We have discussed the big three (model checking, and satisfiability and validity checking) but this by no means exhausts the reasoning tasks of interest. To conclude this section, let’s briefly consider some others.
Although we have stressed the locality of modal logic, some problems demand a global per- spective. In particular, if we view a modal formula as a general background constraint, we will typically want it to be globally satisfied: that is, we will be interested in modelsMsuch that
M|=ϕ. The importance of the global satisfiability problem has been strongly emphasised by the description logic community. Indeed, description logic builds into its architecture the idea of a TBox, a collection of formulas that encode background knowledge about some domain (for example, that all men are mortal, that all Martians own flying saucers, or that each employee has a social security number). Description logicians are interested in models in which the TBox is globally satisfied, for these are the models that reflect all the background assumptions.
Once the importance of background constraints is realised, it becomes clear that it is not the pure global satisfiability task itself that is of primary interest. Rather, it is the local-global
satisfiability task: given formulasϕandψ, is there a model which locally satisfiesϕand globally satisfiesψ? That is, is it possible to satisfyϕsubject to the global constraintψ?
Here’s an example. Suppose we’re working in a zoological setting, and are interested in the interaction of maternal love and professional responsibility on the feeding of our furry ursine
bretheren. To put it another way, suppose we have the following TBox:
bear∨human bear→ hMOTHERibear
bear→ ¬human bear→[FEDBY](zoo-keeper∨mother)
Let’s call this TBoxBEAR-CARE. The sort of queries we might be interested in posing are: is it possible to globally satisfyBEAR-CAREand, simultaneously, to locally satisfy
hMOTHERi(bear∧human)?
(No, it’s not.) And is it possible to globally satisfyBEAR-CAREand simultaneously to locally satisfy
hFEDBYi(¬human∧ ¬mother)?
(Yes, it is:BEAR-CAREdoesn’t rule out having bears as zoo-keepers. This may well be a bug in the TBox.)
Local-global satisfiability problems are also natural in the setting of parsing problems. It is possible to encode various kinds of grammars (such as regular grammars or context-free gram- mars) as modal formulas (see Chapter 19 of this handbook for a discussion of such approaches). Then, given a string of symbols, the parsing problem is to decide whether it is possible to find a model which embodies all the constraints encoded in the grammar, and which simultaneously satisfies the formula encoding the input string. That is, we would like to globally satisfy the modal formulaGRAMMARand simultaneously locally satisfyINPUT-STRING.
Unsurprisingly, both the global, and the local-global satisfiability tasks are tougher than the ordinary satisfiability problem:
THEOREM 23. The global satisfiability and the local-global satisfiability tasks for basic modal
languages are both EXPTIME-complete.
Proof. The stated result is an immediate consequence of Hemaspaandra’s [107, 59] complexity
results for the universal modality (we introduce the universal modality in Section 6.1). But the result holds for even stronger languages; see De Giacomo and Lenzerini [26] for related results
for more expressive description logics. a
EXPTIME-complete problems are decidable but provably intractable: they contain problem in- stances that will require time exponential in the size of the input to solve (which can mean that they require more time than the expected lifetime of the universe). This, however, is a worst- case measure. One of the most important recent developments in computational logic has come from the description logic community, who have shown it is possible to specify and implement tableaux-based algorithms for such problems that are remarkably efficient in practice. Moreover, interesting work exists on performing modal theorem proving via (non-standard) translations into first-order logic, so that optimised first-order resolution provers can be applied to the task. For a detailed discussion and comparison of these methods, see Chapter 4 of this handbook, and for a deeper examination of the complexity of modal logic, see Chapter 3.
We conclude with a remark on the model comparison task. As bisimulation is the modally fundamental notion of graph equivalence, it is natural to wonder how difficult it is to determine when two models are bisimilar. The corresponding problems for first-order logic (namely, testing for graph isomorphism) is thought to be difficult: there is no known polynomial algorithm for testing for graph isomorphism, though the problem has not been shown to be NP-complete either.
In fact, the problem of identifying isomorphic graphs is sometimes regarded as giving rise to a special complexity class of its own.
Testing for bisimulation, however, turns out to be computationally tractable, and there are el- egant polynomial algorithms which work by discarding pairs of point that cannot make it into any bisimulation (see Dovier, Piazza and Policriti [30]). Again an expressivity result lies be- hind this result: the maximal bisimulation between two modelsMandNis explicitly definable in a first-order fixed-point language over the disjoint unionM]N of the two models. Such languages have been studied extensively in computer science, and they are known to have good computational behaviour.
5 RICHER LOGICS
Until now, we have deliberately said rather little about modal logics and what they are. Instead we have acted as if there was only one modal logic of any interest, namely the set of valid formulas (that is, the set of formulas satisfied at all points in all models) or, to put it syntactically, the set of formulas generated by the minimal proof system K (which we defined at the start of Section 2.2). But traditional presentations of modal logic tend to emphasise the multiplicity of modal logics, and devote a great deal of attention to logics richer than K, logics with such names as T, K4, S4,
S5, GL, and Grz. Where do richer modal logics come from?
As a first approximation (we’ll shortly see why it’s only an approximation) we might say that richer logics emerge at the level of frames, via the concept of frame validity. Letϕ(p1, . . . , pn)be a basic modal formula built out of the proposition lettersp1, . . . , pn. We say thatϕ(p1, . . . , pn) is valid on a frameF= (W, R)at a pointwif, for each valuationV for its proposition symbols
p1,. . . ,pn, we have thatϕis satisfied in the resulting model atw; in such a case we writeF, w|=
ϕ. We sayϕis valid onFif it is valid at each point inF, and we write this asF|=ϕ. Moreover, we say that a modal formula is valid on a class of framesFif it is valid on each frameFinF. Note that a valid formula (as defined in Section 2.1) is simply a formula that is valid on the class of all frames.
The starting point for this section is the observation that different applications of modal logic typically validate different modal axioms, axioms over and above those to be found in the mini- mal system K. For example, if we view our models as flows of time, it is natural to assume that the accessibility relation is transitive, and (as the reader should check) any instance of the schema 2ϕ→22ϕis valid on the class of transitive frames (for example, the formula2p→22pis valid on such frames, and2(p∨q)→22(p∨q)is too). However no instance of this schema (which for historical reasons is called 4) is provable in K, so if we want a logic for working with temporal flows we should add all its instances as extra axioms, and doing so yields the logic known as K4. Or suppose we are modeling situations where the frame relation has to be treated as a partial function. As the reader should check, all instances of the schema3ϕ→2ϕare valid on the class of such frames, and none of them can be proved in K, so once again we should add them as extra axioms. Doing so yields the logic called KAlt1.
We begin this section by briefly discussing such axiomatic extensions of K a little further. But our real interest is not the richer logics that arise by adding extra axioms (for an introduction to this topic, see Chapter 2 of this handbook) rather it centres on the following semantic questions: what can modal formulas say about frames, and how do they say it? As we shall see, there is a fundamental expressivity distinction between the level of models and the level of frames: whereas modal logic at the level of models is the bisimulation invariant fragment of first-order logic, at the level of frames it is a fragment of second-order logic.