Shallow Models for Non-Iterative Modal Logics

Shallow Models for Non-Iterative Modal Logics

Lutz Schröder
DFKI-Lab Bremen
and Department of Computer Science, Universität Bremen
Dirk Pattinson
Department of Computing
Imperial College London
Abstract

The methods used to establish -bounds for modal logics can roughly be grouped into two classes: syntax driven methods establish that exhaustive proof search can be performed in polynomial space whereas semantic approaches directly construct shallow models. In this paper, we follow the latter approach and establish generic -bounds for a large and heterogeneous class of modal logics in a coalgebraic framework. In particular, no complete axiomatisation of the logic under scrutiny is needed. This does not only complement our earlier, syntactic, approach conceptually, but also covers a wide variety of new examples which are difficult to harness by purely syntactic means. Apart from re-proving known complexity bounds for a large variety of structurally different logics, we apply our method to obtain previously unknown -bounds for Elgesem’s logic of agency and for graded modal logic over reflexive frames.

1 Introduction

Special purpose modal logics often combine expressivity and decidability, usually in a low complexity class. In the absence of fixed point operators, these logics are frequently decidable in PSPACE, i.e. not dramatically worse than propositional logic. While lower PSPACE bounds for modal logics can typically be obtained directly from seminal results of Ladner [20] by embedding a PSPACE-hard logic such as or , upper bounds are often non-trivial to establish. In particular PSPACE upper bounds for non-normal logics have recently received much attention:

• A PSPACE upper bound for graded modal logic [11] is obtained using a constraint set algorithm in [30]. This corrects a previously published incorrect algorithm and refutes a previous EXPTIME hardness conjecture.

• More recently, a upper bound for Presburger modal logic (which contains graded modal logic and majority logic [23]) has been established using a Ladner-type algorithm [6] .

• Using a variant of a shallow neighbourhood frame construction from [31], a PSPACE upper bound for coalition logic is established in [26].

• PSPACE upper bounds for and related conditional logics [3] are obtained in [22] by a detailed proof-theoretic analysis of a labelled sequent calculus.

The methods used to obtain these results can be broadly grouped into two classes. Syntactic approaches presuppose a complete tableaux or Gentzen system and establish that proof search can be performed in polynomial space. Semantics-driven approaches, on the other hand, directly construct shallow tree models. Both approaches are intimately connected in the case of normal modal logics interpreted over Kripke frames: counter models can usually be derived directly from search trees [16]. It should be noted that this method is not immediately applicable in the non-normal case, where the structure of models often goes far beyond mere graphs.

Using coalgebraic techniques, we have previously shown [29] that the syntactic approach uniformly generalises to a large class of modal logics: starting from a one-step complete axiomatisation, we have applied resolution closure to obtain complete tableaux systems. Generic PSPACE-bounds follow if the ensuing rule set is PSPACE-tractable. Here, we present a different, semantic, set of methods to establish uniform PSPACE bounds by directly constructing shallow models for logics subject to the one-step polysize model property, or a variant of the latter. In particular, no axiomatisation of the logic itself is needed.

Apart from the fact that both methods use substantially different techniques, they apply to different classes of examples. While it is e.g. relatively easy to obtain a resolution closed rule set for coalition logic [26], proving the one-step polysize model property for (the coalgebraic semantics of) coalition logic is a non-trivial task. On the other hand, small one-step models are comparatively easy to construct for complex modal logics such as probabilistic modal logic [8] or Presburger modal logic [6] that are not straightforwardly amenable to the syntactic approach via resolution closure, either because no axiomatisation has been given or because the complexity of the axiomatisation makes the resolution closure hard to harness.

Moreover, the present semantic approach to PSPACE-bounds takes a significant step to overcome an important barrier in the coalgebraic treatment of modal logics. Existing decidability and completeness results [25, 5, 28, 29] are limited to rank-1 logics, given by axioms whose modal nesting depth is uniformly equal to one. While this already encompasses a large class of examples (including all logics mentioned so far), the semantic model construction in the present paper applies to non-iterative logics [21], i.e. logics axiomatised without nested modalities (rank-1 logics additionally exclude top-level propositional variables). Despite the seemingly minute difference between the two classes of logics, this generalisation is not only technically non-trivial but also substantially extends the scope of the coalgebraic method. Besides the modal logic , the class of non-iterative logics includes e.g. all conditional logics covered in [22] (of which only 4 are rank-1), in particular  [3], as well as Elgesem’s logic of agency [7, 12] and the graded version of  [11].

