Circular (Yet Sound) Proofs in Propositional Logic

Proofs in propositional logic are typically presented as trees of derived formulas or, alternatively, directed acyclic graphs formulas. This distinction between tree-like vs. dag-like structure is particularly relevant when making quantitative considerations regarding, for example, proof size. Here we analyze a more general type structural restriction proofs rule-based systems. In this definition, which cycles allowed long every formula at least many times it required premise. We call such “circular”. show that, all sets standard inference rules with single or multiple conclusions, circular sound. start the study complexity Circular Resolution, version Resolution. immediately see that Resolution stronger than since, show, encoding pigeonhole principle has polynomial Furthermore, derivations clauses from clauses, is, surprisingly, equivalent to Sherali-Adams, system reasoning through inequalities linear programming its base. As corollaries get: (1) polynomial-time (LP-based) algorithms find constant width, (2) examples separate and variants, (3) exponentially hard cases Contrary case Frege can be converted into most overhead.