Discriminant solver for polynomial time

Abstract

The invention belongs to the technical field of digital circuit design, and particularly relates to a polynomial time discriminant solver, which comprises the following steps of: S1, researching, applying and developing a Boolean function and a Boolean satisfiability discriminant solver in digital circuit design; comprising the following steps: judging whether a satisfiability condition exists or not by utilizing a Boolean satisfiability judgment solver and quickly finding out a working condition; and S2, selecting a proven NPC problem SAT, giving a solution of the SAT, and proving that the solution is completed in polynomial time, thereby proving that P is equal to NP. According to the method, by utilizing examples of three groups of circuit algebras, the high efficiency and usefulness of the given SAT solver are explained, and a complete mathematical proof is given, so that the method has the advantages of high efficiency, high efficiency and high efficiency. Therefore, the cracking of the Pvs NP problem, which is one of seven unsolved problems, is declared. The solver is specially designed for a circuit, and the major problems in the aspects of computer science and application are solved.

Description

Technical Field

[0001] The invention relates to the technical field of digital circuit design, in particular to a polynomial time discriminant solver. Background Art

[0002] Switching algebra refers to a special Boolean algebra that discusses switching circuits. Let K be a set of (abstract) switches, "1, 0" represent a switch that is always on and a switch that is always off, respectively; "∪, ∩, -" represent the parallel, series, and reverse connection of switches, respectively. The Boolean algebra 〈K, ∪, ∩, -, 0, 1〉 formed by this is called switching algebra (sometimes A∩B is simply written as AB, and A is written as A′ with a horizontal line above it). Switching algebra has very important application value. It is a powerful tool for circuit analysis and circuit design.

[0003] Switch algebra is a binary algebra, that is, all variables and constants can only take one of two values: 0 or 1; quantities that are not naturally binary must be encoded in binary format. Physically, they can represent a light on and off, a switch on and off, a low voltage and a high voltage, a magnetic field in one direction or another. Switch algebra is an application of logical algebra, studying the on and off conditions of circuits composed of parallel and series switches, with variables representing switches; the numbers 0 and 1 represent the disconnection and connection of switches respectively; the three basic operations "-", "∧", "∨" (i.e.: -, ∩, ∪) represent the basic connection methods of switches: inversion, series, parallel (reverse can be represented by a letter with a short dash or '). Using algebraic methods to solve the analysis and design problems of switch circuits is widely used in electronic computers, automatic program control, and telephone exchange systems. From the point of view of algebra, it doesn't matter what the physical meaning is. In the laboratory, one of the physical representations will be selected to represent one of the values.

[0004] The basic unit that constitutes the switch circuit, there is a certain logical relationship between the input and output of each gate circuit. Here, logic refers to the relationship between "conditions" and "results". The input signal of the circuit reflects the "conditions", and the output of the circuit reflects the "results", so that the output and input of the circuit represent a certain logical relationship. The most basic logical relationship can be summarized as AND, OR, and NOT. Other complex logical relationships are composed of these three basic logical relationships. Therefore, basic gate circuits (see AND gate, OR gate, NOT gate, XOR gate) can be used to form digital circuits with various logical functions. In addition to basic gate circuits, commonly used gate circuits include AND-NON, OR-NON, AND-OR-NON and other gate circuits. All gate circuits have series of integrated circuit products for selection.

[0005] In logic circuits, there are two opposing logic states, such as "high" and "low" potential, "presence" and "absence" of pulses, "closed" and "opened" switches, "true" and "false" things, etc. The symbols 0 and 1 are usually used to represent two opposing logic states, called logic 0 and logic 1, so that various components with only two states can be selected to form circuits with various logic functions, such as relays, switches, diodes and triodes, integrated gate circuits, etc.

[0006] In logic circuits, there are two logic systems: one uses 1 to represent high potential and 0 to represent low potential, which is the "positive logic system". The other uses 1 to represent low potential and 0 to represent high potential, which is the "negative logic system". For the same circuit, you can use the positive logic system or the negative logic system. Depending on the positive and negative logic systems selected, even the same circuit has different logical functions.

[0007] In logic circuits, the high and low potentials are often described by high level and low level, with the unit being V (volt). Due to the influence of factors such as temperature changes, power supply voltage fluctuations, interference, and changes in component characteristics, the actual high and low levels are not fixed values, so a level variation range is usually considered. If the level is within this range, it is judged as 1 state or 0 state.

[0008] The Boolean satisfiability problem (SAT) is the first proven NP-complete problem in the field of computational complexity. This difficult combinatorial problem has always attracted the attention of researchers due to its wide range of applications and the wide variety of problems that can be reduced to SAT. In terms of theoretical research, many combinatorial problems can be expressed as propositional formulas and solved by running SAT solvers, such as graph coloring, vertex cover, and clique detection. In addition, SAT solving has many industrial applications, such as bounded model checking, configuration management, and equivalence checking in circuit design. It is also an important component in logic synthesis, and many SAT solvers are designed specifically for this task. Therefore, SAT solving is not only of great research significance, but can also help the industry achieve a more economical workflow.

[0009] Since the P / NP problem remains unsolved, researchers are in awe of the difficulty of the SAT problem and work hard to design efficient SAT solvers. At the same time, machine learning, especially the emerging deep learning technology, has been applied to the field of combinatorial optimization and has stimulated many promising new research directions. In recent years, fields such as graph matching and mixed integer programming have been studied in depth, and the SAT field is no exception. As one of the most cutting-edge events for SAT solvers, in past SAT competitions, there have been several solvers that have won awards using machine learning technology, including Kissat_MAB (ranked first in the 2021 SAT competition) and MapleCOMSPS (ranked first in the 2016 SAT competition). In addition to using machine learning technology to assist traditional solvers, end-to-end frameworks such as NeuroSAT have also achieved the integration of machine learning and SAT solving, thereby achieving automatic derivation and saving manpower.

[0010] To this end, a polynomial time discriminant solver is proposed to solve the above problems. Summary of the invention

[0011] The object of the present invention is to provide a polynomial time discriminant solver to solve the problems raised in the above background technology.

[0012] To achieve the above object, the present invention provides the following technical solution: a polynomial time discriminant solver, comprising:

[0013] S1. Research, application and development of Boolean functions and Boolean satisfiability solvers in digital circuit design, including using Boolean satisfiability solvers to determine whether there are satisfiable conditions and quickly find their working conditions;

[0014] S2. Choose a proven NPC problem SAT, give a solution to SAT, and prove that the solution can be completed in polynomial time, thus proving that P = NP;

[0015] S3. Determine whether there is an interpretation problem that satisfies a given Boolean formula; determine propositional logic and Boolean satisfiability problems; determine paradigms and propositional logic;

[0016] S4, Unified theory to solve the P vs NP problem, including unified theory mapping criteria, specific working methods, relevant algorithm descriptions and simple examples of SAT solvers;

[0017] S5. Determine the technical details and technical derivation, including using minimal terms to prove the validity and completeness of the algorithm, using power set lattices to prove the validity and completeness of the algorithm, and using multi-products to prove the validity and completeness of the algorithm;

[0018] S6. Identify Boolean functions and their expressions, and understand the two dual standard forms of Boolean algebra;

[0019] S7. List practical examples of digital circuits;

[0020] S8. Conduct a comprehensive analysis of the switching circuit.

[0021] Preferably, in S1, the combination of the wires connecting the two endpoints T1 and T2 and the switches is called a switch network, a, b, c, ... are used to represent the switches, 1 and 0 are used to represent the "connected" and "disconnected" of the switches and the "connected" and "disconnected" of the entire circuit respectively, then whether the circuit on this network can be connected depends entirely on the state of these switches (0 or 1) and their connection method (parallel, series, or inverted), and is therefore a function of the switches.

[0022] Preferably, in S3, it is asked whether the variables of a given Boolean formula can be consistently replaced by the values ​​TRUE or FALSE, and the formula calculation result is TRUE. If this is the case, the formula is called satisfiable; on the other hand, if there is no such assignment, then for all possible variable assignments, the function represented by the formula is FALSE, and the formula is not satisfiable. For example, the formula "a AND NOT b" is satisfiable because values ​​a=TRU and b=FALSE can be found, which makes (a AND NOTb)=TRUE. On the contrary, "a AND NOT a" is not satisfiable.