As in [29], we work in the framework of coalgebraic modal logic [25] to obtain results that are parametric in the underlying semantics of particular logics. While normal modal logics are usually interpreted over Kripke frames, non-normal logics see a large variety of different semantics, e.g. probabilistic systems [8], frames with ordered branching [6], game frames [26], or conditional frames [3]. The coalgebraic treatment allows us to encapsulate the semantics in the choice of a signature functor, whose coalgebras then play the role of models, leading to results that are unformly applicable to a large class of different logics.

Since the class of all coalgebras for a given signature functor can always be completely axiomatised in rank 1 [28], in analogy to the fact that the -axioms are complete for the class of all Kripke frames, the standard coalgebraic approach is not directly applicable to non-iterative logics. To overcome this limitation, we introduce the new concept of interpreting modal logics over coalgebras for copointed functors, i.e. functors equipped with a natural transformation of type .

In this setting, our main technical tool is to cut back model constructions from modal logics to the level of one-step logics which semantically do not involve state transitions, and then amalgamate the corresponding one-step models into shallow models for the full modal logic, which ideally can be traversed in polynomial space. For this approach to work, the logic at hand needs to support a small model property for its one-step fragment, the one-step polysize model property (OSPMP). Our first main theorem shows that the OSPMP guarantees decidability in polynomial space. Crucially, the OSPMP is much easier to establish than a shallow model property for the logic itself. To reprove e.g. Ladner’s PSPACE upper bound for , one just observes that to construct a set that intersects given sets, one needs at most elements. For the conditional logics , , and , the OSPMP is similarly easy to check. For other logics, in particular various logics of quantitative uncertainty, the OSPMP can be obtained by sharpening known off-the-shelf results. As a new result, we establish the OSPMP for Elgesem’s logic of agency to obtain a previously unknown PSPACE upper bound.

As a by-product of our construction, we obtain NP-bounds for the bounded rank fragments of all logics with the OSPMP, generalizing the corresponding result for the logics and from [13] to a large variety of structurally different (non-iterative) logics.

While the OSPMP is usually easy to establish, a weaker property, the one-step pointwise polysize model property (OSPPMP), can be used in cases where the OSPMP fails, provided that the signature functor supports a notion of pointwise smallness for overall exponential-sized one-step models. This allows traversing exponentially branching shallow models in polynomial space by dealing with the successor structures of single states in a pointwise fashion. Our second main result, which yields PSPACE upper bounds for logics with the OSPPMP, is applied to reprove the known PSPACE bound for Presburger modal logic [6] and to derive a new PSPACE bound for Presburger modal logic over reflexive frames, and hence for  [11] (which was so far only known to be decidable [10]). The latter result extends straightforwardly to a description logic with role hierarchies, qualified number restrictions, and reflexive roles.

2 Coalgebraic Modal Logic

We recall the coalgebraic interpretation of modal logic and extend it to non-iterative logics using copointed functors.

A modal signature is a set of modal operators with associated finite arity. The signature determines two languages: firstly, the one-step logic of , whose formulas (the one-step formulas) over a set of propositional variables are defined by the grammar

 ψ::=⊥ ∣ψ1∧ψ2 ∣ ¬ψ ∣ L(ϕ1,…,ϕn),

where is -ary and the are propositional formulas over ; and secondly, the modal logic of , whose set of -formulas is defined by the grammar

 ψ::=⊥ ∣ψ1∧ψ2 ∣ ¬ψ ∣ L(ψ1,…,ψn).

Thus, the modal logic of is distinguished from the one-step logic in that it admits nested modalities. The boolean operations , , , are defined as usual. The rank of is the maximal nesting depth of modalities in (note however that the notion of rank- logic [28, 29] is stricter than suggested by this definition, as it excludes top-level propositional variables in axioms; the latter are allowed only in non-iterative logics). We denote by the set of formulas of rank at most ; we refer to the languages as bounded-rank fragments.

We treat one-step logics as a technical tool in the study of modal logics. However, one-step logics also appear as logics of independent interest in the literature [9, 14, 15]. One of the central ideas of coalgebraic modal logic is that properties of the full modal logic, such as soundness, completeness, and decidability, can be reduced to properties of the much simpler one-step logic. This is also the spirit of the present work, whose core is a construction of polynomially branching shallow models for the modal logic assuming a small model property for the one-step logic.

