Kleene algebra with tests (KAT) is an equational system that extends Kleene algebra, the algebra of regular expressions, and that is specially suited to capture and verify properties of simple imperative programs. In this paper we study two constructions of automata from KAT expressions: the Glushkov automaton ($\mathcal{A}_{\mathsf{pos}}$), and a new construction based on the notion of prebase (equation automata, $\mathcal{A}_{\mathsf{eq}}$). Contrary to other automata constructions from KAT expressions, these two constructions enjoy the same descriptional complexity behaviour as their counterparts for regular expressions, both in the worst-case as well as in the average-case. In particular, our main result is to show that, asymptotically and on average the number of transitions of the $\mathcal{A}_{{\mathsf{pos}}}$ is linear in the size of the KAT expression.

On the average size of glushkov and equation automata for KAT expressions

We present efficient algorithms to reduce the size of nondeterministic Buchi word automata (NBA) and nondeterministic finite word automata (NFA), while retaining their languages. Additionally, we describe methods to solve PSPACE-complete automata problems like language universality, equivalence, and inclusion for much larger instances than was previously possible ($\ge 1000$ states instead of 10-100). This can be used to scale up applications of automata in formal verification tools and decision procedures for logical theories. The algorithms are based on new techniques for removing transitions (pruning) and adding transitions (saturation), as well as extensions of classic quotienting of the state space. These techniques use criteria based on combinations of backward and forward trace inclusions and simulation relations. Since trace inclusion relations are themselves PSPACE-complete, we introduce lookahead simulations as good polynomial time computable approximations thereof. Extensive experiments show that the average-case time complexity of our algorithms scales slightly above quadratically. (The space complexity is worst-case quadratic.) The size reduction of the automata depends very much on the class of instances, but our algorithm consistently reduces the size far more than all previous techniques. We tested our algorithms on NBA derived from LTL-formulae, NBA derived from mutual exclusion protocols and many classes of random NBA and NFA, and compared their performance to the well-known automata tool GOAL.

Efficient reduction of nondeterministic automata with application to language inclusion testing

Synchronous Kleene algebra (SKA) is a decidable framework that combines Kleene algebra (KA) with a synchrony model of concurrency. Elements of SKA can be seen as processes taking place within a fixed discrete time frame and that, at each time step, may execute one or more basic actions or then come to a halt. The synchronous Kleene algebra with tests (SKAT) combines SKA with a Boolean algebra. Both algebras were introduced by Prisacariu, who proved the decidability of the equational theory, through a Kleene theorem based on the classical Thompson $\varepsilon $-NFA construction. Using the notion of partial derivatives, we present a new decision procedure for equivalence between SKA terms. The results are extended for SKAT considering automata with transitions labeled by Boolean expressions instead of atoms. This work continous previous research done for KA and KAT, where derivative based methods were used in feasible algorithms for testing terms equivalence.

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Deciding Synchronous Kleene Algebra with Derivatives

We consider the problem of state-space reduction for nondeterministic weakly-hierarchical visibly pushdown automata (Vpa). Vpa recognize a robust and algorithmically tractable fragment of context-free languages that is natural for modeling programs.

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Minimization of Visibly Pushdown Automata Using Partial Max-SAT

We consider the problem of state-space reduction for nondeterministic weakly-hierarchical visibly pushdown automata (VPA). VPA recognize a robust and algorithmically tractable fragment of context-free languages that is natural for modeling programs. 
We define an equivalence relation that is sufficient for language-preserving quotienting of VPA. Our definition allows to merge states that have different behavior, as long as they show the same behavior for reachable equivalent stacks. We encode the existence of such a relation as a Boolean partial maximum satisfiability (PMax-SAT) problem and present an algorithm that quickly finds satisfying assignments. These assignments are sub-optimal solutions to the PMax-SAT problem but can still lead to a significant reduction of states. 
We integrated our method in the automata-based software verifier Ultimate Automizer and show performance improvements on benchmarks from the software verification competition SV-COMP.

Kleene algebra with tests (KAT) is an equational system for program verification, which is the combination of Boolean algebra (BA) and Kleene algebra (KA), the algebra of regular expressions. In particular, KAT subsumes the propositional fragment of Hoare logic (PHL) which is a formal system for the specification and verification of programs, and that is currently the base of most tools for checking program correctness. Both the equational theory of KAT and the encoding of PHL in KAT are known to be decidable. In this paper we present a new decision procedure for the equivalence of two KAT expressions based on the notion of partial derivatives. We also introduce the notion of derivative modulo particular sets of equations. With this we extend the previous procedure for deciding PHL. Some experimental results are also presented.

Deciding KAT and Hoare Logic with Derivatives

We present an efficient algorithm to reduce the size of nondeterministic tree automata, while retaining their language. It is based on new transition pruning techniques, and quotienting of the state space w.r.t. suitable equivalences. It uses criteria based on combinations of downward and upward simulation preorder on trees, and the more general downward and upward language inclusions. Since tree-language inclusion is EXPTIME-complete, we describe methods to compute good approximations in polynomial time.

We implemented our algorithm as a module of the well-known libvata tree automata library, and tested its performance on a given collection of tree automata from various applications of libvata in regular model checking and shape analysis, as well as on various classes of randomly generated tree automata. Our algorithm yields substantially smaller and sparser automata than all previously known reduction techniques, and it is still fast enough to handle large instances.

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Reduction of Nondeterministic Tree Automata

We introduce saturation of nondeterministic tree automata, a technique that consists of adding new transitions to an automaton while preserving its language. We implemented our algorithm on minotaut - a module of the tree automata library libvata that reduces the size of automata by merging states and removing superfluous transitions - and we show how saturation can make subsequent merge and transition-removal operations more effective. Thus we obtain a Ptime algorithm that reduces the size of tree automata even more than before. Additionally, we explore how minotaut alone can play an important role when performing hard operations like complementation, allowing to both obtain smaller complement automata and lower computation times. We then show how saturation can extend this contribution even further. We tested our algorithms on a large collection of automata from applications of libvata in shape analysis, and on different classes of randomly generated automata.

Ricardo Almeida

Papers

Deciding KAT and Hoare Logic with Derivatives

Reduction of Nondeterministic Tree Automata

Deciding KAT and Hoare Logic with Derivatives

Reducing Nondeterministic Tree Automata by Adding Transitions.

Reducing Nondeterministic Tree Automata by Adding Transitions