3,194 research outputs found
The Satisfiability Problem for Boolean Set Theory with a Choice Correspondence
Given a set U of alternatives, a choice (correspondence) on U is a
contractive map c defined on a family Omega of nonempty subsets of U.
Semantically, a choice c associates to each menu A in Omega a nonempty subset
c(A) of A comprising all elements of A that are deemed selectable by an agent.
A choice on U is total if its domain is the powerset of U minus the empty set,
and partial otherwise. According to the theory of revealed preferences, a
choice is rationalizable if it can be retrieved from a binary relation on U by
taking all maximal elements of each menu. It is well-known that rationalizable
choices are characterized by the satisfaction of suitable axioms of
consistency, which codify logical rules of selection within menus. For
instance, WARP (Weak Axiom of Revealed Preference) characterizes choices
rationalizable by a transitive relation. Here we study the satisfiability
problem for unquantified formulae of an elementary fragment of set theory
involving a choice function symbol c, the Boolean set operators and the
singleton, the equality and inclusion predicates, and the propositional
connectives. In particular, we consider the cases in which the interpretation
of c satisfies any combination of two specific axioms of consistency, whose
conjunction is equivalent to WARP. In two cases we prove that the related
satisfiability problem is NP-complete, whereas in the remaining cases we obtain
NP-completeness under the additional assumption that the number of choice terms
is constant.Comment: In Proceedings GandALF 2017, arXiv:1709.01761. "extended" version at
arXiv:1708.0612
The satisfiability problem for Boolean set theory with a choice correspondence (Extended version)
Given a set of alternatives, a choice (correspondence) on is a
contractive map defined on a family of nonempty subsets of .
Semantically, a choice associates to each menu a nonempty
subset comprising all elements of that are deemed
selectable by an agent. A choice on is total if its domain is the powerset
of minus the empty set, and partial otherwise. According to the theory of
revealed preferences, a choice is rationalizable if it can be retrieved from a
binary relation on by taking all maximal elements of each menu. It is
well-known that rationalizable choices are characterized by the satisfaction of
suitable axioms of consistency, which codify logical rules of selection within
menus. For instance, WARP (Weak Axiom of Revealed Preference) characterizes
choices rationalizable by a transitive relation. Here we study the
satisfiability problem for unquantified formulae of an elementary fragment of
set theory involving a choice function symbol , the Boolean set
operators and the singleton, the equality and inclusion predicates, and the
propositional connectives. In particular, we consider the cases in which the
interpretation of satisfies any combination of two specific axioms
of consistency, whose conjunction is equivalent to WARP. In two cases we prove
that the related satisfiability problem is NP-complete, whereas in the
remaining cases we obtain NP-completeness under the additional assumption that
the number of choice terms is constant
An exactly solvable random satisfiability problem
We introduce a new model for the generation of random satisfiability
problems. It is an extension of the hyper-SAT model of Ricci-Tersenghi, Weigt
and Zecchina, which is a variant of the famous K-SAT model: it is extended to
q-state variables and relates to a different choice of the statistical
ensemble. The model has an exactly solvable statistic: the critical exponents
and scaling functions of the SAT/UNSAT transition are calculable at zero
temperature, with no need of replicas, also with exact finite-size corrections.
We also introduce an exact duality of the model, and show an analogy of
thermodynamic properties with the Random Energy Model of disordered spin
systems theory. Relations with Error-Correcting Codes are also discussed.Comment: 31 pages, 1 figur
Do Hard SAT-Related Reasoning Tasks Become Easier in the Krom Fragment?
Many reasoning problems are based on the problem of satisfiability (SAT).
While SAT itself becomes easy when restricting the structure of the formulas in
a certain way, the situation is more opaque for more involved decision
problems. We consider here the CardMinSat problem which asks, given a
propositional formula and an atom , whether is true in some
cardinality-minimal model of . This problem is easy for the Horn
fragment, but, as we will show in this paper, remains -complete (and
thus -hard) for the Krom fragment (which is given by formulas in
CNF where clauses have at most two literals). We will make use of this fact to
study the complexity of reasoning tasks in belief revision and logic-based
abduction and show that, while in some cases the restriction to Krom formulas
leads to a decrease of complexity, in others it does not. We thus also consider
the CardMinSat problem with respect to additional restrictions to Krom formulas
towards a better understanding of the tractability frontier of such problems
The Complexity of Satisfiability for Sub-Boolean Fragments of ALC
The standard reasoning problem, concept satisfiability, in the basic
description logic ALC is PSPACE-complete, and it is EXPTIME-complete in the
presence of unrestricted axioms. Several fragments of ALC, notably logics in
the FL, EL, and DL-Lite family, have an easier satisfiability problem;
sometimes it is even tractable. All these fragments restrict the use of Boolean
operators in one way or another. We look at systematic and more general
restrictions of the Boolean operators and establish the complexity of the
concept satisfiability problem in the presence of axioms. We separate tractable
from intractable cases.Comment: 17 pages, accepted (in short version) to Description Logic Workshop
201
(Un)Decidability Results for Word Equations with Length and Regular Expression Constraints
We prove several decidability and undecidability results for the
satisfiability and validity problems for languages that can express solutions
to word equations with length constraints. The atomic formulas over this
language are equality over string terms (word equations), linear inequality
over the length function (length constraints), and membership in regular sets.
These questions are important in logic, program analysis, and formal
verification. Variants of these questions have been studied for many decades by
mathematicians. More recently, practical satisfiability procedures (aka SMT
solvers) for these formulas have become increasingly important in the context
of security analysis for string-manipulating programs such as web applications.
We prove three main theorems. First, we give a new proof of undecidability
for the validity problem for the set of sentences written as a forall-exists
quantifier alternation applied to positive word equations. A corollary of this
undecidability result is that this set is undecidable even with sentences with
at most two occurrences of a string variable. Second, we consider Boolean
combinations of quantifier-free formulas constructed out of word equations and
length constraints. We show that if word equations can be converted to a solved
form, a form relevant in practice, then the satisfiability problem for Boolean
combinations of word equations and length constraints is decidable. Third, we
show that the satisfiability problem for quantifier-free formulas over word
equations in regular solved form, length constraints, and the membership
predicate over regular expressions is also decidable.Comment: Invited Paper at ADDCT Workshop 2013 (co-located with CADE 2013
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