Journal of Logic, Language, and Information9: 129–132, 2000. 129

Book Review

Multi-Dimensional Modal Logic, Maarten Marx and Yde Venema, Applied Logic Series, Volume 4,Dordrecht: Kluwer Academic Publishers, 1997. Price: Dfl. 171.00/USD 104.00/GBP 68.50, xiii +239 pages, ISBN 0-7923-4345-X.

No doubt, this book is a significant event in modal logic. Its publication sets up one more milestone inthe development of the discipline. Originated as the logic of a single unary necessity operator, modallogic has turned to considering languages with several polyadic modalities, and now it becomesmulti-dimensional(in the geometrical sense).

The idea of using modal languages for talking about multi-dimensional relational structuresis not new. Both in pure and applied modal logic it has been developed since the 1970s. Indeed,if, say, epistemic logics can represent knowledge of certain agents, and temporal logics can beused for reasoning about time, then why can’t we combine these two kinds of logics to capturethe development of a multi-agent system in time? The intended semantical structures can then beregarded as the Cartesian products ofS5-frames (or some other frames modeling knowledge) and〈ω,<〉 (or some other frame modeling time), i.e., as two-dimensional frames. Many other examplesof combined multi-dimensional logics of this sort have been constructed in computer science, artifi-cial intelligence, and computational linguistics: multi-dimensional temporal logics, spatial temporallogics, epistemic, dynamic and temporal description logics, arrow logic, to mention just a few. Inpure modal logic multi-dimensional systems, e.g., Cartesian products of standard monomodal logics,are constructed with the aim of designing languages balancing expressive power and complexity, forinstance, maximally expressive and still decidable.

The book written by Maarten Marx and Yde Venema is the first attempt in the monographic literatureto present sufficiently general mathematical methods for studying various multi-dimensional modalformalisms.

There is no general concept of a modal logic covering all existing systems that can be regardedas multi-dimensional. The choice of logics for this book was actually determined by the authors’dissertations written at the University of Amsterdam. With only one exception, all the logics con-sidered in the book have their origins in algebraic logic (relation, cylindric, and polyadic algebras,to be more precise); in a sense the content of the book can be described as algebraic logic treated asmulti-dimensional modal logic.

The general structure of the book is as follows. The authors consider several types of multi-dimensional formalisms. For each of them, they focus their attention mainly on the following fourimportant properties:

1. the expressive power, namely

− the corresponding fragment of first-order logic interpreted on the intended models, and

− the ability of the formalism to characterize the class of intended frames;

130 BOOK REVIEW

2. axiomatizability;3. decidability;4. interpolation.

It is well known that even two-dimensional combinations of rather simple standard (one-dimensional)modal logics may be quite complex (e.g., the Cartesian square of the logic determined by〈ω,<〉 isnot recursively enumerable). So it is not a surprise that quite often the logics presented in this bookare not finitely axiomatizable, are not decidable, and do not enjoy interpolation. In these cases theauthors try to modify the semantics of the logic under consideration (by means of “relativizing,” i.e.,enlarging the class of models, or “flattening,” i.e., restricting possible valuations), widen the notion ofacceptable axiomatizations (by allowing the non-structural Gabbay-type inference rules), and expandthe language with new operators (say, the difference operator) in order to obtain a related system withthe desired behavior.

Let us now have a closer look at the contents of the book. After a brief (too brief, in my opinion)introduction to the subject in Chapter 1, the authors analyze simple (two-dimensional) versions ofthe logics to be studied later on in the book in Chapter 2. These are:

− S5× S5 (i.e., the bimodal logic interpreted in Cartesian products ofS5-frames) extended withthe identity constant and the converse operator (see below); the language of this logic has thesame expressive power as the first-order language with equality, binary predicates and only twovariables. The logic can be regarded as a modal counterpart of representable cylindric algebrasof dimension 2.

− The modal logic of composition interpreted on squares, corresponding to the compositionreduct of representable relation algebras.

− A simple two-dimensional temporal (or “compass”) logic, which is the product of two lineartemporal logics.

The pedagogical aim of this “appetizer” chapter is to illustrate by means of those simple examplesthe concepts and techniques that are applied later to much more complex logics. (Unfortunately, thelogics are defined in Chapter 2, in a rather abstract way; for motivations the reader is referred to the“main dish.”)

The next three chapters form the core of the book. To give the reader an impression of theirorganization and content, I will analyze one of them, Chapter 3, in some detail.

Chapter 3 deals with arrow logic, the basic modal logic of transitions closely connected withrelation algebras. Arrow logic is based on a language with three modal operators: a dyadic◦ (com-position), a monadic⊗ (converse), and a constantιδ (identity). In accordance with the topic of thisbook, as the intended structures for the arrow language the authors take two-dimensional pair frames,and in particular, squares and local squares (i.e., unions of squares) in which the modal operators areinterpreted as follows:(u, v) |= ϕ ◦ ψ iff there is w such that(u, w) |= ϕ and (w, v) |= ψ ;(u, v) |= ⊗ϕ iff (v, u) |= ϕ; (u, v) |= ιδ iff u = v. The arrow logic of the squares can be regarded asa modal counterpart of representable relation algebras.

