An Internal Semantics for Modal Logic: Preliminary Report Ronald Fagin IBM Research Laboratory, 5600 Cottle Road, San Jose, CA 95193 Moshe Y.Vardi CSLI, Ventura Hall, Stanford University, Stanford, CA 94305 Abstract: In Kripke semantics for modal logic, “pos- and culminating in the book Symbolic Logic with sible world^" and the possibility relation are both Langford in 1932. Since then modal logic has been primitive notions. This has both technical and con- extensively studied by logicians and philosophers ceptual shortcomings. From a technical point of ([Chl is a good textbook). More recently, modal view, the mathematics associated with Kripke. se- logic has been applied in several areas of computer mantics is often quite complicated. From a concep- science, such as artificial intelligence [MH]. program tual point of view, it is not clear how to use Kripke verification and synthesis [MW.Pn,Pral]. hardware structures to model know!edge and belief, where one specification [Bo.RS], protocol specification and ver- wants a clearer understanding of the notions that ification [CES.SM], database theory [ CCF,Li], and are primitive in Kripke semantics. We introduce distributed computing [HMl]. modal structures as models for modal logic. We use Lewis’ semantics for modal logic was of an al- the idea of possible worlds, but by directly describing gebraic cast. Algebraic semantics, however, though the “internal semantics” of each possible world. It technically adequate (cf. [Gu,Mc,McT,Ts]), is nev- is much easier to study the standard logical questions, ertheless not very intuitive. In 194.6 Carnap such as completeness, decidability, and compactness, [Cal,Ca2] suggested using the more intuitive ap- ushg modal structures. Furthermore, modal struc- proach of possible wort% to assign semantics to mo- tures offer a much more intuitive approach to mod- dalities. According to this approach, one starts with elling knowledge and belief. a set of possible worlds. Then statements of the form up(i.e., p 3 wessuri/y true) are interpreted in 1. Introduction the following way: upis true if p is true in every Modal bgic can be described briefly as the logic “possible world”. (The idea that necessity is truth of necessity and possibility, of “must be” and “may in all possible worlds is often credited to Leibniz be”. (One should not take “necessity” and “possi- [Bat], though this is historically debatable.) Possible- bility” literally. “Necessarily” can mean “according worlds semantics was further developed indepen- to the laws of physics” or “according to my beliefs”, dently by several researchers [Bay, Hil, Hi2, Ka, or even “after the program terminates”.) Modal Krl, Me, Mol, Pril, reaching its current form with logic was discussed by several authors in ancient Kripke [Kr21. The basic idea of the development is times, notably Aristotle in De Inrerpretatioone and Prior to consider, instead of a set of worlds that are Amlystia, and by medieval logicians, but like most possible outright. a set of worlds that are, or are work before the modern period. it was non-symbolic, not. possible with respect to each other. Kripke structures and not particularly systematic in approach. The capture this intuitive idea. A Kripke structure can first symbolic and systematic approach to the subject be viewed as a labeled directed graph: the nodes are appears to be the work of Lewis beginning in 1912 the possible worlds labeled by truth assignments, and a world Y is possible with respect to a world u if there is an edge from I( to v. Permission to copy without fee all or part of this material is granted Kripke structures were immensely successful provided that the copies are not made or distributed for direct mathematical. tools and served as the basis for ex- commercial advantage, the ACM copyright notice and the title of the tremely fertile research. Nevertheless, they do suffer publication and its date appear, and notice is given that copying is by jxrrnission of the Association for Computing Machinery. To copy from both technical and conceptual shortcomings. otherwise, or to republish, requires a fee and/or specific permission. From a technical point of view, the mathematics associated with them is often quite complicated. For @ 1985 ACM 0-89791-151-2/85/005/0305 $00.75 example, completeness proofs are either non-elegant 305 [Kr2] (Kripke himself described his completeness of depth 2 is essentially a set of worlds of depth 1; proof as “rather messy”), or non-constructive etc. Modal structures are worlds of depth O, and [Ma,Lem2]. (Only in 1975 did Fine come with an their recursive structure enables us to assign meaning elegant and constructive proof [Fi], but his proof is to iterated modalities. far from straightforward.) Also. the standard tech- Having introduced modal structures, we investi- nique for proving decidability is to show that the gate their relationship to Kripke structures. It