In Section 4 we contrasted the PSPACE decidability of modal logic with the undecidability of first-order logic. But these results concerned satisfiability and validity on the class of all frames. Suppose we restrict attention to particular classes of frames defined by basic modal formulas. There is no reason to suppose that modal satisfiability and validity problems over such frame classes will always be in PSPACE, or even that they will be decidable. And indeed, in many cases they are not not.
In some cases, restricting attention to a certain class of frames may lower the computational complexity. For example, suppose we restrict attention to the frames defined by3p→2p, that is, the class of frames in which Ris a partial function. Then the task of testing basic modal formulas for satisfiability becomes NP-complete, that is, no worse than the satisfiability problem for propositional logic. This is because (as the reader can easily check) if a basic modal formula
ϕhas a model based on a frame in this class, then it has not only has a finite model in this class, but a model containing at mostm+ 1points, wheremis the number of modalities inϕ. Thus
a non-deterministic algorithm which guesses a model, checks that it belongs to the frame class, and verifies that the formula is satisfied on it, runs in time polynomial in the size ofϕ.
But restricting attention to particular frame classes can easily results in undecidable problems. Moreover, undecidable problems arise even when attention is restricted to finite frames; see, for example, Urquhart [115]. And indeed, failures of decidability turn out to be the norm. It is not difficult to show that there are non-denumerably many distinct frame satisfiability problems (a particularly elegant demonstration of this, due to Spaan [107], is given as Exercise 6.2.4 of Black- burn, de Rijke and Venema [13]). As there are only denumerably many computable functions, undecidability is guaranteed in most cases.
So what about recursive enumerability? That is, if we restrict attention to a class of framesF that is defined by a modal formula, is the theory of this frame class (that is, the set of formulasϕ
valid on all framesF) recursively enumerable? Well, ifFis elementary, the answer is yes: PROPOSITION 34. Suppose thatFis an elementary class of frames defined by a basic modal formulaϕ. Then the (basic modal) theory ofFis recursively enumerable.
Proof. AsF is an elementary class thatϕdefines, ϕcorresponds to some first-order formula
α. Now a basic modal formula ψ is valid on frames for ϕ iff its second-order translation ∀P1· · ·Pn∀xSTx(ψ)is true in all models of the first-order formulaα, that is, iff
α|=∀P1· · ·Pn∀xSTx(ψ),
where|=is classical entailment. But asαis first-order, the predicatesP1· · ·Pndo not occur in
αand hence this is equivalent to
α|=∀xSTx(ψ).
But this is a first-order entailment, and as such entailments are recursively enumerable the result
follows. a
However once we move beyond the elementary frame classes, even recursive enumerability is lost. A key result here is Thomason’s [114] reduction of the standard consequence relation for the second-order correspondence language to the global frame consequence relation for a basic modal language with one modality. A basic modal formulaϕis a global frame consequence of Γif for all framesF, ifF |= Γ, thenF |=ϕ. It follows that global frame consequence is not recursively enumerable. For further discussion of Thomasons’s work in this area, see Chapter 7 of this handbook.
6 RICHER LANGUAGES
The purpose of this section is to discuss a typical, but not yet widely appreciated, aspect of contemporary modal logic: flexible language (re-)design. As we have seen, the basic modal language has a number of attractive properties, and as the bisimulation invariant fragment of the first-order correspondence language it is a special tool when it comes to talking about graphs. Nonetheless, many of its design parameters were fixed by historical accident. Perhaps judicious experimentation could lead to improvements, or at least to interesting variants? Modal logicians have been carrying our such experiments for years, and in this section we survey some of their work.
But what should count as a richer modal language? It’s easier to explain what shouldn’t. Here’s an obvious example. It is straightforward to extend our basic definitions to cover polyadic
modalities (that is,n-place diamonds and boxes). Simply work with models in which there is ann+ 1-place relationRmfor everyn-place diamondhmi. We interpret using the following satisfaction clause:
M, w|=hmi(ϕ1, . . . , ϕn) iff for somev1, . . . , vn ∈W such thatRmwvi. . . vn we haveM, v1|=ϕ1and . . . andM, vn |=ϕn. Now, suchn-place modalities are undeniably useful for certain purposes, but developing their theory (standard translation, bisimulation, and so on) is essentially a matter of sprinkling our earlier work with additional indices. These operators don’t give rise to richer languages in any logically interesting sense.
As we shall see, the richer languages explored in this section offer more. Moreover, their richness arises from different sources. Sometimes the enrichment consists of taking a standard language and insisting that a modality be interpreted by some mathematically fundamental re- lation (the universal modality is a good example). Sometimes the enrichment takes the form of more complex satisfaction definitions (both temporal logic with Until and Since and conditional logic are examples of this). In other cases, syntactic enhancements are introduced to support novel semantic capabilities (hybrid logic, propositional dynamic logic, and the modalµ-calculus all do this) and in one case (the guarded fragment) we enrich by abandoning modal syntax and using first-order syntax instead. Moreover, it is also possible to enrich by combining logics. For example, we might combine two propositional modal logics to enable some application domain to be more accurately modeled, or we might combine modal logic with first-order logic, a move which takes us to the historical heartland of philosophical applications of modal logic.
This variety raises a question of its own: what, if anything, do all these richer languages have in common? That is, what makes them all modal? This is not an easy question to answer. Nonetheless, as we work our way through this landscape a number of themes will recur: robust decidability, the importance of bisimulations, and characterisations of fragments of first- and second-order logic. As we shall see at the end of the section, the idea of restricted quantification that underlies the guarded fragment goes a long way towards accounting for these properties, for both first- and second-order enrichments. Moreover, it is possible to draw on ideas from abstract model theory and prove Lindstr¨om-style characterisation results.