Fuzzy mutual simulation formal verification and modeling method based on theorem prover

Abstract

The invention provides a fuzzy mutual simulation formalized verification and modeling method based on a theorem prover, and particularly relates to a fuzzy mutual simulation equivalent model based on an uncertain fuzzy migration system by utilizing a functional language and the theorem prover, so that whether a fuzzy mutual simulation relationship exists between fuzzy systems or not is quickly acquired. Furthermore, correctness can be strictly guaranteed, and behavior equivalence of different fuzzy migration systems is effectively guaranteed.

Description

Technical Field

[0001] The invention belongs to the field of computers, and in particular relates to a fuzzy mutual simulation formal verification and modeling method based on a theorem prover. Background Art

[0002] The importance of computers in social life is becoming more and more prominent. Any computer error may cause losses, especially in key areas such as transportation, medical and other sensitive systems, the correctness and stability of the hardware and software systems are crucial. In order to ensure the correctness and stability of the system, computer scientists have proposed formal methods. The reliability and security of software are the core issues of software development, especially with the enhancement of functions and computing speed, the complexity of algorithm programs is also increasing. How to develop reliable software systems has become a huge challenge. With the continuous application of fuzzy mathematics in the industrial field, different fuzzy systems have also been developed in industrial systems. Fuzzy theory also plays an important role in the quantitative analysis of computers, such as in automatic control, artificial intelligence, weather forecasting, etc. When analyzing fuzzy systems, for the same fuzzy system, a variety of different fuzzy migration systems will be established. The fuzzy mutual simulation algorithm will be a good judgement of whether these fuzzy migrations can be simplified to the same fuzzy migration system, thereby ensuring their consistency. With the in-depth research and wide application of algorithms, the requirements for the credibility of algorithms are also constantly increasing.

[0003] Bisimulation is an important concept in concurrency theory, which is used to test the behavioral equivalence between systems or system states. Bisimulation is now used in many fields of computer science. This method can identify migration systems with similar behaviors, that is, systems that satisfy the bisimulation relationship can imitate each other's actions. In recent years, the concept of bisimulation has been introduced into fuzzy migration systems, called fuzzy bisimulation. Fuzzy bisimulation introduces the concept of fuzziness theory on the basis of bisimulation to deal with the uncertainty of system behavior, and further considers the fuzziness of states and transitions. The fuzzy bisimulation algorithm is used to determine whether fuzzy systems are equivalent, thereby simplifying the fuzzy system and representing it as a model with a higher degree of abstraction, effectively dealing with uncertainty, avoiding the problem of state space explosion, and improving the efficiency of system verification and analysis. With the continuous application of fuzzy mathematics in the industrial field, different fuzzy systems have also been developed in industrial systems. Fuzzy theory also plays an important role in the quantitative analysis of computers, such as in automatic control, artificial intelligence, weather forecasting, etc.

[0004] At present, the common definitions of mutual simulation based on fuzzy migration systems are mainly divided into two categories: based on relational closure and based on relational promotion. On this basis, the research related to fuzzy mutual simulation has been continuously deepened and fruitful results have been achieved. The credibility requirements for fuzzy mutual simulation and its related algorithms are also constantly increasing. However, the existing research focuses on the theoretical definition of fuzzy mutual simulation, the optimization of algorithms and modal logic. The definition of fuzzy mutual simulation involves mathematical properties, and the termination and correctness of fuzzy mutual simulation algorithms have not been strictly proved. They are only manually deduced and proved through natural language, or the algorithms are tested through examples, resulting in low credibility, low degree of automation, and the inability to guarantee the correctness of the algorithms. Compared with manual mathematical proofs, mechanized theorem proofs can ensure strict correctness, which is of great significance to improving the security and reliability of fuzzy mutual simulation judgment algorithms.

