We prove that every monadic second-order property of the unfolding of a transition system is a monadic second-order property of the system itself. An unfolding is an instance of the general notion of graph covering. We consider two more instances of this notion. A similar result is possible for one of them but not for the other.

In the automata-theoretic approach to verification, we translate specifications to automata. Complexity considerations motivate the distinction between different types of automata. Already in the 60s, it was known that deterministic Büchi word automata are less expressive than nondeterministic Büchi word automata. The proof is easy and can be stated in a few lines. In the late 60s, Rabin proved that Büchi tree automata are less expressive than Rabin tree automata. This proof is much harder. In this work we relate the (...) expressiveness gap between deterministic and nondeterministic Büchi word automata and the expressiveness gap between Büchi and Rabin tree automata. We consider tree automata that recognize derived languages. For a word language L, the derived language of L, denoted L, is the set of all trees all of whose paths are in L. Since often we want to specify that all the computations of the program satisfy some property, the interest in derived languages is clear. Our main result shows that L is recognizable by a nondeterministic Büchi word automaton but not by a deterministic Büchi word automaton iff L is recognizable by a Rabin tree automaton and not by a Büchi tree automaton. Our result provides a simple explanation for the expressiveness gap between Büchi and Rabin tree automata. Since the gap between deterministic and nondeterministic Büchi word automata is well understood, our result also provides a characterization of derived languages that can be recognized by Büchi tree automata. Finally, it also provides an exponential determinization of Büchi tree automata that recognize derived languages. (shrink)

We say that a first order sentence A defines a graph G if A is true on G but false on any graph non-isomorphic to G. Let L ) denote the minimum length of such a sentence. We define the succinctness function s ) to be the minimum L ) over all graphs on n vertices.We prove that s and q may be so small that for no general recursive function f we can have f)≥n for all n. However, for (...) the function q*=maxi≤nq, which is the least nondecreasing function bounding q from above, we have q*=)log*n, where log*n equals the minimum number of iterations of the binary logarithm sufficient to lower n to 1 or below.We show an upper bound qshrink)

A new method for obtaining lower bounds on the computational complexity of logical theories is presented. It extends widely used techniques for proving the undecidability of theories by interpreting models of a theory already known to be undecidable. New inseparability results related to the well known inseparability result of Trakhtenbrot and Vaught are the foundation of the method. Their use yields hereditary lower bounds . By means of interpretations lower bounds can be transferred from one theory to another. Complicated machine (...) codings are replaced by much simpler definability considerations, viz., the kinds of binary relations definable with short formulas on large finite sets. Numerous examples are given, including new proofs of essentially all previously known lower bounds for theories, and lower bounds for various theories of finite trees, which turn out to be particularly useful. (shrink)

In many logics dealing with information one needs to make statements not only about cognitive states, but also about transitions between them. In this paper we analyze a dynamic modal logic that has been designed with this purpose in mind. On top of an abstract information ordering on states it has instructions to move forward or backward along this ordering, to states where a certain assertion holds or fails, while it also allows combinations of such instructions by means of operations (...) from relation algebra. In addition, the logic has devices for expressing whether in a given state a certain instruction can be carried out, and whether that state can be arrived at by carrying out a certain instruction.This paper deals mainly with technical aspects of our dynamic modal logic. It gives an exact description of the expressive power of this language; it also contains results on decidability for the language with arbitrary structures and for the special case with a restricted class of admissible structures. In addition, a complete axiomatization is given. The paper concludes with a remark about the modal algebras appropriate for our dynamic modal logic, and some questions for further work. (shrink)

In this article the structure of the models of decidable monadic theories of planar graphs is investigated. It is shown that if the monadic theory of a class K of planar graphs is decidable, then the tree-width in the sense of Robertson and Seymour of the elements of K is universally bounded and there is a class T of trees such that the monadic theory of K is interpretable in the monadic theory of T.

It is shown that the decision problem for the temporal logic with until and since connectives over real-numbers time is PSPACE-complete. This is the most practically useful dense time temporal logic.

We show that the complete first order theory of an MV algebra has equation image countable models unless the MV algebra is finitely valued. So, Vaught's Conjecture holds for all MV algebras except, possibly, for finitely valued ones. Additionally, we show that the complete theories of finitely valued MV algebras are equation image and that all ω-categorical complete theories of MV algebras are finitely axiomatizable and decidable. As a final result we prove that the free algebra on countably many generators (...) of any locally finite variety of MV algebras is ω-categorical. (shrink)

The classical Feferman–Vaught Theorem for First Order Logic explains how to compute the truth value of a first order sentence in a generalized product of first order structures by reducing this computation to the computation of truth values of other first order sentences in the factors and evaluation of a monadic second order sentence in the index structure. This technique was later extended by Läuchli, Shelah and Gurevich to monadic second order logic. The technique has wide applications in decidability and (...) definability theory. Here we give a unified presentation, including some new results, of how to use the Feferman–Vaught Theorem, and some new variations thereof, algorithmically in the case of Monadic Second Order Logic MSOL . We then extend the technique to graph polynomials where the range of the summation of the monomials is definable in MSOL . Here the Feferman–Vaught Theorem for these polynomials generalizes well known splitting theorems for graph polynomials. Again, these can be used algorithmically. Finally, we discuss extensions of MSOL for which the Feferman–Vaught Theorem holds as well. (shrink)

