The Two-Valued Iterative Systems of Mathematical Logic. (AM-5)

Post, Emil L.

doi:10.1515/9781400882366

Cited by 463 publications

(446 citation statements)

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“…The complete proof is available in the original paper [44]. The idea is based on the decomposition theory of Boolean functions [5,12,34]. Namely, we can decompose the majority function with an odd number of input bits into compositions of 3-bit majority functions.…”

Section: Lp Bound With Clique Inequalitiesmentioning

confidence: 99%

Exploring the Limits of Subadditive Approaches: Parallels between Optimization and Complexity Theory

Ueno

2015

IIS

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In this paper, we review subadditive approaches which arise in the theory of mathematical programming and computational complexity. In particular, we explain the duality theorem of integer programming and techniques to prove formula-size lower bounds as fundamental subjects in mathematical programming and computational complexity, respectively. We discuss parallel visions of these two different areas by showing some connections between them.

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Section: Lp Bound With Clique Inequalitiesmentioning

confidence: 99%

Exploring the Limits of Subadditive Approaches: Parallels between Optimization and Complexity Theory

Ueno

2015

IIS

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“…We denote by [B] the smallest clone containing B and call B a base for [B]. Post (1941) identified, the set of all clones of Boolean functions. He gave a finite base for each of the clones and showed that they form a lattice under the usual ⊆-relation, hence the name Post's lattice (Figure 1).…”

Section: Post's Latticementioning

confidence: 99%

Complexity of logic-based argumentation in Post's framework

Creignou¹,

Schmidt²,

Thomas³

et al. 2011

Argument & Computation

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Many proposals for logic-based formalisations of argumentation consider an argument as a pair ( , α), where the support is understood as a minimal consistent subset of a given knowledge base which has to entail the claim α. In case the arguments are given in the full language of classical propositional logic reasoning in such frameworks becomes a computationally costly task. For instance, the problem of deciding whether there exists a support for a given claim has been shown to be p 2 -complete. In order to better understand the sources of complexity (and to identify tractable fragments), we focus on arguments given over formulae in which the allowed connectives are taken from certain sets of Boolean functions. We provide a complexity classification for four different decision problems (existence of a support, checking the validity of an argument, relevance and dispensability) with respect to all possible sets of Boolean functions. Moreover, we make use of a general schema to enumerate all arguments to show that certain restricted fragments permit polynomial delay. Finally, we give a classification also in terms of counting complexity.

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“…In fact it can be shown that, for this constraint language Γ, the set Γ consists of precisely those Boolean relations (of any arity) that can be expressed as a conjunction of unary or binary Boolean relations [81,87]. This is equivalent to saying that the constraint language Γ expresses precisely this set of relations.…”

Section: Example 15 Consider the Boolean Constraint Languagementioning

confidence: 99%

The Complexity of Constraint Languages

Cohen¹,

Jeavons²

2006

Foundations of Artificial Intelligence

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One of the most fundamental challenges in constraint programming is to understand the computational complexity of problems involving constraints. It has been shown that the class of all constraint satisfaction problem instances is NP-hard [71], so it is unlikely that efficient general-purpose algorithms exist for solving all forms of constraint problem. However, in many practical applications the instances that arise have special forms that enable them to be solved more efficiently [11,25,69,82].One way in which this occurs is that there is some special structure in the way that the constraints overlap and intersect each other. The natural theory for discussing the structure of such interaction between constraints is the mathematical theory of hypergraphs. Much work has been done in this area, and many tractable classes of constraint problems have been identified based on structural properties (see Chapter 5). There are strong parallels between this work and similar investigations into the structure of so-called conjunctive queries in relational databases [41,58].Another way in which constraint problems can be defined which are easier to solve than in the general case is when the types of constraints are limited. The natural theory for discussing the properties of constraint types is the mathematical theory of relations and their associated algebras. Again considerable progress has been made in this investigation over the past few years. For example, a complete characterisation of tractable constraint types is now known for both 2-element domains [85] and 3-element domains [14]. In addition, a number of novel efficient algorithms have been developed for solving particular types of constraint problems over both finite and infinite domains [3,8,16,25,26,28,63].In this chapter we will focus on the second approach. That is, we will investigate how the complexity of solving constraint problems varies with the types of constraints which are allowed. One fundamental open research problem in this area is to characterise exactly which types of constraints give rise to constraint problems which can be solved

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The Two-Valued Iterative Systems of Mathematical Logic. (AM-5)

Cited by 463 publications

References 0 publications

Exploring the Limits of Subadditive Approaches: Parallels between Optimization and Complexity Theory

Exploring the Limits of Subadditive Approaches: Parallels between Optimization and Complexity Theory

Complexity of logic-based argumentation in Post's framework

The Complexity of Constraint Languages

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