Studying the four properties mentioned above, the authors proceed as follows. The expressivepower of arrow logic interpreted on squares is proved to coincide with that of the three-variablefragment of first-order logic with only dyadic predicates (the theorem and its proof are due to Tarski).The classes of squares and local squares are characterized by sets of first-order formulas and then it isshown that some of these formulas are not expressible by formulas of arrow logic. However, this canbe done by extending its language with the difference operatorD. According to a result of Monk, thelogic of squares is not finitely axiomatizable in the standard way. It is shown, however, that it has anon-orthodox axiomatization using the irreflexivity rule for the operatorD′ϕ = >◦¬ιδ∨¬ιδ◦ϕ◦>:

(p ∧ ¬D′)→ ϕ

ϕprovided thatp does not occur inϕ.

BOOK REVIEW 131

The authors also provide a finite (orthodox) axiomatization of the logic of local squares (in itsalgebraic form this result was first obtained by Maddux). By results of Tarski and McKenzie, thearrow logic of squares is undecidable and lacks interpolation. It is shown, however, that it becomesdecidable and enjoys interpolation if interpreted on local squares (algebraic versions of these resultswere given by Németi and Sain). The chapter ends with a study of a temporal version of arrow logicand an outline of other directions in the field.

Chapter 4 considers the interval-based temporal logic of Halpern and Shoham. It is representedas a two-dimensional modal logic interpreted on so-called HS-frames of the form〈INT(T ), B,E〉,whereINT(T ) is the set of intervals of some linear flow of timeT , andB andE are the beginningand end interval relations (e.g.,[x, y]B[x′, y′] iff x = x′ andy < y′). The modal operators of

this logic are interpreted in the following way:i |= 3ϕ holds iff there is an intervalj ∈ INT(T )such thatiBj and j |= ϕ, etc. The authors show that this language is more expressive than thatof point-based temporal logics, and they construct a (non-orthodox) axiomatization of the class oftwo-dimensional HS-frames. Besides, an orthodox axiom system for HS-frames based on〈ω,<〉 isconstructed, provided that only “flat” (i.e., depending only on the first coordinate) valuations on suchframes are allowed.

Chapter 5 studies first-order classical logic as a multi-dimensional modal logic (in terms of algeb-raic logic this idea has been developed by Quine, Tarski, Németi and others starting from the 1970s).Roughly, first-order models are represented as Kripke models of the formM = 〈A, {Rx }x∈var, I 〉,where “worlds” inA are all possible assignments to the variables invar, aRxb holds iff assign-mentsa andb differ only on variablex, andI gives truth-values to predicates at every world. ThenM |= ∃xϕ[a] iff there isb ∈ A such thataRxb andM |= ϕ[b]. Now, what is the modal counterpartof atomic formulas in this framework? To begin with, one can consider the fragment of the first-orderlanguage withn variablesv1, . . . , vn andn-ary predicates of the formP(v1, . . . , vn) correspondingto propositional variables of the modal language. Thus we arrive atcylindric modal logic CMLn withconstantsιδij and unary diamonds3i , for i, j < n. Another option is to allow predicates with arbit-rarily ordered (and possibly repeating) variables; then the modal language should be extended withsubstitution operators�σ , for every substitutionσ . The resulting logic is calledcylindric substitutionmodal logic MLRn; its algebraic counterpart is the class of polyadic algebras introduced by Halmos.

The authors consider these languages interpreted on cubes and local cubes of dimensionn, re-spectively, and provide strongly sound and complete axiomatizations (non-orthodox for cubes andorthodox for local cubes). As concerns decidability and interpolation, it follows from well-knowntheorems of Tarski and Comer that the modalCMLn-theory of the cubes is undecidable forn > 2and does not have interpolation. It is shown, however, that the modalMLRn-theory of local cubesis decidable (this result is due to Mikulás) and enjoys Craig’s interpolation theorem. These resultsare then generalized toCML andMLR of infinite dimension. The chapter ends with a discussion ofapplications to algebraic and some other logics. In general, I found this chapter, perhaps the mostinteresting in the book, somewhat chaotic. It contains a large number of results, but in my opinionthey don’t form a complete picture.

Finally, Chapter 6 shows that, in principle, the multi-dimensional approach developed in thisbook can be applied to any modal language in the sense that the basic derivation systemKS of agiven similarity typeS is strongly sound and complete with respect to a class of multi-dimensionalframes depending only onS.

In a conclusion, the authors provide the reader with an outline of possible directions for furtherresearch and a number of open problems.

Although thoroughly written from the mathematical point of view, the book is far from being ele-mentary, it “is of a rather technical character.” First, the reader is supposed to know quite a lotfrom modal, algebraic and first-order logics. Two appendices “Modal similarity types” and “A modaltoolkit” can certainly make the reading easier, though I found it annoying to consult appendicessearching for the meaning of the unexplained notation. Second, in general the style of the book is

132 BOOK REVIEW

rather “dry.” In particular, a more detailed and illustrated motivation of the subject could make theintended audience of this book wider by attracting, for instance, people from computer science andartificial intelligence working with multi-dimensional modal logics. As it is, the book is orientedmainly to mathematicians with “some working knowledge of modal logic.” Actually, the choice oflogics for the book with such a general title looks rather limited. I wonder, for instance, if it is possibleto extend the developed machinery to other types of multi-dimensional modal systems. It would bealso helpful to make the connections with the existing literature on algebraic logic more transparent,in particular as concerns the notation.

To sum up: The book is a research monograph studying various systems of algebraic (and tem-poral) logic as multi-dimensional modal logics. It contains a large amount of interesting material onthose logics and it links two traditions in mathematical logic and algebra – modal logic and algebraiclogic – which, for a long time, have been developed separately.

Michael ZakharyaschevKeldysh Institute for Applied Mathematics4 Miusskaya Square125047 MoscowRussiaE-mail: mz@cs.msu.su