On the Proof Theory of the Modal mu-Calculus

Abstract

We study the proof-theoretic relationship between two deductive systems for the modal mu-calculus. First we recall an infinitary system which contains an omega rule allowing to derive the truth of a greatest fixed point from the truth of each of its (infinitely many) approximations. Then we recall a second infinitary calculus which is based on non-well-founded trees. In this system proofs are finitely branching but may contain infinite branches as long as some greatest fixed point is unfolded infinitely often along every branch. The main contribution of our paper is a translation from proofs in the first system to proofs in the second system. Completeness of the second system then follows from completeness of the first, and a new proof of the finite model property also follows as a corollary.

A Proof System for Compositional Verification of Probabilistic Concurrent Processes

“…This is essential for the potential applicability of the approach. It is proved along the lines of similar results for non-probabilistic cyclic proofs (see, e.g., [15], [21] and [3]), using decidability results for one-player stochastic parity games established in [5].…”

“…The rules {∨ 1 , ∨ 2 , ∧, a , [a] , µ, ν} are called logical rules. The rules ∨ 1 , ∨ 2 , ∧, µ, ν are standard and also the rules a , [a] for reasoning about modalities are natural counterparts to the analogous rules adopted in proof systems for modal (fixed point) logics appeared in the literature (see, e.g., [19], [21] and [15]). The rules {δ, + λ , { }} are called distribution rules and constitute a crucial aspect of the system.…”

“…[15,21,3]) in which some leaves of the tree are identified with sequents internal to the tree, with the proof looping back to that point. Technically, it is convenient to view such cyclic trees as the infinite trees they unfold to, and to work with general infinite trees, with the finite cyclic ones corresponding exactly to the regular trees (those with only finitely many subtrees).…”

“…In the literature on infinitary proof systems for fixed-point logics (see, e.g., [15,21,3]), valid proofs are defined as those preproofs whose infinite branches all contain at least one legitimate sequence (called a valid trace) of fixed-point unfoldings, along which a greatest fixed-point is unfolded infinitely often. This can equivalently be reformulated in terms of a single player game.…”

“…This approach is familiar from proof systems for the ordinary (non-probabilistic) modal µ-calculus [15,21] and other fixed-point logics [3]. However, there is a twist in our probabilistic setting.…”