Nondeterministic Buchi Automata Research Articles

An Automata-Theoretic Approach to Model-Checking Systems and Specifications Over Infinite Data Domains

Data-parameterized systems model systems with finite control over an infinite data domain. VLTL is an extension of LTL that uses variables in order to specify properties of computations over infinite data, and as such, VLTL is suitable for specifying properties of data-parameterized systems. We present alternating variable Buchi word automata (AVBWs), a new model of automata over infinite alphabets, capable of modeling a significant fragment of VLTL. While alternating and non-deterministic Buchi automata over finite alphabets have the same expressive power, we show that this is not the case for infinite data domains, as we prove that AVBWs are strictly stronger than the previously defined non-deterministic variable Buchi word automata (NVBWs). However, while the emptiness problem is easy for NVBWs, it is undecidable for AVBWs. We present an algorithm for translating AVBWs to NVBWs in cases where such a translation is possible. Additionally, we characterize the structure of AVBWs that can be translated to NVBWs with our algorithm. We then rely on the natural iterative behavior of our translation algorithm to describe a bounded model-checking procedure for the logic that we consider. Furthermore, we present several fragments of the logic that can be expressed by NVBWs, as well as a fragment that cannot be expressed by NVBWs, yet whose satisfiability is decidable.

On relative and probabilistic finite counterability

A counterexample to the satisfaction of a linear property $$\psi $$ in a system $${\mathcal {S}}$$ is an infinite computation of $${\mathcal {S}}$$ that violates $$\psi $$ . When $$\psi $$ is a safety property, a counterexample to its satisfaction need not be infinite. Rather, it is a bad-prefix for $$\psi $$ : a finite word all whose extensions violate $$\psi $$ . The existence of finite counterexamples is very helpful in practice. Liveness properties do not have bad-prefixes and thus do not have finite counterexamples. We extend the notion of finite counterexamples to non-safety properties. We study counterable languages—ones that have at least one bad-prefix. Thus, a language is counterable iff it is not liveness. Three natural problems arise: (1) given a language, decide whether it is counterable, (2) study the length of minimal bad-prefixes for counterable languages, and (3) develop algorithms for detecting bad-prefixes for counterable languages. We solve the problems for languages given by means of LTL formulas or nondeterministic Buchi automata. In particular, our EXPSPACE-completeness proof for the problem of deciding whether a given LTL formula is counterable, and hence also for deciding liveness, settles a long-standing open problem. In addition, we make finite counterexamples more relevant and helpful by introducing two variants of the traditional definition of bad-prefixes. The first adds a probabilistic component to the definition. There, a prefix is bad if almost all its extensions violate the property. The second makes it relative to the system. There, a prefix is bad if all its extensions in the system violate the property. We also study the combination of the probabilistic and relative variants. Our framework suggests new variants also for safety and liveness languages. We solve the above three problems for the different variants. Interestingly, the probabilistic variant not only increases the chances to return finite counterexamples, but also makes the solution of the three problems exponentially easier.

Use of Büchi automata and randomness for the description of biological processes

The main topic of this study is the modelling and verification of biological systems using ω -automata. This work focuses on the protein folding problem and the infinite behaviour it features in many cases. Specifically, stochastic computational models with infinite input are used in this paper’s approach and indicative aspects of a biological problem are presented using both ω -automata and probabilistic Buchi automata (PBAs), making a novel attempt to establish their use in reasoning about biological processes. Necessary preliminary definitions and background towards this direction are provided. Finally, the pros and cons of each model are shown through examples. Overall, this work contributes by combining for the very first time PBAs with real biological mechanisms, with indicative examples, both with PBAs and NBAs (non-deterministic Buchi automata).

Practical CTL * model checking: Should SPIN be extended?

We describe an efficient CTL* model checking algorithm based on alternating automata and games. A CTL* formula, expressing a correctness property, is first translated to a hesitant alternating automaton and then composed with a Kripke structure representing the model to be checked, after which this resulting automaton is then checked for nonemptiness. We introduce the nonemptiness game that checks the nonemptiness of a hesitant alternating automaton (HAA). In the same way that alternating automata generalise nondeterministic automata, we show that this game for checking the nonemptiness of HAA, generalises the nested depth-first algorithm used to check the nonemptiness of nondeterministic Buchi automata (used in Spin).

Complementing deterministic Büchi automata in polynomial time

For any Buchi automaton Γ with n states which accepts the (ω-regular) language L (Γ), an explicit construction is given for a Büchi automaton Γ with 2n states which, when Γ is deterministic, accepts exactly the complementary language L (Γ)′. It follows that the nonemptiness of complement problem for deterministic Buchi automata ( i.e., whether L (Γ)′ = ⊘) is solvable in polynomial time. The best previously known construction for complementing a deterministic Büchi automaton with n states has O(2 4 n 2) states; for nondeterministic Γ, determining whether L (Γ)′ = ⊘, is known to be PSPACE-complete. Interest in deterministic Büchi automata arises from the suitability of deterministic automata in general to describe properties of physical systems; such properties have been found to be more naturally expressible by deterministic automata than by nondeterministic automata. However, if Γ is nondeterministic, then Γ provides a “poor man's” approximate inverse to Γ in the following sense: L (Γ)′ ⊂ L ( Γ ), and as nondeterministic branches of T are removed, the two languages become closer. Hence, for example, given two nondeterministic Buchi automata Λ and Γ, one may test for containment of their associated languages through use of the corollary that L (Λ ∗ Γ = ⊘ ⇒ L (Λ) ⊂ L (Γ) (where Γ ∗ Γ is one of the standard constructions satisfying L (Λ ∗ Γ ) = L (Λ) ∩ L ( Γ )). The “error term” L = L ( Γ ) ⧹ L (Γ)′ may be deter exactly, and whether L = ⊘ may be determined in time O( e 2), where e is the number of edges of Γ.