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

@article{Das2016OnLR,
  title={On linear rewriting systems for Boolean logic and some applications to proof theory},
  author={Anupam Das and Lutz Stra{\ss}burger},
  journal={Log. Methods Comput. Sci.},
  year={2016},
  volume={12}
}
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 systems consisting only of linear inferences. Our main result is that the length of any 'nontrivial' derivation in such a system is bound by a polynomial. As a… 

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References

SHOWING 1-10 OF 38 REFERENCES

No complete linear term rewriting system for propositional logic

It is shown that, as long as reduction steps are polynomial-time decidable, such a rewriting system does not exist unless coNP=NP, i.e. that every Boolean tautology can be written as a (left- and right-) linear rewrite rule.

Rewriting with Linear Inferences in Propositional Logic

A specific set of linear inferences, MS, is considered, corresponding to the logical rules in deep inference proof theory, and it is shown that there is noPolynomial-time characterisation of MS, assuming that integers cannot be factorised in polynomial time.

On the Axiomatisation of Boolean Categories with and without Medial

This work will see a series (with increasing strength) of possible such axiomatisations, all based on the notion of ⁄-autonomous category, and particularly focus on the medial map, which has its origin in an inference rule in KS, a cut-free deductive system for Boolean logic in the calculus of structures.

A Local System for Classical Logic

The calculus of structures is a framework for specifying logical systems, which is similar to the one-sided sequent calculus but more general, and a system of inference rules for propositional classical logic is presented, with the main novelty that all the rules are local.

Proof complexity in algebraic systems and bounded depth Frege systems with modular counting

It is shown that a lower bound for proofs in a bounded depth Frege system in the language with the modular counting connectiveMODp follows from a lower Bound on the degree of Nullstellensatz proofs with a constant number of levels of extension axioms.

On the Proof Complexity of Cut-Free Bounded Deep Inference

This work improves on the proof search side by demonstrating that the same exponential speed-up in proof size can be obtained in bounded-depth cut-free systems, which retain the top-down symmetry of deep inference, but can be designed at the same depth level of sequent systems.

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

  • Anupam Das
  • Computer Science, Mathematics
    CSL-LICS
  • 2014
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

On the relative proof complexity of deep inference via atomic flows

  • Anupam Das
  • Mathematics, Computer Science
    Log. Methods Comput. Sci.
  • 2015
It is shown that standard deep inference systems, as well as bounded-depth Frege systems, cannot polynomially simulate KS, by giving polynomial-size proofs of certain variants of the propositional pigeonhole principle in KS.

Breaking Paths in Atomic Flows for Classical Logic

This paper contains an original 2-dimensional-diagram exposition of atomic flows, which helps to connect atomic flows with other known formalisms, and makes crucial use of the `path breaker', an atomic flow construction that can be used in any proof system with sufficient symmetry.

Extension without cut