Artificial intelligence (deterministic reasoning methods)

3. Deterministic reasoning method

3.1 Basic concepts of reasoning

3.1.1 Definition of reasoning

3.1.2 Reasoning methods and their classification

3.1.3 Direction of reasoning

3.1.4 Conflict resolution strategies

• Targeted sorting

• Sort known facts by freshness

• Sort by match

• Sort by number of conditions

• Sort by contextual constraints

• Sort by redundancy limit

• Sort according to the characteristics of domain problems

3.2 Natural deductive reasoning

    从一组已知为真的事实出发，直接运用经典逻辑中的推理规则推出结论的过程

Inference rules: P rule, T rule, hypothetical reasoning, rejection reasoning

3.3 Method of formulating predicates into clause sets

3.4 Herbrand’s theorem

1) H domain and Hebron’s theorem

Assume that the clause set of the predicate formula G is S, then the individual variable domain H constructed according to the following method is called the Hebron domain (Herbrand domain, referred to as H domain) of the formula G or the clause set S:

(1) Let H0 be the set of constants appearing in S. If no constant appears in S, just take a constant a∈D and stipulate H0=a.

(2) Let Hi+1=Hi∪{all elements of the form f(t1,...,tn) in S), where f(t1,...,tn) is any function symbol appearing in G , and t1,…,tn are elements in Hi. i=0, 1, 2,….

3.5 Robinson reduction principle

two keys

• There is a conjunctive relationship between clauses in concentrated clauses

• The empty clause is unsatisfiable

Basic idea

• Negate the conclusion of the problem to be proved and add the clause set to obtain an expanded clause set S

• Check whether the clause set S contains an empty clause. If it does, S cannot be satisfied.

• If it is not included, select an appropriate clause in S to summarize. Once an empty clause is summarized, it means that S cannot satisfy

1) Principle of reduction in propositional logic (reduction of base clauses)

    设C1与C2是子句集中的任意两个子句，如果 C1中的文字L1与 C2中的文字L2互补，那么从C1和 C2中分别消去L1和L2，并将二个子句中余下的部分析取，构成一个新子句C12 。

$$Corollary1:\; Let{C}_{1}with{C}_{2}aretwo\; clauses\; in\; the\; clause\; setS\; ,\; andC12is\; their\; reduction\; formula.\; IfC12\; is\; usedinstead\; ofC{}_{1}with{C}_{2}Then\; we\; get\; the\; new\; clause\; set{S}_1,\; then\; by{S}_{1}The\; unsatisfiability\; can\; be\; derived\; fromthe\; unsatisfiability\; of\; the\; atomic\; sentence\; setS.$$

$$Corollary2:\; Suppose{C}_{1}with{C}_{2}are\; two\; clauses\; inthe\; clause\; setS\; ,\; andC12is\; their\; reduction\; formula.\; IfC12is\; added\; to\; the\; atomic\; sentence\; setS,\; a\; new\; clause\; setS\; is\; obtained.{}_1,\; thenSand{S}_{1}Equivalent\; in\; the\; sense\; of\; being\; unsatisfiable.$$

2) The principle of resolution in predicate logic (resolution of clauses containing variables)

3.6 Reductive inversion

3.7 Apply the principle of reduction to solve problems