[0023] Preferably, in S4, the unified theory mapping criterion is to convert the language into propositional logic, unify the propositional logic into CNF form, the logical value of the clause must be positive, if the propositional logic is in disjunctive normal form, then negate the disjunctive normal form and convert it into conjunctive normal form, if the clause is in conjunctive normal form but the logical value is not positive, then transform and split it so that the clause corresponding to the semantics satisfies the conjunctive normal form and is in a positive normal form;

[0024] The specific working method is to convert the problem of judging whether a circuit can be connected into a Boolean satisfiability judgment problem, and develop a polynomial time judgment solver. When the circuit can be connected, the truth table is used to find out its working conditions;

[0025] The related algorithm description starts from partially ordered sets and then goes to lattice knowledge, and then uses Boolean algebra and set theory to solve the problem.

[0026] Preferably, the comprehensive analysis of the switch circuit in S8 refers to finding out the conditions for connecting and disconnecting the circuit, synthesizing the automation equipment, and designing the circuit according to the given conditions so that the circuit can meet these conditions.

[0027] Compared with the prior art, the present invention has the following beneficial effects:

[0028] The present invention uses three sets of circuit algebra examples to illustrate the efficiency and usefulness of the proposed SAT solver, and gives a complete mathematical proof, thereby declaring the solution of the P vs NP problem, one of the seven Millennium Unsolved Problems. The solver is specifically designed for circuit design and solves major problems in computer science and applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Figure 1 This is a schematic diagram of the three-step solution to SAT;

[0030] Figure 2 This is a Hasse diagram. Figure 1 ;

[0031] Figure 3 This is a Hasse diagram. Figure 2 ;

[0032] Figure 4 It is the circuit diagram in circuit analysis;

[0033] Figure 5 A circuit diagram of Embodiment 2 of circuit analysis;

[0034] Figure 6 It is the circuit diagram in circuit synthesis;

[0035] Figure 7 is a station diagram of Embodiment 2 of the circuit synthesis;

[0036] Figure 8 It is a circuit diagram of Embodiment 2 of circuit synthesis. DETAILED DESCRIPTION

[0037] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0038] In the description of the present invention, it is necessary to understand that the terms "upper", "lower", "front", "back", "left", "right", "top", "bottom", "inside", "outside", etc., indicating the orientation or position relationship are based on the orientation or position relationship shown in the drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as a limitation on the present invention.

[0039] Example:

[0040] See also Figure 1-8 The present invention provides a technical solution: a polynomial time discriminant solver, comprising:

[0041] S1. Research, application and development of Boolean functions and Boolean satisfiability solvers in digital circuit design, including using Boolean satisfiability solvers to determine whether there are satisfiable conditions and quickly find their working conditions;

[0042] Boolean algebra and Boolean satisfiability solvers are widely used, most notably in automation and computer technology, and also in artificial intelligence, machine proof, mathematical logic, probability theory, measure theory, topology, and functional analysis. This patent proposes a new method for using Boolean algebra and Boolean satisfiability solvers in the analysis, synthesis, and design of logic circuits.

[0043] S2. Choose a proven NPC problem SAT, give a solution to SAT, and prove that the solution can be completed in polynomial time, thus proving that P = NP;

[0044] S3. Determine whether there is an interpretation problem that satisfies a given Boolean formula; determine propositional logic and Boolean satisfiability problems; determine paradigms and propositional logic;

[0045] S4, unified theory to solve the P vs NP problem, including unified theory mapping criteria;

[0046] Example:

[0047] 1. You can wear a shirt only. The CNF in the conversion cost patent is

[0048] 2. Use conjunction between rows and analysis between columns to get F.

[0049] 3. In circuit algebra, missing characters are not considered;

[0050] 4. Multiply each row and column vertically to get a row; vertical multiplication rule: Get a line of text F*; determine whether they are all negative text. If they are all negative text, then it is unsatisfactory. Otherwise, it is satisfactorily satisfied.

[0051] 5. When negative text intersects with negative text, it becomes negative text; when positive text intersects with positive text, it becomes positive text; when positive text intersects with negative text, it becomes negative text.

[0052] If the original circuit is satisfiable, then the feasible circuit is solved according to the algorithm of this patent. The specific process of finding a satisfiable solution is: convert the circuit algebra into the form of disjunctive normal form, and then calculate the minimum terms of the disjunctive normal form. If each minimum term can take a positive logical value, then it is a satisfiable solution;

[0053] The specific working method is to convert the problem of judging whether the circuit can be connected into a Boolean satisfiability judgment problem, and develop a polynomial time judgment solver. When the circuit can be connected, the truth table is used to find out its working conditions;

[0054] Related algorithm description: This algorithm starts from partially ordered sets and then goes to lattice knowledge, and then uses Boolean algebra and set theory to solve the problem. Based on the idea of ​​propositional logic that integrates algebra and geometry, it has gradually become an important tool for studying the laws of human thinking since it was publicly published in the mid-19th century. After the 1930s, it was widely used in electronic circuit and chip design. At present, the original results proposed by this algorithm have become one of the important basic theories of computer science. This patent will prove the completeness and effectiveness of the algorithm from two aspects;

[0055] A simple example of a SAT solver is as follows:

[0056] 1. Only one variable

[0057] (1)Judgment Can it be satisfied?

[0058] (2) Convert to CNF, that is, negate the original formula =

[0059] (3) The unified theoretical calculation table is as follows

[0060] F* is a line of all negative text, so F is unsatisfiable. The original formula is whether F is satisfiable.

[0061]

[0062] 2. Only one variable

[0063] (1)Judgment Is it satisfyable?

[0064] (2) Converted into CNF, i.e.

[0065] (3) The unified theoretical calculation table is as follows:

[0066]

[0067] F* is a line of all negative text, so F is unsatisfiable;

[0068] S5. Determine the technical details and technical derivation, including using minimal terms to prove the effectiveness and completeness of the algorithm; using the relevant theorems of Boolean algebra: Let<B;+,·,';0,1> is a Boolean algebra, F n is the set of all n-dimensional Boolean functions on it. If F n Any two functions f,g∈F n The three operations are as follows:

[0069] (f+g)(x 1 ,x 2 ,…,x n )=f(x 1 ,x 2 ,…,x n )+g(x 1 ,x 2 ,…,x n ),

[0070] (f·g)(x 1 ,x 2 ,…,x n )=f(x 1 ,x 2 ,…,x n )·g(x 1 ,x 2 ,…,x n ),

[0071] f'(x 1 ,x 2 ,…,x n )=(f(x 1 ,x 2 ,…,x n ))'

[0072] Then F n These operations constitute a A Boolean algebra with 1 elements whose zero element is a function that always takes the value 0 and whose unit element is a function that always takes the value 1.

[0073] There are two standard forms for Boolean functions, the miniterm normal form ("disjunctive normal form") and the maximal normal form ("conjunctive normal form").

[0074] Define n-ary Boolean expressions of the following form:

[0075]

[0076] It is called a minterm or fundamental conjunctive form, where a i ∈{0,1}, and stipulate

[0077]

[0078] In other words, the n-dimensional minimum term m k is the product of n factors, where the i-th factor is x i or x' i ; This factor actually corresponds to the positive and negative characters in the algorithm.

[0079] There are 2 minimum terms of n variables. n , and denote them as m 0 ,m 1 ,m 2 ,…, The following method is usually used to minimize the term m k ; First, express the subscript k in decimal and binary notation respectively, let

[0080] (k) 10 =(a 1 ·a 1 …a n ) 2 ,

[0081]

[0082] Now let n=1,2,3 write out all n-variable minimal terms as shown in the following table.

[0083] n=1,m k Number = 2

[0084]

[0085] n=2,m k Number = 4

[0086]

[0087] n=2,m k Number = 8

[0088]

[0089] Theorem 1: Minimal terms have the following properties:

[0090] (1) When i,j=0,1,…,2 n-1 hour,

[0091] (That is, the product of two different minima is 0, and the product of two identical minima is itself.)

