Researcher in proof theory, logic and complexity
Intuitionistic logic and the polynomial hierarchy We conduct a complexity-theoretic study of focussed proof systems, relating the alternation of synchronous and asynchronous phases in a proof to an appropriate alternating time hierarchy, for instance the polynomial hierarchy. We propose a notion of ‘over-focussing’ that admits non-branching invertible rules during synchronous phases, due to the fact …
Linear rules have played an increasing role in structural proof theory in recent years. It has been observed that the set of all sound linear inference rules in Boolean logic is already coNP-complete, i.e. that every Boolean tautology can be written as a (left- and right-)linear rewrite rule. In this paper we study properties of …
We prove a general form of ‘free-cut elimination’ for first-order theories in linear logic, yielding normal forms of proofs where cuts are anchored to nonlogical steps. To demonstrate the usefulness of this result, we consider a version of arithmetic in linear logic, based on a previous axiomatisation by Bellantoni and Hofmann. We prove a witnessing …
We study versions of second-order bounded arithmetic where induction and comprehension formulae are positive or where the underlying logic is intuitionistic, examining their relationships to monotone and deep inference proof systems for propositional logic. In the positive setting a restriction of a Paris-Wilkie (PW) style translation yields quasipolynomial-size monotone propositional proofs from Π01 arithmetic theorems, …
We present deductive systems for various modal logics that can be obtained from the constructive variant of the normal modal logic CK by adding combinations of the axioms d, t, b, 4, and 5. This includes the constructive variants of the standard modal logics K4, S4, and S5. We use for our presentation the formalism …
We show that an adaptation of Peano’s axioms for second-order arithmetic to the language of MSO completely axiomatizes the theory over infinite trees. This continues a line of work begun by Büchi and Siefkes with axiomatizations of MSO over various classes of linear orders. Our proof formalizes, in the axiomatic theory, a translation of MSO …
Recently it has been observed that the set of all sound linear inference rules in propositional logic is already coNP-complete, i.e. that every Boolean tautology can be written as a (left- and right-) linear rewrite rule. This raises the question of whether there is a rewriting system on linear terms of propositional logic that is …
We consider the proof complexity of the minimal complete fragment, KS, of standard deep inference systems for propositional logic. To examine the size of proofs we employ atomic flows, diagrams that trace structural changes through a proof but ignore logical information. As results we obtain a polynomial simulation of versions of Resolution, along with some …