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34
Decidability and Expressiveness for FirstOrder Logics of Probability
 Information and Computation
, 1989
"... We consider decidability and expressiveness issues for two firstorder 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 t ..."
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Cited by 40 (6 self)
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We consider decidability and expressiveness issues for two firstorder 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...
Constraint Databases: A Survey
 Semantics in Databases, number 1358 in LNCS
, 1998
"... . 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 inte ..."
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Cited by 23 (3 self)
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. 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...
Extended OrderGeneric Queries
, 1998
"... We consider relational databases organized over an ordered domain with some additional relationsa typical example is the ordered domain of rational numbers together with the operation of addition. In the focus of our study are the firstorder (FO) queries that are invariant under orderpreser ..."
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Cited by 19 (2 self)
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We consider relational databases organized over an ordered domain with some additional relationsa typical example is the ordered domain of rational numbers together with the operation of addition. In the focus of our study are the firstorder (FO) queries that are invariant under orderpreserving "permutations"such queries are called ordergeneric. It has recently been discovered that for some domains ordergeneric FO queries fail to express more than pure order queries. For example, every ordergeneric FO query over rational numbers with + can be rewritten without +. For some other domains, however, this is not the case. We provide very general conditions on the FO theory of the domain that ensure the collapse of ordergeneric extended FO queries to pure order queries over this domain: the Pseudofinite Homogeneity Property and a stronger Isolation Property. We further distinguish one broad class of domains satisfying the Isolation Property, the socalled quasio ...
UNIFORM FAMILIES OF POLYNOMIAL EQUATIONS OVER A FINITE FIELD AND STRUCTURES ADMITTING AN EULER CHARACTERISTIC OF DEFINABLE SETS
, 2000
"... ..."
The Crane Beach Conjecture
 In Proc. 16th Symp. on Logic in Comp. Sci. (LICS01
, 2001
"... A language L over an alphabet A is said to have a neutral letter if there is a letter e # A such that inserting or deleting e's from any word in A # does not change its membership (or nonmembership) in L. The presence of a neutral letter affects the definability of a language in firstorder lo ..."
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Cited by 7 (1 self)
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A language L over an alphabet A is said to have a neutral letter if there is a letter e # A such that inserting or deleting e's from any word in A # does not change its membership (or nonmembership) in L. The presence of a neutral letter affects the definability of a language in firstorder logic. It was conjectured that it renders all numerical predicates apart from the order predicate useless, i.e., that if a language L with a neutral letter is not definable in firstorder logic with linear order, then it is not definable in firstorder logic with any set N of numerical predicates. We investigate this conjecture in detail, showing that it fails already for N = {+, #}, or, possibly stronger, for any set N that allows counting up to the m times iterated logarithm, lg (m) , for any constant m. On the positive side, we prove the conjecture for the case of all monadic numerical predicates, for N = {+}, for the fragment BC(# 1 ) of firstorder logic, and for binary alphabets. # Supported by NSF grant CCR9988260. + Supported by NSF grant CCR9877078. # Supported by NSERC and FCAR. 1
CHARACTERIZING INTEGERS AMONG RATIONAL NUMBERS WITH A UNIVERSALEXISTENTIAL FORMULA
"... Abstract. We prove that Z in definable in Q by a formula with two universal quantifiers followed by seven existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Qmorphisms, whether there exists one that is surjective on rational points. We also giv ..."
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Cited by 5 (0 self)
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Abstract. We prove that Z in definable in Q by a formula with two universal quantifiers followed by seven existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Qmorphisms, whether there exists one that is surjective on rational points. We also give a formula, again with universal quantifiers followed by existential quantifiers, that in any number field defines the ring of integers. 1. Introduction. 1.1. Background. D. Hilbert, in the 10th of his famous list of 23 problems, asked for an algorithm for deciding the solvability of any multivariable polynomial equation in integers. Thanks to the work of M. Davis, H. Putnam, J. Robinson [DPR61], and Y. Matijasevič [Mat70], we know that no such algorithm
On diophantine definability and decidability in some infinite totally real extensions
 of Q, Trans. Amer. Math. Soc
"... Abstract. Let M be a number field, and WM a set of its nonArchimedean primes. Then let OM,W = {x ∈ M  ordt x ≥ 0, ∀t � ∈ WM}. Let{p1,...,pr} M be a finite set of prime numbers. Let Finf be the field generated by all the p j ith roots of unity as j → ∞ and i = 1,...,r. Let Kinf be the largest tot ..."
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Cited by 5 (5 self)
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Abstract. Let M be a number field, and WM a set of its nonArchimedean primes. Then let OM,W = {x ∈ M  ordt x ≥ 0, ∀t � ∈ WM}. Let{p1,...,pr} M be a finite set of prime numbers. Let Finf be the field generated by all the p j ith roots of unity as j → ∞ and i = 1,...,r. Let Kinf be the largest totally real subfield of Finf. Then for any ε>0, there exist a number field M ⊂ Kinf,andasetWMof nonArchimedean primes of M such that WM has density greater than 1 − ε, andZhas a Diophantine definition over the integral closure of OM,W in Kinf. M 1.
On Flat Programs with Lists
"... In this paper we analyze the complexity of checking safety and termination properties, for a very simple, yet nontrivial, class of programs with singlylinked list data structures. Since, in general, programs with lists are knownto have the power of Turing machines, we restrict the control struct ..."
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Cited by 5 (0 self)
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In this paper we analyze the complexity of checking safety and termination properties, for a very simple, yet nontrivial, class of programs with singlylinked list data structures. Since, in general, programs with lists are knownto have the power of Turing machines, we restrict the control structure, by forbidding nested loops and destructive updates. Surprisingly, even with these simplifying conditions, verifying safety and termination for programs working on heaps with more than one cycle are undecidable, whereas decidability can be established when the input heap may have at most one loop. The proofs for both the undecidability and the decidability results rely on nontrivial numbertheoreticresults.
Automating elementary numbertheoretic proofs using Gröbner bases
"... Abstract. We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain numbertheoretic predicates ..."
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Cited by 4 (0 self)
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Abstract. We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain numbertheoretic predicates such as ‘divisible by’, ‘congruent ’ and ‘coprime’; one notable example in this class is the Chinese Remainder Theorem (for a specific number of moduli). The method is based on a reduction to ideal membership assertions that are then solved using Gröbner bases. As well as illustrating the usefulness of the procedure on examples, and considering some extensions, we prove a limited form of completeness for properties that hold in all rings. 1