[0092] (2) (That is, the sum of all n-dimensional minima is 1).

[0093] It can be seen from this that the total minimum term {m i} is the Boolean algebra F n of atoms.

[0094] Definition: Atom;<B;+,·,';0,1> is a Boolean algebra, a∈B, if for There are: (1) a≠0,

[0095] (2) x·a=a or x·a=0,

[0096] Then a is called an atom of B (they are the elements directly above the zero element in the Hasse diagram). According to the atomic representation theorem, F n Every Boolean algebra of can be expressed as their sum.

[0097] Definition 2: If for Boolean algebra<B;+,·,';0,1> If for every non-zero element x there is an atom a≤x, then B is called an atomic Boole algebra.

[0098] Theorem 3: Any finite Boolean algebra is an atomic Boolean algebra; the elements of an atomic Boolean algebra (especially a finite Boolean algebra) can be represented using atoms, and the structure of a finite Boolean algebra can be seen through the power set of the atomic set.

[0099] Theorem 4: Atomic Representation Theorem; Let<B;+,·,';0,1> is an atomic Boolean algebra, A is its atomic set, let π(x) = {a|a∈A and a≤x}, then x = ∑ a∈π(x) a. And regardless of the order, the representation of this atomic sum of x is unique.

[0100] Theorem 5: Stone representation theorem for finite Boolean algebras.

[0101] This explains why the algorithm only uses vertical multiplication, that is, the positive and negative products of the same characters, rather than the cross products of different characters, because the result of the cross product is zero. This sum of products actually corresponds to the F in this algorithm. * This also proves the completeness and reliability of the algorithm of this patent.

[0102] These operations constitute a n The Boolean algebra of elements, whose zero element is a function that always takes the value 0, and whose unit element is a function that always takes the value 1. In the Boolean function of this patent, there are two standard forms, namely, the miniterm normal form and the maximal term normal form.

[0103] The effectiveness and completeness of this algorithm are proved by using power set lattices;

[0104] Partially ordered sets and lattices

[0105] Definition: Let P be a set. If a binary relation “≤” on P satisfies the following three conditions, then “≤” is called a partial order relation (or partial order relation) on P:

[0106] (1) Reflexivity: a≤a;

[0107] (2) Antisymmetry: If a≤b and b≤a, then a=b;

[0108] (3) Transitivity: If a≤b and b≤c, then a=c;

[0109] A set P with a partial order relation “≤” is called a partial order set, denoted by<P;≤> . In particular, let S be a partially ordered set, if for If a≤b or b≤a exists, then S is called a total order set or a chain.

[0110] For a finite set P, a partial order relation is usually represented by a Hasse diagram, which uses the endpoints of a line segment to represent the elements of P, and the line segment connecting the upper and lower endpoints represents the relationship "≤". If endpoint a is below b, it means a≤b.

[0111] Example 1: Let A = {a, b}, the power set of A Then the partially ordered set The graph is as follows Figure 2 As shown;

[0112] Definition:<P;≤> is a partially ordered set if the element m of P is If m≤x, then m is called the least element of P; conversely, if the element n of P is If x≤n, then n is called the greatest element of P.

[0113] Definition:<P;≤> is a partially ordered set, T is a subset of P, if for If x≤u (or u≤x) holds, then the element u of P is called the upper (lower) bound of T.

[0114] Definition:<P;≤> is a partially ordered set, T is a subset of P, if for any upper bound x of T, u≤x, then the upper bound u of T is called the least upper bound, denoted by u=supT; on the other hand, if for any lower bound x≤v, then the lower bound v of T is called the greatest lower bound, denoted by v=infT;

[0115] For example: Let set A = {a, b, c}, P(A) be its power set, partial order set as follows Figure 3 It is easy to see that the largest element is A and the smallest element is If we take a subset T of P(A) 1 ={{b,c},{b},{c}}, then A and {b,c} are T 1 The upper bound of , and supT 1 ={b,c}; It is T 1 The lower bound of is also inf. If we take a subset T of P(A) 2 ={{a,b},{a},{b}}, then supT 2 ={a,b}, and

[0116] Definition:<L;≤> is a partially ordered set. If any two elements in L have a minimum upper bound and a maximum lower bound, then it is called<L;≤> It is lattic.

[0117] Example: Let A be any set, whose power set P(A) contains Composition Because it is a partially ordered set, and for sup{S,T}=S∪T,inf{S,T}=S∩T

[0118] This definition is the key to the proof of the algorithm later. * Isomorphic to F.

[0119] Definition: lattice<L;≤> Binary computation in :

[0120]

[0121]

[0122] They are called the union and intersection of elements a and b respectively, and are denoted as:

[0123] a∨b=sup{a,b},

[0124] a∧b=inf{a,b}.

[0125] This definition is the key to the proof of the algorithm later. * Isomorphic to F.

[0126] Theorem: Assume<L;≤> is a lattice, on which the union and intersection have the following properties: Let a, b∈L, then:

[0127] (1)a,b≤a∨b,a∧b≤a,b;

[0128] (2)

[0129] In (1), a,b≤a∨b can be written as a≤a∨b and b≤a∨b,a∧b≤a,b can be written as a∧b≤a,a∧b≤b. This is an important theorem when proving connectivity.

[0130] Theorem: (1) Assume<L;≤> It is a lattice. After defining the binary operations ∨ and ∧ on it, the following rules must be followed:

[0131] 1) Idempotence law a∨a=a,a∧a=a;

[0132] 2) Commutative law a∨b=b∨a,a∧b=b∧a

[0133] 3) Associative law (a∨b)∨c=a∨(b∨c),

[0134] (a∧b)∧c=a∧(b∧c).

[0135] 4) Absorption rate a∨(a∧b)=a,a∧(a∨b)=a

[0136] (2) Define two binary operations ∨ and ∧ on the set L so that they satisfy the four rules in (1), then<L;∨,∧> From this, we can get another definition of the lattice (another definition can also be used to prove the completeness and reliability of the algorithm, because these two definitions are equivalent, so this patent can use the operations and partial order relations on the lattice at the same time when discussing the lattice):

[0137] Definition: Let the set L have two binary operations ∨ and ∧ that satisfy the rules 1)-4) of Theorem 5.2.2, then we say<L;∨,∧> For the grid.

[0138] Theorem: Assume<L;≤> It is grid. Then the following are established:

[0139] (1)

[0140] (2)

[0141] Theorem: Duality Theorem: Let S be a true proposition about partially ordered sets. Then the dual proposition obtained by swapping all "≤" and "≥" (or swapping ∨ and ∧) is also a true proposition. (For example, the two idempotent laws in the theorem are dual equations of each other, and they are both true propositions; in addition, the transformation laws, associative laws, and absorption rates that appear in pairs also illustrate this phenomenon.)

[0142] Define subgrid: Let<L;∨,∧> is a lattice, T is a non-empty subset of L if T is closed with respect to ∨, ∧ (i.e. and a∧b∈T), then<T;∨,∧> yes<L;∨,∧> sub lattice.

[0143] Define lattice homomorphism: Let<L;∨,∧> and<L';+,·> are two lattices, if there exists a mapping σ from L to L' such that have:

[0144] σ(a∨b)=σ(a)+σ(b),

[0145] σ(a∧b)=σ(a)·σ(b),

[0146] Then σ is called a lattice homomorphism from L to L'.

[0147] This patent can also imitate the aforementioned algebraic structures to give the concepts of exceptional single homomorphism, full homomorphism, bimorphism (isomorphism), automorphism, and automorphism. Due to space limitations, they are no longer listed here one by one.

[0148] Theorem: Set the grid<L;∨,∧> Pass<L';+,·> If a, b∈L and a≤b, then σ(a)a≤'σ(b), where ≤ and ≤' are partial order relations in L and L' respectively.

