Consistent Consequence for Boolean Equation Systems

Abstraction in Fixpoint Logic

“…In fact, in Gazda and Willemse [2012a], it was shown that the consistent consequence relation coincides with direct simulation [Gazda and Willemse 2012a;Keiren 2013] for parity games, but only for the fragment of equations in standard recursive form: equations in which all right-hand sides are purely disjunctive or purely conjunctive. Moving beyond that fragment, it quickly becomes unclear which parity game relation (if any) coincides with consistent consequence.…”

“…One way in which the relation between predicate variable instances such as X(v) and X (v ) can be established is through the notion of a consistent consequence, a coinductively defined relation on the signatures of equation systems. This notion was previously introduced in Gazda and Willemse [2012a] for the fragment of Boolean equation systems. The absence of data in that setting greatly simplifies its formalisation; here, we generalise the definition to arbitrary equation systems.…”

“…The problem of checking whether there is a consistent consequence between signatures of two Boolean equation systems is coNP-complete already [Gazda and Willemse 2012a]. We show that our abstraction framework can nevertheless lead to effective tooling.…”

“…It shows that abstraction can, in a sense, be seen as a coinductive generalisation of logical consequence. Indeed, in Gazda and Willemse [2012a], it was shown that consistent consequence for Boolean equation systems is essentially captured by adding the below rule to the standard axiomatisation of logical consequence for negation-free propositional logic:…”

“…The abstraction framework we introduce here captures such approximations through a coinductive relation between equation systems called consistent consequence; this relation is essentially the standard logical implication, lifted to equation systems. Consistent consequence was introduced in Gazda and Willemse [2012a] for the fragment of Boolean equation systems (and it is generalised to arbitrary equation systems in this article) with the purpose of proving that a consistent correlation [Willemse 2010] admits a sound and complete proof system, and to study the latter's computational complexity. Note that consistent correlations, in turn, were introduced to simplify many of the involved and cumbersome correctness proofs associated with the aforementioned syntax-based transformations on equation systems.…”