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Finite-valued Logics for Information Processing

Avron A., Konikowska B.

Fundamenta Informaticae

2012

Vol. 114, nr 1

1-30

We examine the issue of collecting and processing information from various sources, which involves handling incomplete and inconsistent information. Inspired by the framework first proposed by Belnap, we consider structures consisting of information sources which provide information about the values of formulas of classical propositional logic, and a processor which collects that information and extends it by deriving conclusions following from it according to the truth tables of classical logic, applied forward and backward. Our model extends Belnap’s in allowing the sources to provide information also about complex formulas. As that framework cannot be captured using finite ordinary logical matrices, if we want to represent each of the relevant logics with a single matrix, we employ Nmatrices for that purpose. In opposition to the approach proposed in our earlier work, we assume that the information sources are reasonable, i.e. that they provide information consistent with certain coherence rules. We provide sound and complete sequent calculi admitting strong cut elimination for the logic of a single information source, and (several variants of) the logic generated by the source and processor structures described above. In doing this, we also provide new characterizations for some known logics. We prove that, in opposition to the variantwith unconstrained information sources considered earlier, the latter logic cannot be generated by structures with any bounded number of sources.

Proof Systems for Logics Based on Non-deterministic Multiple-valued Structures

Avron A., Konikowska B.

Prace Instytutu Podstaw Informatyki Polskiej Akademii Nauk

2004

Nr 976

1-26

Non-deterministic matrices (Nmatrices) are multiple-valued structures in which the value assigned by a valuation to a complex formula can be chosen non-deterministically out of a certain nonempty set of options. We consider two different types of semantics which are based on Nmatrices: the dynamic one and the static one (the latter is new here). We use the Rasiowa-Sikorski (R-S) decomposition methodology to get sound and complete proof systems employing finite sets of mv-signed formulas for all propositional logics based on such structures with either of the above types of semantics. Later we demonstrate how these systems can be converted into cut-free ordinary Gentzen calculi which are also sound and complete for the corresponding non-deterministic semantics. As a by-product, we get new semantic characterizations for some well-known logics (like the logic CAR from [18, 28]).

Macierze niedeterministyczne to wielowartościowe struktury, w których wartość przypisaną przez wartościowanie złożonej formule można wybrać w niedeterministyczny sposób z pewnego niepustego zbioru opcji. Opracowujemy ogólne dedukcyjne systemy dowodzenia dla logik opartych na tych strukturach z użyciem mechanizmu n-sekwentów oraz dekompozycyjnej metodologii R-S. Później pokazujemy, jak można użyć tych systemów do uzyskania zwykłych rachunków Gentzenowskich bez reguły cięcia dla pewnych dobrze znanych logik tolerujących sprzeczność.