Boolean unification — The story so far

Martin, Ursula; Nipkow, Tobias

doi:10.1016/s0747-7171(89)80013-6

Cited by 71 publications

(50 citation statements)

References 6 publications

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“…In particular, we show there that the theory of Boolean algebras is Gaussian. This result is obtained as a consequence of results for unification in Boolean rings [21].…”

Section: Introductionsupporting

confidence: 54%

“…We will use the following general result, adapted from [21], on the computation of most general BR-unifiers based on Löwnheim's formula. fresh variables z, the substitution {y → z + t(c, z) * (z + r(c))} is a most general BR-unifier of t(c, y) ≈ 0.…”

Section: Boolean Solved Formsmentioning

confidence: 99%

“…Let V be such an assignment and for every term u let V The next lemma is related to Boole's method for computing most general BR-unifiers [21].…”

Section: Boolean Solved Formsmentioning

confidence: 99%

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A New Combination Procedure for the Word Problem That Generalizes Fusion Decidability Results in Modal Logics

Baader

Ghilardi

Tinelli

2004

Automated Reasoning

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Previous results for combining decision procedures for the word problem in the nondisjoint case do not apply to equational theories induced by modal logics-which are not disjoint for sharing the theory of Boolean algebras. Conversely, decidability results for the fusion of modal logics are strongly tailored towards the special theories at hand, and thus do not generalize to other types of equational theories.In this paper, we present a new approach for combining decision procedures for the word problem in the non-disjoint case that applies to equational theories induced by modal logics, but is not restricted to them. The known fusion decidability results for modal logics are instances of our approach. However, even for equational theories induced by modal logics our results are more general since they are not restricted to so-called normal modal logics.

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“…In particular, we show there that the theory of Boolean algebras is Gaussian. This result is obtained as a consequence of results for unification in Boolean rings [21].…”

Section: Introductionsupporting

confidence: 54%

Section: Boolean Solved Formsmentioning

confidence: 99%

See 1 more Smart Citation

A New Combination Procedure for the Word Problem That Generalizes Fusion Decidability Results in Modal Logics

Baader

Ghilardi

Tinelli

2004

Automated Reasoning

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“…The definition of a projective formula translates into symbolic logic the statement "the free algebra over V ar(A) divided by the congruence generated by the equation A = is projective": in fact, since projective algebras can be equivalently defined as retract of free algebras, being a retract means precisely that there is a projective unifier. On the other hand, projective unifiers corresponds to the so-called "reproductive solutions" known from the research on Boolean unification [50] and are quite close to the "transparent unifiers" introduced by Wronski [74].…”

Section: Boolean Unification and The Universal Modalitymentioning

confidence: 79%

“…The formula (6) is due to Löwenheim: it is precisely the Löwenheim formula taken from [50] (we slightly modified it because we do not use Boolean ring notation and because the standard meaning of our unification problems is A ↔ ? rather than A ↔ ?…”

Section: Boolean Unification and The Universal Modalitymentioning

confidence: 99%

Unification in modal and description logics

Baader

Ghilardi

2010

Logic Journal of IGPL

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Unification Algorithms Cannot Be Combined in Polynomial Time

Hermann

Kolaitis

2000

Information and Computation

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We establish that there is no polynomial-time general combination algorithm for uni cation in nitary equational theories, unless the complexity class #P of counting problems is contained in the class FP of function problems solvable in polynomial-time. The prevalent view in complexity theory is that such a collapse is extremely unlikely for a number of reasons, including the fact that the containment of #P in FP implies that P = NP. Our main result is obtained by establishing the intractrability of the counting problem for general AG-uni cation, where AG is the equational theory of Abelian groups. Speci cally, we show that computing the cardinality of a minimal complete set of uni ers for general AG-uni cation is a #P-hard problem. In contrast, AG-uni cation with constants is solvable in polynomial time. Since an algorithm for general AG-uni cation can be obtained as a combination of a polynomialtime algorithm for AG-uni cation with constants and a polynomial-time algorithm for syntactic uni cation, it follows that no polynomial-time general combination algorithm exists, unless #P is contained in FP.

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Boolean unification — The story so far

Cited by 71 publications

References 6 publications

A New Combination Procedure for the Word Problem That Generalizes Fusion Decidability Results in Modal Logics

A New Combination Procedure for the Word Problem That Generalizes Fusion Decidability Results in Modal Logics

Unification in modal and description logics

Unification Algorithms Cannot Be Combined in Polynomial Time

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