I'm a Senior 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.

In this project we investigate the computational potential of deep-inference proof theory, via the Curry-Howard correspondence between intuitionistic logic and simple types for the lambda-calculus.

We investigate intersection types and resource lambda-calculi from
the perspective of deep-inference proof theory. We give a single
underlying type system that is *parameterized* in a choice of
algebraic laws to represent different flavours of quantitative
reasoning about lambda-calculi.

Probabilistic lambda-calculi are generally non-confluent, with
call-by-name and call-by-value providing different results. Via a
decomposition of the usual probabilistic sum into a *generator*
that generates a probabilistic event and a *choice* that
evaluates differently depending on a given event, we capture both
reduction regimes, as well as many intermediate ones, in a single,
confluent probabilistic lambda-calculus.

We investigate the computational side of the *switch* rule of
deep-inference proof theory, and link it to *end-of-scope*
operators and *director strings* in lambda-calculus. We use
this to extend the atomic lambda-calculus, obtaining a more refined
version of full laziness, *spine duplication*.

We give two flavours of proof nets for additive linear logic with first-order quantification: one where existential witnesses are recorded via explicit substitutions, attached to axiom links; and one where witness information is omitted, and reconstructed by unification.

We extend Hughes's classical Combinatorial Proofs to intuitionistic
logic. We obtain a purely geometric, concrete semantics that has
*polynomial full completeness* (no size explosion when
translating from sequent calculus) and *local canonicity* (all
non-duplicating permutations are factored out).

Bi-intuitionistic linear logic (BILL) combines the tensor and linear
implication of intuitionistic linear logic (ILL) with their duals,
*par* and *subtraction*, relating tensor and par through a
linear distributivity. We give canonical proof nets for this
proof-theoretically challenging logic, with correctness through both
contractibility and switching.

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 A deep quantitative type system.

Superseded by Atomic lambda-calculus: a typed lambda-calculus with explicit sharing.

Superseded by Classical Proof Forestry.

Doctoral thesis: A lambda-calculus that achieves full laziness with spine duplication

A collection of educational computer games for logic in computer science, made by final-year undergraduate students.

A LaTeX package for drawing derivations in open deduction.

Folding instructions for a good-looking and reasonably-flying stealth paper airplane.

Department of Computer Science

University of Bath

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