Sasha Rubin

Automatic structures: richness and limitations

This paper studies the existence of automatic presentations for various algebraic structures. The automatic Boolean algebras are characterised, and it is proven that the free Abelian group of infinite rank and many Fraisse limits do not have automatic presentations. In particular, the countably infinite random graph and the universal partial order do not have automatic presentations. Furthermore, no infinite integral domain is automatic. The second topic of the paper is the isomorphism problem. We prove that the complexity of the isomorphism problem for the class of all automatic structures is /spl Sigma//sub 1//sup 1/-complete.

Automata presenting structures: A survey of the finite string case

A structure has a (finite-string) automatic presentation if the elements of its domain can be named by finite strings in such a way that the coded domain and the coded atomic operations are recognised by synchronous multitape automata. Consequently, every structure with an automatic presentation has a decidable first-order theory. The problems surveyed here include the classification of classes of structures with automatic presentations, the complexity of the isomorphism problem, and the relationship between definability and recognisability.

Automatic linear orders and trees

We investigate partial orders that are computable, in a precise sense, by finite automata. Our emphasis is on trees and linear orders. We study the relationship between automatic linear orders and trees in terms of rank functions that are related to Cantor--Bendixson rank. We prove that automatic linear orders and automatic trees have finite rank. As an application we provide a procedure for deciding the isomorphism problem for automatic ordinals. We also investigate the complexity and definability of infinite paths in automatic trees. In particular, we show that every infinite path in an automatic tree with countably many infinite paths is a regular language.

Parameterized Model Checking of Rendezvous Systems

A standard technique for solving the parameterized model checking problem is to reduce it to the classic model checking problem of finitely many finite-state systems. This work considers some of the theoretical power and limitations of this technique. We focus on concurrent systems in which processes communicate via pairwise rendezvous, as well as the special cases of disjunctive guards and token passing; specifications are expressed in indexed temporal logic without the next operator; and the underlying network topologies are generated by suitable Monadic Second Order Logic formulas and graph operations. First, we settle the exact computational complexity of the parameterized model checking problem for some of our concurrent systems, and establish new decidability results for others. Second, we consider the cases that model checking the parameterized system can be reduced to model checking some fixed number of processes, the number is known as a cutoff. We provide many cases for when such cutoffs can be computed, establish lower bounds on the size of such cutoffs, and identify cases where no cutoff exists. Third, we consider cases for which the parameterized system is equivalent to a single finite-state system (more precisely a Buchi word automaton), and establish tight bounds on the sizes of such automata.

Automatic Structures: Richness and Limitations

This paper studies the existence of automatic presentations for various algebraic structures. The automatic Boolean algebras are characterised, and it is proven that the free Abelian group of infinite rank and many Fraisse limits do not have automatic presentations. In particular, the countably infinite random graph and the universal partial order do not have automatic presentations. Furthermore, no infinite integral domain is automatic. The second topic of the paper is the isomorphism problem. We prove that the complexity of the isomorphism problem for the class of all automatic structures is /spl Sigma//sub 1//sup 1/-complete.

Parameterized Model Checking of Token-Passing Systems

We revisit the parameterized model checking problem for token-passing systems and specifications in indexed CTL i?ź\X. Emerson and Namjoshi 1995, 2003 have shown that parameterized model checking of indexed CTL i?ź\X in uni-directional token rings can be reduced to checking rings up to some cutoff size. Clarke et al. 2004 have shown a similar result for general topologies and indexed LTL \X, provided processes cannot choose the directions for sending or receiving the token. We unify and substantially extend these results by systematically exploring fragments of indexed CTL i?ź\X with respect to general topologies. For each fragment we establish whether a cutoff exists, and for some concrete topologies, such as rings, cliques and stars, we infer small cutoffs. Finally, we show that the problem becomes undecidable, and thus no cutoffs exist, if processes are allowed to choose the directions in which they send or from which they receive the token.

On automatic partial orders

We investigate partial orders that are computable, in a precise sense, by finite automata. Our emphasis is on trees and linear orders. We study the relationship between automatic linear orders and trees in terms of rank functions that are versions of Cantor-Bendixson rank. We prove that automatic linear orders and automatic trees have finite rank. As an application we provide a procedure for deciding the isomorphism problem for automatic ordinals. We also investigate the complexity and definability of infinite paths in automatic trees. In particular, we show that every infinite path in an automatic tree with countably many infinite paths is a regular language.

Definability and Regularity in Automatic Structures

An automatic structure \(\mathcal{A}\) is one whose domain A and atomic relations are finite automaton (FA) recognisable. A structure isomorphic to \(\mathcal{A}\) is called automatically presentable. Suppose R is an FA recognisable relation on A. This paper concerns questions of the following type. For which automatic presentations of \(\mathcal{A}\) is (the image of) R also FA recognisable? To this end we say that a relation R is intrinsically regular in a structure \(\mathcal{A}\) if it is FA recognisable in every automatic presentation of the structure. For example, in every automatic structure all relations definable in first order logic are intrinsically regular. We characterise the intrinsically regular relations of some automatic fragments of arithmetic in the first order logic extended with quantifiers ∃ ∞ interpreted as ‘there exists infinitely many’, and ∃ (i) interpreted as ‘there exists a multiple of i many’.

Verifying ω-Regular Properties of Markov Chains

In this work we focus on model checking of probabilistic models. Probabilistic models are widely used to describe randomized protocols. A Markov chain induces a probability measure on sets of computations. The notion of correctness now becomes probabilistic. We solve here the general problem of linear-time probabilistic model checking with respect to ω-regular specifications. As specification formalism, we use alternating Buchi infinite-word automata, which have emerged recently as a generic specification formalism for developing model checking algorithms. Thus, the problem we solve is: given a Markov chain \({\cal {M}}\) and automaton \({\cal {A}}\), check whether the probability induced by \({\cal {M}}\) of \(L({\cal {A}})\) is one (or compute the probability precisely). We show that these problem can be solved within the same complexity bounds as model checking of Markov chains with respect to LTL formulas. Thus, the additional expressive power comes at no penalty.

Finite and Algorithmic Model Theory: Automata-based presentations of infinite structures

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