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Pseudo-Boolean Constraints from a Knowledge Representation Perspective

Daniel Le Berre1,2 Pierre Marquis1,2,3 Stefan Mengel1 Romain Wallon1,2 July 18, 2018

1CRIL-CNRS UMR 8188, Lens, France 2Universit´

e d’Artois

3Institut Universitaire de France

Linear Pseudo-Boolean Constraints

A linear pseudo-Boolean constraint is of the form: ÿ

j

ajlj ⊲ k where:

• @j, aj P Z
• @j, lj is a literal (i.e. a boolean value)
• ⊲ P tă, ď, “, ě, ąu
• k P Z is the degree (threshold) of the constraint

Example: 3a ´ 2b ` c ´ 4d ď ´1

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PBC and CARD

Normalized pseudo-Boolean constraints are of the form: ÿ

j

ajlj ě k @j, aj P N, k P N@i ‰ j, li ‰ ljli ‰ lj Example: 3a ` 2b ` c ě 3 Cardinality constraints are of the form: ÿ

j

lj ě k k P N Example: a ` b ` c ě 2 A formula of PBC (resp. CARD) is a conjunction of normalized constraints (resp. cardinality constraints)

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Why using PBC and CARD?

Representing a problem with pseudo-Boolean constraints requires less constraints than CNF [Dixon, 2004] The cutting-planes proof system, used together with PBC and CARD, is more powerful than resolution (inconsistent pigeonhole formulae have a polynomial size refutation proof) With PBC or CARD, modeling problems is more natural: subset-sum and knapsack require two normalized pseudo-Boolean constraints to be modeled

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Why using PBC and CARD?

Representing a problem with pseudo-Boolean constraints requires less constraints than CNF [Dixon, 2004] The cutting-planes proof system, used together with PBC and CARD, is more powerful than resolution (inconsistent pigeonhole formulae have a polynomial size refutation proof) With PBC or CARD, modeling problems is more natural: subset-sum and knapsack require two normalized pseudo-Boolean constraints to be modeled Is there a subset of t9, 4, 3, 2, 2, 1, 1u which sums to 12?

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Why using PBC and CARD?

Representing a problem with pseudo-Boolean constraints requires less constraints than CNF [Dixon, 2004] The cutting-planes proof system, used together with PBC and CARD, is more powerful than resolution (inconsistent pigeonhole formulae have a polynomial size refutation proof) With PBC or CARD, modeling problems is more natural: subset-sum and knapsack require two normalized pseudo-Boolean constraints to be modeled Is there a subset of t9, 4, 3, 2, 2, 1, 1u which sums to 12? 9a ` 4b ` 3c ` 2d ` 2e ` f ` g “ 12

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Why using PBC and CARD?

Representing a problem with pseudo-Boolean constraints requires less constraints than CNF [Dixon, 2004] The cutting-planes proof system, used together with PBC and CARD, is more powerful than resolution (inconsistent pigeonhole formulae have a polynomial size refutation proof) With PBC or CARD, modeling problems is more natural: subset-sum and knapsack require two normalized pseudo-Boolean constraints to be modeled Is there a subset of t9, 4, 3, 2, 2, 1, 1u which sums to 12? 9a ` 4b ` 3c ` 2d ` 2e ` f ` g “ 12 9a ` 4b ` 3c ` 2d ` 2e ` f ` g ě 12 9¯ a ` 4¯ b ` 3¯ c ` 2 ¯ d ` 2¯ e ` ¯ f ` ¯ g ě 10

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Why using PBC and CARD?

Representing a problem with pseudo-Boolean constraints requires less constraints than CNF [Dixon, 2004] The cutting-planes proof system, used together with PBC and CARD, is more powerful than resolution (inconsistent pigeonhole formulae have a polynomial size refutation proof) With PBC or CARD, modeling problems is more natural: subset-sum and knapsack require two normalized pseudo-Boolean constraints to be modeled Can these languages be good alternatives for knowledge representation? Let us answer this by looking at PBC and CARD from a knowledge representation perspective

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A Knowledge Compilation Map [DM02]

Lots of languages exist to represent propositional information, and choosing one requires to identify which operations you want to perform with the information To help in making this choice, Adnan Darwiche and Pierre Marquis proposed in 2002 a knowledge compilation map Three criteria are taken into account to identify which language is the best to use w.r.t. the wanted operations

• succinctness
• queries
• transformations

The study proposed in the map only considers equivalent formulae: adding new variables is not permitted

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Succinctness: some results

Succinctness captures the ability of a language to represent information using little space Let us consider two examples:

• PBC and CARD are strictly more succinct than CNF, since a clause

is a cardinality constraint of degree 1

• PBC and CARD are less succinct than OBDD, since representing the

parity function requires an exponential number of pseudo-Boolean constraints, whereas a polynomial size OBDD is enough

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Queries: some (not surprising) results

Given a formula, does this formula has a certain property? Let us consider two examples:

• PBC and CARD do not satisfy consistency (CO) unless P “ NP,
• therwise CNF would
• PBC and CARD both satisfy validity (VA), since checking the

validity of a formula is checking that all the degrees are equal to 0

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Queries: all the results (compared to CNF)

CO VA CE IM EQ SE CT ME CNF ˝

• ˝
• ˝

˝ ˝ ˝ CARD ˝

• ˝
• ˝

˝ ˝ ˝ PBC ˝

• ˝
• ˝

˝ ˝ ˝ CO (COnsistency) VA (VAlidity) CE (Clausal Entailment) IM (IMplication) EQ (EQuivalence) SE (Sentential Entailment) CT (CounTing) ME (Model Enumeration)

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Transformations: some results

Given a formula, can this formula be transformed in a certain way? Let us consider two examples:

• PBC and CARD both satisfy conditionning (CD), since it just

consists in replacing a variable by a boolean constant

• PBC and CARD do not satisfy bounded disjunction (_BC), because

representing an inequality (‰) requires exponentially many constraints.

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Transformations: all the results (compared to CNF)

CD FO SFO ^C ^BC _C _BC C CNF

• ˝
• CARD
• ˝

?

• PBC
• CD (ConDitioning)

SFO (Singleton FOrgetting) FO (FOrgetting) ^C (Closure under ^) ^BC (Bounded Closure under ^) _C (Closure under _) _BC (Bounded Closure under _) C (Closure under )

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Conclusion

• Pseudo-Boolean constraints have been studied from a knowledge

representation perspective

• PBC and CARD are strictly more succinct than CNF
• Less transformations are tractable, and no query is gained compared

to CNF

• Not good compilation languages, but interesting alternatives to CNF

when space matters

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Pseudo-Boolean Constraints from a Knowledge Representation Perspective

Daniel Le Berre1,2 Pierre Marquis1,2,3 Stefan Mengel1 Romain Wallon1,2 July 18, 2018

1CRIL-CNRS UMR 8188, Lens, France 2Universit´

e d’Artois

3Institut Universitaire de France