Abstract
Chapter 3 studies the expressive power of various classes of valued constraints. It contains several results of the following form: let \(\mathcal{C}\) be a class of valued constraints with functions of unbounded arities; then \(\mathcal{C}\) can be expressed by a subset of \(\mathcal{C}\) consisting of valued constraints with functions of a fixed bounded arity. The only known class for which this is not true is the class of finite-valued max-closed functions of different arities. This chapter also presents some results on the algebraic properties of finite-valued max-closed functions.
I just wondered how things were put together.
Claude Shannon
Most of the material in this chapter is reprinted from Theoretical Computer Science, 409(1), D.A. Cohen, P.G. Jeavons, and S. Živný, The Expressive Power of Valued Constraints: Hierarchies and Collapses, 137–153, Copyright (2008), with permission from Elsevier and from Information Processing Letters, 109(11), B. Zanuttini and S. Živný, A Note on Some Collapse Results of Valued Constraints, 534–538, Copyright (2009), with permission from Elsevier.
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Notes
- 1.
This is the only place where we use the condition that |D|≥3.
- 2.
Note that a “finite variant” of ϕ 1, defined by ϕ 1(〈0,1〉)=K for some finite K<∞ and ϕ 1(〈⋅,⋅〉)=0 otherwise, is not max-closed. The infinite cost is necessary.
- 3.
Least common multiple.
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Živný, S. (2012). Expressibility of Fixed-Arity Languages. In: The Complexity of Valued Constraint Satisfaction Problems. Cognitive Technologies. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33974-5_3
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