Basic Hybrid Logic Research Articles

Completeness and termination for a Seligman-style tableau system

Proof systems for hybrid logic typically use @-operators to access information hidden behind modalities; this labelling approach lies at the heart of the best known hybrid resolution, natural deduction, and tableau systems. But there is another approach, which we have come to believe is conceptually clearer. We call this Seligman-style inference, as it was rst introduced and explored by Jerry Seligman in natural deduction [31] and sequent calculus [32] in the 1990s. The purpose of this paper is to introduce a Seligman-style tableau system, to prove its completeness, and to show how it can be made to terminate. The most obvious feature of Seligman-style systems is that they work with arbitrary formulas, not just statements prexed by @-operators. They do so by introducing machinery for switching to other proof contexts. We capture this idea in the setting of tableaus by introducing a rule called GoTo which allows us to \jump to a named world on a tableau branch. We rst develop a Seligman-style tableau system for basic hybrid logic and prove its completeness. We then prove termination of a restricted version of the system without resorting to loop checking, and show that the restrictions do not eect completeness. Both completeness and termination results are proved by explicit translations that transform tableaus in a standard labelled system into Seligman-style tableaus and vice-versa.

Hybrid logic with the difference modality for generalisations of graphs

We discuss recent work generalising the basic hybrid logic with the difference modality to any reasonable notion of transition. This applies equally to both subrelational transitions such as monotone neighbourhood frames or selection function models as well as those with more structure such as Markov chains and alternating temporal frames. We provide a generic canonical cut-free sequent system and a terminating proof-search strategy for the fragment without the difference modality but including the global modality.

An efficient approach to nominal equalities in hybrid logic tableaux

Basic hybrid logic extends modal logic with the possibility of naming worlds by means of a distinguished class of atoms (called nominals) and the so-called satisfaction operator, that allows one to state that a given formula holds at the world named a, for some nominal a. Hence, in particular, hybrid formulae include “equality” assertions, stating that two nominals are distinct names for the same world. The treatment of such nominal equalities in proof systems for hybrid logics may induce many redundancies. This paper introduces an internalized tableau system for basic hybrid logic, significantly reducing such redundancies. The calculus enjoys a strong termination property: tableau construction terminates without relying on any specific rule application strategy, and no loop-checking is needed. The treatment of nominal equalities specific of the proposed calculus is briefly compared to other approaches. Its practical advantages are demonstrated by empirical results obtained by use of implemented systems. Finally, it is briefly shown how to extend the calculus to include the global and converse modalities.

Using modal logics to express and check global graph properties

Graphs are among the most frequently used structures in Computer Science. Some of the properties that must be checked in many applications are connectivity, acyclicity and the Eulerian and Hamiltonian properties. In this work, we analyze how we can express these four properties with modal logics. This involves two issues: whether each of the modal languages under consideration has enough expressive power to describe these properties and how complex (computationally) it is to use these logics to actually test whether a given graph has some desired property. First, we show that these properties are not definable in a basic modal logic or in any bisimulation-invariant extension of it, like the modal μ-calculus. We then show that it is possible to express some of the above properties in a basic hybrid logic. Unfortunately, the Hamiltonian and Eulerian properties still cannot be efficiently checked. In a second attempt, we propose an extension of CTL∗ with nominals and show that the Hamiltonian property can be more efficiently checked in this logic than in the previous one. In a third attempt, we extend the basic hybrid logic with the ↓ operator and show that we can check the Hamiltonian property with optimal (NP) complexity in this logic. Finally, we tackle the Eulerian property in two different ways. First, we develop a generic method to express edge-related properties in hybrid logics and use it to express the Eulerian property. Second, we express a necessary and sufficient condition for the Eulerian property to hold using a graded modal logic.

A Logical Approach to Hamiltonian Graphs

Graphs are among the most frequently used structures in computer science. A lot of problems can be modelled using a graph and can then be solved by checking whether the graph satisfies some property. In this work, we are interested in how to use logical frameworks as a generic tool to express and efficiently check graph properties. In order to reason about this, we choose to analyze the Hamiltonian property and choose the family of modal logics as our framework. Our analysis has to deal with two central issues: whether each of the modal languages under consideration has enough expressive power to describe this property and how complex (computationally) it is to use these logics to actually test whether a given graph has this property. First, we show that this property is not definable in a basic modal logic or in any bisimulation-invariant extension of it, like the modal μ -calculus. We then show that it is possible to express it in a basic hybrid logic. Unfortunately, the Hamiltonian property still cannot be efficiently checked in this logic. In a second attempt, we extend this basic hybrid logic with the ↓ operator and show that we can check the Hamiltonian property with optimal (NP-Complete) complexity in this logic.

Hybrid Tableaux for the Difference Modality

We present the first tableau-based decision procedure for basic hybrid logic with the difference modality. The decision procedure is gracefully degrading in that the less expressive constructs don't pay for the computationally expensive difference modality. The procedure can be specialized to reflexive and transitive frames. Key features of our approach are nominal elimination, pattern-based blocking, and expansion control.

Modal Expressiveness of Graph Properties

Graphs are among the most frequently used structures in Computer Science. In this work, we analyze how we can express some important graph properties such as connectivity, acyclicity and the Eulerian and Hamiltonian properties in a modal logic. First, we show that these graph properties are not definable in a basic modal language. Second, we discuss an extension of the basic modal language with fix-point operators, the modal μ-calculus. Unfortunately, even with all its expressive power, the μ-calculus fails to express these properties. This happens because μ-calculus formulas are invariant under bisimulations. Third, we show that it is possible to express some of the above properties in a basic hybrid logic. Fourth, we propose an extension of CTL* with nominals, that we call hybrid-CTL*, and then show that it can express the Hamiltonian property in a better way than the basic hybrid logic. Finally, we introduce a promising way of expressing properties related to edges and use it to express the Eulerian property.

Termination for Hybrid Tableaus

This article extends and improves work on tableau-based decision methods for hybrid logic by Bolander and Braüner. Their paper gives tableau-based decision procedures for basic hybrid logic (with unary modalities) and the basic logic extended with the global modality. All their proof procedures make use of loop-checks to ensure termination. Here we take a closer look at termination for hybrid tableaus. We cover both types of system used in hybrid logic: prefixed tableaus and internalized tableaus. We first treat prefixed tableaus. We prove a termination result for the basic language (with n-ary operators) that does not involve loop-checks. We then successively add the global modality and n-ary inverse modalities, show why various different types of loop-check are required in these cases, and then re-prove termination. Following this we consider internalized tableaus. At first sight, such systems seem to be more complex. However, we define a internalized system which terminates without loop-checks. It is simpler than previously known internalized systems (all of which require loop-checks to terminate) and simpler than our prefix systems (no non-local side conditions on rules are required).

A proof–theoretic study of the correspondence of hybrid logic and classical logic

In this paper, we show the equivalence between the provability of a proof system of basic hybrid logic and that of translated formulas of the classical predicate logic with equality and explicit substitution by a purely proof---theoretic method. Then we show the equivalence of two groups of proof systems of hybrid logic: the group of labelled deduction systems and the group of modal logic-based systems.

Hybrid logics with Sahlqvist axioms

We show that every extension of the basic hybrid logic with modal Sahlqvist axioms is complete. As a corollary of our approach, we also obtain the Beth property for a large class of hybrid logics. Finally, we show that the new completeness result cannot be combined with the existing general completeness result for pure axioms.