turns logic has thefinitemdelpropzny, which is not straight- out that modal structures model individual nodes in forward at all. Furthermore, in order to model dif- Kripke structures, while Kripke structures model ferent modal logics, certain graph-theoretic contraints collections of modal structures. Thus, modal struc- on the possibility relation between possible worlds tures can be seen as duals to Kripke structures. have to be imposed; these constraints are very often Nevertheless, it is much easier to study the standard far from intuitive, and sometimes they are not even logical questions, such as completeness, decidability, first-order definable [Go]. and compactness, using modal structures. The crucial From a conceptual point of view, it is not clear point is that satisfaction of a formula in a modal that Kripke structures are as intuitive as they are structure depends only on a certain finite part of supposed to be. The basic problem is that in Kripke that structure. Furthermore, the “size” of that part semantics the notion of a possible world is a pimitin? depends on the “size” of the formula. Thus, the notion. (Indeed possible worlds are called reference proofs of decidability and compactness are almost pints in [Mo2] and indices in [Sc].) This works well straightforward, and the completeness proof is both in applications where it is intuitively clear what a elegant and constructive. We urge the reader to possible world is. For example, m dmicbgic a compare our proofs to previous proofs (e.g., [Fr. possible world is just a program state [Prall, i.e.. an Kr2, Lem2, Ma, Mc]) in order to appreciate their assignment of values to the variables and to the elegance. location counter, and in temporal bgic a possible Beyond the technical usefulness of modal struc- world is just a point in time [Bu]. But in applications tures, we claim that they are more intuitive and where it is not clear what a possible world is, e.g., more appropriate to conceptual modelling. For ex- in eNtemic logic, the logic of knowledge and belief, ample, the simple scenarios in distributed environ- how can we construct a Kripke structure without ments mentioned above can be modelled by modal understandir.g its basic constituents? Indeed, in dy- structures in a straightforward way [FHV]. We also r:amic and temporal logic one constructs first the demonstrate the intuitiveness of modal structures by structures, and then proceeds to find the axioms modelling belief and by modelling joint knowledge, [Bu,KP], while in epistemic logic one first selects then using our techniques to prove decidability and axioms and then tailors the structures to the axioms completeness in these cases (and compactness in the [HM2.Re]. case of belief; compactness fails for joint knowledge). Furthermore, if we want to upp& modal logic it The double perspective that we have now on is often necessary to construct models for particular modal logic, namely. Kripke structures and modal situations. But if we have no means of explicitly structures, turns out to be very useful in proving describing the possible worlds, how can we construct optimal upper bounds for the complexity of the Kripke structures to model particular situations? decision problem. By their graph-theoretic nature, Indeed, as pointed out in [FHV], there are simple Kripke structures are amenable to automata-theoretic scenarios in distributed environments that cannot techniques [ES,Str,VW]. By combining our results be easily modelled by Kripke structures. for modal structures with a new automata-theoretic We believe that our approach, which is to describe technique for Kripke structures, we prove that sev- explicitly the “internal” semantics of a possible eral of the logics that we study are complete in world, is a much more intuitive approach to mod- SPACE. elling. We introduce ~aklstructuresas models for 2. Modal structures modal logic. We use the idea of possible worlds, but in Carnap’s style rather than Kripke’s style. hic definifions. We now define structures that Thus, we define a modal structure to be essentially capture the essence of the “possible worlds” ap a set of modal structures. This is of course a circular proach. (For the sake of simplicity, we restrict ourselves here to mml modal logics [Ch]. Never- definition, and to make it meaningful, we define work& inductively, by constructing worlds of greater theless, this is not an inherent limitation of our approach.) In anticipation of subsequent develop- and greater depth. A world of depth 0 is a description of reality. i.e., a truth assignment; a world of depth ments where modalities correspond to “attitudes” of agents or players, we allow multiple modalities 1 is essentially a set of worlds of depth 0; a world 306 01, ..., 0,. A good way to interpret the statement world and <gQ..,.&-2 > E fk-l(i)] for each j 2 k and Dip is “after program i terminates, p must be true”.