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27
A Survey of Computational Complexity Results in Systems and Control
, 2000
"... The purpose of this paper is twofold: (a) to provide a tutorial introduction to some key concepts from the theory of computational complexity, highlighting their relevance to systems and control theory, and (b) to survey the relatively recent research activity lying at the interface between these fi ..."
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Cited by 116 (21 self)
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The purpose of this paper is twofold: (a) to provide a tutorial introduction to some key concepts from the theory of computational complexity, highlighting their relevance to systems and control theory, and (b) to survey the relatively recent research activity lying at the interface between these fields. We begin with a brief introduction to models of computation, the concepts of undecidability, polynomial time algorithms, NPcompleteness, and the implications of intractability results. We then survey a number of problems that arise in systems and control theory, some of them classical, some of them related to current research. We discuss them from the point of view of computational complexity and also point out many open problems. In particular, we consider problems related to stability or stabilizability of linear systems with parametric uncertainty, robust control, timevarying linear systems, nonlinear and hybrid systems, and stochastic optimal control.
On Diophantine Complexity and Statistical ZeroKnowledge Arguments
 Advances on Cryptology — ASIACRYPT 2003
, 2003
"... Abstract. We show how to construct practical honestverifier statistical zeroknowledge Diophantine arguments of knowledge (HVSZK AoK) that a committed tuple of integers belongs to an arbitrary language in bounded arithmetic. While doing this, we propose a new algorithm for computing the Lagrange re ..."
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Cited by 28 (6 self)
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Abstract. We show how to construct practical honestverifier statistical zeroknowledge Diophantine arguments of knowledge (HVSZK AoK) that a committed tuple of integers belongs to an arbitrary language in bounded arithmetic. While doing this, we propose a new algorithm for computing the Lagrange representation of nonnegative integers and a new efficient representing polynomial for the exponential relation. We apply our results by constructing the most efficient known HVSZK AoK for nonnegativity and the first constantround practical HVSZK AoK for exponential relation. Finally, we propose the outsourcing model for cryptographic protocols and design communicationefficient versions of the Damg˚ardJurik multicandidate voting scheme and of the LipmaaAsokanNiemi (b + 1)stprice auction scheme that work in this model.
A Quantifier Elimination Algorithm for a Fragment of Set Theory Involving the Cardinality Operator
 In 18th International Workshop on Unification
, 2004
"... We present a decision procedure based on quantifier elimination for a fragment of set theory involving elements, integers, and finite sets of elements in the presence of the cardinality operator. The language allows quantification on element variables and integer variables, but not on set variables. ..."
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Cited by 13 (2 self)
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We present a decision procedure based on quantifier elimination for a fragment of set theory involving elements, integers, and finite sets of elements in the presence of the cardinality operator. The language allows quantification on element variables and integer variables, but not on set variables. We also show that if we identify the sort of elements with the sort of integers, thus considering the case of finite sets of integers, then the resulting language becomes undecidable. 1
Constraint Satisfaction Problems with Countable Homogeneous Templates
"... Allowing templates with infinite domains greatly expands the range of problems that can be formulated as a nonuniform constraint satisfaction problem. It turns out that many CSPs over infinite templates can be formulated with templates that are ωcategorical. We survey examples of such problems in ..."
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Cited by 13 (7 self)
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Allowing templates with infinite domains greatly expands the range of problems that can be formulated as a nonuniform constraint satisfaction problem. It turns out that many CSPs over infinite templates can be formulated with templates that are ωcategorical. We survey examples of such problems in temporal and spatial reasoning, infinitedimensional algebra, acyclic colorings in graph theory, artificial intelligence, phylogenetic reconstruction in computational biology, and tree descriptions in computational linguistics. We then give an introduction to the universalalgebraic approach to infinitedomain constraint satisfaction, and discuss how cores, polymorphism clones, and pseudovarieties can be used to study the computational complexity of CSPs with ωcategorical templates. The theoretical results will be illustrated by examples from the mentioned application areas. We close with a series of open problems and promising directions of future research.
On Herbrand Skeletons
, 1995
"... . Herbrand's theorem plays an important role both in proof theory and in computer science. Given a Herbrand skeleton, which is basically a number specifying the count of disjunctions of the matrix, we would like to get a computable bound on the size of terms which make the disjunction into a qua ..."
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Cited by 12 (0 self)
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. Herbrand's theorem plays an important role both in proof theory and in computer science. Given a Herbrand skeleton, which is basically a number specifying the count of disjunctions of the matrix, we would like to get a computable bound on the size of terms which make the disjunction into a quasitautology. This is an important problem in logic, specifically in the complexity of proofs. In computer science, specifically in automated theorem proving, one hopes for an algorithm which avoids the guesses of existential substitution axioms involved in proving a theorem. Herbrand's theorem forms the very basis of automated theorem proving where for a given number n we would like to have an algorithm which finds the terms in the n disjunctions of matrices solely from the shape of the matrix. The main result of this paper is that both problems have negative solutions. 1 Introduction By the theorem of Herbrand we have for a quantifierfree OE: j= 9 x OE( x) iff j= OE( a 1 ) OE( ...
