Maarten Marx

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

Multi-Dimensional Modal Logic

We start with informally defining the subject matter of this book: multi-dimensional modal logic (MDML). First let us briefly consider what we understand by the notion of “modal logic”. The last decade has seen a development in modal logic towards a more abstract and technical approach. In this perspective of what one might call abstract modal logic, arbitrary relational structures can be seen as models for an (extended) modal language: any relation is a potential accessibility relation of some suitably defined modal operator. As the essentially modal aspect of the framework one could point out that the mechanism for evaluating formulas forces certain moves along the accessibility relations. Thus, for instance quantification over a model is restricted to an “accessible” part of the structure.

Complexity of products of modal logics

A relatively new way of combining modal logics is to consider their products. The main application of these product logics lies in the description of parallel computing processes. Axiomatics and decidability of the validity problem have been rather extensively investigated and many logics behave well in these respects. In this paper we look at the product construction from a computational complexity point of view. We show that in many cases there is a drastic increase in complexity, e.g., all products containing the finite S5×S5 products as models have an nexptime-hard satisfaction problem. Products with a functional modality however do not lead to an increase in complexity. For the products K× S5 and S5× S5, we provide a matching upper bound. Combining (modal) logics is a very active area, witness e.g., [4] and the book [1]. A rather special way of combining two modal logics is to consider their products. This approach started with [20], and has recently been developed in great detail in [5]. In temporal logic, products of two logics have been used to describe the temporal logic of intervals (cf. the “product treatment” of the system HS from [8] in [15]: Chapter 4 and the references therein). Almost all products of temporal logics are undecidable, sometimes the validities are not even recursively enumerable [8, 22]. Products of modal (and modal and temporal) logics have applications in the theory of parallel computing [18]. Here we are concerned with the general mathematical theory of products of modal logics, in particular the complexity of several natural decision problems, like the validity problem. With respect to (Hilbert style) axiomatizability a lot of general results are obtained in [5], cf e.g., Theorem 5.7. That paper also contains decidability results for a large number of cases. The general trend for these results is that they are rather hard to prove, but become a lot easier if one of the logics is S5, though even then the filtration arguments are rather involved, and lead to models whose size is in general double exponential in the length of the formula which is to be satisfied. The upper bounds we obtain from these proofs are very bad, in the general case (when none of the logics is S5), the decision-algorithm is non-elementary, and when one of the logics is S5 we only obtain a non-deterministic double exponential time upper bound for the satisfaction problem. Questions concerning computational complexity, have hitherto not been addressed, and we will make a start here. The overall trend is that these logics have a very bad complexity for the satisfaction problem: in many simple cases it is nexptime-hard. Also, even if the satisfaction problem is decidable, the problem whether for a formula φ there exists a model The author is supported by UK EPSRC grant No. GR/K54946.

Undecidability of compass logic.

It is known that the tiling technique can be used to give simple proofs of undecidability of various two-dimensional modal and temporal logics. However, up until now, the simplest two-dimensional temporal logic, the compass logic of Venema, has eluded such treatment. We present a new coding of an enumeration of the tiling plane which enables us to show that the compass logic is undecidable.

Two-Dimensional Modal Logics

This chapter contains a technical introduction to the world of multi-dimensional modal logics. We will treat some relatively simple logics with a two-dimensional semantics. In section 2.1, we introduce the family of modal operators we are going to study, with their two-dimensional semantics. In sections 2.2 and 2.3, we study two-dimensional modal logic with unary operators. These sections can be seen as an appetizer for the α-dimensional case which is treated in chapter 5. Section 2.4 deals with the modal logic of composition. This section is an introduction to chapter 3, which is completely devoted to logics with composition as their main connective. Section 2.5, finally, is about two-dimensional tense logic, a subject which is taken up again in chapter 4. We conclude this chapter with some historical notes on the logics described here.

