computer system 900. However, in some embodiments, the virtualization manager implemented by the offload card(s) 970A or 970B may accommodate requests from other entities (e.g., from compute instances themselves) and may not need to coordinate with (or service) a separate hypervisor.

[0073] In some embodiments, system memory 920 may be an embodiment of a computer-accessible medium configured to store program instructions and data as described above. However, in other embodiments, program instructions and / or data may be received, sent, or stored on different types of computer-accessible media. In general, a computer-accessible medium may include non-transitory storage media or storage media such as magnetic or optical media, e.g., a hard disk or DVD / CD, connected to computer system 900 via I / O interface 930. A non-transitory computer-accessible storage medium may also include any volatile or non-volatile media such as RAM (e.g., SDRAM, Double Data Rate (DDR) SDRAM, SRAM, etc.), read-only memory (ROM), etc.which, in some embodiments of computer system 900, may be included as system memory 920 or another type of memory. Furthermore, a computer-accessible medium may include transmission media or signals, such as electrical, electromagnetic, or digital signals, conveyed over a communication medium, such as a network and / or a wireless link, such as may be implemented via network interface 940.

[0074] Various embodiments discussed or suggested herein may be implemented in a wide variety of operating environments, which in some cases may include one or more user computers, computing devices, or processing devices that may be used to run any of a variety of applications. User or client devices may include any number of general-purpose computers, such as desktop or laptop computers running a standard operating system, as well as cellular, wireless, and wearable devices running mobile software and capable of supporting a variety of networking and messaging protocols.Such a system may also include a number of workstations running any of a variety of commercially available operating systems and other known applications for purposes such as development and database management. These devices may also include other electronic devices, such as dummy terminals, thin clients, gaming systems, and / or other devices capable of communicating over a network.

[0075] Most embodiments utilize at least one network familiar to those skilled in the art to support communication using a variety of commonly available protocols, such as Transmission Control Protocol / Internet Protocol (TCP / IP), File Transfer Protocol (FTP), Universal Plug and Play (UPnP), Network File System (NFS), Common Internet File System (CIFS), Extensible Messaging and Presence Protocol (XMPP), AppleTalk, etc. The network(s) may include, for example, a local area network (LAN), a wide-area network (WAN), a virtual private network (VPN), the Internet, an intranet, an extranet, a public switched telephone network (PSTN), an infrared network, a wireless network, and any combination thereof.

[0076] In embodiments utilizing a web server, the web server may run a variety of server or mid-tier applications, including HTTP servers, File Transfer Protocol (FTP) servers, Common Gateway Interface (CGI) servers, data servers, Java servers, enterprise application servers, etc. The server(s) may also be capable of executing programs or scripts in response to requests from user devices, such as by executing one or more web applications, which may be implemented as one or more scripts or programs written in any programming language, such as Java®, C, C#, or C++, or any scripting language, such as Ruby, PHP, Perl, Python, or TCL, as well as combinations thereof. The server(s) may also include database servers, including, but not limited to, those commercially available from Oracle®, Microsoft®, Sybase®, IBM®, etc.The database servers can be relational or non-relational (e.g., “NoSQL”), distributed or non-distributed, etc.

[0077] Environments disclosed herein may include a variety of data stores and other storage and other storage media, as discussed above. These may be located in a variety of locations, such as on a storage medium local to (and / or located within) one or more of the computers or remote from any or all of the computers across the network. In a particular set of embodiments, the information may be located in a storage area network (SAN), which is familiar to those skilled in the art. Likewise, any files required to perform the functions associated with the computers, servers, or other network devices may be stored locally and / or remotely, as appropriate.When a system includes computer-based devices, each such device may include hardware elements that may be electrically coupled via a bus, where the elements may include, for example, at least one central processing unit (CPU), at least one input device (e.g., a mouse, a keyboard, a controller, a touchscreen, or a keypad), and / or at least one output device (e.g., a display device, a printer, or a speaker). Such a system may also include one or more storage devices, such as disk drives, optical storage devices, and solid-state storage devices such as random access memory (RAM) or read-only memory (ROM), as well as removable media devices, memory cards, flash cards, etc.

