Carlos Areces

Hybrid Logics: Characterization, Interpolation and Complexity

Hybrid languages are expansions of propositional modal languages which can refer to (or even quantify over) worlds. The use of strong hybrid languages dates back to the work of Arthur Prior in the 1960s, but recent work has focussed on a more constrained system in which variables can only be bound to the current state. We show in detail that the constrained system is modally natural. We begin by studying its expressivity, and provide model theoretic characterizations (via a restricted notion of Ehrenfeucht-Fraisse game, and an enriched notion of bisimulation) and a syntactic characterization (in terms of bounded formulas). The key result to emerge is that the constrained system corresponds to the fragment of first-order logic which is invariant for generated submodels. We then show that the system enjoys (strong) interpolation, provide counterexamples for its finite variable fragments, and show that weak interpolation holds for an important subsystem called H(@). Finally, we provide complexity results for H(@) and other fragments and variants, and sharpen known undecidability results for the full fragment

A Road-Map on Complexity for Hybrid Logics

Hybrid languages are extended modal languages which can refer to (or even quantify over) states. Such languages are better behaved proof theoretically than ordinary modal languages for they internalize the apparatus of labeled deduction. Moreover, they arise naturally in a variety of applications, including description logic and temporal reasoning. Thus it would be useful to have a map of their complexity-theoretic properties, and this paper provides one. Our work falls into two parts. We first examine the basic hybrid language and its multi-modal and tense logical cousins. We show that the basic hybrid language (and indeed, multi-modal hybrid languages) are no more complex than ordinary uni-modal logic: all have PSPACE-complete K-satisfiability problems. We then show that adding even one nominal to tense logic raises complexity from pspace to EXPTIME. In the second part we turn to stronger hybrid languages in which it is possible to bind nominals. We prove a general expressivity result showing that even the weak form of binding offered by the ↓ operator easily leads to undecidability.

The computational complexity of hybrid temporal logics

In their simplest form, hybrid languages are propositional modal languages which can refer to states. They were introduced by Arthur Prior, the inventor of tense logic, and played an important role in his work: because they make reference to specific times possible, they remove the most serious obstacle to developing modal approaches to temporal representation and reasoning. However very little is known about the computational complexity of hybrid temporal logics. In this paper we analyze the complexity of the satisfiability problem of a number of hybrid temporal logics: the basic hybrid language over transitive frames; nominal tense logic over transitive frames, strict total orders, and transitive trees; nominal Until logic; and referential interval logic. We discuss the effects of including nominals, the @ operator, the somewhere modality E, and the difference operator D. Adding nominals to tense logic leads for several frame–classes to an increase in complexity of the satisfiability problem from pspace to exptime. On transitive trees, however, the satisfiability problem for this language can be decided in pspace. Along the way we make a detour through hybrid propositional dynamic logic: we establish upper bounds for a number of temporal logics by generalizing results due to Passy and Tinchev [PT91] and De Giacomo [De 95]. We conclude with some remarks on the relevance of our results to description logic, and draw attention to the utility of the spypoint technique for proving upper and lower bounds.

Keys, nominals, and concrete domains

Many description logics (DLs) combine knowledge representation on an abstract, logical level with an interface to “concrete” domains like numbers and strings with built-in predicates such as <, +, and prefix-of. These hybrid DLs have turned out to be useful in several application areas, such as reasoning about conceptual database models. We propose to further extend such DLs with key constraints that allow the expression of statements like “US citizens are uniquely identified by their social security number”. Based on this idea, we introduce a number of natural description logics and perform a detailed analysis of their decidability and computational complexity. It turns out that naive extensions with key constraints easily lead to undecidability, whereas more careful extensions yield NExpTime-complete DLs for a variety of useful concrete domains.

Resolution in modal, description and hybrid logic

We provide a resolution-based proof procedure for modal, description and hybrid logic that improves on previous proposals in important ways. It avoids translations into large undecidable logics, and works directly on modal, description or hybrid logic formulas instead. In addition, by using the hybrid machinery it avoids the complexities of earlier propositional resolution-based methods for modal logic. It combines ideas from the method of prefixes used in tableaux, and resolution ideas in such a way that some of the heuristics and optimizations devised in either field are applicable.

Referring expressions as formulas of description logic

In this paper, we propose to reinterpret the problem of generating referring expressions (GRE) as the problem of computing a formula in a description logic that is only satisfied by the referent. This view offers a new unifying perspective under which existing GRE algorithms can be compared. We also show that by applying existing algorithms for computing simulation classes in description logic, we can obtain extremely efficient algorithms for relational referring expressions without any danger of running into infinite regress.

Repairing the interpolation theorem in quantified modal logic

Abstract Quantified hybrid logic is quantified modal logic extended with apparatus for naming states and asserting that a formula is true at a named state. While interpolation and Beths definability theorem fail in a number of well-known quantified modal logics (for example in quantified modal K, T, D, S4, S4.3 and S5 with constant domains), their counterparts in quantified hybrid logic have these properties. These are special cases of the main result of the paper: the quantified hybrid logic of any class of frames definable in the bounded fragment of first-order logic has the interpolation property, irrespective of whether varying, constant, expanding, or contracting domains are assumed.

Bringing them all Together

[. . . ] No way to say warm in French. There was only hot and tepid. If there’s no word for it, how do you think about it? [. . . ] Imagine, in Spanish having to assign a gender to every object: dog, table, tree, can-opener. Imagine, in Hungarian, not being able to assign a gender to anything: he, she, it all the same word. Thou art my friend, but you are my king; thus the distinctions of Elizabeth the First’s English. But with some oriental languages, which all but dispense with gender and number, you are my friend, you are my parent, and YOU are my priest, and YOU are my king, and You are my servant, and You are my servant whom I’m going to fire tomorrow if You don’t watch it, and YOU are my king whose policies I totally disagree with and have sawdust in YOUR head instead of brains, YOUR highness, and YOU may be my friend, but I’m still gonna smack YOU up side the head if YOU ever say that to me again: and who the hell are you anyway. . . ? Babel-17 Samuel R. Delany

Expressive Power and Decidability for Memory Logics

Taking as inspiration the hybrid logic

Relation-changing modal operators

\mathcal{HL}({\downarrow})