[0005] In general, how to quickly determine whether there is a fuzzy mutual simulation relationship between fuzzy systems while ensuring strict accuracy is a technical problem that needs to be solved urgently in this field. Summary of the invention

[0006] To solve the above problems, the present invention proposes a fuzzy bisimulation formal verification and modeling method based on theorem prover, which can model and verify a variety of fuzzy bisimulation definitions and algorithms. The specific steps are as follows:

[0007] A fuzzy bisimulation formal verification and modeling method based on a theorem prover comprises the following steps:

[0008] Step S1: construct a fuzzy bisimulation system through Isabelle / HOL theorem prover;

[0009] In the construction process, the fuzzy mutual simulation relationship is defined through the fuzzy system, specifically: the system defines the fuzzy mutual simulation relationship based on the closed set of relations or based on the relation enhancement;

[0010] The fuzzy bisimulation system is formalized as a weighted directed graph consisting of several triples, expressed as:

[0011] (S, Act, →);

[0012] in, (S, Act, →) represents a triple, S, Act represent the storage state set and action respectively, → represents the transition set, F(S) represents all fuzzy sets of the storage state set S, each transition represents a storage state state reaching a fuzzy set through an action, × represents the Cartesian product;

[0013] During the construction process, the fuzzy bisimulation relations based on relation closure or relation lifting are formally verified;

[0014] Step S2: importing a pair of systems to be tested or verified into the fuzzy bisimulation system in step S1, and predicting whether there is a fuzzy bisimulation relationship between the pair of systems.

[0015] Furthermore, the fuzzy mutual simulation relationship is defined by the fuzzy system in step S1, specifically:

[0016] Define (S, Act, →) as a fuzzy system and define the relationship between states If for any (s, t)∈R, for any There is a migration If μ(B) = ν(B) holds for any closed set of relations B∈S / R, then the relation R is called a fuzzy bisimulation relation;

[0017] Among them, the state pair (s, t)∈R represents belongs to the relation R, It means that a transition state s reaches the fuzzy set μ through action α. It means that state t reaches fuzzy set ν through action α; S / R represents all relational closed sets obtained by dividing the state set S by the state-to-state relation R, μ(B)=ν(B) means that for any relational closed set B, the maximum membership value under the fuzzy set μ is equal to the maximum membership value under the fuzzy set ν.

[0018] Furthermore, the fuzzy mutual simulation relationship is defined by the fuzzy system in step S1, specifically:

[0019] Based on the definition of relation-based lifting fuzzy bisimulation: define (S, Act, →) as a fuzzy transition system and define the relationship between states If for any (s, t)∈R, for any There is a transfer So that (μ, ν)∈R F Established, where α∈Act, then the relation R is called a fuzzy bisimulation relation;

[0020] Where (μ, ν)∈R F The condition is: let S be a state set, μ, ν∈F(S), if and only if for any s∈S, such that Then (μ, ν)∈R F ;

[0021] where (μ, ν)∈R F Indicates that the fuzzy set pair (μ, ν) belongs to the lifting relation R F ; μ, ν∈F(S) means μ and ν are fuzzy sets; and Denotes the sets {s′|(s, s′)∈R} and {s′|(s′, s)∈R}, where {s′|(s, s′)∈R} means that if the state pair (s, s′) belongs to the relation R, then the state s′ belongs to the set

[0022] Indicates that the membership of state s in the fuzzy set μ is less than or equal to the state set The maximum membership degree in the fuzzy set ν, and the membership degree of state s in the fuzzy set ν is less than or equal to the state set The maximum membership value under the fuzzy set μ.

[0023] Furthermore, in the construction process of step S1, the fuzzy mutual simulation relationship is defined by the fuzzy system, and the two mutual simulation relationships are formally verified as follows:

[0024] Construct Theorem 1: Based on the verification of the relational closure theorem based on relational lifting, it is proved that when the input relation satisfies the fuzzy bisimulation based on relational lifting, it also satisfies the fuzzy bisimulation based on relational closure.

[0025] Construct Theorem 2: Based on the verification of the relation lifting theorem based on relation closure implication, it is proved that when the input relation satisfies the fuzzy bisimulation of relation closure, it also satisfies the fuzzy bisimulation of relation lifting;

[0026] Construct Theorem 3 to assist in proving Theorem 1: When the input relation satisfies the fuzzy bisimulation based on relation lifting, if the fuzzy set (μ, ν) belongs to the lifting relation, that is, (μ, ν)∈R F , then for any relational closed set B, μ(B) = ν(B), formalized as:

[0027] Several lemmas are constructed to assist in proving Theorem 3, specifically:

[0028] Lemma 1: When the input relation satisfies the fuzzy bisimulation based on relation lifting, for any state s1 in relation r, there exists a relation closed set B partitioned by relation r, such that state s1 belongs to B; that is, the states in the relation all exist in the set partitioned by the relation;

[0029] Lemma 2: When the input relation satisfies the fuzzy bisimulation based on relation lifting, the conclusion is that for any relation closed set B, any state s is a state in relation r, which can be formalized as:

