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Complexity and Expressive Power of Logic Programming
, 1997
"... This paper surveys various complexity results on different forms of logic programming. The main focus is on decidable forms of logic programming, in particular, propositional logic programming and datalog, but we also mention general logic programming with function symbols. Next to classical results ..."
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Cited by 281 (57 self)
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This paper surveys various complexity results on different forms of logic programming. The main focus is on decidable forms of logic programming, in particular, propositional logic programming and datalog, but we also mention general logic programming with function symbols. Next to classical results on plain logic programming (pure Horn clause programs), more recent results on various important extensions of logic programming are surveyed. These include logic programming with different forms of negation, disjunctive logic programming, logic programming with equality, and constraint logic programming. The complexity of the unification problem is also addressed.
A Logic for Reasoning about Probabilities
 Information and Computation
, 1990
"... We consider a language for reasoning about probability which allows us to make statements such as “the probability of E, is less than f ” and “the probability of E, is at least twice the probability of E,, ” where E, and EZ are arbitrary events. We consider the case where all events are measurable ( ..."
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Cited by 214 (19 self)
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We consider a language for reasoning about probability which allows us to make statements such as “the probability of E, is less than f ” and “the probability of E, is at least twice the probability of E,, ” where E, and EZ are arbitrary events. We consider the case where all events are measurable (i.e., represent measurable sets) and the more general case, which is also of interest in practice, where they may not be measurable. The measurable case is essentially a formalization of (the propositional fragment of) Nilsson’s probabilistic logic. As we show elsewhere, the general (nonmeasurable) case corresponds precisely to replacing probability measures by DempsterShafer belief functions. In both cases, we provide a complete axiomatization and show that the problem of deciding satistiability is NPcomplete, no worse than that of propositional logic. As a tool for proving our complete axiomatizations, we give a complete axiomatization for reasoning about Boolean combinations of linear inequalities, which is of independent interest. This proof and others make crucial use of results from the theory of linear programming. We then extend the language to allow reasoning about conditional probability and show that the resulting logic is decidable and completely axiomatizable, by making use of the theory of real closed fields. ( 1990 Academic Press. Inc 1.
A classification of symbolic transition systems
 ACM TRANSACTIONS ON COMPUTATIONAL LOGIC
, 2005
"... We define five increasingly comprehensive classes of infinitestate systems, called STS1STS5, whose state spaces have finitary structure. For four of these classes, we provide examples from hybrid systems.STS1 These are the systems with finite bisimilarity quotients. They can be analyzed symbolica ..."
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Cited by 44 (5 self)
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We define five increasingly comprehensive classes of infinitestate systems, called STS1STS5, whose state spaces have finitary structure. For four of these classes, we provide examples from hybrid systems.STS1 These are the systems with finite bisimilarity quotients. They can be analyzed symbolically by iteratively applying predecessor and Boolean operations on state sets, starting from a finite number of observable state sets. Any such iteration is guaranteed to terminate in that only a finite number of state sets can be generated. This enables model checking of the μcalculus.STS2 These are the systems with finite similarity quotients. They can be analyzed symbolically by iterating the predecessor and positive Boolean operations. This enables model checking of the existential and universal fragments of the μcalculus.STS3 These are the systems with finite traceequivalence quotients. They can be analyzed symbolically by iterating the predecessor operation and a restricted form of positive Boolean operations (intersection is restricted to intersection with observables). This enables model checking of all ωregular properties, including linear temporal logic.STS4 These are the systems with finite distanceequivalence quotients (two states are equivalent if for every distance d, the same observables can be reached in d transitions). The systems in this class can be analyzed symbolically by iterating the predecessor operation and terminating when no new state sets are generated. This enables model checking of the existential conjunctionfree and universal disjunctionfree fragments of the μcalculus.STS5 These are the systems with finite boundedreachability quotients (two states are equivalent if for every distance d, the same observables can be reached in d or fewer transitions). The systems in this class can be analyzed symbolically by iterating the predecessor operation and terminating when no new states are encountered (this is a weaker termination condition than above). This enables model checking of reachability properties.
On the Complexity of Nonrecursive XQuery and Functional Query Languages on Complex Values
 In Proc. PODS’05
"... This article studies the complexity of evaluating functional query languages for complex values such as monad algebra and the recursionfree fragment of XQuery. We show that monad algebra with equality restricted to atomic values is complete for the class TA[2O(n) , O(n)] of problems solvable in lin ..."
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Cited by 40 (1 self)
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This article studies the complexity of evaluating functional query languages for complex values such as monad algebra and the recursionfree fragment of XQuery. We show that monad algebra with equality restricted to atomic values is complete for the class TA[2O(n) , O(n)] of problems solvable in linear exponential time with a linear number of alternations. The monotone fragment of monad algebra with atomic value equality but without negation is complete for nondeterministic exponential time. For monad algebra with deep equality, we establish TA[2O(n) , O(n)] lower and exponentialspace upper bounds. We also study a fragment of XQuery, Core XQuery, that seems to incorporate all the features of a query language on complex values that are traditionally deemed essential. A close connection between monad algebra on lists and Core XQuery (with “child ” as the only axis) is exhibited, and it is shown that these languages are expressively equivalent up to representation issues. We show that Core XQuery is just as hard as monad algebra w.r.t. query and combined complexity, and that it is in TC0 if the query is assumed fixed. As Core XQuery is NEXPTIMEhard, it is commonly believed that any algorithm for evaluating Core XQuery has to require exponential amounts of working memory and doubly exponential time in the worst case. We present a property of queries – the lack of a certain form of composition – that virtually all realworld XQueries have and that allows for query evaluation in singly exponential time and polynomial space. Still, we are able to show for an important special case – Core XQuery with equality testing restricted to atomic values – that the compositionfree language is just as expressive as the language with composition. Thus, under widelyheld complexitytheoretic assumptions, the compositionfree language is an exponentially less succinct version of the language with composition.
