Automata-based presentations of infinite structures

Abstract

The model theory of finite structures is intimately connected to various fields in computer science, including complexity theory, databases, and verification. In particular, there is a close relationship between complexity classes and the expressive power of logical languages, as witnessed by the fundamental theorems of descriptive complexity theory, such as Fagin's Theorem and the ImmermanVardi Theorem (see [78, Chapter 3] for a survey).However, for many applications, the strict limitation to finite structures has turned out to be too restrictive, and there have been considerable efforts to extend the relevant logical and algorithmic methodologies from finite structures to suitable classes of infinite ones. In particular this is the case for databases and verification where infinite structures are of crucial importance [130]. Algorithmic model theory aims to extend in a systematic fashion the approach and methods of finite model theory, and its interactions with computer science, from finite structures to finitely-presentable infinite ones.There are many possibilities to present infinite structures in a finite manner. A classical approach in model theory concerns the class of computable structures; these are countable structures, on the domain of natural numbers, say, with a finite collection of computable functions and relations. Such structures can be finitely presented by a collection of algorithms, and they have been intensively studied in model theory since the 1960s. However, from the point of view of algorithmic model theory the class of computable structures is problematic. Indeed, one of the central issues in algorithmic model theory is the effective evaluation of logical formulae, from a suitable logic such as, for instance, first-order logic (FO), monadic second-order logic (MSO), or a fixed point logic like LFP or the modal µ-calculus. But on computable structures, only the quantifier-free formulae generally admit effective evaluation, and already the existential fragment of first-order logic is undecidable, for instance on the computable structure (N, +, · ).This leads us to the central requirement that for a suitable logic L (depending on the intended application) the model-checking problem for the class C of finitely presented structures should be algorithmically solvable. At the very 1

Isomorphisms of scattered automatic linear orders

“…The hardness results from (1) are based on a sharpening of Krob's result (see [9]): there is a fixed weighted automata such that the set of equivalent weighted automata is Π 0 1 -hard (and therefore undecidable). A closer analysis of this proof, together with the techniques for proving (1) and (2), finally yields (3).…”

“…Over the last decade, a fair amount of results have been obtained, see e.g. the surveys [28,1] as well as the list of open questions [19], for very recent results not covered by the mentioned articles, see e.g. [5,10,17,16].…”

“…The proof of (2) uses an encoding of polynomials similarly to [23] but avoids the use of shuffle sums. The technique for proving (1) and (3) is genuinely new: One can understand a weighted automaton over the semiring (N ∪ {−∞}; max, +) as a classical automaton with a partition of the set of transitions into two sets T 0 and T 1 . The behavior of such a weighted automaton assigns numbers to words w, namely the maximal number of transitions from T 1 in an accepting run on the word w. Krob [22] showed that the equivalence problem for such weighted automata is Π 0 1complete.…”

Isomorphism of Regular Trees and Words

“…Following [2, proof of Theorem 65 & 66] we will now give an algorithm that produces a proper 2-level system. We will proceed along the hierarchical order of the variables in Up where in each step we possibly add a constant number of new variables and change the right-hand sides of the old variables such that all variables are proper and of the form (1)- (5) and, moreover, all old variables X are of type (1,2) and fulfill the following technical condition (TEC):…”

Tree-Automatic Well-Founded Trees

“…Various classes of infinite but finitely presented structures received a lot of attention in algorithmic model theory [2]. Among the most important such classes of structures is the class of string-automatic structures [17].…”