[0149] Example: Let set A = {a, b}, its power set but<P(A);∪,∩> It is a grid. Let B={0,1}, then B 2 ={(0,0),(0,1),(1,0),(1,1)}, and stipulate that (a 1 ,a 2 )∨(b 1 ,b 2 )=(c 1 ,c 2 ), where c 1 =a 1 ∨b 1 ,c 2=a 2 ∨b 2 On the other hand (a 1 ,a 2 )∧(b 1 ,b 2 )=(d 1 ,d 2 ), where d 1 =a 1 ∧b 1 ,d 2 =a 2 ∧b 2 . This proves that 2 ; ∨,∧> is a lattice, which is denoted as σ:P(A)→B in this patent 2 , from which we can get Then σ is P(A) to B 2 isomorphic mapping.

[0150] This theorem fully proves that from F to F * is an isomorphic mapping.

[0151] In the algorithm based on unified theory in this patent, if the semantic system of the entire proposition is L, then according to the above definition, it can be proved that the CNF form F is a homomorphic mapping, from F to F * It is also a homomorphic mapping. It also proves the compactness from the language system to propositional logic and then to CNF. It also proves that from F to F * The compactness.

[0152] The mapping from the language system to the propositional calculus CNF is P(A) to B 2 isomorphic mapping.

[0153] In addition, according to Stone's topological representation theorem, it can be concluded that the mapping σ (i.e., from F to F) under the unified theory * ) is an isomorphic mapping. This mapping σ is also P(A) to B n isomorphic mapping of , where n is also the number of characters in this patent.

[0154] Next, we need to introduce the concept of complement lattice.

[0155] Classification of lattices: According to the definition of lattices, any two elements of a lattice have a minimum upper bound and a maximum lower bound. By mathematical induction, it can be proved that every finite sub-lattice of a lattice has a minimum upper bound and a maximum lower bound, but this property does not necessarily apply to infinite subsets of a lattice. Therefore, the following definition is given.

[0156] ​Definition: If every non-empty subset of a lattice L has sup and inf, then it is called a complete lattice. Obviously, every finite lattice is a complete lattice.

[0157] definition:

[0158] (1) If there is a zero element “0” in the lattice L, the following conditions are met:

[0159] Then 0 is called the zero element or universal lower bound of L.

[0160] (2) If there is element 1 in the lattice L, the following conditions are met:

[0161] Then 1 is called the unity element or universal upper bound of L.

[0162] It can be proved that if there are zero elements or units in the lattice L, then they must be unique.

[0163] For example: Let A = {a, b, c}, then The zero yuan is Unit is A;

[0164] A lattice with zero elements and units is called a bounded lattice. To highlight the boundedness, the general representation of a bounded lattice is<L;∨,∧;0,1> .

[0165] Definition:<L;∨,∧;0,1> is a bounded lattice, x∈L, if If x∨y=1, x∧y=0, then y is said to be the complement of x or x is the complement element of y. y is the complement element of x and is represented by y=x'.

[0166] Definition:<L;∨,∧;0,1> is a bounded lattice, if each element a has at least one complement a', then it is called<L;∨,∧;0,1> It is a complemented lattice.

[0167] Definition:<L;∨,∧> Yes, if The following dichotomy holds:

[0168] x∨(y∧z)=(x∨y)∧(x∨z),

[0169] x∧(y∨z)=(x∧y)∨(x∧z),

[0170] Then it is called<L;∨,∧> It is a distributive lattice.

[0171] Definition: If the lattice L is both a complemented lattice and a distributive lattice, then the element is called a Boole lattice.

[0172] Definition, examples and properties of Boolean algebra: Boolean algebra is Boolean, which can be defined using partial order relations or operations;

[0173] Definition: A distributive lattice with complement is called Boole algebra.

[0174] Since the complement of each element a' exists uniquely, and (a')' = a, we can regard complementation as a unary operation of order 2. Combining the results of defining binary operations on Boolean lattices gives another definition of Boolean algebra using algebraic operations.

[0175] Definition: Let B be a set with two binary operations ∨ and ∧ (for convenience, hereafter, they are represented by + and · according to convention) and a unary operation "'" and the elements 0 and 1. If the following five axioms are satisfied, then B is called a Boolean algebra, denoted by B=<B;+,·,,‘;0,1> :

[0176] (1) Closure:

[0177]

[0178] (2) Existence of zero element and unit:

[0179] for

[0180] for

[0181] (3) Existence of complement:

[0182]

[0183] (4) Commutative law:

[0184]

[0185] x+y=y+x,x·y=y·x;

[0186] (5) Distributive Law:

[0187]

[0188] x·(y+z)=x·y+x·z;

[0189] x+(y·z)=(x+y)·(x+z).

[0190] For example: It is known that the power set P(x) of a set X forms a lattice about ∪, ∩,'. Let X be an arbitrary set, P(x) be its power set, and its zero element and unit are empty sets respectively. and the complete set X, then<P(X);+,·,';0,1> This constitutes Boolean algebra, often called the dense algebra of X.

[0191] For example, the three operations are shown in the following table:

[0192]

[0193] It can be directly verified that it is Boolean algebra, which is also called binary algebra or switch algebra. Among them, switch algebra is commonly used in this patent.

[0194] Example: Let S be a set of propositional formulas, ∨, ∧ and Respectively represent the disjunction (p∨q), conjunction (p∧q) and negation of propositions p and q In addition, T represents a true proposition p→p, and F represents a false proposition. It can be proved that the propositional calculus system satisfies all the axioms of Boolean algebra, so it is also Boolean algebra. This algebra is called propositional algebra or formula algebra, and is expressed by express.

[0195] Since Boolean algebra is a complement distributive lattice, it has the properties of the corresponding algebraic structure. For ease of use, its operational properties and partial order properties are listed below.

[0196] Theorem: Let be a Boolean algebra, then it satisfies the following laws:

[0197] (1) Idempotence law: a+a=a, a·a=a;

[0198] (2) Commutative law: a+b=b+a, a·b=b·a;

[0199] (3) Associative law: (a+b)+c=a+(b+c);

[0200] (a·b)·c=a·(b·c);

[0201] (4) Absorption rate: a+(a·b)=a, a·(a+b)=a;

[0202] (5) Distributive law: a+(b·c)=(a+b)·(b+c),a·(b+c)=(a·b)+(b·c);

[0203] (6)(a·b)+(b·c)+(c·a)=(a+b)·(b+c)·(c+a);

[0204] (7) Elimination law:

[0205] (8) Boundedness: 0≤a≤1.

[0206] (9) 0-1 law: a+0=a, a+1=1; a·0=0, a·1=a;

[0207] (10) Complementarity law: a+a'=1, a·a'=0; 0'=1, 1'=0;

[0208] (11) DeMorgan's law: (a+b)'=a'·b', (a·b)'=a'+b';

[0209] (12)

[0210] (13)a+b=sup(a,b),a·b=inf(a,b);

[0211] (14)

[0212] (15) Duality Principle For an equation in Boolean algebra B, if we swap the “+”, “·”, “0”, and “1” in it, the result is still an equation.

[0213] In this way, (15) proves the isomorphic mapping relationship between the language system, propositional formula set, and propositional calculus system in this patent, and (13) proves that F and F* are in a mapping isomorphism relationship.

[0214] The construction and generation of CNF in Boolean algebra and algorithm are the same as other algebraic structures. In Boolean algebra, concepts such as sub-Boolean algebra and Boolean isomorphism should be established. In order to study its construction, the concept of atom should be introduced here.

[0215] Definition:<B;+,·,';0,1> is Boolean algebra, B 0 is a subset of B, if<B;+,·,';0,1> Itself is a Boolean algebra, then we call B 0 is a subalgebra of B. In other words, B 0 If the following properties are met, then it constitutes a subalgebra of B:

[0216] (1)0,1∈B 0 ;

[0217] (2)

[0218] Definition:<B;+,·,';0,1> and * ;∨,∧,-;0 * ,1 * > are two Boolean algebras, mapping It is called a Boole homomorphism if it satisfies the following conditions:

[0219] (1)

[0220] (2)

[0221] Just like other algebraic structures, concepts such as simple homomorphism, full homomorphism, isomorphism, automorphism, and automorphism can also be given.