[0078] Such devices may also include a computer-readable storage media reader, a communication device (e.g., a modem, a network card (wireless or wired), an infrared communication device, etc.), and memory, as described above. The computer-readable storage media reader may be connected to or configured to receive a computer-readable storage medium, which may include remote, local, fixed, and / or removable storage devices, as well as storage media for temporarily and / or permanently containing, storing, transmitting, and retrieving computer-readable information.The system and the various devices also typically include a number of software applications, modules, services, or other elements located within at least one memory device, including an operating system and application programs, such as a client application or a web browser. It should be understood that alternative embodiments may include numerous variations from that described above. For example, custom hardware may also be used and / or specific elements may be implemented in hardware, software (including portable software such as applets), or both. Furthermore, connection to other computing devices, such as network input / output devices, may be employed.

[0079] Storage media and computer-readable media containing code or portions of code may include any convenient media known or used in the art, including storage media and communication media, such as, but not limited to, volatile and non-volatile, removable and non-removable media, implemented in any method or technique for storing and / or transmitting information, such as computer-readable instructions, data structures, program modules, or other data, including RAM, ROM, electrically erasable programmable read-only memory (EEPROM), flash memory or other storage technology, compact disc read-only memory (CD-ROM), digital versatile disk (DVD) or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium that can be used toto store the desired information and which can be accessed by a system device. Based on the disclosure and teachings provided herein, one of ordinary skill in the art will understand that other possibilities and / or methods exist to implement the various embodiments.

[0080] In the foregoing description, various embodiments have been described. For purposes of explanation, specific configurations and details are set forth in order to provide a thorough understanding of the embodiments. However, it will also be apparent to those skilled in the art that the embodiments may be practiced without the specific details. Furthermore, well-known features may be omitted or simplified so as not to obscure the described embodiment.

[0081] Text in parentheses and blocks with dashed borders (e.g., capital dashes, small dashes, dot-dash, and dots) are used herein to illustrate optional operations that add additional features to some embodiments. However, such notation should not be understood to imply that these are the only options or optional operations and / or that blocks with solid borders are not optional in certain embodiments.

[0082] Reference numerals with appended letters (e.g., 818A-818N) can be used to indicate that there may be one or more instances of the referenced entity in different embodiments. Where there are multiple instances, they need not all be identical but may instead share some common properties or behave in a common manner. Furthermore, the suffixes used do not imply that a concrete quantity of the entity exists unless explicitly stated otherwise. Thus, two entities using the same or different suffix letters may or may not have the same number of instances in different embodiments.

[0083] References to "one embodiment," "an embodiment," etc., indicate that the described embodiment may include a particular feature, structure, or characteristic, but not every embodiment necessarily includes that particular feature, structure, or characteristic. Furthermore, such language does not necessarily refer to the same embodiment. Further, it should be understood that where a particular feature, structure, or characteristic is described in connection with one embodiment, it is within the skill of one skilled in the art to produce such a feature, structure, or characteristic in connection with other embodiments, whether explicitly described or not.

[0084] Furthermore, in the various embodiments described above, unless explicitly stated otherwise, disjunctive language such as the phrase "at least one of A, B, or C" is intended to mean either A, B, or C, or any combination thereof (e.g., A, B, and / or C). Therefore, disjunctive language is not intended, nor should it be understood, to imply that at least one of A, at least one of B, or at least one of C must be present for a given embodiment.

[0085] Unless explicitly stated otherwise, articles such as "a" or "an" should generally be interpreted to include one or more of the described items. Accordingly, phrases such as "a device configured to" include one or more of the recited devices. Such one or more of the recited devices may also be collectively configured to perform the recited recitations. For example, "a processor configured to perform recitations A, B, and C" may include a first processor configured to perform recitations A operating in conjunction with a second processor configured to perform recitations B and C.