[0030] Lemma 3: When the input relation satisfies the fuzzy bisimulation based on relation lifting, the conclusion is that any state s1 in the equivalence relation r, if s1 belongs to the relation closed set B, is also equal to B under the corresponding action;

[0031] Lemma 4: Prove that equal lists or sets of states have equal maximum membership under fuzzy sets;

[0032] Lemma 5: Prove that the relational closure is a finite set;

[0033] Lemma 6: Prove that under the same set premise, the relation promotion operation implies the relation closure operation, specifically:

[0034] If the relation is finite, then for any state s1 belonging to B, This implies that μ(B) = ν(B); formally, it is expressed as:

[0035] Theorem 4 is constructed to assist in proving Theorem 2. Specifically, when the input relation satisfies the fuzzy bisimulation based on relation lifting, the conclusion shows that if for any relation closed set B, μ(B) = ν(B), then the fuzzy sets μ and ν belong to the lifting relation, that is, (μ, ν)∈R F ;

[0036] Lemma 1, Lemma 2, Lemma 3 and Lemma 4 are used to assist in proving Theorem 4.

[0037] Furthermore, in step S2, a pair of systems to be tested or verified is introduced into the fuzzy bisimulation system in step S1 to predict whether there is a fuzzy bisimulation relationship between the pair of systems, specifically:

[0038] For all state pairs (s, t) in the input relation r, if for any There is a migration So that (μ, ν)∈R F If it holds, keep the state pair, otherwise delete it; if the state pair in r is deleted, perform the above operation on the output result again until no state pair is deleted, then directly output the result;

[0039] During this process, build the BIS_induct function to output the results.

[0040] Furthermore, the prediction result in step S2 is formally verified as follows:

[0041] Theorem 5 is constructed on the basis of Theorem 3: The output of the model is a fuzzy bisimulation relationship based on the closed set of relations; specifically:

[0042] When the input relation satisfies the fuzzy bisimulation based on relation lifting, the conclusion is that if the output result r of the BIS_induct function and the result of the input discriminant function are True, then r is a fuzzy bisimulation relation of relation closed set;

[0043] Several lemmas are constructed to assist in proving Theorem 5, specifically:

[0044] Lemma 7: Prove that the constructed function satisfies the algorithm correctness; specifically:

[0045] When the input relation satisfies the fuzzy bisimulation based on relation lifting, the conclusion is that no state pair is deleted in the output result r of BIS_induct, that is, r is a fuzzy bisimulation relation of relation closed set;

[0046] Lemma 8: The output result is the fuzzy bisimulation relation of the relation closure;

[0047] Lemma 9: The output result is a fuzzy bisimulation relation of relation promotion.

[0048] Compared with the prior art, the present invention has the following advantages: Based on the fuzzy mutual simulation modeling and verification of uncertain fuzzy migration systems, the present method can quickly obtain whether there is a fuzzy mutual simulation relationship between fuzzy systems. The present method ensures the strict correctness of the algorithm, and uses theorem proving tools to define the properties involved and mechanical verification of the correctness of the algorithm, thereby effectively ensuring the behavioral equivalence of different uncertain fuzzy migration systems. At the same time, the present method can be extended to the fuzzy mutual simulation definition of other uncertain fuzzy migration systems and the modeling and verification of the algorithm. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 A flowchart of the steps of a fuzzy bisimulation formal verification and modeling method based on theorem prover.

[0050] Figure 2 The figure is a flowchart of the steps of a formal verification method based on fuzzy mutual simulation formal verification and modeling method of theorem prover. DETAILED DESCRIPTION

[0051] The following describes the embodiments of the present invention through specific examples, and those skilled in the art can easily understand other advantages and effects of the present invention from the contents disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed in various ways based on different viewpoints and applications without departing from the spirit of the present invention.