[0222] Example: In any Boolean algebra<B;+,·,';0,1> In , the subset {0, 1} is its subalgebra.

[0223] Let A = {a}, its power set is is Boolean algebra, which is isomorphic to binary algebra. That can be concluded.

[0224] The validity and completeness of this algorithm are proved by using the multi-product.

[0225] In this patent, the language system, the propositional formula set, and the propositional calculus system are in a Boolean homomorphism relationship, and F and F* are also in a Boolean homomorphism relationship, so the compactness, effectiveness, and completeness of the algorithm in this article can be proved from three perspectives: power set algebra, Stone's Boolean representation theorem and its corollary, and Boolean homomorphism. This patent mainly proves this algorithm from the perspective of Boolean homomorphism.

[0226] For the construction of Boolean algebra and CNF, the concept of atoms is introduced below;

[0227] Define atom; let<B;+,·,';0,1> is a Boolean algebra, a∈B, if for have:

[0228] (1)a≠0; (2)x·a=a or x·a=0,

[0229] Then a is called an atom of B (they are the elements directly above the zero element in the Hasse diagram). According to the atomic representation theorem, F n Every Boolean algebra of can be expressed as their sum.

[0230] ​Definition: If for Boolean algebra<B;+,·,';0,1> If for every non-zero element x there is an atom a≤x, then B is called an atomic Boole algebra.

[0231] Theorem: Every finite Boolean algebra is an atomic Boolean algebra:

[0232] Atoms can be used to represent the elements of atomic Boolean algebra (especially finite Boolean algebra), and the structure of finite Boolean algebra can be seen through the power set of atomic sets.

[0233] Theorem: Atomic Representation Theorem<B;+,·,';0,1> is an atomic Boolean algebra, A is its atomic set, let π(x) = {a|a∈A and a≤x}, then x = ∑ a∈π(x) a. And regardless of the order, the representation of this atomic sum of x is unique.

[0234] Theorem: The sum of all atoms in an atomic Boolean algebra is equal to its largest element 1:

[0235] It can also be shown that any finite Boolean algebra is isomorphic to the power set algebra of its atomic set. A more precise statement is the following theorem:

[0236] Theorem: Stone Representation Theorem for Finite Boolean Algebras:

[0237] set up<B;+,·,';0,1> is a finite Boolean algebra, A is an atomic set of B, then the Boolean algebra B to the power set algebra Mapping And a≤x} constitutes an isomorphism between B and P(A). This theorem is consistent with the concept of maximal filter in Stone's topological representation theorem in this patent, and both prove that F and F* are also isomorphic mappings.

[0238] As a corollary, there is the following theorem:

[0239] Theorem 1: The number of elements of every finite Boolean algebra B must be a power of 2 n , where n is the number of atoms of B;

[0240] Theorem 2: If two Boolean algebras have the same number of elements, then they must be isomorphic.

[0241] For general Boolean algebra (whether or not it is atomic Boolean algebra) the following theorem holds.

[0242] Theorem: Stone states theorem that every Boolean algebra is isomorphic to a set algebra.

[0243] This theorem is consistent with the concept of maximal filter in Stone's topological representation theorem in this patent. It also proves that F and F* are also isomorphic mappings.

[0244] First, we discuss two special types of Boolean algebra: and

[0245] (1) From the above theorem, we can see that the number of elements of any finite Boolean algebra = 2 r , so the smallest Boolean algebra is the binary algebra B 2 ; 2 A slightly larger Boolean algebra is B 2 ={0,1,α,β}, its operation table is shown in the following table:

[0246]

[0247]

[0248] (2) In B 2 Based on B 2 The r-multiple product set

[0249] And taking it as the base set, three operations are defined as follows:

[0250] set up Regulation

[0251] (β 1 ,β 2 ,…β r-1 ,β r )+(β' 1 ,β' 2 ,…β' r-1 ,β' r )=(γ 1 ,γ 2 ,…,γ r-1 ,γ r ),

[0252] (β 1 ,β 2 ,…β r-1 ,β r )·(β' 1 ,β' 2 ,…β' r-1 ,β' r )=(δ 1 ,δ 2 ,…,δ r-1 ,δ r ),

[0253] (β 1 ,β2 ,…,β r-1 ,β r )'=(β' 1 ,β' 2 ,…β' r-1 ,β' r )

[0254] Among them, γ i =max(β i ,β' i ),δ i =min(β i ,β' i )

[0255] β′ i =1-β i .

[0256] Among them, max represents the maximum term and min represents the minimum term.

[0257] It has been verified that These three operations constitute Boolean algebra Its zero element 0 and unit 1 are <0,0,…,0> and <1,1,…,1> respectively.

[0258] There are also the following inferences about the above two types of algebra. Theorem Boolean algebra Must go with isomorphic, so any Boolean algebra must be isomorphic to some Isomorphism.

[0259] Example: Take B 4 ={0,α,β,1} and Take an example to illustrate the isomorphism between them and make a mapping:

[0260]

[0261] That is, we know that σ is an isomorphic mapping, so

[0262] From the perspective of direct product (multi-product set), this theorem proves that F and F* are also isomorphic mappings. It also reveals the isomorphism between Boolean algebra and sets. Sets are a kind of geometric form, so the integration of algebra and geometry proves the completeness and reliability of this theorem.

[0263] S6. Identify Boolean functions and their expressions, and understand the two dual standard forms of Boolean algebra;

[0264] In general, functions on Boolean algebra B may not be represented by the formula on it, but in binary Boolean algebra 2 ; +, ·, '; 0, 1>, each of which is from​ To B 2 All functions can be expressed by expressions containing +,·,' (the completeness theorem of two-valued Boolean algebra). The completeness theorem of two-valued Boolean algebra is also one of the proofs of this algorithm. It shows the completeness from propositional logic to propositional calculus.

[0265] In this patent, for simplicity, Boolean expressions are directly regarded as Boolean functions; after giving its recursive definition, the paradigm theorem will be introduced for use in designing automation equipment and digital circuits.

[0266] Definition:<B;+,·,';0,1> is a Boolean algebra. The following expression is called a Boolean function on B:

[0267] (1) The elements in the basis set B are Boolean functions;

[0268] (2) Variable x 1 ,x 2 ,…,x n is a Boolean function.

[0269] (3) If f and g are Boolean functions, then f+g, f·g, f', g' are also Boolean functions.

[0270] (4) Only expressions formed by (1), (2), and (3) are Boolean algebra.

[0271] Using the relevant theorems of Boolean algebra: Let<B;+,·,';0,1> is a Boolean algebra, F n is the set of all n-dimensional Boolean functions on it. If F n Any two functions f,g∈F n The three operations are as follows:

[0272] (f+g)(x 1 ,x 2 ,…,x n )=f(x 1 ,x 2 ,…,x n )+g(x 1 ,x 2 ,…,x n );

[0273] (f·g)(x 1 ,x 2 ,…,x n )=f(x 1 ,x 2 ,…,x n )·g(x 1 ,x 2 ,…,xn );

[0274] f'(x 1 ,x 2 ,…,x n )=(f(x 1 ,x 2 ,…,x n ))'.

[0275] Then F n These operations constitute a The Boolean algebra of 1 elements has a zero element that is a function that always takes the value 0, and a unit element that is a function that always takes the value 1. This theorem also partially proves the isomorphic mapping relationship between the language system, the propositional formula set, and the propositional calculus system in this patent.

[0276] There are two standard forms for Boolean functions, the miniterm normal form ("disjunctive normal form") and the maximal normal form ("conjunctive normal form").

[0277] Define an n-ary Boolean expression of the form

[0278]

[0279] It is called a minterm or fundamental conjunctive form, where a i ∈{0,1}, and stipulate

[0280]

[0281] In other words, the n-dimensional minimum term m k is the product of n factors, where the i-th factor is x i or x' i . This factor actually corresponds to the positive and negative characters in the algorithm.

[0282] There are 2 minimum terms of n variables. n , and denote them as m 0 ,m 1 ,m 2 ,…, The following method is usually used to minimize the term m k ; First, express the subscript k in decimal and binary notation respectively, let

[0283] Now let n=1,2,3 write out all n-variable minimal terms as shown in the following table.

