Closed nominal rewriting and efficiently computable nominal algebra equality

We analyse the relationship between nominal algebra and nominal rewriting, giving a new and concise presentation of equational deduction in nominal theories. With some new results, we characterise a subclass of equational theories for which nominal rewriting provides a complete procedure to check nominal algebra equality. This subclass includes specifications of the lambda-calculus and first-order logic.Comment: In Proceedings LFMTP 2010, arXiv:1009.218

Automatic Generation of Proof Tactics for Finite-Valued Logics

A number of flexible tactic-based logical frameworks are nowadays available that can implement a wide range of mathematical theories using a common higher-order metalanguage. Used as proof assistants, one of the advantages of such powerful systems resides in their responsiveness to extensibility of their reasoning capabilities, being designed over rule-based programming languages that allow the user to build her own `programs to construct proofs' - the so-called proof tactics. The present contribution discusses the implementation of an algorithm that generates sound and complete tableau systems for a very inclusive class of sufficiently expressive finite-valued propositional logics, and then illustrates some of the challenges and difficulties related to the algorithmic formation of automated theorem proving tactics for such logics. The procedure on whose implementation we will report is based on a generalized notion of analyticity of proof systems that is intended to guarantee termination of the corresponding automated tactics on what concerns theoremhood in our targeted logics

Expressivity Within Second-Order Transitive-Closure Logic

Second-order transitive-closure logic, SO(TC), is an expressive declarative language that captures the complexity class PSPACE. Already its monadic fragment, MSO(TC), allows the expression of various NP-hard and even PSPACE-hard problems in a natural and elegant manner. As SO(TC) offers an attractive framework for expressing properties in terms of declaratively specified computations, it is interesting to understand the expressivity of different features of the language. This paper focuses on the fragment MSO(TC), as well on the purely existential fragment SO(2TC)(exists); in 2TC, the TC operator binds only tuples of relation variables. We establish that, with respect to expressive power, SO(2TC)(exists) collapses to existential first-order logic. In addition we study the relationship of MSO(TC) to an extension of MSO(TC) with counting features (CMSO(TC)) as well as to order-invariant MSO. We show that the expressive powers of CMSO(TC) and MSO(TC) coincide. Moreover we establish that, over unary vocabularies, MSO(TC) strictly subsumes order-invariant MSO