Author:
Dušan Guller
Affiliation:
Comenius University, Slovak Republic
Keyword(s):
Gödel Logic, Resolution, Many-valued Logics, Automated Deduction.
Related
Ontology
Subjects/Areas/Topics:
Approximate Reasoning and Fuzzy Inference
;
Artificial Intelligence
;
Computational Intelligence
;
Fuzzy Systems
;
Mathematical Foundations: Fuzzy Set Theory and Fuzzy Logic
;
Soft Computing
Abstract:
This paper addresses the deduction problem of a formula from a countable theory in the first-order G\"{o}del logic from a perspective of automated deduction. Our approach is based on the translation of a formula to an equivalent satisfiable CNF one, which contains literals of the augmented form: either a or a → b or (a→b) →b or Qx c→ a or a→Qx c where a, c are atoms different from 0 (the false), 1 (the true); b is an atom different from 1; Q ∈ {∀,∃}; x is a variable occurring in c. A CNF formula is further translated to an equivalent satisfiable finite order clausal theory, which consists of order clauses - finite sets of order literals of the form: either a ≖ b or Qx c ≖ a or a ≖ Qx c or a ≺ b or Qx c ≺ a or a ≺ Qx c where a, b, c are atoms; Q ∈ {∀,∃}; x is a variable occurring in c. ≖ and ≺ are interpreted by the equality and strict linear order on [0,1], respectively. For an input theory, the proposed translation produces a so-called semantically admissible order clausal theory. A
n order hyperresolution calculus, operating on semantically admissible order clausal theories, is devised. The calculus is proved to be refutation sound and complete for the countable case.
(More)