Counter Simulations via Higher Order Quantifier Elimination: a preliminary report

Abstract

Quite often, verification tasks for distributed systems are accomplished via counter abstractions. Such abstractions can sometimes be justified via simulations and bisimulations. In this work, we supply logical foundations to this practice, by a specifically designed technique for second order quantifier elimination. Our method, once applied to specifications of verification problems for parameterized distributed systems, produces integer variables systems that are ready to be model-checked by current SMT-based tools. We demonstrate the feasibility of the approach with a prototype implementation and first experiments. * The first authar was supported by the INdAM's GNSAGA group. System Specifications in Higher Order LogicThe behavior of a computer system can be modeled through a transition system, which is a tuple T = (W,W 0 , R, AP,V ) such that (i) W is the set of possible configurations, (ii) W 0 ⊆ W is the set of initial configurations, (iii) AP is a set of 'atomic propositions', (iv) V : W −→ AP is a function labeling each state with the set of propositions 'true in it', (v) R ⊆ W ×W is the transition relation: w 1 Rw 2 describes how the system can 'evolve in one step'.

Higher-Order Quantifier Elimination, Counter Simulations and Fault-Tolerant Systems

“…In the electronic Appendix A in the supplementary material we run three representative examples in full detail; the electronic Appendix is available on the publisher website. Preliminary versions of the material included in this paper were presented in the Workshops [26,27].…”

“…), the user can only introduce static counters, whereas the z f are dynamic counters: they count how many processes make each possible change of values for their enumerated array-ids. 12 The original counters are expressed as linear combinations of these new dynamic counters; in addition, in each disjunct (25), the universally quantified formula ∀x θ(x) is replaced by the equation N = f z f , where f is 0 or 1 depending on whether the formula defining the set (26) counted by z f is consistent or not with θ . (iv) In the final steps, all arithmetic atoms involving old and new counters are collected for each disjunct (25); the new dynamic counters are eliminated by quantifier elimination and the resulting formulae give the disjuncts of the transition of the desired arithmetic projection of the input system.…”