7,795 research outputs found
On the Complexity of Boolean Unification
Unification modulo the theory of Boolean algebras has been investigated by several autors. Nevertheless, the exact complexity of the decision problem for unification with constants and general unification was not known. In this research note, we show that the decision problem is complete for unification with constants and PSPACE-complete for general unification. In contrast, the decision problem for elementary unification (where the terms to be unified contain only symbols of the signature of Boolean algebras) is 'only' NP-complete
Set Unification
The unification problem in algebras capable of describing sets has been
tackled, directly or indirectly, by many researchers and it finds important
applications in various research areas--e.g., deductive databases, theorem
proving, static analysis, rapid software prototyping. The various solutions
proposed are spread across a large literature. In this paper we provide a
uniform presentation of unification of sets, formalizing it at the level of set
theory. We address the problem of deciding existence of solutions at an
abstract level. This provides also the ability to classify different types of
set unification problems. Unification algorithms are uniformly proposed to
solve the unification problem in each of such classes.
The algorithms presented are partly drawn from the literature--and properly
revisited and analyzed--and partly novel proposals. In particular, we present a
new goal-driven algorithm for general ACI1 unification and a new simpler
algorithm for general (Ab)(Cl) unification.Comment: 58 pages, 9 figures, 1 table. To appear in Theory and Practice of
Logic Programming (TPLP
Undecidability of the unification and admissibility problems for modal and description logics
We show that the unification problem `is there a substitution instance of a
given formula that is provable in a given logic?' is undecidable for basic
modal logics K and K4 extended with the universal modality. It follows that the
admissibility problem for inference rules is undecidable for these logics as
well. These are the first examples of standard decidable modal logics for which
the unification and admissibility problems are undecidable. We also prove
undecidability of the unification and admissibility problems for K and K4 with
at least two modal operators and nominals (instead of the universal modality),
thereby showing that these problems are undecidable for basic hybrid logics.
Recently, unification has been introduced as an important reasoning service for
description logics. The undecidability proof for K with nominals can be used to
show the undecidability of unification for boolean description logics with
nominals (such as ALCO and SHIQO). The undecidability proof for K with the
universal modality can be used to show that the unification problem relative to
role boxes is undecidable for Boolean description logic with transitive roles,
inverse roles, and role hierarchies (such as SHI and SHIQ)
Carnap: an Open Framework for Formal Reasoning in the Browser
This paper presents an overview of Carnap, a free and open framework for the development of formal reasoning applications. Carnap’s design emphasizes flexibility, extensibility, and rapid prototyping. Carnap-based applications are written in Haskell, but can be compiled to JavaScript to run in standard web browsers. This combination of features makes Carnap ideally suited for educational applications, where ease-of-use is crucial for students and adaptability to different teaching strategies and classroom needs is crucial for instructors. The paper describes Carnap’s implementation, along with its current and projected pedagogical applications
Efficient Groundness Analysis in Prolog
Boolean functions can be used to express the groundness of, and trace
grounding dependencies between, program variables in (constraint) logic
programs. In this paper, a variety of issues pertaining to the efficient Prolog
implementation of groundness analysis are investigated, focusing on the domain
of definite Boolean functions, Def. The systematic design of the representation
of an abstract domain is discussed in relation to its impact on the algorithmic
complexity of the domain operations; the most frequently called operations
should be the most lightweight. This methodology is applied to Def, resulting
in a new representation, together with new algorithms for its domain operations
utilising previously unexploited properties of Def -- for instance,
quadratic-time entailment checking. The iteration strategy driving the analysis
is also discussed and a simple, but very effective, optimisation of induced
magic is described. The analysis can be implemented straightforwardly in Prolog
and the use of a non-ground representation results in an efficient, scalable
tool which does not require widening to be invoked, even on the largest
benchmarks. An extensive experimental evaluation is givenComment: 31 pages To appear in Theory and Practice of Logic Programmin
Enhanced sharing analysis techniques: a comprehensive evaluation
Sharing, an abstract domain developed by D. Jacobs and A. Langen for the analysis of logic
programs, derives useful aliasing information. It is well-known that a commonly used core
of techniques, such as the integration of Sharing with freeness and linearity information, can
significantly improve the precision of the analysis. However, a number of other proposals for
refined domain combinations have been circulating for years. One feature that is common
to these proposals is that they do not seem to have undergone a thorough experimental
evaluation even with respect to the expected precision gains.
In this paper we experimentally
evaluate: helping Sharing with the definitely ground variables found using Pos, the domain
of positive Boolean formulas; the incorporation of explicit structural information; a full
implementation of the reduced product of Sharing and Pos; the issue of reordering the
bindings in the computation of the abstract mgu; an original proposal for the addition of
a new mode recording the set of variables that are deemed to be ground or free; a refined
way of using linearity to improve the analysis; the recovery of hidden information in the
combination of Sharing with freeness information. Finally, we discuss the issue of whether
tracking compoundness allows the computation of more sharing information
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