[0086] At least some embodiments of the present technologies may be described with respect to the following paragraphs: 1. Computer-implemented method comprising: Receiving, by a policy analysis service of a cloud provider network, a request to determine whether a formula is satisfiable, wherein the formula relates to an analysis of one or more policies applicable to one or more computing resources associated with a user of the cloud provider network; identifying a set of predicates in the formula based on string variables included in the formula, wherein each predicate of the set of predicates is a binary-valued function of at least one string variable included in the formula; partitioning a search space associated with the formula into a plurality of subformulas, each subformula of the plurality of subformulas including the formula under an assumption that a respective predicate of the set of predicates is either true or false; sending a respective subformula of the plurality of subformulas to a satisfiability modulo theory (SMT) solver executing on each of a plurality of computational resources; Receiving an indication from the SMT solver executing on a computing resource of the plurality of computing resources that the respective subformula analyzed by the SMT solver executing on the computing resource is satisfiable; and Display information indicating that the formula is satisfiable. 2. The computer-implemented method of paragraph 1, further comprising identifying a most frequently occurring string variable in the formula, and wherein each predicate of the set of predicates includes an instance of the string variable. 3. The computer-implemented method of any one of paragraphs 1-2, wherein each computational resource of the plurality of computational resources is one of the following: an independently executable thread, a computational instance, a container, or an on-demand executable function. 4. Computer-implemented method comprising: Receiving a request by a coordinator process executing on a first computing resource to determine whether a formula expressed in propositional logic is satisfiable; identifying a set of predicates in the formula based on a type of theory associated with the set of predicates, wherein each predicate of the set of predicates is a binary-valued function of at least one theory variable included in the formula; partitioning a search space associated with the formula into a plurality of subformulas, each subformula of the plurality of subformulas including the formula under an assumption that a respective predicate of the set of predicates is either true or false; Sending a respective subformula of the plurality of subformulas to a satisfiability modulo theory (SMT) solver executing on each of a respective plurality of second computing resources; Receiving an indication from the SMT solver executing on a compute resource of the plurality of compute resources that the formula is satisfiable; and Transmit a message indicating that the formula is satisfiable. 5. The computer-implemented method of paragraph 4, wherein the type of theory is a theory of strings and the theory variable is a string variable, and wherein the method further comprises identifying a most frequently occurring theory variable in the formula, and wherein each predicate of the set of predicates includes an instance of the most frequently occurring theory variable. 6. The computer-implemented method of any one of paragraphs 4-5, wherein each computing resource of the plurality of computing resources is an independently executable thread of a plurality of threads, and wherein the plurality of threads execute on one or more computing devices. 7. A computer-implemented method according to any one of paragraphs 4-6, wherein the type of theory is at least one of the following: a theory of strings, a theory of integers, a theory of real arithmetic, a theory of bit vectors, a theory of arrays, or a theory of list structures. 8. The computer-implemented method of any one of paragraphs 4-7, wherein each computing resource of the plurality of computing resources is one of: a computing instance provided by a hardware virtualization service of a cloud provider network, a container provided by a container service of the cloud provider network, or an on-demand function provided by an on-demand code service of the cloud provider network. 9. The computer-implemented method of any of paragraphs 4-8, wherein the formula relates to at least one of the following: an analysis of policies applicable to one or more computing resources associated with a user of a cloud provider network, an analysis of the correctness of a computer program, or an analysis of the correctness of a computer network configuration. 10. The computer-implemented method of any one of paragraphs 4-9, wherein the satisfiability of the formula determines whether a computing resource associated with an account or organization defined by a cloud provider network is accessible to an entity outside the account or organization, and wherein the message indicating that the formula is satisfiable indicates that the computing resource is accessible to an entity outside the account or organization. 11. Computer-implemented methods according to any one of paragraphs 4-10, wherein the request identifies a policy managed by an identity and access management service of a cloud provider network, and wherein the method further comprises generating the formula by encoding the policy in a first-order logic format. 