[0052] Reference Figure 1 , a fuzzy bisimulation formal verification and modeling method based on theorem prover, in one example, includes the following steps:

[0053] Step S1: construct a fuzzy bisimulation system through Isabelle / HOL theorem prover;

[0054] In the construction process, the fuzzy mutual simulation relationship is defined through the fuzzy system, specifically: the system defines the fuzzy mutual simulation relationship based on the closed set of relations or based on the relation enhancement;

[0055] The fuzzy bisimulation system is formalized as a weighted directed graph consisting of several triples, expressed as:

[0056] (S, Act, →);

[0057] in, (S, Act, →) represents a triple, S, Act represent the storage state set and action respectively, → represents the transition set, F(S) represents all fuzzy sets of the storage state set S, each transition represents a storage state state reaching a fuzzy set through an action, and × represents the Cartesian product:

[0058] During the construction process, the fuzzy bisimulation relations based on relation closure or relation lifting are formally verified;

[0059] Step S2: importing a pair of systems to be tested or verified into the fuzzy bisimulation system in step S1, and predicting whether there is a fuzzy bisimulation relationship between the pair of systems.

[0060] Furthermore, the fuzzy mutual simulation relationship is defined by the fuzzy system in step S1, specifically:

[0061] Define (S, Act, →) as a fuzzy system and define the relationship between states If for any (s, t)∈R, for any There is a migration If μ(B) = ν(B) holds for any closed set of relations B∈S / R, then the relation R is called a fuzzy bisimulation relation;

[0062] Among them, the state pair (s, t)∈R represents belongs to the relation R, It means that a transition state s reaches the fuzzy set μ through action α. It means that state t reaches fuzzy set ν through action α; S / R represents all relational closed sets obtained by dividing the state set S by the state-to-state relation R, μ(B)=ν(B) means that for any relational closed set B, the maximum membership value under the fuzzy set μ is equal to the maximum membership value under the fuzzy set ν.

[0063] Furthermore, the fuzzy mutual simulation relationship is defined by the fuzzy system in step S1, specifically:

[0064] Based on the definition of relation-based lifting fuzzy bisimulation: define (S, Act, →) as a fuzzy transition system and define the relationship between states If for any (s, t)∈R, for any There is a transfer So that (μ, ν)∈R F Established, where α∈Act, then the relation R is called a fuzzy bisimulation relation;

[0065] Where (μ, ν)∈R F The condition is: let S be a state set, μ, ν∈F(S), if and only if for any s∈S, such that Then (μ, ν)∈R F ;

[0066] where (μ, ν)∈R F Indicates that the fuzzy set pair (μ, ν) belongs to the lifting relation R F ; μ, ν∈F(S) means μ and ν are fuzzy sets; t Denotes the sets {s′|(s, s′)∈R} and {s′|(s′, s)∈R}, where {s′|(s, s′)∈R} means that if the state pair (s, s′) belongs to the relation R, then the state s′ belongs to the set

[0067] Indicates that the membership of state s in the fuzzy set μ is less than or equal to the state set The maximum membership degree in the fuzzy set ν, and the membership degree of state s in the fuzzy set ν is less than or equal to the state set The maximum membership value under the fuzzy set μ.

[0068] Furthermore, in the construction process of step S1, the fuzzy mutual simulation relationship is defined by the fuzzy system, and the two mutual simulation relationships are formally verified, so as to ensure the accuracy of the inference result: specifically:

[0069] Construct Theorem 1, specifically: Verification of the relation lifting implies relation closure theorem, proving that when the input relation r satisfies the fuzzy bisimulation of relation lifting, it also satisfies the fuzzy bisimulation of relation closure; formalized as:

[0070] theorem RL__R__closed_eq:

[0071] assumes″f≠[]″;

[0072] "equiv(allstates f)(set r)";

[0073] "(get_R_states_set r)=(get_NFTS_states_set f)";

[0074] "r = r2";

[0075] shows

[0076] theorem is a keyword for constructing theorem, indicating that this is a theorem. RL_R_closed_eq is the formal name of Theorem 1. assumes means assumption, i.e. premise, followed by specific conditions. shows means conclusion.

[0077] Construction Theorem 2: Verification of the relation lifting theorem implied by relation closure. It proves that when the input relation r satisfies the fuzzy bisimulation of relation closure, it also satisfies the fuzzy bisimulation of relation lifting. It can be formalized as:

[0078] theorem R_closed_RL_eq:

[0079] assumes "f ≠ []";

[0080] "equiv(allstates f)(set r)";

[0081] "(get_R_states_set r)=(get_NFTS_states_set f)";

[0082] "r=r2";

[0083] shows

[0084] Where, f = [] indicates that NFTS is not an empty system; equiv indicates the definition in the Equiv_Relations package of the Isabelle / HOL standard library, indicating that relation r is an equivalence relation; (get_R_states_setr) = (get_NFTS_states_setf) indicates that the states in relation r are consistent with the states in the migration system f, thereby ensuring that relation r is related to the migration system f;