[0284] n=1,m k Number = 2

[0285]

[0286] n=2,m k Number = 4

[0287]

[0288] n=2,m k Number = 8

[0289]

[0290] Theorem: Minimal terms have the following properties:

[0291] (1) When i,j=0,1,…,2 n-1 hour,

[0292] (That is, the product of two different minima is 0, and the product of two identical minima is itself.)

[0293] (2)

[0294] (That is, the sum of all n-dimensional minima is 1).

[0295] It can be seen from this that the total minimum term {m i} is the Boolean algebra F n atoms, according to the previous atomic representation theorem, F n Every Boolean function of can be expressed as their sum.

[0296] Definition: The sum of one or more n-dimensional miniterms is called the miniterm normal form or disjunctive normal form.

[0297] Theorem: Minimal Term Normal Form Theorem;

[0298] Assume B 2 = 2 ;+,·,';0,1> is a binary Boolean algebra, then each of its non-zero mappings (functions): This theorem also partially explains the compactness and completeness of the process from linguistic logic to propositional logic and then to CNF.

[0299] According to the duality principle, corresponding to the minimal term and its normal form theorem, Boolean functions have another standard form. Definition: The following form of n-variable Boolean algebraic expression,

[0300] ​It is called the n-ary maximal term or fundamental disjunctive form, where a i ∈{0,1}, and stipulate

[0301]

[0302] In other words, an n-variable maximal term is the sum of n variables, where each variable is x i or x' i , if the subscript (k) 10 =(a 1 ·a 1 …a n ) 2 ,but, The m and n here are consistent with the definition of m rows and n columns in the algorithm.

[0303] Now let's write down all the n-dimensional maximal terms M for n=1,2,3. k As shown in the following table:

[0304] n=1,M k Number = 2

[0305]

[0306] n=2,M k Number = 4

[0307]

[0308]

[0309] n=2,m k Number = 8

[0310]

[0311] Definition: The product of one or more n-ary maximal terms is called the maxterm normal form or conjunctive normal form.

[0312] Theorem: The maximal term paradigm theorem

[0313] Assume B 2 = 2 ;+,·,';0,1> is a binary Boolean algebra, then every mapping (function) that is not always equal to 1 is: This theorem also partially explains the compactness and completeness of the process from linguistic logic to propositional logic and then to CNF. ​

[0314] The comprehensive minimal term theorem also proves the isomorphic mapping relationship between the language system, propositional formula set, and propositional calculus system in this patent. Through this algorithm, if it is judged to be Boolean satisfiable, when the logical value is positive, when the binary number at the beginning of each minimum term is substituted, that is, the positive word is positive and the negative word is negative, it is a satisfiable solution. On the contrary, when the logical value is negative, when the binary number at the beginning of each minimum term is substituted, that is, the positive word is negative and the negative word is positive, it is a satisfiable solution.

[0315] S7. List practical examples of digital circuits;

[0316] S8. Conduct a comprehensive analysis of the switching circuit.

[0317] Example: Circuit Analysis

[0318] Given a circuit diagram, it is required to find out the conditions for connecting and disconnecting them. To do this, first make its construction formula based on the circuit diagram. In most cases, the construction formula is a disjunctive normal form. Find the negation of the disjunctive normal form and use the algorithm of this patent to first determine whether it can be satisfied. If it can be satisfied, use the disjunctive normal form to find the case where the minimum term is logically positive, which is all the connection conditions. Then the conditions other than the other connection conditions are the disconnection conditions. Finally, the solution result is verified using the function value table.

[0319] Solution: (1) Circuit diagram

[0320] (like Figure 4 )

[0321] (2) The circuit diagram is as shown above. Analyze it and find out its working conditions.

[0322] The circuit is constructed as a+a'b'+ab

[0323] (3) According to the algorithm of this patent, first convert the construct into CNF. f(a,b,c)=a+a'b'+ab, first negate f and convert it into conjunctive normal form F. F=a'·ab·a'b'

[0324]

[0325] If F is unsatisfiable, then the original circuit algebra f is satisfiable.

[0326] (3) Find the conditions for making and breaking the circuit. The disjunctive normal form of the circuit algebra f is f(a,b,c)=a+a'b'+ab, so the minimum term is:

[0327]

[0328] The condition that can be satisfied is that the minimum term is a logical positive value. Then the connection conditions are 1, 3, 4, and the disconnection conditions are verified using the truth table. The construction formula of the circuit is a+a'b'+ab. In order to facilitate the truth table analysis, it is first simplified: a+a'b'+ab=(a+b')+ab=(a+ab)+b'=a+b'

[0329] By making a function value table of a+b', we know that the circuit connected by this current is:

[0330] 1) a = b = 0;

[0331] 2) a=1, b=0;

[0332] 3)a=b=1.

[0333] By comparison, it can be seen that the value solved by this algorithm is consistent with the enumeration method (truth table method). However, the complexity of the truth table algorithm is very high, so the polynomial time solution proposed in this patent greatly improves the efficiency.

[0334] Then, the function value table (or other methods) is used to find the function value corresponding to the state of each switch, so that the function value of 1 is the state of each switch when the circuit is connected. The verification result is consistent with the solution result, which proves the effectiveness of this algorithm.

[0335] Example 2:

[0336] (1) Circuit diagram (such as Figure 5 )

[0337] (2) The circuit is as shown above, analyze its working conditions;

[0338] (3) The circuit algebra construction is f(a,b,c)=ab+ac+bc. According to the algorithm of this patent, first convert f(a,b)=ab+ac+bc into CNF. Find the negation of f(a,b)=ab+ac+bc. The unified theoretical calculation table based on this algorithm is as follows:

[0339] Unified Theory Calculation Table

[0340]

[0341] Then the original circuit can be satisfied.

[0342] (4) Find the conditions for connection and disconnection. The disjunctive normal form of circuit algebra f is f(a,b,c)=ab+ac+bc, so the minimum term is:

[0343]

[0344]

[0345] It turns out that 4, 6, 7, and 8 are the connection conditions, and 1, 2, 3, and 5 are the disconnection conditions.

[0346] The construction formula for the above figure is f(a,b,c)=ab+ac+bc. By making a function value table, we know that the condition for the circuit to be connected is:

[0347] 1) a=0, b=c=1;

[0348] 2) a=c=1, b=0;

[0349] 3) a=b=1, c=0;

[0350] 4)a=b=c=1.

[0351] Circuit synthesis example: Circuit synthesis puts forward working conditions for the equipment and requires the circuit to be designed to meet the given conditions. From the perspective of the switch function, it is to find the function expression when the function value is known. This patent is based on the minimal term paradigm and adds the patent's unique algorithm to achieve a fast solution.

[0352] (1) Circuit diagram (such as Figure 6 )

[0353] (2) You need to design a circuit for a three-person group to conduct a secret vote. The circuit must light up when two or more people press the switch to indicate their agreement, and remain off otherwise.

[0354] (3) Assume that three people control switches a, b, and c respectively. According to the question, construct the switch function f(a, b, c) = a'bc + ab'c + abc' + abc, and negate the disjunctive normal form.

[0355] The unified theoretical calculation table based on this algorithm is as follows:

[0356]

[0357] Then the original circuit can be satisfied.

[0358] (4) Solve the connectivity and disconnection conditions.

[0359] Assume that three people control switches a, b, and c respectively. According to the question, make a function value table of the switch function f(a, b, c) and the minimum values ​​of each row where the function value is 1 as shown in the table. Add up the minimum terms to get the product-sum normal form of the circuit:

[0360] f(a,b,c)=a′bc+ab′c+abc′+abc.

[0361]

[0362]

[0363] The connection conditions are 4, 6, 7, 8, and the disconnection conditions are 1, 2, 3, 5.

[0364] (5) Verification using truth table

[0365] In order to more conveniently calculate the truth table, this disjunctive normal form is simplified to: f(a,b,c)=a(b+c)+bc. That is, the condition for the circuit to be connected is:

[0366] 1) a=0, b=c=1;

[0367] 2) b = 0, a = c = 1;

[0368] 3) a=b=1, c=0;

[0369] 4)a=b=c=1.

