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 …

The complexity of propositional proofs in deep inference

Anupam Das. PhD thesis. Supervised by Alessio Guglielmi and John Power, University of Bath. PDF

Deep inference is a relatively recent proof methodology whose systems differ from traditional systems by allowing inference rules to operate on any connective appearing in a formula, rather than just the main connective. Its distinguishing feature, from a structural proof theoretic point of view, is that its systems are local : inference steps can be …

Rewriting with linear inferences in propositional logic

Anupam Das. In proceedings of RTA '13. DOI PDF

Linear inferences are sound implications of propositional logic where each variable appears exactly once in the premiss and conclusion. We consider a specific set of these inferences, MS, first studied by Straßburger, corresponding to the logical rules in deep inference proof theory. Despite previous results characterising the individual rules of MS, we show that there …