[0085] Construct Theorem 3 to assist in proving Theorem 1: The premise is consistent with Theorem 1. If the fuzzy set (μ, ν) belongs to a lifting relation, that is, (μ, ν)∈RF, then for any relation closed set B, μ(B)=ν(B), which can be formalized as:

[0086] The specific definitions are as follows:

[0087] theorem RL_R_closed:

[0088]

[0089] Several lemmas are constructed to assist in proving Theorem 3, specifically:

[0090] Lemma 1: The premise remains unchanged, that is, when the input relation satisfies the fuzzy bisimulation based on relation lifting, for any state s1 in relation r, there exists a relation closed set B partitioned by relation r, so that state s1 belongs to B. That is, the states in the relation all exist in the set partitioned by the relation, which can be formalized as:

[0091] lemmaallstate_inR_closed_set:

[0092] f≠[](get_R_states_setr)=(get_NFTS_states_setf);

[0093]

[0094] Among them, lemma is the lemma construction keyword, indicating that this is a lemma;

[0095] Lemma 2: The premise remains unchanged. The conclusion is that for any closed set of relations B, any state s is a state in relation r, which can be formalized as:

[0096] lemma allstate_inR_closed_set:

[0097] Lemma 3: The premise remains unchanged, and the conclusion is that any state s1 in the equivalence relation r, if s1 belongs to the relation closed set B, then under the action of R_right and R_left functions, it is also equal to B, that is, Formally:

[0098] lemma R_closed_set_relation:

[0099] Lemma 4: Prove that equal lists or sets of states have equal maximum membership under fuzzy sets. Specifically, if the elements in list L and set B are the same, then the values ​​returned by get_values ​​and get_values_set are equal. This can be formalized as:

[0100] lemma get_values_set_eq:

[0101] Lemma 5: Prove that the relational closure set is a finite set. Specifically, the relational closure set partitioned by the function R_closed_set is finite. Formally, it can be expressed as:

[0102] lemma R_closed_set_finite:

[0103] Lemma 6: Prove that under the same set of premises, the relation promotion operation implies the relation closure operation. Specifically, if the relation closure is finite, then for any state s1 belonging to B, μ(s1)≤ This implies μ(B) = ν(B). The formal expression is: Formally:

[0104] lemma RL_R_closed_lemma:

[0105] Theorem 4 is constructed to assist in proving Theorem 2. Specifically, the premise is consistent with Theorem 1. The conclusion shows that if for any closed relation set B, μ(B) = ν(B), then the fuzzy sets μ and ν belong to a lifting relation, that is, (μ, ν)∈R F ; formalized as:

[0106] The specific definitions are as follows:

[0107] theorem R_closed_RL:

[0108] We prove Theorem 4 by Lemma 1, Lemma 2, Lemma 3 and Lemma 4.

[0109] Further, in one example, the step S2 imports a pair of systems to be tested or verified into the fuzzy bisimulation system in step S1 to predict whether there is a fuzzy bisimulation relationship between the pair of systems, specifically:

[0110] For all state pairs (s, t) in the input r, if for any There is a migration So that (μ, ν)∈R F If it holds, keep the state pair, otherwise delete it; if the state pair in r is deleted, perform the above operation on the output result again until no state pair is deleted, then directly output the result;

[0111] During this process, build the BIS_induct function to output the results.

[0112] Furthermore, the prediction result in step S2 is verified as follows:

[0113] Construct Theorem 5: The premise is consistent with the previous theorem. The conclusion is the result r of the BIS_induct function. The BIS_R_closed function, which is the discriminant function, is input to obtain the prediction result. The prediction result is True or false. If it is True, r is a fuzzy bisimulation relation of the relational closed set. It can be formalized as:

[0114] theorem BIS_induct_correct:

[0115] assumes"f≠[]""equiv(allstates f)(set r)";

[0116] "(get_R_states_set r)=(get_NFTS_states_set f)";

[0117] shows

[0118] Several lemmas are constructed to prove Theorem 5:

[0119] Lemma 7 The premise is consistent with Theorem 5. The conclusion is that no state pair is deleted in the result r of BIS, that is, r = BISrr2f, which implies that the result of r input BIS_R_closed function is True, that is, r is a fuzzy bisimulation relation of relation closure; formalized as:

[0120] lemma BIS_correct:

[0121] assumes"f≠[]""equiv(allstates f)(set r)";

