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Deductive reasoning method for separating contradiction based on extended triangle in first-order logic

A reasoning method and triangular technology, applied in the direction of reasoning methods, etc., can solve the problems that the ability and efficiency cannot meet the objective needs, cannot reflect the cooperative logical relationship of multiple clauses, and the clause set S cannot be satisfied, etc.

Inactive Publication Date: 2018-11-23
SOUTHWEST JIAOTONG UNIV
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AI Technical Summary

Problems solved by technology

Binary resolution selects two clauses with complementary pairs from the clause set S to be determined each time, deletes the complementary pair from the two clauses, and adds the new clauses formed by the disjunction of the remaining words into the atomic sentence set, and repeats as above process, until an empty clause is obtained, it can be determined that the clause set S is unsatisfiable
[0005] Although the automatic reasoning system based on the resolution principle has made many achievements, it is still far from meeting the objective needs in terms of ability and efficiency. Only in the internationally renowned problem bank TPTP in the field of automatic reasoning, there are a large number of the most difficult (Rating 1) problems so far. solve
In the resolution method, because each deduction has and can only have two clauses involved, it cannot reflect the cooperative logical relationship between multiple clauses
The reasoning mechanism of binary resolution seriously restricts the further improvement of the ability and efficiency of automatic reasoning

Method used

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  • Deductive reasoning method for separating contradiction based on extended triangle in first-order logic
  • Deductive reasoning method for separating contradiction based on extended triangle in first-order logic
  • Deductive reasoning method for separating contradiction based on extended triangle in first-order logic

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Embodiment 1

[0058] based on figure 1 In the extended triangle method shown, set the clause set S={C in the first-order logic 1 ,C 2 ,C 3 ,C 4 ,C 5 ,C 6 ,C 7}: C 1 =P 1 (a), C 2 =~P 2 (a, b), C 3 =P 3 (a,f(c),f(b)),C 4 =P 3 (x 1 ,x 1 ,f(x 1 )),C 5 =~P 3 (x 2 ,x 3 ,x 4 )∨P 3 (x 3 ,x 2 ,x 4 ),C 6 =~P 3 (x 5 ,x 6 ,x 7 )∨P 2 (x 5 ,x 7 ),C 7 =~P 1 (x8 )∨~P 3 (x 9 ,x 10 ,x 11 )∨~P 2 (x 8 ,x 11 )∨P 2 (x 8 ,x 9 )∨P 2 (x 8 ,x 10 ). Among them, P 1 ,P 2 ,P 3 is the predicate symbol, x 1 ,x 2 ,...,x 11 are variable symbols, a, b, c are constant symbols, and f is a function symbol. The attribute of S is judged by the extended triangle automatic deductive reasoning method.

[0059] Practice: select clause C from S 1 ,C 2 ,C 3 ,C 4 ,C 6 ,C 7 , respectively replace σ 1 ,σ 2 ,σ 3 ,σ 4 ,σ 6 ,σ 7 , under substitution, the extended triangular contradiction is constructed from the clauses (as in Table 1), where,

[0060]

[0061]

[...

Embodiment 2

[0066] Based on the extended triangle method, set the set of clauses in first-order logic S={C 1 ,C 2 ,C 3 ,C 4 ,C 5 ,C 6 ,C 7}:

[0067] C 1 =~P 1 (x 11 ,x 12 ,x 13 )∨~P 2 (x 11 ,x 13 ),

[0068] C 2 =P 1 (x 22 ,x 21 ,x 23 )∨~P 1 (x 21 ,x 22 ,x 23 ),

[0069] C 3 =P 2 (x 31 ,x 34 )∨~P 3 (x 31 )∨~P 1 (x 32 ,x 33 ,x 34 )∨~P 2 (x 31 ,x 32 )∨~P 2 (x 31 ,x 33 ),

[0070] C 4 =P 1 (x 41 ,x 41 , f 1 (x 41 )),

[0071] C 5 =P 1 (a 1 , f 1 (a 1 ), f 1 (a 3 )),

[0072] C 6 =P 3 (a 1 ),

[0073] C 7 =P 2 (a 1 ,a 3 ).

[0074] Among them, P 1 ,P 2 ,P 3 is the predicate symbol, x 1 ,x 2 ,...,x 41 is the variable symbol, a 1 ,a 2 ,a 3 is a constant symbol, f 1 is a function symbol. The attribute of S is judged by the extended triangle automatic deductive reasoning method.

[0075] (1) For the clause set S:

[0076] select clause C from S 1 ,C 2 ,C 3 ,C 4 ,C 5 ,C 6 ,C 7 , respectively replace σ 1 ,σ 2 ,σ ...

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Abstract

The invention discloses a deductive reasoning method for separating a contradiction based on an extended triangle in first-order logic. The method sequentially constructs an extended triangle standardcontradiction for a clause set in the first-order logic, forms a standard contradiction separating formula, and judges the unsatisfiability of the clause set according to the standard contradiction separating formula. If it can be judged that the set of clauses is not satisfiable, stop operation is performed, and the conclusion can be drawn: the judged set of clauses S is not satisfiable; otherwise, the obtained standard contradiction is added into the clause set S to form a new clause set, and then the extended triangle standard contradiction is repeatedly constructed, and the extended triangle standard contradiction is separated until the judgment conclusion of the attribute of the clause set S is obtained or the set condition is satisfied and stop operation is performed. The inventionprovides an effective dynamic automatic deductive reasoning method based on the contradiction separation in the first-order logic, which can effectively characterize the cooperative logic relationshipamong multiple clauses, and can be applied to the fields of program verification, theorem machine proving and the like.

Description

technical field [0001] The invention belongs to the technical field of automatic deductive reasoning based on logic, and relates to a multivariate dynamic automatic deductive reasoning method based on extended triangle-based contradiction separation in first-order logic. Background technique [0002] A large number of scientific problems in the fields of logic, mathematics, system optimization, artificial intelligence, computer science, etc. can be formalized into logical representations. One of the essences of solving these problems is to analyze the attributes of the corresponding logical formulas (satisfiability or unsatisfiability (constant False)), but because of its abstraction, complexity, and scale, it is impossible to effectively implement logical reasoning and solution artificially, so it needs to be automatically judged by a computer. Automatic reasoning is the process of formalizing the reasoning process through a series of symbols, and making the computer automa...

Claims

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Application Information

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IPC IPC(8): G06N5/04
CPCG06N5/042G06N5/045
Inventor 徐扬陈树伟刘军钟小梅何星星
Owner SOUTHWEST JIAOTONG UNIV
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