On linear rewriting systems for Boolean logic and some applications to proof theory

Anupam Das and Lutz Straßburger. Logical Methods in Computer Science. Special issue of selected papers from RTA and TLCA '15. DOI arXiv PDF

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 …

Free-cut elimination in linear logic and an application to a feasible arithmetic

Patrick Baillot and Anupam Das. In proceedings of CSL '16. DOI HAL PDF

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 …

From positive and intuitionistic bounded arithmetic to monotone proof complexity

Anupam Das. In proceedings of LICS '16. DOI PDF

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, …

On nested sequents for constructive modal logics

Ryuta Arisaka, Anupam Das and Lutz Straßburger. Logical Methods in Computer Science. DOI arXiv PDF

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 …

A complete axiomatization of MSO on infinite trees

Anupam Das and Colin Riba. In proceedings of LICS '15. DOI PDF

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 …

No complete linear term rewriting system for propositional logic

Anupam Das and Lutz Straßburger. In proceedings of RTA '15. DOI PDF

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 …

On the relative proof complexity of deep inference via atomic flows

Anupam Das. Logical Methods in Computer Science. Special issue of selected papers of the Turing Centenary Conference: CiE 2012. DOI arXiv PDF

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 …

On the pigeonhole and related principles in deep inference and monotone systems

Anupam Das. In proceedings of CSL-LICS '14. DOI PDF

We construct quasipolynomial-size proofs of the propositional pigeonhole principle in the deep inference system KS, addressing an open problem raised in previous works and matching the best known upper bound for the more general class of monotone proofs. We make significant use of monotone formulae computing boolean threshold functions, an idea previously considered in works …