[0122] "(get_R_states_set r)=(get_NFTS_states_set f)";

[0123] "r=r2";

[0124] shows

[0125] Lemma 8: All state pairs in the result r output by the BIS function can satisfy the condition of the if expression in the BIS_R_closed function; formalized as:

[0126] lemma BIS_Distr_equal:

[0127] Lemma 9: All state pairs in r output by the BIS function can satisfy the conditions of the if expression in the BIS function; formalized as:

[0128] lemma BIS_R_correct:

[0129] After the auxiliary lemmas are mechanically proved in Isabelle / HOL, the five theorems can be verified by interactively proving them with the Isabelle / HOL theorem prover and using Isabelle / HOL's Sledgehammer tool to automatically find other theorems for auxiliary proof.

[0130] Through the above-mentioned mechanical formal proof, the accuracy of the prediction results obtained by the fuzzy mutual simulation system modeled in this application can be strictly ensured.

[0131] The present invention is described in detail above in conjunction with the embodiments of the accompanying drawings. A person skilled in the art can make various variations of the present invention according to the above description. Therefore, some details in the embodiments should not be construed as limiting the present invention, and the present invention shall be protected by the scope defined by the attached claims.

Claims

1. A fuzzy bisimulation formal verification and modeling method based on theorem prover, characterized in that: The following steps are involved: Step S1: Construct a fuzzy bisimulation system through the Isabelle / HOL theorem prover, In the construction process, the fuzzy mutual simulation relationship is defined through the fuzzy system, specifically: the system defines the fuzzy mutual simulation relationship based on the closed set of relations or based on the relation promotion; The fuzzy bisimulation system is formalized as a weighted directed graph consisting of several triplets, expressed as: (S, Act, →); in, (S, Act, →) represents a triple, S, Act represent the storage state set and action respectively, → represents the transition set, F(S) represents all fuzzy sets of the storage state set S, each transition represents a storage state state reaching a fuzzy set through an action, × represents the Cartesian product; During the construction process, the fuzzy bisimulation relations based on relation closure or relation lifting are formally verified; Step S2: importing a pair of systems to be tested or verified into the fuzzy bisimulation system in step S1, and predicting whether there is a fuzzy bisimulation relationship between the pair of systems.

2. A method for modeling a mutual simulation verification model based on a theorem prover as claimed in claim 1, characterized in that: The fuzzy mutual simulation relationship is defined by the fuzzy system in step S1, specifically: Define (S, Act, →) as a fuzzy system and define the relationship between states If for any (s, t)∈R, for any There is a migration If for any closed set of relations B∈s / R, μ(B)=v(B) holds, then the relation R is called a fuzzy bisimulation relation; Among them, the state pair (s, t)∈R represents belongs to the relation R, It means that a transition state s reaches the fuzzy set μ through action α. It means that state t reaches fuzzy set v through action α; S / R represents all relational closed sets obtained by dividing the state set S by the state-to-state relation R, μ(B)=v(B) means that for any relational closed set B, the maximum membership value under the fuzzy set μ is equal to the maximum membership value under the fuzzy set v.

3. A method for modeling a mutual simulation verification model based on a theorem prover as claimed in claim 2, characterized in that: The fuzzy mutual simulation relationship is defined by the fuzzy system in step S1, specifically: Based on the definition of relation-based lifting fuzzy bisimulation: define (S, Act, →) as a fuzzy transition system and define the relationship between states If for any (s, t)∈R, for any There is a transfer So that (μ, ν)∈R F Established, where α∈Act, then the relation R is called a fuzzy bisimulation relation; Where (μ, v)∈R F The condition is: let S be a state set, μ,v∈F(S), if and only if for any s∈S, such that Then (μ, v)∈R F ; where (μ, v)∈R F Indicates that the fuzzy set pair (μ, ν) belongs to the lifting relation R F ; μ, ν∈F(S) means μ and ν are fuzzy sets; and Denotes the sets {s′|(s, s′)∈R} and {s′|(s′, s)∈R}, where {s′|(s, s′)∈R} means that if the state pair (s, s′) belongs to the relation R, then the state s′ belongs to the set Indicates that the membership of state s in the fuzzy set μ is less than or equal to the state set The membership degree of state s in the fuzzy set v is the maximum, and the membership degree of state s in the fuzzy set v is less than or equal to the state set The maximum membership value under the fuzzy set μ.

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