LIRA: Handling Constraints of Linear Arithmetics over the Integers and the Reals

The mechanization of many verification tasks relies on efficient implementations of decision procedures for fragments of first-order logic. Interactive theorem provers like pvs also make use of such decision procedures to increase the level of automation. Our tool lira3 implements decision procedures based on automata-theoretic techniques for first-order logics with linear arithmetic, namely, for FO(N,+), FO(Z,+, <), and FO(R, Z,+, <). The theoretical foundations for using automata to decide logics like Presburger arithmetic, i.e., FO(N,+) were laid in the 1960s [4]: For Presburger arithmetic, the elements of the domain are represented by finite words, and for a given formula, one constructs recursively over the formula structure an automaton that accepts precisely the words that represent the natural numbers that satisfy the formula. Automata constructions handle the logical connectives and quantifiers. A similar approach works for FO(Z,+, <) and FO(R, Z,+, <). To represent reals, one uses infinite words. In [2], it is shown that weak deterministic Büchi automata (wdbas) suffice to decide FO(R, Z,+, <). wdbas are a restricted class of Büchi automata, which can be handled algorithmically almost as efficiently as deterministic finite automata (dfas). lira also provides an automata library that efficiently represents and manipulates dfas and wdbas. lira’s automata library can be compared to a bdd library for representing and manipulating finite sets encoded by booleans. Instead of bdds, lira uses dfas to represent and manipulate sets that are definable in FO(N,+) and FO(Z,+, <), and uses wdbas for sets definable in FO(R, Z,+, <). Efficiently representing and manipulating such definable sets has applications beyond deciding these logics efficiently. For instance, in the safety verification of integer-counter systems and hybrid systems one has to cope with such sets. Furthermore,