The semantics of both the one-step logic and the modal logic of are parametrized coalgebraically by the choice of a set functor. The standard setup of coalgebraic modal logic using all coalgebras for a plain set functor covers only rank-1 logics, i.e. logics axiomatised one-step formulas [28] (a typical example is the -axiom ). Here, we improve on this by considering the class of coalgebras for a given copointed set functor, which enables us to cover the more general class of non-iterative logics, axiomatised by arbitrary formulas without nested modalities (such as the -axiom ). We follow a purely semantic approach and hence do not formally consider axiomatisations in the present work (where we do mention axioms, this is for solely explanatory purposes). However, the extended scope of the new framework and its relation to non-iterative modal logics (which can be made precise in the same way as for plain functors and rank-1 logics [28]) will become clear in the examples.

In general, a copointed functor consists of a functor , where is the category of sets, and a natural transformation . For our present purposes, a slightly restricted notion is more convenient:

Definition 2.1.

A (restricted) copointed functor with signature functor is a subfunctor of (where ). We say that is trivially copointed if . An -coalgebra  consists of a set of states and a transition function such that for all .

Remark 2.2.

The modal logic does not explicitly include propositional variables. These may be regarded as nullary modal operators in ; their semantics is then defined over coalgebras for , where is the set of variables (cf. also e.g. [28]). We omit discussion of propositional variables in the examples, even in cases like the modal logic of probability that become trivial in the absence of variables; our treatment extends straightforwardly to the case with variables in the manner just indicated.

We view coalgebras as generalized transition systems: the transition function maps a state to a structured set of successors and observations, with the structure prescribed by the signature functor. Thus, the latter encapsulates the branching type of the underlying transition systems. Copointed functors additionally impose local frame conditions that relate a state to the collection of its successors.

Assumption 2.3.

We assume w.l.o.g. that preserves injective maps [2], and even in case , and that is non-trivial, i.e. .

Generalising earlier work (e.g. [17, 19]), coalgebraic modal logic abstractly captures the interpretation of modal operators as polyadic predicate liftings [25, 27],

Definition 2.4.

An -ary predicate lifting () for is a natural transformation

 λ:Qn→Q∘Sop0,

where  denotes the contravariant powerset functor (i.e.  is the powerset , and ), and is defined by .

A coalgebraic semantics for is formally defined as a -structure (over ) consisting of a copointed functor with signature functor and an assignment of an -ary predicate lifting for to every -ary modal operator . When is trivially copointed, we will also call a simple -structure (over ). We fix the notation , , , throughout the paper. The semantics of the modal language is then given in terms of a satisfaction relation between states of -coalgebras and -formulas over . The relation is defined inductively, with the usual clauses for boolean operators. The clause for an -ary modal operator is

 x⊨CL(ϕ1,…,ϕn)⇔ξ(x)∈[[L]]([[ϕ1]]C,…,[[ϕn]]C)

where . We drop the subscripts when clear from the context. Our main interest is in the (local) satisfiability problem over :

Definition 2.5.

An -formula is satisfiable if there exist an -coalgebra and a state in such that . Dually, is valid if for all .

Contrastingly, the semantics of the one-step logic is given in terms of satisfaction relations between elements and one-step formulas over , where is a set and is a -valuation for , i.e. a map . The valuation canonically induces an interpretation of of propositional formulas over . We write if . The relation is then defined by the usual clauses for boolean operators, and

 t⊨1X,τL(ϕ1,…,ϕn)⇔t∈[[L]]([[ϕ1]]0τ,…,[[ϕn]]0τ).

Note in particular that the semantics of the one-step logic does not involve a notion of state transition.

Definition 2.6.

A one-step model over consists of a set , a -valuation for , , and such that . The latter condition is vacuous if is trivially copointed, in which case we omit the mention of . For a one-step formula over , is a one-step model of if .

We recall some basic notation:

Definition 2.7.

We denote the set of propositional formulas over a set , generated by the basic connectives  and , by . We use variables etc. to denote either nothing or . Thus, a literal over  is a formula of the form , with . A (conjunctive) clause is a finite, possibly empty, disjunction (conjunction) of literals. We denote by the set .

In the above notation, the set of one-step formulas over is . The one-step logic may alternatively be presented in terms of pairs of formulas separating out the lower propositional layer:

Definition 2.8.

A one-step pair over consists of formulas and . A one-step model is a one-step model of if and .

In analogy to the equivalence between axioms and one-step rules described in [28], one-step pairs and one-step formulas may replace each other for purposes of satisfiability:

Lemma 2.9.

For every one-step pair over with satisfiable, there exists a -substitution such that the one-step formula is equivalent to in the sense that if then , and if then (and ). Here, denotes the -valuation taking to .

Conversely, we have, for , an equivalent one-step pair over , where decomposes as , with , a -substitution, and , and where is the conjunction of the formulas , . Here, restricting valuations to induces a bijection between one-step models of and one-step models of .

Proof.

