By Nadia Creignou, Phokion G. Kolaitis, Heribert Vollmer

Nowadays constraint pride difficulties (CSPs) are ubiquitous in lots of various components of computing device technological know-how, from synthetic intelligence and database structures to circuit layout, community optimization, and conception of programming languages. hence, you will need to research and pinpoint the computational complexity of definite algorithmic projects with regards to constraint delight. The complexity-theoretic result of those initiatives can have an instantaneous effect on, for example, the layout and processing of database question languages, or innovations in data-mining, or the layout and implementation of planners.

This state of the art survey includes the papers that have been invited by means of the organizers after end of a global Dagstuhl-Seminar on Complexity of Constraints, held in Dagstuhl citadel, Germany, in October 2006. a few audio system have been solicited to jot down surveys featuring the cutting-edge of their forte. those contributions have been peer-reviewed via specialists within the box and revised sooner than they have been collated to the nine papers of this quantity. furthermore, the amount features a reprint of a survey by means of Kolaitis and Vardi at the logical method of constraint delight that first seemed in 'Finite version idea and its Applications', released by way of Springer in 2007.

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**Extra info for Complexity of Constraints: An Overview of Current Research Themes**

**Example text**

Xin ). – Φ(∃, ∧, =) consists of all primitive positive formulas. – If ϕ ≡ σ1 (x1 , . . , xm ) ∧ σ2 (x1 , . . , xm ), then Lϕ describes the intersection of two m-ary relations, Lϕ (σ1 , σ2 ) = σ1 ∩ σ2 . – For the formula f (considered with m free variables) we obtain as Lf the (m) (m-ary) empty relation Lf = ∅(m) ∈ RD . Similarly, the formula t (considered with m free variables) describes the full (m) relation Lt = Dm ∈ RD . 48 F. B¨ orner – If ϕ ≡ (x1 = x2 ), then Lϕ is the binary equality relation ΔD on D.

Xri ) = ∃y1 . . ∃yqi φi (x1 , . . , xri , y1 , . . , yqi ), where φi is a Γ -formula, then for each tuple aj = (a1 , . . , ari ) ∈ Ri there are several tuples (b1 , . . , bqi ) such that φi (a1 , . . , ari , b1 , . . , bqi ) holds (we denote by uj the number of such extensions of aj ). Therefore, the number of satisfying assignments of φ cannot be directly computed from the number of satisfying assignments of φ . The trick is not to compute the formula φ but p such formulas φ (l) for l = 1, .

The dual isomorphisms are the operators α : CX → CY and β : CY → CX . This shows a great advantage of Galois connections. The “large” elements of CX are determined by the “small” elements of CY and vice versa. C. often can be carried over to information on the other side. Derived Galois Connections There is an easy way to derive new Galois connections from α–β. Let M ⊆ X be a so called restriction set and consider the new relation ΞM := Ξ ∩ (M × Y) ⊆ M × Y. Then this relation ΞM deﬁnes a new Galois connection.