Canonical Proof nets for Classical Logic

Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proof-net, they are in essence the same proof. Providing a convincing proof-net counterpart to proofs in the classical sequent calculus is thus an important step in understanding classical sequent calculus proofs. By convincing, we mean that (a) there should be a canonical function from sequent proofs to proof nets, (b) it should be possible to check the correctness of a net in polynomial time, (c) every correct net should be obtainable from a sequent calculus proof, and (d) there should be a cut-elimination procedure which preserves correctness. Previous attempts to give proof-net-like objects for propositional classical logic have failed at least one of the above conditions. In [23], the author presented a calculus of proof nets (expansion nets) satisfying (a) and (b); the paper defined a sequent calculus corresponding to expansion nets but gave no explicit demonstration of (c). That sequent calculus, called LK\ast in this paper, is a novel one-sided sequent calculus with both additively and multiplicatively formulated disjunction rules. In this paper (a self-contained extended version of [23]), we give a full proof of (c) for expansion nets with respect to LK\ast, and in addition give a cut-elimination procedure internal to expansion nets - this makes expansion nets the first notion of proof-net for classical logic satisfying all four criteria.Comment: Accepted for publication in APAL (Special issue, Classical Logic and Computation

Sequentiality vs. Concurrency in Games and Logic

Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic.Comment: 35 pages, appeared in Mathematical Structures in Computer Scienc

Expansion Trees with Cut

Herbrand's theorem is one of the most fundamental insights in logic. From the syntactic point of view it suggests a compact representation of proofs in classical first- and higher-order logic by recording the information which instances have been chosen for which quantifiers, known in the literature as expansion trees. Such a representation is inherently analytic and hence corresponds to a cut-free sequent calculus proof. Recently several extensions of such proof representations to proofs with cut have been proposed. These extensions are based on graphical formalisms similar to proof nets and are limited to prenex formulas. In this paper we present a new approach that directly extends expansion trees by cuts and covers also non-prenex formulas. We describe a cut-elimination procedure for our expansion trees with cut that is based on the natural reduction steps. We prove that it is weakly normalizing using methods from the epsilon-calculus

A Direct Version of Veldman's Proof of Open Induction on Cantor Space via Delimited Control Operators

First, we reconstruct Wim Veldman's result that Open Induction on Cantor space can be derived from Double-negation Shift and Markov's Principle. In doing this, we notice that one has to use a countable choice axiom in the proof and that Markov's Principle is replaceable by slightly strengthening the Double-negation Shift schema. We show that this strengthened version of Double-negation Shift can nonetheless be derived in a constructive intermediate logic based on delimited control operators, extended with axioms for higher-type Heyting Arithmetic. We formalize the argument and thus obtain a proof term that directly derives Open Induction on Cantor space by the shift and reset delimited control operators of Danvy and Filinski

Formal logic: Classical problems and proofs

Not focusing on the history of classical logic, this book provides discussions and quotes central passages on its origins and development, namely from a philosophical perspective. Not being a book in mathematical logic, it takes formal logic from an essentially mathematical perspective. Biased towards a computational approach, with SAT and VAL as its backbone, this is an introduction to logic that covers essential aspects of the three branches of logic, to wit, philosophical, mathematical, and computational

On Affine Logic and {\L}ukasiewicz Logic

The multi-valued logic of {\L}ukasiewicz is a substructural logic that has been widely studied and has many interesting properties. It is classical, in the sense that it admits the axiom schema of double negation, [DNE]. However, our understanding of {\L}ukasiewicz logic can be improved by separating its classical and intuitionistic aspects. The intuitionistic aspect of {\L}ukasiewicz logic is captured in an axiom schema, [CWC], which asserts the commutativity of a weak form of conjunction. This is equivalent to a very restricted form of contraction. We show how {\L}ukasiewicz Logic can be viewed both as an extension of classical affine logic with [CWC], or as an extension of what we call \emph{intuitionistic} {\L}ukasiewicz logic with [DNE], intuitionistic {\L}ukasiewicz logic being the extension of intuitionistic affine logic by the schema [CWC]. At first glance, intuitionistic affine logic seems very weak, but, in fact, [CWC] is surprisingly powerful, implying results such as intuitionistic analogues of De Morgan's laws. However the proofs can be very intricate. We present these results using derived connectives to clarify and motivate the proofs and give several applications. We give an analysis of the applicability to these logics of the well-known methods that use negation to translate classical logic into intuitionistic logic. The usual proofs of correctness for these translations make much use of contraction. Nonetheless, we show that all the usual negative translations are already correct for intuitionistic {\L}ukasiewicz logic, where only the limited amount of contraction given by [CWC] is allowed. This is in contrast with affine logic for which we show, by appeal to results on semantics proved in a companion paper, that both the Gentzen and the Glivenko translations fail.Comment: 28 page