Total reflections, partial products, and hereditary factorizations

AbstractThis paper shows that many, but not all, reflective subcategories of Top have a certain property, here called total reflectivity, hitherto studied in some special cases, such as for compactness. It is related to Pasynkov's partial topological products and to the stability of topological factorizations under pullback along open inclusions

Call-by-Value Lambda-calculus and LJQ

Accepté pour publication dans J. Logic Comput. ; 24 pagesLJQ is a focused sequent calculus for intuitionistic logic, with a simple restriction on the first premiss of the usual left introduction rule for implication. In a previous paper we discussed its history (going back to about 1950, or beyond) and presented its basic theory and some applications; here we discuss in detail its relation to call-by-value reduction in lambda calculus, establishing a connection between LJQ and the CBV calculus Lambda_C of Moggi. In particular, we present an equational correspondence between these two calculi forming a bijection between the two sets of normal terms, and allowing reductions in each to be simulated by reductions in the other

Cut-elimination and a permutation-free sequent calculus for intuitionistic logic

We describe a sequent calculus, based on work of Herbelin's, of which the cut-free derivations are in 1-1 correspondence with normal natural deduction proofs of intuitionistic logic. We present a simple proof of Herbelin's strong cut-elimination theorem for the calculus, using the recursive path oredering theorem of Dershowitz.Junta Nacional de Investigação Científica e Tecnológica (JNICT).União Europeia (UE) - Programa ESPRIT - grant BRA 7232 GENTZEN

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

Loop-free construction of counter-models for intuitionistic propositional logic

We present a non-looping method to construct Kripke trees refuting the non-theorems of intuitionistic propositional logic, using a contraction-free sequent calculus.União Europeia (UE) - project ESPRIT BRA 7232 GENTZEN.Junta Nacional de Investigação Científica e Tecnológica (JNICT)

Permutability of proofs in intuitionistic sequent calculi

We prove a folklore theorem, that two derivations in a cut-free sequent calculus for intuitionistic propositional logic (based on Kleene's {\bf G3}) are inter-permutable (using a set of basic "permutation reduction rules'' derived from Kleene's work in 1952) iff they determine the same natural deduction. The basic rules form a confluent and weakly normalising rewriting system. We refer to Schwichtenberg's proof elsewhere that a modification of this system is strongly normalising.União Europeia (UE) - Programa ESPRIT BRA 7232 GENTZEN.Centro de Matemática da Universidade do Minho (CMAT)

POSIX lexing with derivatives of regular expressions (proof pearl)

Brzozowski introduced the notion of derivatives for regular expressions. They can be used for a very simple regular expression matching algorithm. Sulzmann and Lu cleverly extended this algorithm in order to deal with POSIX matching, which is the underlying disambiguation strategy for regular expressions needed in lexers. Sulzmann and Lu have made available on-line what they call a “rigorous proof” of the correctness of their algorithm w.r.t. their specification; regrettably, it appears to us to have unfillable gaps. In the first part of this paper we give our inductive definition of what a POSIX value is and show (i) that such a value is unique (for given regular expression and string being matched) and (ii) that Sulzmann and Lu’s algorithm always generates such a value (provided that the regular expression matches the string). We also prove the correctness of an optimised version of the POSIX matching algorithm. Our definitions and proof are much simpler than those by Sulzmann and Lu and can be easily formalised in Isabelle/HOL. In the second part we analyse the correctness argument by Sulzmann and Lu and explain why the gaps in this argument cannot be filled easily.Postprin

Contraction-free sequent calculi in intuitionistic logic : a correction

We present a much-shortened proof of a major result (originally due to Vorob’ev) about intuitionistic propositional logic: in essence, a correction of our 1992 article, avoiding several unnecessary definitionsPublisher PDFPeer reviewe

Commentary on Grigori Mints’ “Classical and Intuitionistic Geometric Logic”

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Analyticity, balance and non-admissibility of Cut in Stoic Logic

This paper shows that, for the Hertz-Gentzen Systems of 1933 (without Thinning), extended by a classical rule T1 (from the Stoics) and using certain axioms (also from the Stoics), all derivations are analytic: every cut formula occurs as a subformula in the cut’s conclusion. Since the Stoic cut rules are instances of Gentzen’s Cut rule of 1933, from this we infer the decidability of the propositional logic of the Stoics. We infer the correctness for this logic of a “relevance criterion” and of two “balance criteria”, and hence (in contrast to one of Gentzen’s 1933 results) that a particular derivable sequent has no derivation that is “normal” in the sense that the first premiss of each cut is cut-free. We also infer that Cut is not admissible in the Stoic system, based on the standard Stoic axioms, the T1 rule and the instances of Cut with just two antecedent formulae in the first premiss.Publisher PDFPeer reviewe