Automata vs. Logics on Data Words

Abstract

Abstract. The relationship between automata and logics has been investigated since the 1960s. In particular, it was shown how to determine, given an automaton, whether or not it is definable in first-order logic with label tests and the order relation, and for first-order logic with the successor relation. In recent years, there has been much interest in languages over an infinite alphabet. Kaminski and Francez introduced a class of automata called finite memory automata (FMA), that represent a natural analog of finite state machines. A FMA can use, in addition to its control state, a (bounded) number of registers to store and compare values from the input word. The class of data languages recognized by FMA is incomparable with the class of data languages defined by firstorder formulas with the order relation and an additional binary relation for data equality. We first compare the expressive power of several variants of FMA with several data word logics. Then we consider the corresponding decision problem: given an automaton A and a logic, can the language recognized by A be defined in the logic? We show that it is undecidable for several variants of FMA, and then investigate the issue in detail for deterministic FMA. We show the problem is decidable for first-order logic with local data comparisons -an analog of first-order logic with successor. We also show instances of the problem for richer classes of first-order logic that are decidable.Logics are natural ways of specifying decision problems on discrete structures, while automata represent natural processing models. On finite words from a fixed (finite) alphabet, Büchi [1] showed that monadic second-order logic has the same expressiveness as deterministic finite state automata, while results of Schützenberger and McNaughton and Papert showed that first-order logic with the label and order predicates has the same expressiveness as counter-free automata [2,3]. The latter theorem gives a decidable characterization of which automata correspond to first-order sentences. Decidable characterizations have also been given for first-order logic with the label and successor predicates [4]. These characterizations have been extended to many other contexts; for example there are characterizations of the tree automata that correspond to sentences in logics on trees [5].Automata processing finite words over infinite alphabets (so called data words) are attracting significant interest from the database and verification communities, since they can be often used as low-level formalisms for representing and reasoning about data streams, program traces, and serializations of structured documents. Moreover, properties specified using high-level formalisms (for

Nominal Monoids

“…In [1], they show that definability in first-order logic is undecidable if the input is a nondeterministic register automaton. Also, they provide some decidable characterizations, including a characterization of first-order logic with local data comparisons within the class of languages recognized by deterministic register automata.…”

“…Intuitively speaking, two elements of a syntactic monoid are considered to be in the same orbit if there is a renaming of data values that maps one element to the other. 1 For instance, in the running example L dd , the elements of the syntactic monoid that correspond to the words 1 · 7 and 2 · 3 · 4 · 8 are not equal, but they are in the same orbit, because the renaming i → i + 1 maps 1 · 7 to 2 · 8, which corresponds to the same element of the syntactic monoid as 2 · 3 · 4 · 8. It is not difficult to see that the syntactic monoid of L dd has four orbits: one for the empty word, one for words outside L dd where the first and last letters are not equal, and for words outside L dd where the first and last letters are equal, and one for words inside L dd .…”

Weighted Recognizability over Infinite Alphabets

“…All these models refer to qualitative aspects of infinite state systems. Furthermore, rational [1,25] and logic definable languages [4,36] have been studied over infinite alphabets.…”

Logics with rigidly guarded data tests

“…We then show that rigidly guarded MSO ∼ is exactly as expressive as orbit-finite data monoids, and that its first-order fragment corresponds to aperiodic orbit-finite data monoids. (3) We show that an extension of rigidly guarded MSO ∼ is decidable, even on general classes of structures with data (e.g., data trees). We show that the same extension of rigidly guarded MSO ∼ defines a proper subclass of data languages recognized by non-deterministic (in fact, unambiguous) finite memory automata.…”

“…The differences between all such formalisms are reflected in the fact that it is difficult to obtain algebraic characterizations for robust classes of data languages. In [3,8,10,13] some preliminary results on relating automata to logics are given. However, the algebraic theory for these automaton models is not fully developed yet.…”