Comparative similarity, tree automata, and Diophantine equations
 In Proceedings of LPAR 2005, volume 3835 of LNAI
, 2005
"... Abstract. The notion of comparative similarity ‘X is more similar or closer to Y than to Z ’ has been investigated in both foundational and applied areas of knowledge representation and reasoning, e.g., in concept formation, similaritybased reasoning and areas of bioinformatics such as protein sequ ..."
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Cited by 12 (8 self)
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Abstract. The notion of comparative similarity ‘X is more similar or closer to Y than to Z ’ has been investigated in both foundational and applied areas of knowledge representation and reasoning, e.g., in concept formation, similaritybased reasoning and areas of bioinformatics such as protein sequence alignment. In this paper we analyse the computational behaviour of the ‘propositional ’ logic with the binary operator ‘closer to a set τ1 than to a set τ2 ’ and nominals interpreted over various classes of distance (or similarity) spaces. In particular, using a reduction to the emptiness problem for certain tree automata, we show that the satisfiability problem for this logic is ExpTimecomplete for the classes of all finite symmetric and all finite (possibly nonsymmetric) distance spaces. For finite subspaces of the real line (and higher dimensional Euclidean spaces) we prove the undecidability of satisfiability by a reduction of the solvability problem for Diophantine equations. As our ‘closer ’ operator has the same expressive power as the standard operator> of conditional logic, these results may have interesting implications for conditional logic as well. 1
Decision Procedures for Multisets with Cardinality Constraints
"... Abstract. Applications in software verification and interactive theorem proving often involve reasoning about sets of objects. Cardinality constraints on such collections also arise in these applications. Multisets arise in these applications for analogous reasons as sets: abstracting the content of ..."
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Cited by 11 (7 self)
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Abstract. Applications in software verification and interactive theorem proving often involve reasoning about sets of objects. Cardinality constraints on such collections also arise in these applications. Multisets arise in these applications for analogous reasons as sets: abstracting the content of linked data structure with duplicate elements leads to multisets. Interactive theorem provers such as Isabelle specify theories of multisets and prove a number of theorems about them to enable their use in interactive verification. However, the decidability and complexity of constraints on multisets is much less understood than for constraints on sets. The first contribution of this paper is a polynomialspace algorithm for deciding expressive quantifierfree constraints on multisets with cardinality operators. Our decision procedure reduces in polynomial time constraints on multisets to constraints in an extension of quantifierfree Presburger arithmetic with certain “unbounded sum ” expressions. We prove bounds on solutions of resulting constraints and describe a polynomialspace decision procedure for these constraints. The second contribution of this paper is a proof that adding quantifiers to a constraint language containing subset and cardinality operators yields undecidable constraints. The result follows by reduction from Hilbert’s 10th problem. 1
Statistical ZeroKnowledge Proofs from Diophantine Equations. Cryptology ePrint Archive
, 2001
"... Abstract. A familyËØof sets isÔbounded Diophantine ifËØhas a representingÔbounded polynomialÊË�Ø, s.t.ÜËØ �Ý�ÊËÜ�Ý�℄. We say thatËØis unbounded Diophantine if additionally,ÊË�Øis a fixedØindependent polynomial. We show thatÔbounded (resp., unbounded) Diophantine set has a polynomialsize (resp., ..."
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Cited by 5 (2 self)
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Abstract. A familyËØof sets isÔbounded Diophantine ifËØhas a representingÔbounded polynomialÊË�Ø, s.t.ÜËØ �Ý�ÊËÜ�Ý�℄. We say thatËØis unbounded Diophantine if additionally,ÊË�Øis a fixedØindependent polynomial. We show thatÔbounded (resp., unbounded) Diophantine set has a polynomialsize (resp., constantsize) statistical zeroknowledge proof system that a committed tupleÜbelongs toË. We describe efficient SZK proof systems for several cryptographically interesting sets. Finally, we show how to prove in SZK that an encrypted number belongs toË.
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.
Compositional information flow security for concurrent programs
 Journal of Computer Security
, 2007
"... We present a general unwinding framework for the definition of information flow security properties of concurrent programs, described in a standard imperative language enriched with parallelism. We study different classes of programs obtained by instantiating the general framework and we prove that ..."
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Cited by 4 (1 self)
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We present a general unwinding framework for the definition of information flow security properties of concurrent programs, described in a standard imperative language enriched with parallelism. We study different classes of programs obtained by instantiating the general framework and we prove that they entail the noninterference principle. Accurate proof techniques for the verification of such properties are defined by exploiting the Tarski decidability result for first order formulae over the reals. Moreover, we illustrate how the unwinding framework can be instantiated in order to deal with intentional information release and we extend our verification techniques to the analysis of security properties of programs admitting downgrading. 1