Beth Definability for the Guarded Fragment

The gueirded fragment (GF) was introduced in [ABN98] as a fragment of first order logic which combines a great expressive power with nice modal behavior. It consists of relationeil first order formulas whose qucintiiiers are relativized by atoms in a certain way. While GF hsis been established as a particularly well-behaved fragment of first order logic in many respects, interpolation fails in restriction to GF, [HM99]. In this paper we consider the Beth property of first order logic and show that, despite the failure of interpolation, it is retained in restriction to GF. Being a closure property w.r.t. definability, the Beth property is of independent interest, both theoretically and for typical potential appUcations of GF, e.g., in the context of description logics. The Beth property for GF is here established on the basis of a limited form of interpolation, which more closely resembles the interpolation property that is usually studied in modal logics. From this we obtain that, more specifically, even every n-variable guarded fragment with up to n-ary relations has the Beth property.

Undecidable relativizations of algebras of relations.

In this paper we show that relativized versions of relation set algebras and cylindric set alge- bras have undecidable equational theories if we include coordinatewise versions of the counting operations into the similarity type. We apply these results to the guarded fragment of first-order logic. ?1. Introduction. Relativized algebras of relations are extensively investigated in the literature, cf., e.g., (HMT, HMTAN, Ma82, Mo93, Ne9 1). In general, relativized versions of algebras of relations have a nicer behavior from the computational point of view than the original versions. In this paper, we concentrate on (un)decidability. We show that if we include coordinatewise versions of the counting operations into the similarity type, then the expressive power is strong enough to interpret the tiling problem into the equational theories of relativized relation set algebras and cylindric-relativized set algebras of dimension (at least) three. Thus these equational theories must be undecidable. Finally, in the last section, we apply these results to logic: the corresponding versions of the guarded fragment of first-order logic and of arrow logic are unde- cidable. Acknowledgments: Thanks are due to Hajnal Andreka and an anonymous referee for careful reading and valuable suggestions. Special thanks are due to the members of the Group of Algebraic Logicians in London: Robin Hirsch, Ian Hodkinson and Mark Reynolds. 1.1. Relativization. Relativization of an algebra amounts to intersecting all its elements with a fixed set (usually an element of the algebra or a subset of the unit) and to defining the operations using this set as the unit of the new algebra. It turned out that if we relativize (set) algebras of relations with arbitrary, sym- metric and/or reflexive elements, then we get a class of algebras with nice algebraic properties. For instance, while relation (set) algebras and cylindric (set) algebras of dimension at least three have undecidable equational theories, the sets of equations valid in the above relativizations are decidable. Traditionally, during relativization we keep the original similarity type in the case of relation algebras: Booleans, composition, converse, identity. As a consequence, some operations that are definable in the original version are not available after relativization. An example is the global counting operations once

Relation Algebra with Binders

The language of relation algebras is expanded with variables denoting individual elements in the domain and with the binder from hybrid logic. Every elementary property of binary relations is expressible in the resulting language, something which fails for the relation algebraic language. That the new language is natural for speaking about binary relations is indicated by the fact that both Craig’s Interpolation, and Beth’s Definability theorems hold for its set of validities. The paper contains a number of worked examples.

Labelled deduction for the guarded fragment

We present the tableau calculus LC 2-TAB which is sound and complete with respect to local square modal logic. The system is a labelled deduction calculus in the spirit of those for modal S5. The novelty here is that the calculus works in two interacting dimensions. This 2-dimensional modal logic allows one to simulate different other modal logics, like K, KT, KTB or multi-K in quite an elegant way. The calculus is also strong enough to decide an interesting PSPACE complete sub-fragment of the guarded fragment, which is generally conceived of as the true modal fragment of first order logic. A PROLOG implementation of this calculus is available through the WWW.

Relation algebras can tile

Abstract Undecidability of the equational theory of the class RA of relation algebras can easily be proved using the undecidability of the word-problem for semigroups. With some effort and ingenuity, one can push this proof through for the larger class SA . We provide another “cause” for undecidability which works for even larger classes than SA . The reason is that we can encode the tiling problem. In doing so we will meet very simple BAO-varieties with undecidable equational theories which might be useful in other undecidability proofs. Our work is part of the research project which tries to establish the border between undecidability and decidability in relational type algebras, cf. [15] , [16] , [12] , [1] and the references therein. The ultimate goal of this research is to come up with versions of relational algebra which are still suitable for modern dynamic applications but whose equational theory is decidable or even tractable.