[0370] By comparison, we can see that this algorithm is polynomial time, so it is very efficient.

[0371] Example 2:

[0372] (1) Circuit diagram (platform diagram such as Figure 7 , the circuit diagram is as follows Figure 8 )

[0373] (2) The figure above is the required circuit diagram. A certain railway station needs to design an automatic dispatching circuit. The platform shape is shown in the figure. Trains are required to enter the station from the direction of the arrow. There is a signal light at G. When the light is on, the train can enter the station. P 1 represents the proposition "Platform I is empty", P 2 , P 3 Synonym; S 1 , S 2 Indicates that the signal light at that location is green, allowing trains to pass; 1 , t 2 It means that the switch at that location has been moved to the position that allows turning.

[0374] (3) According to the driving regulations, the working conditions for a train to enter the station (the signal light G is on when the train is allowed to enter the station) are one of the following three:

[0375] 1)P 1 =1,S 1 =1,t 1 =0 (P can be ignored at this time 2 ,P 3 ,S 2 ,t 2 The value of

[0376] 2) P 2 =1,S 1 =0,t 1 =1,S 2 =0,t 2 =1 (P can be ignored at this time 1 ,P 3 The value of

[0377] 3) P 3 =1,S 1 =0,t 1 =1,S 2 =1,t 2 =0 (P can be ignored at this time 1 ,P 2 The value of

[0378] According to the question, the constructor is: F G =P 1 S 1 t' 1 +P 2 S' 1 t 1 S' 2 t 2 +P 3 S' 1 t 1 S 2 t' 2

[0379] To negate this disjunctive normal form, we get:

[0380] Unified Theory Calculation Table

[0381]

[0382] So the original circuit is satisfactory.

[0383] (4) Find the connectivity conditions:

[0384] According to the disjunctive normal form, we can directly find the minimum term and quickly get the following three connectivity conditions.

[0385]

[0386] (5) Verify using truth table:

[0387] In disjunctive normal form F G =P 1 S 1 t' 1 +P 2 S' 1 t 1 S'2 t 2 +P 3 S' 1 t 1 S 2 t' 2 Based on this, we can simplify to get F G =P 1 S 1 t' 1 +S' 1 t 1 (P 2 S' 2 t 2 +P 3 S 2 t' 2 ), these conditions are listed in a simplified truth table, and the conditions for the circuit to be connected are:

[0388] 1)P 1 =S 1 =1,t 1 =0;

[0389] 2) P 2 =t 1 =t 2 =1,S 1 =S 2 =0;

[0390] 3) P 3 =t 1 =S 2 =1,S 1 =t 2 =0.

[0391] In S1, the combination of the wires and switches connecting the two endpoints T1 and T2 is called a switch network. A, b, c, ... are used to represent switches, and 1 and 0 are used to represent the "connection" and "disconnection" of the switches and the "connection" and "disconnection" of the entire circuit, respectively. Whether the circuit on this network can be connected depends entirely on the state of these switches (0 or 1) and their connection mode (parallel, series, or anti-phase), and is therefore a function of the switches. This patent uses f(x 1 ,x 2 ,…,x n-1 ,x n ) represents the switching function, and F n Represents a set of n-ary switching functions. If the switching algebra satisfies the Boolean satisfiability condition, then this algorithm can quickly give the working conditions. If not, it can be determined in polynomial time.

[0392] In said S3, it is asked whether the variables of a given Boolean formula can be consistently replaced by the values ​​TRUE or FALSE, and the formula evaluates to TRUE. If this is the case, the formula is called satisfiable. On the other hand, if there is no such assignment, then for all possible variable assignments, the function represented by the formula is FALSE, and the formula is unsatisfiable. For example, the formula "a AND NOT b" is satisfiable because it is possible to find values ​​a=TRUE and b=FALSE, which makes (a AND NOT b)=TRUE. On the contrary, "a AND NOT a" is unsatisfiable, as shown below.

[0393] Use the traversal method to determine whether "a AND NOT a" is satisfied;

[0394] Assign a to FALSE, TRUE AND FALSE->FALSE

[0395] Assign a to TRUE, TRUE AND FALSE -> FALSE

[0396] The idea of ​​the traversal method is to consistently replace the variable a of a given Boolean formula with TRUE or FALSE. The calculation result of the formula cannot be TRUE, so there is no such assignment. Then for all the variable assignments, the function represented by the formula is FALSE, and the formula becomes unsatisfiable. In other words, the formula "a AND NOT a" is unsatisfiable.

[0397] In this example, the Boolean formula has n=1 variables, and the computational complexity is 2^n=2^1=2.

[0398] Propositional logic formulas, also called Boolean expressions, consist of variables and the operators AND (connection, also represented by ∧), OR (disjunction, ∨), NOT (negation, ) and parentheses. A formula is said to be satisfiable if it can be made TRUE by assigning appropriate logical values ​​(i.e., TRUE, FALSE) to its variables. Given a formula, the Boolean satisfiability problem (SAT) is to check whether it is satisfiable. This decision problem is of vital importance in various fields of computer science, including theoretical computer science, complexity theory, algorithms, cryptography, and artificial intelligence.

[0399] The theorem obtained for the Boolean satisfiability problem is as follows:

[0400] If the original proposition is satisfiable, then the negation of the original proposition must be unsatisfiable, and vice versa.

[0401] The proof is as follows:

[0402] If the original proposition is satisfiable, then the negation of the original proposition must be unsatisfiable.

[0403] First of all, it is necessary to clarify whether the proposition must negate both the conditions and the conclusion.

[0404] The negation of “any” is “there is one” or “there is one” or “there is at least one”.

[0405] The negation of “for any X∈A, make P(X) true” is “there exists an X∈A such that P(X) is false”.

[0406] So if the original proposition is "there exists an X∈A such that P(X) is true", its negation is "for any X∈A, P(X) is false". Its contrapositive is "if P(x) is false, then for any X∈A".

[0407] The original proposition is: If a, then b. The contrapositive proposition is: If not b, then not a. If the condition and conclusion of one of the two propositions are respectively the conclusion and the negation of the condition of the other proposition, then the two propositions are called contrapositive propositions. The negation of a proposition only negates the conclusion.

[0408] For example, the negation of the original proposition "For all x, x is greater than 10" (arbitrary proposition) is "There exists an x, x is not greater than 10" (existential proposition). Its contrapositive is "If x is not greater than 10, then x does not exist". If the negation of the original proposition "There exists an x, x is greater than 10" (existential proposition) is "For all x, x is not greater than 10" (universal proposition), your negation is "If x is not greater than 10, then x does not exist" (existential proposition).

[0409] The Boolean satisfiability problem is a core problem in computer science. It asks whether the variables of a given Boolean formula can be consistently replaced with the value TRUE or FALSE so that the formula evaluates to TRUE. If such an assignment exists, the formula is called satisfiable; its negation is that for all Boolean formula variables, it is not possible to consistently replace them with the value TRUE or FALSE so that the formula evaluates to TRUE. That is, it is unsatisfiable. So the original theorem is proved, and its converse is that if the formula is unsatisfiable, then there is no such assignment that can be consistently replaced with the value TRUE or FALSE so that the formula evaluates to TRUE, and the proof is complete;

[0410] The theorem obtained from propositional logic and the Boolean satisfiability problem is as follows:

[0411] Any proposition can be transformed into the conjunctive or disjunctive normal form of a homomorphic bijection.

[0412] The proof is as follows:

[0413] The Boolean satisfiability problem can be expressed in some special forms. These include formulas. Formulas need to have a specific structure. A literal is a variable, called a positive literal, or the negation of a variable, called a negative literal. Clauses are separations of literals (or single literals). If a clause contains at most one positive literal, then the clause is called a Horn clause. If a formula is a conjunction of clauses (or single clauses), then the formula is in conjunctive normal form (CNF). For example, x1 is a positive literal, It is negative text. is a clause, is a formula in joint normal form. The formula is satisfyable by choosing x1 = FALSE, x2 = FALSE, and x3 arbitrarily, because The evaluation is (FALSE∨TRUE)∧(TRUE∨FALSE∨x3)∧FALSE, which is TRUE∧TRUE∧FALSE (which is FALSE). In contrast, the CNF formula consists of two clauses of a literal, is unsatisfiable because for a = TRUE or a = FALSE it evaluates to or (ie, FALSE, respectively).