12. The computer-implemented method of any one of paragraphs 4-11, wherein the computational resource of the plurality of computational resources is a first computational resource and the indication is a first indication, and wherein the first indication that the formula is satisfiable is received from the SMT solver executing on the first computational resource before receiving a second indication from a second computational resource of the plurality of computational resources. 13. A computer-implemented method according to any one of paragraphs 4-12, further comprising: Receiving a second query to determine whether a second formula expressed in first-order logic is satisfiable; Generating a second set of predicates based on the second formula; partitioning a search space associated with the second formula into a second plurality of subformulas; Sending a respective subformula of the second plurality of subformulas to the SMT solver executing on each of the respective plurality of second computing resources; Receiving an indication from each SMT solver executing on a computing resource of the plurality of computing resources that the respective formula is unsatisfiable; and Transmit a message indicating that the second formula is not satisfiable. 14. A computer-implemented method according to any one of paragraphs 4-12, further comprising: Receiving a second query to determine whether a second formula expressed in first-order logic is satisfiable; Generating a second set of predicates based on the second formula; partitioning a search space associated with the second formula into a second plurality of subformulas; Sending a respective subformula of the second plurality of subformulas to the SMT solver executing on each of the respective plurality of second computing resources; Receiving a first indication that a concrete subformula of the second plurality of subformulas is not satisfiable under an assumption that its respective predicate is true, and a second indication that the concrete subformula of the second plurality of subformulas is not satisfiable under an assumption that its respective predicate is false; and Transmit a message indicating that the second formula is not satisfiable. 15. A computer-implemented method according to any one of paragraphs 4-14, further comprising: Grouping the set of predicates into a plurality of predicate groups based on an estimated computational complexity associated with each predicate of the set of predicates, wherein the search space is partitioned based on the plurality of predicate groups. 16. The computer-implemented method of any one of paragraphs 4-15, further comprising causing a graphical user interface (GUI) to be displayed including information indicating that the formula is satisfiable, the information indicating a value for the at least one string variable that causes the formula to be satisfiable. 17. The computer-implemented method of any one of paragraphs 4-16, wherein the message indicating that the formula is satisfiable is used as input to another SMT solver. 18. System comprising: one or more first electronic devices for implementing a policy analysis service in a multi-tenant provider network, the policy analysis service including instructions that, when executed, cause the policy analysis service to: Receiving a request to determine whether a formula is satisfiable, where the formula relates to an analysis of one or more policies applicable to one or more computing resources associated with a user of the multi-tenant provider network, Identifying a set of predicates in the formula based on a type of theory associated with the formula, where each predicate of the set of predicates is a binary-valued function of at least one string variable contained in the formula, Partitioning a search space associated with the formula into a plurality of subformulas, each subformula of the plurality of subformulas including the formula under an assumption that a respective predicate of the set of predicates is either true or false, Sending a respective subformula of the plurality of subformulas to a satisfiability modulo theory (SMT) solver executing on each of a plurality of computing resources, Receiving an indication from the SMT solver executing on a computing resource of the plurality of computing resources that the respective subformula analyzed by the SMT solver executing on the computing resource is satisfiable, and Transmitting a message indicating that the formula is satisfiable; and one or more second electronic devices for implementing the SMT solver in the multi-tenant provider network, the SMT solver including instructions that, when executed, cause the SMT solver to: Receiving the respective subformula of the plurality of subformulas, determining whether the subformula is satisfiable, and Sending the statement that the respective subformula is satisfiable. 19. The system of paragraph 18, wherein the policy analysis service further includes instructions that, when executed, cause the policy analysis service to identify a most frequently occurring string variable in the formula, and wherein each predicate of the set of predicates includes a single instance of the string variable. 20. The system of any one of paragraphs 18-19, wherein each computing resource of the plurality of computing resources is one of: a computing instance provided by a hardware virtualization service of a cloud provider network, a container provided by a container service of the cloud provider network, or an on-demand function provided by an on-demand function service of the cloud provider network.

