Pseudo-Boolean Constraints from a Knowledge Representation Perspective

Berre, Daniel Le; Marquis, Pierre; Mengel, Stefan; Wallon, Romain

doi:10.24963/ijcai.2018/261

Cited by 10 publications

(13 citation statements)

References 7 publications

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“…The diagram in Figure 1, summarizing the succinctness relations of many commonly used propositional languages, appeared in (Darwiche & Marquis, 2002). It is amended here by the results from (Le Berre et al, 2018) dealing with the PBC and CARD languages. The main aim of the next section is to add the languages SL < and SL into the framed part of the diagram in Figure 1 (Darwiche & Marquis, 2002) combined with the results from (Le Berre et al, 2018).…”

Section: Definitions and Notationmentioning

confidence: 99%

“…In (Le Berre, Marquis, Mengel, & Wallon, 2018) the authors included Pseudo-Boolean constraint (PBC) and Cardinality constraint (CARD) languages into KCM by showing succinctness relations among PBC, CARD, and languages already in the map, and by proving the complexity status of all queries and transformations introduced in (Darwiche & Marquis, 2002). In this paper, we aim at achieving exactly the same goal for switch-list representations.…”

Section: Introductionmentioning

confidence: 99%

“…Finding direct algorithms for sentential entailment and equivalence check which would avoid compilation into OBDDs may be a good research topic. Berre et al, 2018) and their polytime queries. means "satisfies" and • means "does not satisfy unless P=NP".…”

Section: Introductionmentioning

confidence: 99%

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Properties of Switch-List Representations of Boolean Functions

Chromý

Čepek

2020

jair

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In this paper, we focus on a less usual way to represent Boolean functions, namely on representations by switch-lists, which are closely related to interval representations. Given a truth table representation of a Boolean function f the switch-list representation of f is a list of Boolean vectors from the truth table which have a different function value than the preceding Boolean vector in the truth table. The main aim of this paper is to include this type of representation in the Knowledge Compilation Map by Darwiche and Marquis and to argue that switch-lists may in certain situations constitute a reasonable choice for a target language in knowledge compilation. First, we compare switch-list representations with a number of standard representations (such as CNF, DNF, and OBDD) with respect to their relative succinctness. As a by-product of this analysis, we also give a short proof of a longstanding open question proposed by Darwiche and Marquis, namely the incomparability of MODS (models) and PI (prime implicates) representations. Next, using the succinctness result between switch-lists and OBDDs, we develop a polynomial time compilation algorithm from switch-lists to OBDDs. Finally, we analyze which standard transformations and queries (those considered by Darwiche and Marquis) can be performed in polynomial time with respect to the size of the input if the input knowledge is represented by a switch-list. We show that this collection is very broad and the combination of polynomial time transformations and queries is quite unique. Some of the queries can be answered directly using the switch-list input, others require a compilation of the input to OBDD representations which are then used to answer the queries.

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Section: Definitions and Notationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Properties of Switch-List Representations of Boolean Functions

Chromý

Čepek

2020

jair

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“…Yet, adapting proofs given in [2,14] for OBDD to DNNF is not straightforward, thus our proof of Theorem 1 bears very little resemblance. It has been shown in [18] that there exist sets of PB-constraints such that the whole set (so a conjunction of PB-constraints) requires exponential size DNNF to represent. Our result is a generalisation to single PB-constraints.…”

Section: Theoremmentioning

confidence: 99%

A Lower Bound on DNNF Encodings of Pseudo-Boolean Constraints

Colnet

2020

Theory and Applications of Satisfiability Testing – SAT 2020

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Two major considerations when encoding pseudo-Boolean (PB) constraints into SAT are the size of the encoding and its propagation strength, that is, the guarantee that it has a good behaviour under unit propagation. Several encodings with propagation strength guarantees rely upon prior compilation of the constraints into DNNF (decomposable negation normal form), BDD (binary decision diagram), or some other sub-variants. However it has been shown that there exist PB-constraints whose ordered BDD (OBDD) representations, and thus the inferred CNF encodings, all have exponential size. Since DNNFs are more succinct than OBDDs, preferring encodings via DNNF to avoid size explosion seems a legitimate choice. Yet in this paper, we prove the existence of PB-constraints whose DNNFs all require exponential size.

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“…In [Le Berre et al, 2018] the autors included Pseudo-Boolean constraint (PBC) and Cardinality constraint (CARD) languages into KCM by showing succinctness relations and the complexity of all queries and transformations introduced in [Darwiche and Marquis, 2002]. In this paper we aim at achieving exactly the same goal for the SL language.…”

Section: Introductionmentioning

confidence: 99%

Switch-List Representations in a Knowledge Compilation Map

Čepek¹,

Chromý²

2020

Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence

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In this paper we focus on a less usual way to represent Boolean functions, namely on representations by switch-lists. Given a truth table representation of a Boolean function f the switch-list representation (SLR) of f is a list of Boolean vectors from the truth table which have a different function value than the preceding Boolean vector in the truth table. The main aim of this paper is to include the language SL of all SLR in the Knowledge Compilation Map [Darwiche and Marquis, 2002] and to argue, that SL may in certain situations constitute a reasonable choice for a target language in knowledge compilation. First we compare SL with a number of standard representation languages (such as CNF, DNF, and OBDD) with respect to their relative succinctness. As a by-product of this analysis we also give a short proof of a long standing open question from [Darwiche and Marquis, 2002], namely the incomparability of MODS (models) and PI (prime implicates) languages. Next we analyze which standard transformations and queries (those considered in [Darwiche and Marquis, 2002] can be performed in poly-time with respect to the size of the input SLR. We show that this collection is quite broad and the combination of poly-time transformations and queries is quite unique.

show abstract

Pseudo-Boolean Constraints from a Knowledge Representation Perspective

Cited by 10 publications

References 7 publications

Properties of Switch-List Representations of Boolean Functions

Properties of Switch-List Representations of Boolean Functions

A Lower Bound on DNNF Encodings of Pseudo-Boolean Constraints

Switch-List Representations in a Knowledge Compilation Map

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