A Strong Completeness Theorem for the Gentzen systems associated with finite algebras
ABSTRACT In this paper we study consequence relations on the set of many sided sequents over a propositional language. We deal with the consequence relations axiomatized by the sequent calculi defined in [2] and associated with arbitrary finite algebras. These consequence relations are examples of what we call Gentzen systems. We define a semantics for these systems and prove a Strong Completeness Theorem, which is an extension of the Completeness Theorem for provable sequents stated in [2]. For the special case of the finite linear MV-algebras, the Strong Completeness Theorem was proved in [10], as a consequence of McNaughton's Theorem. The main tool to prove this result for arbitrary algebras is the deduction-detachment theorem for Gentzen systems.
MUltlog and MUltseq Reanimated and Married ?
MUltlog is a logic engineering tool that produces descriptions of various sound and complete logical calculi for an arbitrary finite-valued first-order logic from a given specification of the semantics of such a logic (see [1]). MUltseq, on the other hand, is a simple, generic, sequent based theorem prover for propositional finite-valued logics (see [6]). From its very beginning, MUltseq was intended to be a ‘companion’ to MUltlog. So far, however, MUltseq does not directly use the representation of sequent rules as generated by MUltlog. Moreover (due to lack of funding, personnel and time), further development and maintenance of both system has been stalled for some time now. It is the purpose of this abstract to shortly describe the two systems and the current efforts to integrate them.
Approximating propositional calculi by finite-valued logics
The problem of approximating a propositional calculus is to find many-valued logics which are sound for the calculus (i.e., all theorems of the calculus are tautologies) with as few tautologies as possible. This has potential applications for representing (computationally complex) logics used in AI by (computationally easy) many-valued logics. It is investigated how far this method can be carried using (1) one or (2) an infinite sequence of many-valued logics. It is shown that the optimal candidate matrices for (1) can be computed from the calculus.<<ETX>>