[0414] Through Stone's topological representation theorem and the theory of unification, we exploit the notion of a generalized conjunctive normal form formula, i.e., a concatenation of any number of generalized clauses, the latter of the form R(l1, ..., ln) for some Boolean operator R and a (normal) literal l. Different sets of allowed Boolean operators lead to different versions of the problem. For example (Example 2.3), is a general clause, is a generalized joint normal form.

[0415] Using the laws of Boolean algebra, every propositional logic formula can be converted into an equivalent connected normal form, which, however, can be exponentially longer. For example, converting the formula (x 1 ∧y 1 )∨(x 2 ∧y 2 )∨...∨(x n ∧y n ) to the connected normal form.

[0416] (x 1 ∨x 2 ∨…∨x n )∧

[0417] (y 1 ∨x 2 ∨…∨x n)∧

[0418] (x 1 ∨y 2 ∨…∨x n )∧

[0419] (y 1 ∨y 2 ∨…∨x n )∧...∧

[0420] (x 1 ∨x 2 ∨…∨y n )∧

[0421] (y 1 ∨x 2 ∨…∨y n )∧

[0422] (x 1 ∨y 2 ∨…∨y n )∧

[0423] (y 1 ∨y 2 ∨…∨y n );

[0424] The former is the separation of n connections of 2 variables, which is the disjunctive normal form; the latter consists of 2n clauses of n variables, which is the conjunctive normal form. The proof is complete.

[0425] Theorems of paradigm and propositional logic are as follows:

[0426] Definition 1 (Variable Set): For a propositional logic formula, all propositional variables x 1 ,x 2 ,…,x n-1 ,x n The set of 1 ,x 2 ,…,x n-1 ,x n} is called the variable set of the formula.

[0427] Definition 2 (literal): For any variable x i , shaped like x i or is called a literal, which is the smallest unit of a logical formula. i For the correct text, For negative text, we usually use l i To represent a text.

[0428] Definition 3 (Clause and Clause Length) Disjunction l of a finite number of literals on X 1∨l 2 ∨…∨l n-1 ∨l n It is called a clause, denoted by C. The number of words contained in the clause is called the clause length. A clause with a clause length of k is denoted by a k-clause.

[0429] Definition 4 (Conjunction Normal Form) Conjunction C of a finite number of clauses on X 1 ∧C 2 ∧…C n-1 ∧C n It is called a conjunctive normal form (CNF) on L, and can be expressed as F(X) denoted by a CNF.

[0430] A CNF can be represented as

[0431] Definition 5 (Disjunctive Normal Form) Disjunction C of a finite number of clauses on X 1 ∨C 2 ∨…C n-1 ∨C n It is called a disjunctive normal form (DNF) on L, and a DNF can be expressed as D(X).

[0432] A DNF can be represented as

[0433] In this patent, F* is the mapping of F calculated according to the unified theory.

[0434] In S4, the unified theory mapping criterion is to convert the language into propositional logic, unify the propositional logic into CNF form, and the logical value of the clause must be positive. If the propositional logic is in disjunctive normal form, the disjunctive normal form is negated and converted to conjunctive normal form. If the clause is in conjunctive normal form but the logical value is not positive, the clause corresponding to the semantics is transformed and split so that it satisfies the conjunctive normal form and is in a positive normal form.

[0435] The specific working method is to convert the problem of judging whether a circuit can be connected into a Boolean satisfiability judgment problem, and develop a polynomial time judgment solver. When the circuit can be connected, the truth table is used to find out its working conditions;

[0436] The related algorithm description starts from partially ordered sets and then goes to lattice knowledge, and then uses Boolean algebra and set theory to solve the problem.

[0437] The comprehensive analysis of the switch circuit in S8 refers to finding out the conditions for connecting and disconnecting the circuit, synthesizing the automation equipment, and designing the circuit according to the given conditions so that the circuit can meet these conditions.

[0438] The above shows and describes the basic principles and main features of the present invention and the advantages of the present invention. For those skilled in the art, it is obvious that the present invention is not limited to the details of the above exemplary embodiments, and the present invention can be implemented in other specific forms without departing from the spirit or basic features of the present invention; therefore, no matter from which point of view, the embodiments should be regarded as exemplary and non-restrictive. The scope of the present invention is limited by the attached claims rather than the above description. Therefore, it is intended to include all changes within the meaning and scope of the equivalent elements of the claims in the present invention, and any figure marks in the claims should not be regarded as limiting the claims involved.

[0439] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A polynomial time discriminant solver, characterized in that include: S1. Research, application and development of Boolean functions and Boolean satisfiability solvers in digital circuit design, including using Boolean satisfiability solvers to determine whether there are satisfiable conditions and quickly find their working conditions; S2. Choose a proven NPC problem SAT, give a solution to SAT, and prove that the solution can be completed in polynomial time, thus proving that P = NP; S3. Determine whether there is an interpretation problem that satisfies a given Boolean formula; determine propositional logic and Boolean satisfiability problems; determine paradigms and propositional logic; S4, Unified theory to solve the P vs NP problem, including unified theory mapping criteria, specific working methods, relevant algorithm descriptions and simple examples of SAT solvers; S5. Determine the technical details and technical derivation, including using minimal terms to prove the validity and completeness of the algorithm, using power set lattices to prove the validity and completeness of the algorithm, and using multi-products to prove the validity and completeness of the algorithm; S6. Identify Boolean functions and their expressions, and understand the two dual standard forms of Boolean algebra; S7. List practical examples of digital circuits; S8. Conduct a comprehensive analysis of the switching circuit.

2. A polynomial time discriminant solver according to claim 1, characterized in that: In S1, the combination of the wires connecting the two endpoints T1 and T2 and the switches is called a switch network, a, b, c, ... are used to represent the switches, 1 and 0 are used to represent the "connection" and "disconnection" of the switches and the "connection" and "disconnection" of the entire circuit respectively. Whether the circuit on this network can be connected depends entirely on the state of these switches (0 or 1) and their connection method (parallel, series, or anti-phase), and is therefore a function of the switches.

3. A polynomial time discriminant solver according to claim 1, characterized in that: In S3, it is asked whether the variables of a given Boolean formula can be consistently replaced by the values ​​TRUE or FALSE, and the formula evaluates to TRUE. If this is the case, the formula is called satisfiable; on the other hand, if there is no such assignment, then for all possible variable assignments, the function represented by the formula is FALSE, and the formula is unsatisfiable. For example, the formula "a AND NOTb" is satisfiable because values ​​a=TRU and b=FALSE can be found, which makes (a AND NOTb)=TRUE. On the contrary, "a AND NOTa" is unsatisfiable.

4. A polynomial time discriminant solver according to claim 1, characterized in that: In S4, the unified theory mapping criterion is to convert the language into propositional logic, unify the propositional logic into CNF form, and the logical value of the clause must be positive. If the propositional logic is in disjunctive normal form, the disjunctive normal form is negated and converted to conjunctive normal form. If the clause is in conjunctive normal form but the logical value is not positive, the clause corresponding to the semantics is transformed and split so that it satisfies the conjunctive normal form and is in a positive normal form. The specific working method is to convert the problem of judging whether a circuit can be connected into a Boolean satisfiability judgment problem, and develop a polynomial time judgment solver. When the circuit can be connected, the truth table is used to find out its working conditions; The related algorithm description starts from partially ordered sets and then goes to lattice knowledge, and then uses Boolean algebra and set theory to solve the problem.

5. A polynomial time discriminant solver according to claim 1, characterized in that: The comprehensive analysis of the switch circuit in S8 refers to finding out the conditions for connecting and disconnecting the circuit, synthesizing the automation equipment, and designing the circuit according to the given conditions so that the circuit can meet these conditions.

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