Infinets: The Parallel Syntax for Non-wellfounded Proof-Theory

Abstract

Logics based on the µ-calculus are used to model inductive and coinductive reasoning and to verify reactive systems. A wellstructured proof-theory is needed in order to apply such logics to the study of programming languages with (co)inductive data types and automated (co)inductive theorem proving. While traditional proof system suffers some defects, non-wellfounded (or infinitary) and circular proofs have been recognized as a valuable alternative, and significant progress have been made in this direction in recent years. Such proofs are nonwellfounded sequent derivations together with a global validity condition expressed in terms of progressing threads. The present paper investigates a discrepancy found in such proof systems, between the sequential nature of sequent proofs and the parallel structure of threads: various proof attempts may have the exact threading structure while differing in the order of inference rules applications. The paper introduces infinets, that are proof-nets for non-wellfounded proofs in the setting of multiplicative linear logic with least and greatest fixed-points (µMLL ∞) and study their correctness and sequentialization.

Canonical proof-objects for coinductive programming: infinets with infinitely many cuts

“…5): indeed, in the sequent calculus, an arbitrary formula 𝐹 can be added to a proof of a sequent ⊢ 𝜈𝑋 .𝑋 ; 𝐹 can be seen as justified by the infinite sequence of unfoldings. As such, we need to introduce infinite axioms as in [19] which are sets of finitely many formulas and sequences of unfoldings. Before defining our proof-objects, we thus need to define these sequences of unfolding and how they can participate in infinite axioms.…”

“…Visitable paths Higher order trips (def. 19) Higher order visitable paths In the next section, we will define the notions for proof-structures. Remark 2.…”

“…Infinets. In [19], the authors defined infinets, canonical objects that capture exactly the equivalence classes of pre-proofs under the equivalence by (possibly infinite) permutation of inferences (a.k.a. permutative equivalence).…”

“…Infinitely many cuts. The handling of the cuts in [19] has been rudimentary: only finitely many cuts are considered and cut-elimination is basically interpreted as an infinitary abstract rewriting system with a metric: at first, one guesses the normal form (a.k.a. big-step) then, a transfinite reduction sequence of small steps is shown to converge to the big-step in the limit.…”

“…We state the correctness criterion in section 6, which leads us to develop and adapt the theory of kingdoms to non-wellfounded structures, an additional contribution of the work. This allows us to provide a sequentialisation procedure extending that of [19]. We introduce a new cut reduction system in section 7 and establish the cut-elimination theorem by showing productivity of cut-reduction for valid infinets.…”