Results 1  10
of
74
On the algebraic structure of combinatorial problems
 THEORETICAL COMPUTER SCIENCE
, 1998
"... ..."
Constraint Satisfaction Problems And Finite Algebras
, 1999
"... Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NPcomplete in general, but certain restrictions on the form of the constraints can ensure tractability. In this paper we show that any restricted set of constraint types c ..."
Abstract

Cited by 50 (7 self)
 Add to MetaCart
Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NPcomplete in general, but certain restrictions on the form of the constraints can ensure tractability. In this paper we show that any restricted set of constraint types can be associated with a finite universal algebra. We explore how the computational complexity of a restricted constraint satisfaction problem is connected to properties of the corresponding algebra. For this, we introduce a notion of `tractable algebra' and study how the tractability of an algebra relates to the tractability of its smaller derived algebras, including its subalgebras and homomorphic images. This allows us to significantly reduce the types of algebras which need to be investigated. Using these results we exhibit a common structural property of all known intractable constraint satisfaction problems. Finally, we classify all finite strictly simple surjective algebras wit...
Playing with Boolean blocks, part I: Post’s lattice with applications to complexity theory
 SIGACT News
"... Let us imagine children playing with a box containing a large number of building blocks such as LEGO TM, fischertechnik ® , Polydron, or something similar. Each block belongs to a certain class (given by, e. g., color, shape, or size) and usually the number of different such classes is relatively sm ..."
Abstract

Cited by 40 (12 self)
 Add to MetaCart
Let us imagine children playing with a box containing a large number of building blocks such as LEGO TM, fischertechnik ® , Polydron, or something similar. Each block belongs to a certain class (given by, e. g., color, shape, or size) and usually the number of different such classes is relatively small. It is amazing to see how involved the constructions are that can be built by the kids. From
The Complexity Of Maximal Constraint Languages
, 2001
"... Many combinatorial search problems can be expressed as "constraint satisfaction problems" using an appropriate "constraint language", that is, a set of relations over some fixed finite set of values. It is wellknown that there is a tradeoff between the expressive power of a constraint language and ..."
Abstract

Cited by 35 (8 self)
 Add to MetaCart
Many combinatorial search problems can be expressed as "constraint satisfaction problems" using an appropriate "constraint language", that is, a set of relations over some fixed finite set of values. It is wellknown that there is a tradeoff between the expressive power of a constraint language and the complexity of the problems it can express. In the present paper we systematically study the complexity of all maximal constraint languages, that is, languages whose expressive power is just weaker than that of the language of all constraints. Using the algebraic invariance properties of constraints, we exhibit a strong necessary condition for tractability of such a constraint language. Moreover, we show that, at least for small sets of values, this condition is also sufficient.
Shuffle on Trajectories: Syntactic Constraints
 Theor. Comp. Sci
, 1998
"... We introduce and investigate new methods to define parallel composition of words and languages. The operation of parallel composition leads to new shufflelike operations defined by syntactic constraints on the usual shuffle operation. The approach is applicable to concurrency, providing a method to ..."
Abstract

Cited by 25 (5 self)
 Add to MetaCart
We introduce and investigate new methods to define parallel composition of words and languages. The operation of parallel composition leads to new shufflelike operations defined by syntactic constraints on the usual shuffle operation. The approach is applicable to concurrency, providing a method to define parallel composition of processes. It is also applicable to parallel computation. The operations are introduced using a uniform method based on the notion of trajectory. As a consequence, we obtain a very intuitive geometrical interpretation of the parallel composition operation. These operations lead in a natural way to a large class of semirings. The approach is amazingly flexible, diverse concepts from the theory of concurrency can be introduced and studied in this framework. For instance, we provide examples of applications to fairness property and to parallelization of noncontextfree languages in terms of contextfree and even regular languages. This paper concetrates on syntactic constraints. Semantic constraints will be dealt with in a forthcoming contribution. TUCS Research Group
Constraints and Universal Algebra
 Annals of Mathematics and Artificial Intelligence
, 1998
"... In this paper we explore the links between constraint satisfaction problems and universal algebra. We show that a constraint satisfaction problem instance can be viewed as a pair of relational structures, and the solutions to the problem are then the structure preserving mappings between these two r ..."
Abstract

Cited by 20 (4 self)
 Add to MetaCart
In this paper we explore the links between constraint satisfaction problems and universal algebra. We show that a constraint satisfaction problem instance can be viewed as a pair of relational structures, and the solutions to the problem are then the structure preserving mappings between these two relational structures. We give a number of examples to illustrate how this framework can be used to express a wide variety of combinatorial problems, many of which are not generally considered as constraint satisfaction problems. We also show that certain key aspects of the mathematical structure of constraint satisfaction problems can be precisely described in terms of the notion of a Galois connection, which is a standard notion of universal algebra. Using this result, we obtain an algebraic characterisation of the property of minimality in a constraint satisfaction problem. We also obtain a similar algebraic criterion for determining whether or not a given set of solutions can be expressed...
On the Maximum Tolerable Noise for Reliable Computation by Formulas
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1995
"... It is shown that if a formula is constructed from noisy 2input NAND gates, with each gate failing independently with probability ", then reliable computation can or cannot take place according as "is less than or greater than" 0 = (3 \Gamma p 7)=4 = 0.08856... . ..."
Abstract

Cited by 18 (1 self)
 Add to MetaCart
It is shown that if a formula is constructed from noisy 2input NAND gates, with each gate failing independently with probability ", then reliable computation can or cannot take place according as "is less than or greater than" 0 = (3 \Gamma p 7)=4 = 0.08856... .
Tractable Constraints Closed Under A Binary Operation
 Oxford University
, 2000
"... Many combinatorial search problems can be expressed as instances of the "constraint satisfaction problem" (CSP). This class of problems is known to be NPcomplete in general, so to ensure tractability it is natural to consider restricted subproblems in which the constraints have certain specified fo ..."
Abstract

Cited by 13 (7 self)
 Add to MetaCart
Many combinatorial search problems can be expressed as instances of the "constraint satisfaction problem" (CSP). This class of problems is known to be NPcomplete in general, so to ensure tractability it is natural to consider restricted subproblems in which the constraints have certain specified forms. The algebraic approach to the CSP maintains that certain algebraic invariance properties of constraints can be used to determine the complexity of these restricted problems.
THE COMPLEXITY OF GENERALIZED SATISFIABILITY FOR LINEAR TEMPORAL LOGIC
 VOL. 5 (1:1) 2009, PP. 1–1–21
, 2009
"... ..."
Programmability of Chemical Reaction Networks
"... Summary. Motivated by the intriguing complexity of biochemical circuitry within individual cells we study Stochastic Chemical Reaction Networks (SCRNs), a formal model that considers a set of chemical reactions acting on a finite number of molecules in a wellstirred solution according to standard c ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
Summary. Motivated by the intriguing complexity of biochemical circuitry within individual cells we study Stochastic Chemical Reaction Networks (SCRNs), a formal model that considers a set of chemical reactions acting on a finite number of molecules in a wellstirred solution according to standard chemical kinetics equations. SCRNs have been widely used for describing naturally occurring (bio)chemical systems, and with the advent of synthetic biology they become a promising language for the design of artificial biochemical circuits. Our interest here is the computational power of SCRNs and how they relate to more conventional models of computation. We survey known connections and give new connections between SCRNs and