On connected Boolean functions

Ekin, Oya; Hammer, Peter L.; Kogan, Alexander

doi:10.1016/s0166-218x(99)00098-0

Cited by 11 publications

(10 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We now observe that there is a shortest path between a 0 and c 0 , which is contained in the union of the two cubes. This is a special case of a more general result of Ekin et al [6].…”

Section: Proof Of the Splitting Lemma (Lemma 10)supporting

confidence: 53%

On k-Term DNF with the Largest Number of Prime Implicants

Sloan¹,

Szörényi²,

Turán³

2008

SIAM J. Discrete Math.

View full text Add to dashboard Cite

show abstract

“…We now observe that there is a shortest path between a 0 and c 0 , which is contained in the union of the two cubes. This is a special case of a more general result of Ekin et al [6].…”

Section: Proof Of the Splitting Lemma (Lemma 10)supporting

confidence: 53%

On k-Term DNF with the Largest Number of Prime Implicants

Sloan¹,

Szörényi²,

Turán³

2008

SIAM J. Discrete Math.

View full text Add to dashboard Cite

show abstract

“…This lemma is immediate from the result in Ekin et al [1999] that prime implicates of acyclic Horn theories conflict on at most one atom.…”

Section: Unconstrained Explanations From Acyclic Horn Cnfsmentioning

confidence: 84%

On computing all abductive explanations from a propositional Horn theory

Eiter

Makino

2007

J. ACM

View full text Add to dashboard Cite

Abduction is a fundamental mode of reasoning with applications in many areas of AI and Computer Science. The computation of abductive explanations is an important computational problem, which is at the core of early systems such as the ATMS and Clause Management Systems and is intimately related to prime implicate generation in propositional logic. Many algorithms have been devised for computing some abductive explanation, and the complexity of the problem has been well studied. However, little attention has been paid to the problem of computing multiple explanations, and in particular all explanations for an abductive query. We fill this gap and consider the computation of all explanations of an abductive query from a propositional Horn theory, or of a polynomial subset of them. Our study pays particular attention to the form of the query, ranging from a literal to a compound formula, to whether explanations are based on a set of abducible literals and to the representation of the Horn theory, either by a Horn conjunctive normal form (CNF) or model-based in terms of its characteristic models. For these combinations, we present either tractability results in terms of polynomial total-time algorithms, intractability results in terms of Dedicated to the memory of Marco Cadoli. Most of the results in this article appeared in preliminary form in nonexistence of such algorithms (unless P = NP), or semi-tractability results in terms of solvability in quasi-polynomial time, established by polynomial-time equivalence to the problem of dualizing a monotone CNF expression. Our results complement previous results in the literature, and refute a longstanding conjecture by Selman and Levesque. They elucidate the complexity of generating all abductive explanations and shed light on related problems such as generating sets of restricted prime implicates of a Horn theory. The algorithms for tractable cases can be readily applied for generating a polynomial subset of explanations in polynomial time.

show abstract

“…For example, disjunctive normal forms with special connectivity properties were studied by Ekin et al already in 1997 for their "important role in problems appearing in various areas including in particular discrete optimization, machine learning, automated reasoning, etc." [EHK99].…”

Section: Future Directionsmentioning

confidence: 99%

The Connectivity of Boolean Satisfiability: Dichotomies for Formulas and Circuits

Schwerdtfeger

2015

Theory Comput Syst

View full text Add to dashboard Cite

For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. In 2006, Gopalan et al. studied connectivity properties of the solution graph and related complexity issues for CSPs [GKMP09], motivated mainly by research on satisfiability algorithms and the satisfiability threshold. They proved dichotomies for the diameter of connected components and for the complexity of the st-connectivity question, and conjectured a trichotomy for the connectivity question. Recently, we were able to establish the trichotomy [Sch13].Here, we consider connectivity issues of satisfiability problems defined by Boolean circuits and propositional formulas that use gates, resp. connectives, from a fixed set of Boolean functions. We obtain dichotomies for the diameter and the two connectivity problems: On one side, the diameter is linear in the number of variables, and both problems are in P, while on the other side, the diameter can be exponential, and the problems are PSPACE-complete. For partially quantified formulas, we show an analogous dichotomy.

show abstract

On connected Boolean functions

Cited by 11 publications

References 6 publications

On k-Term DNF with the Largest Number of Prime Implicants

On k-Term DNF with the Largest Number of Prime Implicants

On computing all abductive explanations from a propositional Horn theory

The Connectivity of Boolean Satisfiability: Dichotomies for Formulas and Circuits

Contact Info

Product

Resources

About