Modular Data Structure Verification
 EECS DEPARTMENT, MASSACHUSETTS INSTITUTE OF TECHNOLOGY
, 2007
"... This dissertation describes an approach for automatically verifying data structures, focusing on techniques for automatically proving formulas that arise in such verification. I have implemented this approach with my colleagues in a verification system called Jahob. Jahob verifies properties of Java ..."
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Cited by 36 (21 self)
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This dissertation describes an approach for automatically verifying data structures, focusing on techniques for automatically proving formulas that arise in such verification. I have implemented this approach with my colleagues in a verification system called Jahob. Jahob verifies properties of Java programs with dynamically allocated data structures. Developers write Jahob specifications in classical higherorder logic (HOL); Jahob reduces the verification problem to deciding the validity of HOL formulas. I present a new method for proving HOL formulas by combining automated reasoning techniques. My method consists of 1) splitting formulas into individual HOL conjuncts, 2) soundly approximating each HOL conjunct with a formula in a more tractable fragment and 3) proving the resulting approximation using a decision procedure or a theorem prover. I present three concrete logics; for each logic I show how to use it to approximate HOL formulas, and how to decide the validity of formulas in this logic. First, I present an approximation of HOL based on a translation to firstorder logic, which enables the use of existing resolutionbased theorem provers. Second, I present an approximation of HOL based on field constraint analysis, a new technique that enables
Towards efficient satisfiability checking for boolean algebra with presburger arithmetic
 In CADE21
, 2007
"... 1 Introduction This paper considers the satisfiability problem for a logic that allows reasoning about sets and their cardinalities. We call this logic quantifierfree Boolean Algebra with Presburger Arithmetic and denote it QFBAPA. Our motivationfor QFBAPA is proving the validity of formulas arisi ..."
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Cited by 28 (17 self)
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1 Introduction This paper considers the satisfiability problem for a logic that allows reasoning about sets and their cardinalities. We call this logic quantifierfree Boolean Algebra with Presburger Arithmetic and denote it QFBAPA. Our motivationfor QFBAPA is proving the validity of formulas arising from program verification [12,13,14], but
Complexity Results for FirstOrder Theories of Temporal Constraints
 In Principles of Knowledge Representation and Reasoning: Proceedings of the Fourth International Conference (KR'94
, 1994
"... We study the complexity of quantifier elimination and decision in firstorder theories of temporal constraints. With the exception of Ladkin, AI researchers have largely ignored this problem. We consider the firstorder theories of point and interval constraints over two time structures: the integer ..."
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Cited by 26 (8 self)
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We study the complexity of quantifier elimination and decision in firstorder theories of temporal constraints. With the exception of Ladkin, AI researchers have largely ignored this problem. We consider the firstorder theories of point and interval constraints over two time structures: the integers and the rationals. We show that in all cases quantifierelimination can be done in PSPACE. We also show that the decision problem for arbitrarily quantified sentences is PSPACEcomplete while for 9 k sentences it is \Sigma p k complete. Our results must be of interest to researchers working on temporal constraints, computational complexity of logical theories, constraint databases and constraint logic programming. 1 INTRODUCTION The study of temporal constraints has recently received much attention from the AI community [All83, LM88, Lad88, VKvB89, vBC90, DMP91, KL91, Mei91, vB92, Kou92, GS93, SD93]. Much of this work draws upon concepts and techniques from the literature of general co...
A Decision Procedure for Term Algebras With Queues
, 2000
"... In software verification it is often required to prove statements about heterogeneous domains containing elements of various sorts, such as counters, stacks, lists, trees and queues. Any ..."
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Cited by 24 (0 self)
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In software verification it is often required to prove statements about heterogeneous domains containing elements of various sorts, such as counters, stacks, lists, trees and queues. Any
The Complexity of Query Evaluation in Indefinite Temporal Constraint Databases
 Theoretical Computer Science
, 1997
"... In previous work we have developed the scheme of indefinite Lconstraint databases where L, the parameter, is a firstorder constraint language. This scheme extends the constraint database proposal of Kanellakis, Kuper and Revesz to include indefinite (or uncertain) information in the style of Imiel ..."
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Cited by 21 (7 self)
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In previous work we have developed the scheme of indefinite Lconstraint databases where L, the parameter, is a firstorder constraint language. This scheme extends the constraint database proposal of Kanellakis, Kuper and Revesz to include indefinite (or uncertain) information in the style of Imielinski and Lipski. In this paper we study the complexity of query evaluation in an important instance of this abstract scheme: indefinite temporal constraint databases. Our results indicate that the data/combined complexity of query evaluation does not change when we move from queries in relational calculus over relational databases, to queries in relational calculus with temporal constraints over temporal constraint databases. This fact remains true even when we consider query evaluation in relational databases with indefinite information vs. query evaluation in indefinite temporal constraint databases. In the course of our work, we provide precise bounds on the complexity of decision/quanti...