17 research outputs found
Complexity classification in qualitative temporal constraint reasoning
We study the computational complexity of the qualitative algebra which is a temporal constraint formalism that combines the point algebra, the point-interval algebra and Allen's interval algebra. We identify all tractable fragments and show that every other fragment is NP-complete
Extending uncertainty formalisms to linear constraints and other complex formalisms
Linear constraints occur naturally in many reasoning problems and the information that they represent is often uncertain. There is a difficulty in applying AI uncertainty formalisms to this situation, as their representation of the underlying logic, either as a mutually exclusive and exhaustive set of possibilities, or with a propositional or a predicate logic, is inappropriate (or at least unhelpful). To overcome this difficulty, we express reasoning with linear constraints as a logic, and develop the formalisms based on this different underlying logic. We focus in particular on a possibilistic logic representation of uncertain linear constraints, a lattice-valued possibilistic logic, an assumption-based reasoning formalism and a Dempster-Shafer representation, proving some fundamental results for these extended systems. Our results on extending uncertainty formalisms also apply to a very general class of underlying monotonic logics
Uncertain linear constraints
Linear constraints occur naturally in many reasoning problems and the information that they represent is often uncertain. There is a difficulty in applying many AI uncertainty formalisms to this situation, as their representation of the underlying logic, either as a mutually exclusive and exhaustive set of possibilities, or with a propositional or a predicate logic, is inappropriate (or at least unhelpful). To overcome this, we express reasoning with linear constraints as a logic, and develop the formalisms based on this different underlying logic. We focus in particular on a possibilistic logic representation of uncertain linear constraints, a lattice-valued possibilistic logic, and a Dempster-Shafer representation
Tractability in Constraint Satisfaction Problems: A Survey
International audienceEven though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP
Constraint Satisfaction Problems over Numeric Domains
We present a survey of complexity results for constraint satisfaction problems (CSPs) over the integers, the rationals, the reals, and the complex numbers. Examples of such problems are feasibility of linear programs, integer linear programming, the max-atoms problem, Hilbert\u27s tenth problem, and many more. Our particular focus is to identify those CSPs that can be solved in polynomial time, and to distinguish them from CSPs that are NP-hard. A very helpful tool for obtaining complexity classifications in this context is the concept of a polymorphism from universal algebra
Reasoning about temporal relations : the maximal tractable subalgebras of Allen's interval algebra
Allen's interval algebra is one of the best established formalisms for temporal reasoning. This article provides the final step in the classification of complexity for satisfiability problems over constraints expressed in this algebra. When the constraints are chosen from the full Allen's algebra, this form of satisfiability problem is known to be NP-complete. However, eighteen tractable subalgebras have previously been identified; we show here that these subalgebras include all possible tractable subsets of Allen's algebra. In other words, we show that this algebra contains exactly eighteen maximal tractable subalgebras, and reasoning in any fragment not entirely contained in one of these subalgebras is NP-complete. We obtain this dichotomy result by giving a new uniform description of the known maximal tractable subalgebras, and then systematically using a general algebraic technique for identifying maximal subalgebras with a given property
Complexity Classification Transfer for CSPs via Algebraic Products
We study the complexity of infinite-domain constraint satisfaction problems:
our basic setting is that a complexity classification for the CSPs of
first-order expansions of a structure can be transferred to a
classification of the CSPs of first-order expansions of another structure
. We exploit a product of structures (the algebraic product) that
corresponds to the product of the respective polymorphism clones and present a
complete complexity classification of the CSPs for first-order expansions of
the -fold algebraic power of . This is proved by various
algebraic and logical methods in combination with knowledge of the
polymorphisms of the tractable first-order expansions of and
explicit descriptions of the expressible relations in terms of syntactically
restricted first-order formulas. By combining our classification result with
general classification transfer techniques, we obtain surprisingly strong new
classification results for highly relevant formalisms such as Allen's Interval
Algebra, the -dimensional Block Algebra, and the Cardinal Direction
Calculus, even if higher-arity relations are allowed. Our results confirm the
infinite-domain tractability conjecture for classes of structures that have
been difficult to analyse with older methods. For the special case of
structures with binary signatures, the results can be substantially
strengthened and tightly connected to Ord-Horn formulas; this solves several
longstanding open problems from the AI literature.Comment: 61 pages, 1 figur