Quantitative deduction and its fixpoint theory.

*(English)*Zbl 0609.68068The author introduces Zadeh’s fuzzy method into the study of logic programming. By adding a factor f, a real number in (0,1], to each Horn clause (e.g., the clause \(A\leftarrow B_ 1\&...\&B_ n\) becomes \(A\leftarrow^{f}B_ 1\&...\&B_ n,\) which means A has a fuzzy truth value \(h\geq f\cdot g\) if g is the least fuzzy truth value of \(B_ 1,...,B_ n)\) and by regarding a Herbrand interpretation as a fuzzy subset of the Herbrand base, the usual deduction is extended to quantitative deduction. The author has proved that some results in the semantics of Horn clause rules are still true in the quantitative deduction. For example, in the qualitative case, each member of \(\cap M(P)\) (where M(P) is the set of Herbrand models of a rule set P) is a member of \(T^ n_ P(\phi)\) for an \(n\in N\). In the fuzzy deduction case, the result becomes that the fuzzy truth value of A belonging to \(\cap M(P)\) equals the fuzzy truth value of A belonging to \(T^ n_ P(\phi)\). In terms of the proof theory of the quantitative rules, the author has proved that for every set P of rules with a finite and/or tree and every A in the Herbrand base of P, the value of the root in the and/or tree with A as root equals the value of the membership function for \(\cap M(P)\) at the argument A. Finally, the author reviews briefly the main concepts of two-person games to both, the qualitative and the quantitative case by discussing some examples like the game of Nim and alpha-beta pruning.

Reviewer: Li Xiang

##### MSC:

68T15 | Theorem proving (deduction, resolution, etc.) (MSC2010) |

03B52 | Fuzzy logic; logic of vagueness |

03B35 | Mechanization of proofs and logical operations |

68N01 | General topics in the theory of software |

91A05 | 2-person games |