Harold Schellinx

The Structure of Exponentials: Uncovering the Dynamics of Linear Logic Proofs

We construct the exponential graph of a proof π in (second order) linear logic, an artefact displaying the interdependencies of exponentials in π. Within this graph superfluous exponentials are defined, the removal of which is shown to yield a correct proof π▹ with essentially the same set of reductions.

On the linear decoration of intuitionistic derivations

SummaryWe define an optimal proof-by-proof embedding of intuitionistic sequent calculus into linear logic and analyse the (purely logical) linearity information thus obtained.

Collapsing Graph Models by Preorders

We present a strategy for obtaining extensional (partial) combinatory algebras by slightly modifying the well-known construction of graph models for the untyped lambda calculus. Using the notion of weak cartesian closed category an elegant interpretation of our construction in a category theoretical setting is given.

A Linear Approach to Modal Proof Theory

Linear logic1 bans the structural rules of weakening and contraction from the formulation of classical logic as a sequent calculus, and re-introduces them in modalized form: structural manipulation to the left of the entailment-sign is allowed only for formulas prefixed by “!”, to the right only for those prefixed by “?”. (‘Linear’ logicians refer to !, ? as the exponentials.) The resulting calculus is a proof theoretical jewel, which combines the deep symmetries of classical, with the computational properties (strong normalization, confluence) of intuitionistic logic, without loss of expressive power.

Basic Proof Theory, A.S. Troelstra and H. Schwichtenberg

Structural proof theory originated in the attempts earlier this century to reduce mathematics to the art of mechanical manipulation and combination of (chosen but arbitrary) symbols in formal proofs: start from some set of basic statements ( axioms), and build from there ever on upwards, by a continuing correct application of nothing but the laws of pure reason – that is, logic. Formal proofs then become investigable and manipulable objects in their own right, open to study – maybe even to mathematical study? In their most basic form, formal proofs start from an empty collection of axioms, from bare identity: what we get then are formal proofs – derivations– in pure logic. Now, what kind of “thing” is a formal proof? Obviously some sort of “text,” a – finite or infinite – collection of symbols written and ordered according to the rules of some “grammar,” which (preferably) has been shown sound and complete with respect to some “meaning.” The aim of structural proof theory then is the study of the structure and properties of these “texts.” By its very nature structural proof theory is a “theoretical computing science” avant la lettre. It is in this quality that its study nowadays continues to find its legitimacy. I shall go one better than that: the proper development of structural proof theory indeed is one of themost important scientific tasks in a world that is being re-shaped almost continuously by the efforts and stunning achievements of “automatic reasoning,” and “decision-making” through the manipulation and mechanical transformation of “abstract texts.” A profound understanding of the nature of derivations in plain first order logic is a necessary starting point for the eventual development of a theory of provably correct programs, free of the myriads of uncontrolled and uncontrollable ad hocconstructs that abound these days.

Strong Normalization for All-Style LK

We prove strong normalization of tq-reduction for all standard versions of sequent calculus for classical and intuitionistic (second and first order) logic and give a perspicuous argument for the completeness of the focusing restriction on sequent derivations.

On the Jordan-Hölder decomposition of proof nets

Abstract. Having defined a notion of homology for paired graphs, Métayer ([Ma]) proves a homological correctness criterion for proof nets, and states that for any proof net

A New Deconstructive Logic: Linear Logic

G

LKQ and LKT: sequent calculi for second order logic based upon dual linear decompositions of classical implication

there exists a Jordan-Hölder decomposition of

Some Syntactical Observations on Linear Logic

{\mathsf H}_0(G)