Automated Verification, Synthesis and Correction of Concurrent Systems via MSO Logic

Abstract

In this work we provide algorithmic solutions to five fundamental problems concerning the verification, synthesis and correction of concurrent systems that can be modeled by bounded p/t-nets. We express concurrency via partial orders and assume that behavioral specifications are given via monadic second order logic. A c-partial-order is a partial order whose Hasse diagram can be covered by c paths. For a finite set T of transitions, we let P(c,T,\phi) denote the set of all T-labelled c-partial-orders satisfying \phi. If N=(P,T) is a p/t-net we let P(N,c) denote the set of all c-partially-ordered runs of N. A (b, r)-bounded p/t-net is a b-bounded p/t-net in which each place appears repeated at most r times. We solve the following problems: 1. Verification: given an MSO formula \phi and a bounded p/t-net N determine whether P(N,c)\subseteq P(c,T,\phi), whether P(c,T,\phi)\subseteq P(N,c), or whether P(N,c)\cap P(c,T,\phi)=\emptyset. 2. Synthesis from MSO Specifications: given an MSO formula \phi, synthesize a semantically minimal (b,r)-bounded p/t-net N satisfying P(c,T,\phi)\subseteq P(N, c). 3. Semantically Safest Subsystem: given an MSO formula \phi defining a set of safe partial orders, and a b-bounded p/t-net N, possibly containing unsafe behaviors, synthesize the safest (b,r)-bounded p/t-net N' whose behavior lies in between P(N,c)\cap P(c,T,\phi) and P(N,c). 4. Behavioral Repair: given two MSO formulas \phi and \psi, and a b-bounded p/t-net N, synthesize a semantically minimal (b,r)-bounded p/t net N' whose behavior lies in between P(N,c) \cap P(c,T,\phi) and P(c,T,\psi). 5. Synthesis from Contracts: given an MSO formula \phi^yes specifying a set of good behaviors and an MSO formula \phi^no specifying a set of bad behaviors, synthesize a semantically minimal (b,r)-bounded p/t-net N such that P(c,T,\phi^yes) \subseteq P(N,c) but P(c,T,\phi^no ) \cap P(N,c)=\emptyset.