Author archives

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

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 …

Rewriting with linear inferences in propositional logic

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

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

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

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 …