Symbolic Algorithms for Language Equivalence and Kleene Algebra with Tests

Abstract

We propose algorithms for checking language equivalence of finite automata over a large alphabet. We use symbolic automata, where the transition function is compactly represented using (multi-terminal) binary decision diagrams (BDD). The key idea consists in computing a bisimulation by exploring reachable pairs symbolically, so as to avoid redundancies. This idea can be combined with already existing optimisations, and we show in particular a nice integration with the disjoint sets forest data-structure from Hopcroft and Karp's standard algorithm. Then we consider Kleene algebra with tests (KAT), an algebraic theory that can be used for verification in various domains ranging from compiler optimisation to network programming analysis. This theory is decidable by reduction to language equivalence of automata on guarded strings, a particular kind of automata that have exponentially large alphabets. We propose several methods allowing to construct symbolic automata out of KAT expressions, based either on Brzozowski's derivatives or on standard automata constructions. All in all, this results in efficient algorithms for deciding equivalence of KAT expressions.

Deciding Synchronous Kleene Algebra with Derivatives

“…We ran a similar test considering a sample of 10000 pairs of KAT terms with k = 7, l = 7 and |α| = 100. The values obtained were |α| T = 18.58, H = 41.34, it = 1745 and T ime = 1.9456, which are competitive with the ones in [24] (for Antimirov's algorithm).…”

“…Table 2. Experimental results for uniformly random generated KAT expressions Damien Pous developed an equivalence test for symbolic automata [31] and performed some tests for KAT terms [24]. To ensure equivalence of a pair of KAT terms (α 1 , α 2 ) he added A B to each term.…”

Nominal Kleene Coalgebra

“…Despite the high complexity of the worst-case upper bound, much like the bisimulation-based algorithms for other KA-based systems [4,5,7,22,23], the situation may not be so bad in practice. To actually attain the worst-case bound would seem to require highly pathological examples that would be unlikely to arise in practice.…”

“…The motivation for this investigation is to understand the structure of nominal Kleene algebra from a coalgebraic perspective with an eye toward a more efficient decision procedure for the equational theory in the style of [4,5,22,23] for Kleene algebra and Kleene algebra with tests.…”

“…The preprocessing step also gives a bound on the size of automata, and we use this to prove that the equational theory is decidable in exponential space. The data representation also provides a platform for the implementation of a bisimulationbased decision procedure in the style of [4,5,22,23] for Kleene algebra and Kleene algebra with tests.…”