A Counting Logic for Structure Transition Systems

Abstract

Quantitative questions such as "what is the maximum number of tokens
in a place of a Petri net?" or "what is the maximal reachable height
of the stack of a pushdown automaton?" play a significant role in
understanding models of computation. To study such problems in a
systematic way, we introduce structure transition systems on which
one can define logics that mix temporal expressions (e.g. reachability) with properties of a state (e.g. the height of the stack). We propose a counting logic Qmu[#MSO] which allows to express questions like the ones above, and also many boundedness problems studied so far. We show that Qmu[#MSO] has good algorithmic properties, in particular we generalize two standard methods in model checking, decomposition on trees and model checking through parity games, to this quantitative logic. These properties are used to prove decidability of Qmu[#MSO] on tree-producing pushdown systems, a generalization of both pushdown systems and regular tree grammars.