Cyclic Proofs with Ordering Constraints

Abstract

CLKID ω is a sequent-based cyclic inference system able to reason on first-order logic with inductive definitions. The current approach for verifying the soundness of CLKID ω proofs is based on expensive model-checking techniques leading to an explosion in the number of states. We propose proof strategies that guarantee the soundness of a class of CLKID ω proofs if some ordering and derivability constraints are satisfied. They are inspired from previous works about cyclic well-founded induction reasoning, known to provide effective sets of ordering constraints. A derivability constraint can be checked in linear time. Under certain conditions, one can build proofs that implicitly satisfy the ordering constraints.

Validating Back-links of FOLID Cyclic Pre-proofs

“…The inference system CLKID ω N , introduced in [11], is the restricted version of CLKID ω for which (= L) is replaced by the generalization rule (Gen) that substitutes a term by a variable:…”

“…To highlight the changes w.r.t. [11], we take the same running example (also presented in [6]). Let N and R be two inductive predicates defined by:…”

“…Hence, the total number of minimal cycles in a complete digraph with n nodes is n ∑ k=2 n! (n − k)!k Fortunately, the number of arrows in any digraph built with the approach from [11] is smaller than that for the complete digraphs because each bud node has only one companion. However, a orderingderivability constraint can be checked several times as it may be defined w.r.t.…”

“…different minimal cycles. In [11], it was already suggested that their number can be reduced to the number of buds from the minimal cycles, hence smaller than n. This redundancy can be avoided, for example, by using recording mechanisms.…”

“…Section 3 introduces the soundness criterion for CLKID ω N pre-proofs by detailing the normalisation procedure, the digraph construction and the definition of the ordering and derivability conditions. A comparison is made with the soundness criterion from [11]. The conclusions and future work are given in the last section.…”