Decidability problems in Set/Hyperset Theory and an application to algorithm verification

It is known that the Axiom of Infinity can be stated, relative to the standard ZermeloFraenkel-von Neumann set theory ZF, as an assertion of very low structural complexity: namely, one which involves only universal restricted quantifiers, i.e., belongs to the Bernays-Schönfinkel-Ramsey class BSR. We will exhibit, in Part I, two similar assertions of existence of infinite sets [POT09]: both belonging to BSR, incompatible with ZF’s Axiom of Foundation, but satisfiable in a context such as Aczel’s theory of hypersets. Well beyond these pages, our aim is to enhance into a satisfiability decision algorithm for the BSR class over ill-founded sets the recently discovered semi-decision algorithm for the BSR class over well-founded sets. The latter was achieved by showing that any well-founded infinite model of a BSR formula admits a finite representation in terms of a hereditarily finite family of hypersets (with urelements). As we are about to see, this finite representability result has no straightforward counterpart over the enlarged domain of ill-founded sets. This may be a clue that the collection of satisfiable BSR formulae requires, for hyperset theory—if decidable at all in that context—, a significantly more challenging decision algorithm than for ZF. An application of a decision algorithm, MLSS, for a fragment of ZF set theory has been in the construction of the proof verifier ÆtnaNova/Referee. In Part II, we report on using this proof-verification system to formalize issues regarding the satisfiability of CNF-formulae of propositional logic [OT08]. We specify an “archetype” version of the Davis-Putnam-Logemann-Loveland algorithm through the THEORY of recursive functions based on a well-founded relation, and prove it to be correct.