616 research outputs found
On the relative proof complexity of deep inference via atomic flows
We consider the proof complexity of the minimal complete fragment, KS, of
standard deep inference systems for propositional logic. To examine the size of
proofs we employ atomic flows, diagrams that trace structural changes through a
proof but ignore logical information. As results we obtain a polynomial
simulation of versions of Resolution, along with some extensions. We also show
that these systems, as well as bounded-depth Frege systems, cannot polynomially
simulate KS, by giving polynomial-size proofs of certain variants of the
propositional pigeonhole principle in KS.Comment: 27 pages, 2 figures, full version of conference pape
An Abstract Approach to Consequence Relations
We generalise the Blok-J\'onsson account of structural consequence relations,
later developed by Galatos, Tsinakis and other authors, in such a way as to
naturally accommodate multiset consequence. While Blok and J\'onsson admit, in
place of sheer formulas, a wider range of syntactic units to be manipulated in
deductions (including sequents or equations), these objects are invariably
aggregated via set-theoretical union. Our approach is more general in that
non-idempotent forms of premiss and conclusion aggregation, including multiset
sum and fuzzy set union, are considered. In their abstract form, thus,
deductive relations are defined as additional compatible preorderings over
certain partially ordered monoids. We investigate these relations using
categorical methods, and provide analogues of the main results obtained in the
general theory of consequence relations. Then we focus on the driving example
of multiset deductive relations, providing variations of the methods of matrix
semantics and Hilbert systems in Abstract Algebraic Logic
Double-Negation Elimination in Some Propositional Logics
This article answers two questions (posed in the literature), each concerning
the guaranteed existence of proofs free of double negation. A proof is free of
double negation if none of its deduced steps contains a term of the form
n(n(t)) for some term t, where n denotes negation. The first question asks for
conditions on the hypotheses that, if satisfied, guarantee the existence of a
double-negation-free proof when the conclusion is free of double negation. The
second question asks about the existence of an axiom system for classical
propositional calculus whose use, for theorems with a conclusion free of double
negation, guarantees the existence of a double-negation-free proof. After
giving conditions that answer the first question, we answer the second question
by focusing on the Lukasiewicz three-axiom system. We then extend our studies
to infinite-valued sentential calculus and to intuitionistic logic and
generalize the notion of being double-negation free. The double-negation proofs
of interest rely exclusively on the inference rule condensed detachment, a rule
that combines modus ponens with an appropriately general rule of substitution.
The automated reasoning program OTTER played an indispensable role in this
study.Comment: 32 pages, no figure
A simple sequent calculus for nominal logic
Nominal logic is a variant of first-order logic that provides support for
reasoning about bound names in abstract syntax. A key feature of nominal logic
is the new-quantifier, which quantifies over fresh names (names not appearing
in any values considered so far). Previous attempts have been made to develop
convenient rules for reasoning with the new-quantifier, but we argue that none
of these attempts is completely satisfactory.
In this article we develop a new sequent calculus for nominal logic in which
the rules for the new- quantifier are much simpler than in previous attempts.
We also prove several structural and metatheoretic properties, including
cut-elimination, consistency, and equivalence to Pitts' axiomatization of
nominal logic
Proof search in constructive logics
We present an overview of some sequent calculi organised not for
"theorem-proving" but for proof search, where the proofs themselves
(and the avoidance of known proofs on backtracking) are objects of
interest. The main calculus discussed is that of Herbelin [1994] for
intuitionistic logic, which extends methods used in hereditary
Harrop logic programming; we give a brief discussion of similar
calculi for other logics. We also point out to some related work on
permutations in intuitionistic Gentzen sequent calculi that
clarifies the relationship between such calculi and natural
deduction.Centro de Matemática da Universidade do Minho (CMAT).União Europeia (UE) - Programa ESPRIT - BRA 7232 Gentzen
Dual-Context Calculi for Modal Logic
We present natural deduction systems and associated modal lambda calculi for
the necessity fragments of the normal modal logics K, T, K4, GL and S4. These
systems are in the dual-context style: they feature two distinct zones of
assumptions, one of which can be thought as modal, and the other as
intuitionistic. We show that these calculi have their roots in in sequent
calculi. We then investigate their metatheory, equip them with a confluent and
strongly normalizing notion of reduction, and show that they coincide with the
usual Hilbert systems up to provability. Finally, we investigate a categorical
semantics which interprets the modality as a product-preserving functor.Comment: Full version of article previously presented at LICS 2017 (see
arXiv:1602.04860v4 or doi: 10.1109/LICS.2017.8005089
EARLY STRUCTURAL REASONING. GENTZEN 1932
Abstract. This paper is a study of the opening section of Gentzen’s first publication of 1932, Über die Existenz unabhängiger Axiomensysteme zu unendlichen Satzsystemen, a text which shows the relevance of Hertz’s work of the 1920’s for the young Gentzen. In fact, Gentzen borrowed from Hertz the analysis of the notion of consequence, which was given in terms of the rules of thinning (Verdünnung) and cut (Schnitt) on sequents (there called “sentences”(Sätze)). Moreover, following Hertz again, he also judged it necessary to justify the forms of inference of the system by providing a semantics for them, so that it became possible to make precise the informal notion of consequence, and to show that the inference rules adopted are correct and sufficient
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