I'm a Lecturer and Prize Fellow in the Mathematical Foundations group, Department of Computer Science, University of Bath.

My research is in the area of logic and proof theory, and has focused on canonical graphical representations of proof, commonly known as proof nets.

We give a new notion of proof net for Multiplicative-Additive Linear Logic, that strikes a subtle balance between efficiency and canonicity. Conflict nets are canonical for all local rule commutations, which are those that do not incur a global duplication. As a consequence they have linear size compared to sequent proofs, avoiding the exponential growth of Hughes and Van Glabbeek's Slice Nets.

Proof equivalence in MLL with units is shown to be PSPACE-complete, by a reduction from the graphical formalism called Non-Deterministic Constraint Logic. This effectively rules out a satisfactory notion of proof net with units, as such a notion would constitute a tractable decision algorithm for proof equivalence.

We consider a notion of semi-star-autonomous category: star-autonomous categories without units, corresponding to Girard's proof nets for MLL. (Available online since November 2014)

via additive proof nets and Petri nets

We give an effective correctness criterion for additive proof nets, which is naturally expressed in Petri nets, and is the equivalent of Danos contractibility for MLL. In addition we give simple proof search algorithms for additive linear logic with and without units and an effective correctness algorithm for additive proof nets with units, and we show that first-order additive linear logic is NP-complete.

Proof equivalence in MLL is PSPACE-complete

Proof equivalence in MLL with units is shown to be PSPACE-complete, by a reduction from the graphical formalism called Non-Deterministic Constraint Logic. This effectively rules out a satisfactory notion of proof net with units, as such a notion would constitute a tractable decision algorithm for proof equivalence. (Superseded by the journal version Proof equivalence in MLL is PSPACE-complete)

The atomic lambda-calculus is a typeable lambda-calculus with explicit sharing, which originates in a Curry-Howard interpretation of a deep-inference system for intuitionistic logic. In this paper we prove strong normalization of the typed atomic lambda-calculus using Tait's reducibility method.

a typed lambda-calculus with explicit sharing

The atomic lambda-calculus is a typeable lambda-calculus with explicit sharing, based on a Curry-Howard-style interpretation of the deep inference proof formalism. Duplication of subterms during reduction proceeds `atomically', i.e. on individual constructors, similar to optimal graph reduction in the style of Lamping. The calculus preserves strong normalisation and achieves fully lazy sharing.

The paper describes canonical proof nets for additive linear logic, or sum-product logic, the internal language of categories with free finite products and co-products. Starting from existing proof nets, which disregard the unit laws, canonical nets are obtained by a simple rewriting algorithm, for which a substantial correctness proof is provided.

Awarded the LICS 2011 Kleene award for best student paper

The paper investigates cut-elimination in classical proof forests, a proof formalism for first-order classical logic based on Herbrand's Theorem and backtracking games in the style of Coquand. Cut-free classical proof forests were described by Miller, and are called Expansion Tree Proofs.

Superseded by Atomic lambda-calculus: a typed lambda-calculus with explicit sharing.

Superseded by Classical Proof Forestry.

Laboratory for the Foundations of Computer Science (LFCS)

School of Informatics | University of Edinburgh

Doctoral thesis: *Graphical Representation of Canonical Proof: Two case studies*

Thesis advisor: Professor Alex Simpson

Department of Philosophy | Utrecht University

Master's thesis: *Graph Rewriting for Natural Deduction and the Proper Treatment of Variables*

Winner of the Cognitive Artificial Intelligence thesis award 2007, Department of Philosophy, Utrecht University

Thesis advisors: Professor Albert Visser and Professor Vincent van Oostrom

Department of Computer Science

1 West 4.67

Claverton Down

Bath

BA2 7AY

United Kingdom

w.b.heijltjes@bath.ac.uk