Classical Resolution for Many-Valued Logics

João Marcos, Cláudia Nalon
2016 Electronical Notes in Theoretical Computer Science  
We present a resolution-based proof method for finite-valued propositional logics based on an algorithmic reduction procedure that expresses these logics in terms of bivalent semantics. Our approach is hybrid in using some elements which are internal and others which are external to the many-valued logic under consideration, as we embed its original language into a more expressive metalanguage to deal with the satisfiability problem. In contrast to previous approaches to the same problem, our
more » ... rget language is fully classical, what turns the design of the resolution-based rules for a specific many-valued logic into a straightforward task. Correctness results, which are proved in detail in the present study, follow easily from results on classical resolution. We illustrate the application of the method with examples, and comment on its implementation, readily achievable by direct translation into classical propositional logic, making use of reliable existing automated provers. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). labelled formula. Labels are used to mirror truth-values at the syntactic level, and are intended to represent the semantic conditions under which a formula is satisfied. The resolution inference rule is applied to clauses containing complementary literals and whose labels are unifiable (in the sense that they represent consistent semantic conditions). The main difference between those proof methods reside on the form of the labels: in [1] labels are singletons whilst in [7] labels are sets of signs. In the present investigation, we take a similar route. Formulae to be tested for (un)satisfiability are also transformed into labelled clauses, that is, the inference rules are applied to a more expressive language in which the semantic notions are made explicit. However, labels take in all cases a very simple format, just one out of two possible signs. Thus, differently from the approaches in [1,7], unification on labels can be easily seen as equivalent to the ordinary application of classical propositional resolution. Also, the search for inconsistency (in the language of a given many-valued logic) takes the form of hyper-resolution inference rules in the classical metalanguage, allowing for a uniform classic-like approach to be algorithmically constructed for dealing with any finite-valued logic. The transformation of formulae into the labelled language relies on previous results by one of the authors [4,5], which are briefly surveyed here in order to make the presentation self-contained. The paper is organised as follows. Section 2 introduces logics in abstract and from a semantic viewpoint, explains that they can all be characterised by bivalent semantics, and then focuses on the class of finite-valued truth-functional logics. Emphasis is put on the standard (two-sided) notion of logical consequence, rather than on approaches to many-valued logics that are based on mere combinatorial manipulation of finite-valued algebras. Section 3 explains the algorithm that allows one to describe finite-valued logics in terms of statements written in a fully classical metalanguage in which only two signs are employed. Providing a proof-theoretical perspective on this requires a generalisation of the way that syntactical complexity is measured, in order to allow for analytic calculi to be extracted from such a description. The subsequent section contains our main contribution. Section 4 is dedicated to setting up a generic resolution-based proof method for an arbitrary given finite-valued logic. Subsection 4.1 shows how to transform the mentioned bivalent descriptions into a clausal format that is more appropriate for applying resolution. The corresponding transformation adds new variable symbols that help encoding the structure of the original statements, preserving and reflecting their satisfiability. As output we obtain object-level expressions that take better advantage of the above mentioned generalised notion of complexity. Subsection 4.2 introduces the inference rules of a hyper-resolution proof method that applies to the clauses produced by the latter transformation. This method lies in between internal proof systems that capitalise on syntactic features of the original logics and external proof systems that formalise reasoning about the logics in a classical logical framework. We finish by some comments on what has been achieved and on how the present investigation may be further extended.
doi:10.1016/j.entcs.2016.09.001 fatcat:knq57n63gvg6rkdxe6c7s35ozy