[0087] The description and drawings are accordingly to be considered in an illustrative and not restrictive sense. However, it will be apparent that various modifications and changes may be made thereto without departing from the broader spirit and scope of the disclosure as set forth in the claims.

Claims

[1] Computer-implemented method comprising: Receiving a request by a policy analysis service (146) of a cloud provider network (100) to determine whether a formula is satisfiable, wherein the formula relates to an analysis of one or more policies (126) applicable to one or more computing resources (314A-314N) associated with a user of the cloud provider network (100); identifying (604) a set of predicates (306) in the formula based on string variables included in the formula, each predicate of the set of predicates being a binary-valued function of at least one string variable included in the formula; partitioning (606) a search space associated with the formula into a plurality of subformulas, each subformula of the plurality of subformulas containing the formula under the assumption that a corresponding predicate of the set of predicates (306) is either true or false; Sending (608) a respective subformula of the plurality of subformulas to a satisfiability modulo theories (SMT) solver (152) running on each of a plurality of computing resources (314A-314N); Receiving (610) an indication from the SMT solver (152) running on a computing resource of the plurality of computing resources (314A-314N) that the respective subformula analyzed by the SMT solver (152) running on the computing resource is satisfiable; causing the display of information indicating that the formula is satisfiable, wherein the satisfiability of the formula determines a property of the one or more policies (126) defining permissions applicable to the one or more computing resources (314A-314N) associated with the user of the cloud provider network (100); Controlling access to the one or more computing resources (314A-314N) according to the one or more policies (126) defining permissions applicable to the one or more computing resources associated with the user of the cloud provider network (100), and Grouping the set of predicates (306) into a plurality of predicate groups based on an estimated computational complexity associated with each predicate of the set of predicates (306), wherein lower complexity predicates of the set of predicates are grouped together and higher complexity predicates of the set of predicates are grouped separately, and wherein the search space is partitioned based on the plurality of predicate groups. [2] The computer-implemented method of claim 1, further comprising identifying a most frequently occurring string variable in the formula, and wherein each predicate of the set of predicates (306) includes an instance of the string variable. [3] The computer-implemented method of claim 1, wherein each computational resource of the plurality of computational resources (314A-314N) is one of the following: an independently executable thread, a computational instance, a container, or an on-demand executable function. [4] Computer-implemented method comprising: Receiving a request by a coordinator process executing on first computing resources to determine whether a formula expressed in propositional logic is satisfiable; identifying a set of predicates (306) in the formula based on a type of theory associated with the set of predicates, each predicate of the set of predicates (306) being a binary-valued function of at least one theory variable included in the formula; partitioning a search space associated with the formula into a plurality of subformulas, each subformula of the plurality of subformulas including the formula under an assumption that a respective predicate of the set of predicates (306) is either true or false; sending a respective subformula of the plurality of subformulas to a satisfiability modulo theory solver, SMT solver (152) executing on each of a respective plurality of second computing resources; receiving an indication from the SMT solver (152) executing on a computing resource of the plurality of computing resources (314A-314N) that the formula is satisfiable; Transmitting a message indicating that the formula is satisfiable, wherein the satisfiability of the formula determines a property of a policy (126) defining permissions applicable to a computing resource connected to a cloud provider network (100); Controlling access to the computing resource according to the policy (126) defining permissions for the computing resource connected to the cloud provider network (100); and Grouping the set of predicates (306) into a plurality of predicate groups based on an estimated computational complexity associated with each predicate of the set of predicates (306), wherein lower complexity predicates of the set of predicates are grouped together and higher complexity predicates of the set of predicates are grouped separately, and wherein the search space is partitioned based on the plurality of predicate groups. [5] The computer-implemented method of claim 4, wherein the type of theory is a theory of strings and the theory variable is a string variable, and wherein the method further comprises identifying a most frequently occurring theory variable in the formula, and wherein each predicate of the set of predicates (306) includes an instance of the most frequently occurring theory variable. [6] The computer-implemented method of any of claims 4-5, wherein each computing resource of the plurality of computing resources (314A-314N) is an independently executable thread of a plurality of threads, and wherein the plurality of threads are executed on one or more computing devices. [7] The computer-implemented method of any of claims 4-6, wherein the type of theory is at least one of: a theory of strings, a theory of integers, a theory of real arithmetic, a theory of bit vectors, a theory of arrays, or a theory of list structures. [8] The computer-implemented method of any of claims 4-7, wherein each computing resource of the plurality of computing resources (314A-314N) is one of: a computing instance provided by a hardware virtualization service (118) of the cloud provider network (100), a container provided by a container service of the cloud provider network, or an on-demand executable function provided by an on-demand executable code service of the cloud provider network. [9] The computer-implemented method of any of claims 4-8, wherein the satisfiability of the formula determines whether the computing resource associated with an account or organization defined by the cloud provider network (100) is accessible to an entity outside the account or organization, and wherein the message indicating that the formula is satisfiable indicates that the computing resource is accessible to an entity outside the account or organization. [10] The computer-implemented method of any of claims 4-9, wherein the request identifies the policy (126) managed by an identity and