No complete linear term rewriting system for propositional logic

Anupam Das and Lutz Stra├čburger. 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 Stra├čburger, corresponding to the logical rules in deep inference proof theory. Despite previous results characterising the individual rules of MS, we show that there …

Complexity of deep inference via atomic flows

Anupam Das. In proceedings of Turing Centenary Conference: CiE 2012. DOI PDF

We consider the fragment of deep inference free of compression mechanisms and compare its proof complexity to other systems, utilising ‘atomic flows’ to examine size of proofs. Results include a simulation of Resolution and dag-like cut-free Gentzen, as well as a separation from bounded-depth Frege. NB: There are errors in some statements and proofs of …

Characterising aspects of proof compression

Anupam Das. Note. PDF

We consider deep inference proof formalisms, which are flexible enough to embed many widely used proof systems, and construe compression mechanisms `cut’ and `dag’ in a way that is independent of any particular proof system, but in a way that nonetheless has the same effect from the point of view of proof complexity. The main …

On the proof complexity of cut-free bounded deep inference

Anupam Das. In proceedings of Tableaux '11. DOI PDF

It has recently been shown that cut-free deep inference systems exhibit an exponential speed-up over cut-free sequent systems, in terms of proof size. While this is good for proof complexity, there remains the problem of typically high proof search non-determinism induced by the deep inference methodology: the higher the depth of inference, the higher the …