21 research outputs found
The Complexity of Reasoning with Boolean Modal Logics: Extended Version
Since Modal Logics are an extension of Propositional Logic, they provide Boolean operators for constructing complex formulae. However, most Modal Logics do not admit Boolean operators for constructing complex modal parameters to be used in the box and diamond operators. This asymmetry is not present in Boolean Modal Logics, in which box and diamond quantify over arbitrary Boolean combinations of atomic model parameters.This is an extended version of the article in: Advances in Modal Logic (AiML), Volume
Complexity of Polyadic Boolean Modal Logics: Model Checking and Satisfiability
We study the computational complexity of model checking and satisfiability problems of polyadic modal logics extended with permutations and Boolean operators on accessibility relations. First, we show that the combined complexity of the model checking problem for the resulting logic is PTime-complete. Secondly, we show that the satisfiability problem of polyadic modal logic extended with negation on accessibility relations is ExpTime-complete. Finally, we show that the satisfiability problem of polyadic modal logic with permutations and Boolean operators on accessibility relations is ExpTime-complete, under the assumption that both the number of accessibility relations that can be used and their arities are bounded by a constant. If NExpTime is not contained in ExpTime, then this assumption is necessary, since already the satisfiability problem of modal logic extended with Boolean operators on accessibility relations is NExpTime-hard
One-dimensional fragment of first-order logic
We introduce a novel decidable fragment of first-order logic. The fragment is
one-dimensional in the sense that quantification is limited to applications of
blocks of existential (universal) quantifiers such that at most one variable
remains free in the quantified formula. The fragment is closed under Boolean
operations, but additional restrictions (called uniformity conditions) apply to
combinations of atomic formulae with two or more variables. We argue that the
notions of one-dimensionality and uniformity together offer a novel perspective
on the robust decidability of modal logics. We also establish that minor
modifications to the restrictions of the syntax of the one-dimensional fragment
lead to undecidable formalisms. Namely, the two-dimensional and non-uniform
one-dimensional fragments are shown undecidable. Finally, we prove that with
regard to expressivity, the one-dimensional fragment is incomparable with both
the guarded negation fragment and two-variable logic with counting. Our proof
of the decidability of the one-dimensional fragment is based on a technique
involving a direct reduction to the monadic class of first-order logic. The
novel technique is itself of an independent mathematical interest
A system of relational syllogistic incorporating full Boolean reasoning
We present a system of relational syllogistic, based on classical
propositional logic, having primitives of the following form:
Some A are R-related to some B;
Some A are R-related to all B;
All A are R-related to some B;
All A are R-related to all B.
Such primitives formalize sentences from natural language like `All students
read some textbooks'. Here A and B denote arbitrary sets (of objects), and R
denotes an arbitrary binary relation between objects. The language of the logic
contains only variables denoting sets, determining the class of set terms, and
variables denoting binary relations between objects, determining the class of
relational terms. Both classes of terms are closed under the standard Boolean
operations. The set of relational terms is also closed under taking the
converse of a relation. The results of the paper are the completeness theorem
with respect to the intended semantics and the computational complexity of the
satisfiability problem.Comment: Available at
http://link.springer.com/article/10.1007/s10849-012-9165-
Automated Reasoning over Deontic Action Logics with Finite Vocabularies
In this paper we investigate further the tableaux system for a deontic action
logic we presented in previous work. This tableaux system uses atoms (of a
given boolean algebra of action terms) as labels of formulae, this allows us to
embrace parallel execution of actions and action complement, two action
operators that may present difficulties in their treatment. One of the
restrictions of this logic is that it uses vocabularies with a finite number of
actions. In this article we prove that this restriction does not affect the
coherence of the deduction system; in other words, we prove that the system is
complete with respect to language extension. We also study the computational
complexity of this extended deductive framework and we prove that the
complexity of this system is in PSPACE, which is an improvement with respect to
related systems.Comment: In Proceedings LAFM 2013, arXiv:1401.056
Type-elimination-based reasoning for the description logic SHIQbs using decision diagrams and disjunctive datalog
We propose a novel, type-elimination-based method for reasoning in the
description logic SHIQbs including DL-safe rules. To this end, we first
establish a knowledge compilation method converting the terminological part of
an ALCIb knowledge base into an ordered binary decision diagram (OBDD) which
represents a canonical model. This OBDD can in turn be transformed into
disjunctive Datalog and merged with the assertional part of the knowledge base
in order to perform combined reasoning. In order to leverage our technique for
full SHIQbs, we provide a stepwise reduction from SHIQbs to ALCIb that
preserves satisfiability and entailment of positive and negative ground facts.
The proposed technique is shown to be worst case optimal w.r.t. combined and
data complexity and easily admits extensions with ground conjunctive queries.Comment: 38 pages, 3 figures, camera ready version of paper accepted for
publication in Logical Methods in Computer Scienc
Quine’s Fluted Fragment is Non-elementary
We study the fluted fragment, a decidable fragment of first-order logic with an unbounded number of variables, originally identified by W.V. Quine. We show that the satisfiability problem for this fragment has non-elementary complexity, thus refuting an earlier published claim by W.C. Purdy that it is in NExpTime. More precisely, we consider, for all m greater than 1, the intersection of the fluted fragment and the m-variable fragment of first-order logic. We show that this sub-fragment forces (m/2)-tuply exponentially large models, and that its satisfiability problem is (m/2)-NExpTime-hard. We round off by using a corrected version of Purdy\u27s construction to show that the m-variable fluted fragment has the m-tuply exponential model property, and that its satisfiability problem is in m-NExpTime
Encapsulating deontic and branching time specifications
In this paper, we investigate formal mechanisms to enable designers to decompose specifications (stated in a given logic) into several interacting components in such a way that the composition of these components preserves their encapsulation and internal non-determinism. The preservation of encapsulation (or locality) enables a modular form of reasoning over specifications, while the conservation of the internal non-determinism is important to guarantee that the branching time properties of components are not lost when the entire system is obtained. The basic ideas come from the work of Fiadeiro and Maibaum where notions from category theory are used to structure logical specifications. As the work of Fiadeiro and Maibaum is stated in a linear temporal logic, here we investigate how to extend these notions to a branching time logic, which can be used to reason about systems where non-determinism is present. To illustrate the practical applications of these ideas, we introduce deontic operators in our logic and we show that the modularization of specifications also allows designers to maintain the encapsulation of deontic prescriptions; this is in particular useful to reason about fault-tolerant systems, as we demonstrate with a small example.Fil: Castro, Pablo Francisco. Universidad Nacional de Río Cuarto; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba; ArgentinaFil: Maibaum, Thomas S. E.. Mc Master University; Canad