access management service (102) of the cloud provider network (100), and wherein the method further comprises generating the formula by encoding the policy (126) in a first-order logic format. [11] The computer-implemented method of any of claims 4-10, wherein the computational resource of the plurality of computational resources (314A-314N) is a first computational resource and the indication is a first indication, and wherein the first indication that the formula is satisfiable is received from the SMT solver (152) executing on the first computational resource before receiving a second indication from a second computational resource of the plurality of computational resources (314A-314N). [12] The computer-implemented method of any of claims 4-11, further comprising: Receiving a second query to determine whether a second formula expressed in first-order logic is satisfiable; Generating a second set of predicates (306) based on the second formula; partitioning a search space associated with the second formula into a second plurality of subformulas; sending a respective subformula of the second plurality of subformulas to the SMT solver (152) executing on each of the respective plurality of second computing resources; Receiving an indication from each SMT solver (152) executing on a computing resource of the plurality of computing resources (314A-314N) that the respective formula is not satisfiable; and Transmit a message indicating that the second formula is not satisfiable. [13] The computer-implemented method of any of claims 4-11, further comprising: Receiving a second query to determine whether a second formula expressed in first-order logic is satisfiable; Generating a second set of predicates (306) based on the second formula; partitioning a search space associated with the second formula into a second plurality of subformulas; sending a respective subformula of the second plurality of subformulas to the SMT solver (152) executing on each of the respective plurality of second computing resources; Receiving a first indication that a concrete subformula of the second plurality of subformulas is not satisfiable under an assumption that its respective predicate is true, and a second indication that the concrete subformula of the second plurality of subformulas is not satisfiable under an assumption that its respective predicate is false; and Transmit a message indicating that the second formula is not satisfiable. [14] The computer-implemented method of any of claims 4-13, further comprising causing a display of a graphical user interface, GUI (500) including information indicating that the formula is satisfiable, the information indicating a value for the at least one theory variable that causes the formula to be satisfiable. [15] The computer-implemented method of claim 4, wherein the message indicating that the formula is satisfiable is used as input to another SMT solver (152). [16] System (900) comprising: one or more first electronic devices for implementing a policy analysis service component in a multi-tenant capable provider network (100), the policy analysis service component including instructions that, when executed, cause the policy analysis service component to: Receiving a request to determine whether a formula is satisfiable, wherein the formula relates to an analysis of one or more policies (126) applicable to one or more computing resources (314A-314N) associated with a user of the multi-tenant capable provider network (100), Identifying a set of predicates (306) in the formula based on a type of theory associated with the formula, wherein each predicate of the set of predicates is a binary-valued function of at least one string variable contained in the formula, Partitioning a search space associated with the formula into a plurality of subformulas, each subformula of the plurality of subformulas including the formula under an assumption that a respective predicate of the set of predicates (306) is either true or false, Sending a respective subformula of the plurality of subformulas to a satisfiability modulo theory solver, SMT solver (152) executing on each of a plurality of computing resources (314A-314N), Receiving an indication from the SMT solver (152) executing on a computing resource of the plurality of computing resources (314A-314N) that the respective subformula analyzed by the SMT solver (152) executing on the computing resource is satisfiable, and Transmitting a message indicating that the formula is satisfiable, wherein the satisfiability of the formula determines a property of a policy (126) that defines permissions applicable to a computing resource connected to a cloud provider network (100); and one or more second electronic devices for implementing the SMT solver (152) in the multi-tenant capable provider network (100), wherein the SMT solver (152) includes instructions that, when executed, cause the SMT solver (152) to: Receiving the respective subformula of the plurality of subformulas, determining whether the subformula is satisfiable, and Sending the indication that the respective subformula is satisfiable; and a third electronic device or multiple electronic devices to implement an Identity and Access Management (IAM) service component in the multi-tenant capable provider network, wherein the IAM service component includes instructions that, when executed, cause the IAM service component to: control access to the computing resource according to the policy that defines permissions applicable to the computing resource connected to the cloud provider network (100); and group the set of predicates (306) into a plurality of predicate groups based on an estimated computational complexity associated with each predicate of the set of predicates (306), wherein lower complexity predicates of the set of predicates are grouped together and higher complexity predicates of the set of predicates are grouped separately, and wherein the search space is partitioned based on the plurality of predicate groups. [17] The system (900) of claim 16, wherein the policy analysis service component further includes instructions that, when executed, cause the policy analysis service component to identify a most frequently occurring string variable in the formula, and wherein each predicate of the set of predicates (306) includes a single instance of the string variable. [18] The system (900) of claim 16, wherein each computing resource of the plurality of computing resources (314A-314N) is one of the following: a computing instance provided by a hardware virtualization service (118) of the cloud provider network (100), a container provided by a container service of the cloud provider network, or an on-demand function provided by an on-demand function service of the cloud provider network (100). [19] The computer-implemented method of claim 1 or 4, wherein a plurality of lower complexity predicates are grouped into a single partition, while higher complexity predicates are partitioned separately.

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