In this paper, we introduce a new fragment of the first-order temporal language, called the monodic fragment, in which all formulas beginning with a temporal operator have at most one free variable. We show that the satisfiability problem for monodic formulas in various linear time structures can be reduced to the satisfiability problem for a certain fragment of classical first-order logic. This reduction is then used to single out a number of decidable fragments of first-order temporal logics and of two-sorted (...) first-order logics in which one sort is intended for temporal reasoning. Besides standard first-order time structures, we consider also those that have only finite first-order domains, and extend the results mentioned above to temporal logics of finite domains. We prove decidability in three different ways: using decidability of monadic second-order logic over the intended flows of time, by an explicit analysis of structures with natural numbers time, and by a composition method that builds a model from pieces in finitely many steps. (shrink)

In this paper we introduce a new technique of proving undecidability results. This technique is based on the notion of a Thue tree. We also give examples of applications of this method to term rewriting, Horn implication problem and database dependencies.

We prove that a finitely generated group is context-free whenever its Cayley-graph has a decidable monadic second-order theory. Hence, by the seminal work of Muller and Schupp, our result gives a logical characterization of context-free groups and also proves a conjecture of Schupp. To derive this result, we investigate general graphs and show that a graph of bounded degree with a high degree of symmetry is context-free whenever its monadic second-order theory is decidable. Further, it is shown that the word (...) problem of a finitely generated group is decidable if and only if the first-order theory of its Cayley-graph is decidable. (shrink)

Using the concept of progress measure, we give a new proof of Rabin's fundamental result that the languages defined by tree automata are closed under complementation. To do this we show that for certain infinite games based on tree automata, an immediate determinacy property holds for the player who is trying to win according to a Rabin acceptance condition. Immediate determinancy is stronger than the forgetful determinacy of Gurevich and Harrington, which depends on more information about the past, but applies (...) to another class of games. Next, we show a graph theoretic duality theorem for winning conditions. Finally, we present an extended version of Safra's determinization construction. Together, these ingredients and the determinacy of Borel games yield a straightforward recipe for complementing tree automata. Our construction is almost optimal, i.e. the state space blow-up is essentially exponential- thus roughly the same as for automata on finite or infinite words. To our knowledge, no prior constructions have been better than double exponential. (shrink)

A unary algebra is an algebraic system A = , where ƒ 0 ,…,ƒ n are unary operations on A and n ∈ ω. In the paper we develop the theory of effective unary algebras. We investigate well-known questions of constructive model theory with respect to the class of unary algebras. In the paper we construct unary algebras with a finite number of recursive isomorphism types. We give the notions of program, uniform, and algebraic dimensions of models, and then we (...) investigate these notions on unary algebras. We find connections between algebraic and effective properties of r.e. representable unary algebras. We also deal with finitely generated r.e. unary algebras. We show the connections between trees and unary algebras. Our interests also concern recursive automorphisms groups, r.e. subalgebra and congruence lattices of effective unary algebras. (shrink)

Heidorf, L., Moderate families in Boolean algebras, Annals of Pure and Applied Logic 57 217–250. A subset F of a Boolean algebra B will be called moderate if no element of B splits infinitely many elements of F . Disjoint moderate sets occur in connection with a product construction that is systematically studied in this paper. In contrast to the usual full direct product, these so-called moderate products preserve many properties of their factors. This can be used, for example, to (...) construct big retractive Boolean algebras that are not embeddable into interval algebras. The paper is also concerned with those Boolean algebras whose ideals are generated by moderate sets. The results may be summarized by saying that in many respects these algebras behave like countable ones. (shrink)

We extend some of the classical connections between automata and logic due to Büchi [5] and McNaughton and Papert [12] to languages of finitely varying functions or “signals”. In particular, we introduce a natural class of automata for generating finitely varying functions called ’s, and show that it coincides in terms of language definability with a natural monadic second-order logic interpreted over finitely varying functions Rabinovich [15]. We also identify a “counter-free” subclass of ’s which characterise the first-order definable languages (...) of finitely varying functions. Our proofs mainly factor through the classical results for word languages. These results have applications in automata characterisations for continuously interpreted real-time logics like Metric Temporal Logic Chevalier et al. [6] and [7]. (shrink)

We study tree-like decompositions of models of a theory and a related complexity measure called partition width. We prove a dichotomy concerning partition width and definable pairing functions: either the partition width of models is bounded, or the theory admits definable pairing functions. Our proof rests on structure results concerning indiscernible sequences and finitely satisfiable types for theories without definable pairing functions. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

We provide first-order axioms for the theories of finite trees with bounded branching and finite trees with arbitrary (finite) branching. The signature is chosen to express, in a natural way, those properties of trees most relevant to linguistic theories. These axioms provide a foundation for results in linguistics that are based on reasoning formally about such properties. We include some observations on the expressive power of these theories relative to traditional language complexity classes.