4 research outputs found
A Tour on Ecumenical Systems
Ecumenism can be understood as a pursuit of unity, where diverse thoughts, ideas, or points of view coexist harmoniously. In logic, ecumenical systems refer, in a broad sense, to proof systems for combining logics. One captivating area of research over the past few decades has been the exploration of seamlessly merging classical and intuitionistic connectives, allowing them to coexist peacefully. In this paper, we will embark on a journey through ecumenical systems, drawing inspiration from Prawitz' seminal work [35]. We will begin by elucidating Prawitz' concept of “ecumenism” and present a pure sequent calculus version of his system. Building upon this foundation, we will expand our discussion to incorporate alethic modalities, leveraging Simpson's meta-logical characterization. This will enable us to propose several proof systems for ecumenical modal logics. We will conclude our tour with some discussion towards a term calculus proposal for the implicational propositional fragment of the ecumenical logic, the quest of automation using a framework based in rewriting logic, and an ecumenical view of proof-theoretic semantics
Expressing Ecumenical Systems in the ??-Calculus Modulo Theory
Systems in which classical and intuitionistic logics coexist are called ecumenical. Such a system allows for interoperability and hybridization between classical and constructive propositions and proofs. We study Ecumenical STT, a theory expressed in the logical framework of the ??-calculus modulo theory. We prove soudness and conservativity of four subtheories of Ecumenical STT with respect to constructive and classical predicate logic and simple type theory. We also prove the weak normalization of well-typed terms and thus the consistency of Ecumenical STT
Modularity and Combination of Associative Commutative Congruence Closure Algorithms enriched with Semantic Properties
Algorithms for computing congruence closure of ground equations over
uninterpreted symbols and interpreted symbols satisfying associativity and
commutativity (AC) properties are proposed. The algorithms are based on a
framework for computing a congruence closure by abstracting nonflat terms by
constants as proposed first in Kapur's congruence closure algorithm (RTA97).
The framework is general, flexible, and has been extended also to develop
congruence closure algorithms for the cases when associative-commutative
function symbols can have additional properties including idempotency,
nilpotency, identities, cancellativity and group properties as well as their
various combinations. Algorithms are modular; their correctness and termination
proofs are simple, exploiting modularity. Unlike earlier algorithms, the
proposed algorithms neither rely on complex AC compatible well-founded
orderings on nonvariable terms nor need to use the associative-commutative
unification and extension rules in completion for generating canonical rewrite
systems for congruence closures. They are particularly suited for integrating
into the Satisfiability modulo Theories (SMT) solvers. A new way to view
Groebner basis algorithm for polynomial ideals with integer coefficients as a
combination of the congruence closures over the AC symbol * with the identity 1
and the congruence closure over an Abelian group with + is outlined