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56
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 114 (20 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.
The Lyapunov exponent and joint spectral radius of pairs of matrices are hard  when not impossible  to compute and to approximate
, 1997
"... We analyse the computability and the complexity of various definitions of spectral radii for sets of matrices. We show that the joint and generalized spectral radii of two integer matrices are not approximable in polynomial time, and that two related quantities  the lower spectral radius and th ..."
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Cited by 62 (18 self)
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We analyse the computability and the complexity of various definitions of spectral radii for sets of matrices. We show that the joint and generalized spectral radii of two integer matrices are not approximable in polynomial time, and that two related quantities  the lower spectral radius and the largest Lyapunov exponent  are not algorithmically approximable.
Multiset Rewriting and the Complexity of Bounded Security Protocols
 Journal of Computer Security
, 2002
"... We formalize the DolevYao model of security protocols, using a notation based on multiset rewriting with existentials. The goals are to provide a simple formal notation for describing security protocols, to formalize the assumptions of the DolevYao model using this notation, and to analyze the ..."
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Cited by 56 (5 self)
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We formalize the DolevYao model of security protocols, using a notation based on multiset rewriting with existentials. The goals are to provide a simple formal notation for describing security protocols, to formalize the assumptions of the DolevYao model using this notation, and to analyze the complexity of the secrecy problem under various restrictions. We prove that, even for the case where we restrict the size of messages and the depth of message encryption, the secrecy problem is undecidable for the case of an unrestricted number of protocol roles and an unbounded number of new nonces. We also identify several decidable classes, including a dexpcomplete class when the number of nonces is restricted, and an npcomplete class when both the number of nonces and the number of roles is restricted. We point out a remaining open complexity problem, and discuss the implications these results have on the general topic of protocol analysis.
On the Undecidability of Probabilistic Planning and Related Stochastic Optimization Problems
 Artificial Intelligence
, 2003
"... Automated planning, the problem of how an agent achieves a goal given a repertoire of actions, is one of the foundational and most widely studied problems in the AI literature. The original formulation of the problem makes strong assumptions regarding the agent's knowledge and control over the ..."
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Cited by 49 (0 self)
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Automated planning, the problem of how an agent achieves a goal given a repertoire of actions, is one of the foundational and most widely studied problems in the AI literature. The original formulation of the problem makes strong assumptions regarding the agent's knowledge and control over the world, namely that its information is complete and correct, and that the results of its actions are deterministic and known.
Undecidable Problems for Probabilistic Automata of Fixed Dimension
 Theory of Computing Systems
, 2001
"... We prove that several problems associated to probabilistic finite automata are undecidable for automata whose number of input letters and number of states are fixed. As a corollary of one of our results we prove that the problem of determining if the set of all products of two 47 × 47 matr ..."
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Cited by 34 (5 self)
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We prove that several problems associated to probabilistic finite automata are undecidable for automata whose number of input letters and number of states are fixed. As a corollary of one of our results we prove that the problem of determining if the set of all products of two 47 &times; 47 matrices with nonnegative rational entries is bounded is undecidable.
Complexity of Stability and Controllability of Elementary Hybrid Systems
, 1997
"... this paper, weconsider simple classes of nonlinear systems and provethatbasic questions related to their stabilityandcontrollabilityare either undecidable or computationally intractable (NPhard). As a special case, weconsider a class of hybrid systems in which the state space is partitioned into tw ..."
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Cited by 34 (10 self)
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this paper, weconsider simple classes of nonlinear systems and provethatbasic questions related to their stabilityandcontrollabilityare either undecidable or computationally intractable (NPhard). As a special case, weconsider a class of hybrid systems in which the state space is partitioned into two halfspaces, and the dynamics in eachhalfspace correspond to a differentlinear system
Decidability and undecidability results for planning with numerical state variables
 Proceedings of the Sixth International Conference on Artificial Intelligence Planning and Scheduling
, 2002
"... These days, propositional planning can be considered a quite wellunderstood problem. Good algorithms are known that can solve a wealth of very different and sometimes challenging planning tasks, and theoretical computational properties of both general STRIPSstyle planning and the bestknown benchm ..."
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Cited by 28 (1 self)
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These days, propositional planning can be considered a quite wellunderstood problem. Good algorithms are known that can solve a wealth of very different and sometimes challenging planning tasks, and theoretical computational properties of both general STRIPSstyle planning and the bestknown benchmark problems have been established. However, propositional planning has a major drawback: The formalism is too weak to allow for the easy encoding of many genuinely interesting planning problems, specifically those involving numbers. A recent effort to enhance the PDDL planning language to cope with (among other additions) numerical state variables, to be used at the third international planning competition, has increased interest in these issues. In this contribution, we analyze “STRIPS with numbers” from a theoretical point of view. Specifically, we show that the introduction of numerical state variables makes the planning problem undecidable in the general case and many restrictions thereof and identify special cases for which we can provide decidability results.
When is a Pair of Matrices Mortal?
, 1996
"... A set of matrices over the integers is said to be lengthkmortal (with positive integer) if the zero matrix can be expressed as a product of length of matrices in the set. The set is said to be mortal if it is lengthkmortal for some finite k. ..."
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Cited by 26 (12 self)
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A set of matrices over the integers is said to be lengthkmortal (with positive integer) if the zero matrix can be expressed as a product of length of matrices in the set. The set is said to be mortal if it is lengthkmortal for some finite k.
On the Mortality Problem for Matrices of Low Dimensions
, 1999
"... In this paper, we discuss the existence of an algorithm to decide if a given set of 2 2 matrices is mortal: a set F = fA 1 ; : : : ; Am g of 2 2 matrices is said to be mortal if there exist an integer k 1 and some integers i 1 ; i 2 ; : : : ; i k 2 f1; : : : ; mg with A i 1 A i 2 A i k = 0. We surve ..."
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Cited by 9 (1 self)
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In this paper, we discuss the existence of an algorithm to decide if a given set of 2 2 matrices is mortal: a set F = fA 1 ; : : : ; Am g of 2 2 matrices is said to be mortal if there exist an integer k 1 and some integers i 1 ; i 2 ; : : : ; i k 2 f1; : : : ; mg with A i 1 A i 2 A i k = 0. We survey this problem and propose some new extensions: we prove the problem to be BSSundecidable for real matrices and Turingdecidable for two rational matrices. We relate the problem for rational matrices to the entry equivalence problem, to the zero in the left upper corner problem and to the reachability problem for piecewise a ne functions. Finally, we state some NPcompleteness results.
Undecidability bounds for integer matrices using Claus instances
, 2006
"... There are several known undecidability problems for 3 × 3 integer matrices the proof of which uses a reduction from the Post Correspondence Problem (PCP). We establish new lower bounds in the numbers of matrices for the mortality, zero in left upper corner, vector reachability, matrix reachability, ..."
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Cited by 9 (1 self)
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There are several known undecidability problems for 3 × 3 integer matrices the proof of which uses a reduction from the Post Correspondence Problem (PCP). We establish new lower bounds in the numbers of matrices for the mortality, zero in left upper corner, vector reachability, matrix reachability, scaler reachability and freeness problems. Also, we give a short proof for a strengthened result due to Bell and Potapov stating that the membership problem is undecidable for finitely generated matrix semigroups R ⊆ Z 4×4 whether or not kI4 ∈ R for any given diagonal matrix kI4 with k> 1. These bounds are obtained by using Claus instances of the PCP.