Nowadays computer science is surpassing mathematics as the primary field of logic applications, but logic is not tuned properly to the new role. In particular, classical logic is preoccupied mostly with infinite static structures whereas many objects of interest in computer science are dynamic objects with bounded resources. This chapter consists of two independent parts. The first part is devoted to finite model theory; it is mostly a survey of logics tailored for computational complexity. The second part is devoted to dynamic structures with bounded resources. In particular, we use dynamic structures with bounded resources to model Pascal.

Whereas first-order logic was developed to confront the infinite it is often used in computer science in such a way that infinite models are meaningless. We discuss the first-order theory of finite structures and alternatives to first-order logic, especially polynomial time logic.

: We study classes of infinite but finitely representable databases based on constraints, motivated by new database applications such as geographical databases. We formally define these notions and introduce the concept of query which generalizes queries over classical relational databases. We prove that in this context the basic properties of queries (satisfiability, containment, equivalence, etc.) are nonrecursive. We investigate the theory of finitely representable models and prove that it differs strongly from both classical model theory and finite model theory. In particular, we show that most of the well known theorems of either one fail (compactness, completeness, locality, 0/1 laws, etc.). An immediate consequence is the lack of tools to consider the definability of queries in the relational calculus over finitely representable databases. We illustrate this very challenging problem through some classical examples. We then mainly concentrate on dense order databases, and exhibit...

We consider decidability and expressiveness issues for two first-order logics of probability. In one, the probability is on possible worlds, while in the other, it is on the domain. It turns out that in both cases it takes very little to make reasoning about probability highly undecidable. We show that when the probability is on the domain, if the language contains only unary predicates then the validity problem is decidable. However, if the language contains even one binary predicate, the validity problem is \Pi 2 1 complete, as hard as elementary analysis with free predicate and function symbols. With equality in the language, even with no other symbol, the validity problem is at least as hard as that for elementary analysis, \Pi 1 1 hard. Thus, the logic cannot be axiomatized in either case. When we put the probability on the set of possible worlds, the validity problem is \Pi 2 1 complete with as little as one unary predicate in the language, even without equality. With equalit...

We settle the complexity bounds of the model checking problem for the ambient calculus with public names against the ambient logic. We show that if either the calculus contains replication or the logic contains the guarantee operator, the problem is undecidable. In the case of the replication-free calculus and guarantee-free logic we prove that the problem is PSPACE-complete. For the complexity upper-bound, we devise a new representation of processes that remains of polynomial size during process execution; this allows us to keep the model checking procedure in polynomial space. Moreover, we prove PSPACE-hardness of the problem for several quite simple fragments of the calculus and the logic; this suggests that there are no interesting fragments with polynomial-time model checking algorithms.

this paper. In particular, their comments inspired and gave arguments for the discussion on the value of the classical decision problem after Church's and Turing's results. References

When integrating data coming from multiple different sources we are faced with the possibility of inconsistency in databases. In this paper, we use one of the paraconsistent logics introduced in [9, 7] {\textbf{LFI1}) as a logical framework to model possibly inconsistent database instances obtained by integrating different sources. We propose a method based on the sound and complete tableau proof system of \textbf{LFI1} to treat both the integration process and the evolution of the integrated database submitted to users' updates. In order to treat the integrated database evolution, we introduce a kind of generalized database context, the {\em evolutionary databases}, which are databases having the capability of storing and manipulating inconsistent information and, at the same time, allowing integrity constraints to change in time. We argue that our approach is sufficiently general and can be applied in most circumstances where inconsistency may arise in databases.

. Constraint databases generalize relational databases by finitely representable infinite relations. This paper surveys the state of the art in constraint databases: known results, remaining open problems and current research directions. The paper also describes a new algebra for databases with integer order constraints and a complexity analysis of evaluating queries in this algebra. In memory of Paris C. Kanellakis 1 Introduction There is a growing interest in recent years among database researchers in constraint databases, which are a generalization of relational databases by finitely representable infinite relations. Constraint databases are parametrized by the type of constraint domains and constraint used. The good news is that for many parameters constraint databases leave intact most of the fundamental assumptions of the relational database framework proposed by Codd. In particular, 1. Constraint databases can be queried by constraint query languages that (a) have a semantics ba...

Finite-model theory is a study of the logical properties of finite mathematical structures. This paper is a very personalized view of finite-model theory, where the author focuses on his own personal history, and results and problems of interest to him, especially those springing from work in his Ph.D. thesis. Among the topics discussed are:

It has been shown recently that monodic first-order temporal logic without functional symbols but with equality is incomplete, i.e. the set of the valid formulae of this logic is not recursively enumerable. In this paper we show that an even simpler fragment consisting of monodic monadic two-variable formulae is not recursively enumerable.