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:
In (Guller, 2014), we have generalised the well-known hyperresolution principle to the first-order Godel logic ¨
with truth constants. This paper is a continuation of our work. We propose a hyperresolution calculus suitable
for automated deduction in a useful expansion of Godel logic by intermediate truth constants and the equality, ¨
P, strict order, ≺, projection, ∆, operators. We solve the deduction problem of a formula from a countable
theory in this expansion. We expand Godel logic by a countable set of intermediate truth constants ¯ ¨ c, c ∈
(0,1). Our approach is based on translation of a formula to an equivalent satisfiable finite order clausal theory,
consisting of order clauses. An order clause is a finite set of order literals of the form ε1 ε2 where εi
is
an atom or a quantified atom, and is the connective P or ≺. P and ≺ are interpreted by the equality
and standard strict linear order on [0,1], respectively. We shall investigate the so-called canonical standard
compl
eteness, where the semantics of Godel logic is given by the standard ¨ G-algebra and truth constants are
interpreted by ’themselves’. The hyperresolution calculus is refutation sound and complete for a countable
order clausal theory under a certain condition for the set of truth constants occurring in the theory. As an
interesting consequence, we get an affirmative solution to the open problem of recursive enumerability of
unsatisfiable formulae in Godel logic with truth constants and the equality, ¨ P, strict order, ≺, projection, ∆,
operators.
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