What's Decidable About Sequences?

We present a first-order theory of sequences with integer elements, Presburger arithmetic, and regular constraints, which can model significant properties of data structures such as arrays and lists. We give a decision procedure for the quantifier-free fragment, based on an encoding into the first-order theory of concatenation; the procedure has PSPACE complexity. The quantifier-free fragment of the theory of sequences can express properties such as sortedness and injectivity, as well as Boolean combinations of periodic and arithmetic facts relating the elements of the sequence and their positions (e.g., "for all even i's, the element at position i has value i+3 or 2i"). The resulting expressive power is orthogonal to that of the most expressive decidable logics for arrays. Some examples demonstrate that the fragment is also suitable to reason about sequence-manipulating programs within the standard framework of axiomatic semantics.Comment: Fixed a few lapses in the Mergesort exampl

Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

Two-Way Unary Temporal Logic over Trees

We consider a temporal logic EF+F^-1 for unranked, unordered finite trees. The logic has two operators: EF\phi, which says "in some proper descendant \phi holds", and F^-1\phi, which says "in some proper ancestor \phi holds". We present an algorithm for deciding if a regular language of unranked finite trees can be expressed in EF+F^-1. The algorithm uses a characterization expressed in terms of forest algebras.Comment: 29 pages. Journal version of a LICS 07 pape

Some new results on decidability for elementary algebra and geometry

We carry out a systematic study of decidability for theories of (a) real vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces, Banach spaces and metric spaces, all formalised using a 2-sorted first-order language. The theories for list (a) turn out to be decidable while the theories for list (b) are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic. We find that the purely universal and purely existential fragments of the theory of normed spaces are decidable, as is the AE fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbert's 10th problem show that the EA fragments for metric and normed spaces and the AE fragment for normed spaces are all undecidable.Comment: 79 pages, 9 figures. v2: Numerous minor improvements; neater proofs of Theorems 8 and 29; v3: fixed subscripts in proof of Lemma 3

On Elementary Theories of Ordinal Notation Systems based on Reflection Principles

We consider the constructive ordinal notation system for the ordinal ${\epsilon_0}$ that were introduced by L.D. Beklemishev. There are fragments of this system that are ordinal notation systems for the smaller ordinals ${\omega_n}$ (towers of ${\omega}$-exponentiations of the height $n$). This systems are based on Japaridze's provability logic $\mathbf{GLP}$. They are closely related with the technique of ordinal analysis of $\mathbf{PA}$ and fragments of $\mathbf{PA}$ based on iterated reflection principles. We consider this notation system and it's fragments as structures with the signatures selected in a natural way. We prove that the full notation system and it's fragments, for ordinals ${\ge\omega_4}$, have undecidable elementary theories. We also prove that the fragments of the full system, for ordinals ${\le\omega_3}$, have decidable elementary theories. We obtain some results about decidability of elementary theory, for the ordinal notation systems with weaker signatures.Comment: 23 page

A Bounded Domain Property for an Expressive Fragment of First-Order Linear Temporal Logic

First-Order Linear Temporal Logic (FOLTL) is well-suited to specify infinite-state systems. However, FOLTL satisfiability is not even semi-decidable, thus preventing automated verification. To address this, a possible track is to constrain specifications to a decidable fragment of FOLTL, but known fragments are too restricted to be usable in practice. In this paper, we exhibit various fragments of increasing scope that provide a pertinent basis for abstract specification of infinite-state systems. We show that these fragments enjoy the Bounded Domain Property (any satisfiable FOLTL formula has a model with a finite, bounded FO domain), which provides a basis for complete, automated verification by reduction to LTL satisfiability. Finally, we present a simple case study illustrating the applicability and limitations of our results

Web ontology representation and reasoning via fragments of set theory

In this paper we use results from Computable Set Theory as a means to represent and reason about description logics and rule languages for the semantic web. Specifically, we introduce the description logic \mathcal{DL}\langle 4LQS^R\rangle(\D)--admitting features such as min/max cardinality constructs on the left-hand/right-hand side of inclusion axioms, role chain axioms, and datatypes--which turns out to be quite expressive if compared with \mathcal{SROIQ}(\D), the description logic underpinning the Web Ontology Language OWL. Then we show that the consistency problem for \mathcal{DL}\langle 4LQS^R\rangle(\D)-knowledge bases is decidable by reducing it, through a suitable translation process, to the satisfiability problem of the stratified fragment $4LQS^R$ of set theory, involving variables of four sorts and a restricted form of quantification. We prove also that, under suitable not very restrictive constraints, the consistency problem for \mathcal{DL}\langle 4LQS^R\rangle(\D)-knowledge bases is \textbf{NP}-complete. Finally, we provide a $4LQS^R$-translation of rules belonging to the Semantic Web Rule Language (SWRL)