Fly-Automata, Their Properties and Applications

Journal of Computer and System Sciences, 1992

Automata on directed acyclic graphs are defined and their closure properties and the emptiness problem are studied. It is shown that several types of finite automata on various StrUCtUI'eS are special cases of the above automata.

Graph Transformation

To verify graph programs in the language GP 2, we present a monadic second-order logic with counting and a Hoare-style proof calculus. The logic has quantifiers for GP 2's attributes and for sets of nodes or edges. This allows to specify non-local graph properties such as connectedness, k-colourability, etc. We show how to construct a strongest liberal postcondition for a given graph transformation rule and a precondition. The proof rules establish the total correctness of graph programs and are shown to be sound. They allow to verify more programs than is possible with previous approaches. In particular, many programs with nested loops are covered by the calculus.

International Journal of Foundations of Computer Science

A deterministic finite automaton in which every non-empty set of states occurs as the image of the whole state set under the action of a suitable input word is called completely reachable. We characterize such automata in terms of graphs and trees.

This paper presents a static verification algorithm for a core subset of UnCAL, the query algebra for graph-structured databases proposed by Bunemann et al. Given a query and input/output schemas, our algorithm statically verifies that any graph satisfying the input schema is converted by the query to a graph satisfying the output schema. The basic idea is to reformulate the semantics of UnCAL using monadic second-order logic (MSO). The logic-based foundation allows to express the schema satisfaction of transformations as the validity of MSO formulas over graph structures. Furthermore, with several insights on the established properties of UnCAL, the problem turns out to be reducible to the validity of MSO over trees, which has a sound and complete decision procedure.

2006

Monadic Second Order logic (MSO) has been used as a formalism to describe Government and Binding rules. It also lends itself to be used as a query language. Main arguments are its expressive power and linear data complexity. Particularly in the domain of tree banks, where the size of the data is much bigger than the size of the query, the data complexity is an influential factor. It is known since the late 1960s that an MSO formula can be translated into a tree automaton. These results also predict an more than exponential blowup of the number of states of the tree automaton corresponding to the number of quantifier alternations in the formula. If one refrains from writing complicated formulae and thus avoids the theoretical blowup, one could hope the result becomes practically usable to write a query tool for tree banks. It is shown here that another problem arises: even with few quantifier alternations the transition tables get too big very soon. This is illustrated by a straightf...

HAL (Le Centre pour la Communication Scientifique Directe), 2017

This paper describes a fully automatic technique for verifying safety properties of higherorder functional programs. Tree automata are used to represent sets of reachable states and functional programs are modeled using term rewriting systems. From a tree automaton representing the initial state, a completion algorithm iteratively computes an automaton which over-approximates the output set of the program to verify. We identify a subclass of higher-order functional programs for which the completion is guaranteed to terminate. Precision and termination are obtained conjointly by a careful choice of equations between terms. The verification objective can be used to generate sets of equations automatically. Our experiments show that tree automata are sufficiently expressive to prove intricate safety properties and sufficiently simple for the verification result to be certified in Coq. These results extend state-of-the-art model-checking approaches for higher-order functional programs.

2012

Tree Regular Model Checking (TRMC) is the name of a family of techniques for analyzing infinite-state systems in which states are represented by terms, and sets of states by Tree Automata (TA). The central problem in TRMC is to decide whether a set of bad states is reachable. The problem of computing a TA representing (an over- approximation of) the set of reachable states is undecidable, but efficient solutions based on completion or iteration of tree transducers exist. Unfortunately, the TRMC framework is unable to efficiently capture both the complex structure of a system and of some of its features. As an example, for JAVA programs, the structure of a term is mainly exploited to capture the structure of a state of the system. On the counter part, integers of the java programs have to be encoded with Peano numbers, which means that any algebraic operation is potentially represented by thousands of applications of rewriting rules. In this paper, we propose Lattice Tree Automata (LTAs), an extended version of tree automata whose leaves are equipped with lattices. LTAs allow us to represent possibly infinite sets of interpreted terms. Such terms are capable to represent complex domains and related operations in an efficient manner. We also extend classical Boolean operations to LTAs. Finally, as a major contribution, we introduce a new completion-based algorithm for computing the possibly infinite set of reachable interpreted terms in a finite amount of time.

1997

Computer Science Logic, 2000

The design of a complex system warrants a compositional methodology, i.e., composing simple components to obtain a system that meaningfully exhibits their collective behavior. We propose an automaton-based paradigm for compositional design of such systems where an action is accompanied by one or more preferences. At run-time, these preferences provide a natural fallback mechanism for the component, while at design-time they can be used to reason about the behavior of the component in an uncertain physical world. Using algebraic structures on preferences and actions, we can compose formal representations of individual components or agents to obtain a representation of the composed system, exhibiting intuitively meaningful behavior. We extend linear temporal logic with two unary connectives that reflect the compositional structure of actions, and show that it is decidable whether all behaviors of a given automaton satisfy a formula of this extended logic. We then show how this logic can be used to diagnose undesired behavior by tracing the falsification of a specification back to one or more culpable components. Lastly, we implement a toolchain that compiles our automata to Maude, allowing us to apply the rich model checking capability of Maude to verify agent behavior.

2008