The second claim is clear. The first claim is proved as follows. As in [28], let be a satisfying truth valuation for and put if , and otherwise. Then and the formulas , for , are tautologies [28]. Both directions of the first claim now follow straightforwardly. ∎

The coalgebraic approach subsumes many interesting modal logics, including e.g. graded and probabilistic modal logics and coalition logic [29]. Below, we present the most basic examples, the modal logics and , as well as various conditional logics and logics of quantitative uncertainty. The treatment of Elgesem’s modal logic of agency is deferred to Sect. 4.

Example 2.10.
• Modal logic : Let , with a unary modal operator. We define a simple -structure over the covariant powerset functor (i.e.  is powerset, and ) by putting . Naturality of is just the equivalence .

-coalgebras are Kripke frames, and -models are Kripke models. The modal logic of is precisely the modal logic , equipped with its standard Kripke semantics. Contrastingly, a one-step formula over is a propositional combination of atoms of the form , where . For , we have iff . One easily checks that the one-step logic is NP-complete, while the modal logic is PSPACE-complete [20].

• Modal logic : The logic has the same syntax as . Its coalgebraic semantics is a structure over the copointed functor with signature functor , given by

 RX={(A,x)∈PX×X∣x∈A}.

Thus, -coalgebras are reflexive Kripke frames. The interpretation of is defined as for . (Axiomatically, is determined by the non-iterative axiom .)

• Conditional logic : The signature of conditional logic has a single binary modal operator , written in infix notation. Formulas are read as non-monotonic conditionals. The semantics of the conditional logic  [3] is given by a simple structure over the functor given by , with denoting function space and contravariant powerset, cf. Definition 2.4. -coalgebras are conditional frames [3]. The operator is interpreted over  by

 [[⇒]]X(A,B)={f:QX→PX∣f(A)⊆B}.
• Conditional logic : The conditional logic  [3] extends with the rank-1 axiom , referred to as . The semantics of is modelled by restricting the structure for to the subfunctor of defined by

 CfID(X)={f∈Cf(X)∣∀A∈QX.f(A)⊆A}.
• Conditional Logic : The logic  [3] extends with the non-iterative axiom

 (MP)(a⇒b)→(a→b).

(This axiom is undesirable in default logics, but reasonable in relevance logics.) Semantically, this amounts to passing from the functor to the copointed functor with signature functor , defined by

 CfMP(X)={(f,x)∈Cf(X)×X∣∀A⊆X.x∈A⇒x∈f(A)}.
• Modal logics of quantitative uncertainty: The modal signature of likelihood has -ary modal operators for . The terms are called likelihoods. The interpretation of likelihoods varies. E.g. the semantics of the modal logic of probability [8] is modelled coalgebraically by a structure over the (finite) distribution functor , where is the set of finitely supported probability distributions on , and acts as image measure formation. Coalgebras for are probabilistic transition systems (i.e. Markov chains). Likelihoods are interpreted as probabilities; i.e.

 [[∑ni=1ai⋅l(__)≥b]]X(A1,…,An)={P∈DωX∣∑ni=1aiP(Ai)≥b}.

Alternatively, likelihoods may be interpreted as upper probabilities [14], i.e. the functor is replaced by , and in the above definition, is replaced by , where for , . This setting describes situations where agents are unsure about the actual probability distribution. Further alternative notions of likelihood include Dempster-Shafer belief functions and Dubois-Prade possibility measures [15]. An extension of the modal signature of likelihood is the modal signature of expectation [15], where instead of likelihoods one more generally considers expectations . Here, linear combinations of formulas represent gambles, i.e. real-valued outcome functions, where the payoff of is the characteristic function of . The exact definition of expectation depends on the underlying notion of likelihood.

One-step logics of quantitative uncertainty are often considered to be of independent interest. E.g. the one-step logic of probability, i.e. a logic without nesting of likelihoods that talks only about a single probability distribution, is introduced independently [9] and only later extended to a full modal logic [8]. In fact, logics of expectation [15] and the logic of upper probability [14] so far appear in the literature only as one-step logics; the corresponding modal logics are of interest as natural variations of the modal logic of probability.

Convention 2.11.

We assume that is equipped with a size measure, thus inducing a size measure for . For one-step formulas over , we assume w.l.o.g. that . For finite , we assume given a representation of elements as strings of size over some finite alphabet. We do not require that all elements of are representable, nor that all strings denote elements of . When is trivially copointed, we represent only . We require that inclusions induced according to Assumption 2.3 by inclusions into a finite base set preserve representable elements and increase their size by at most .

We make these issues explicit for the above examples:

Example 2.12.
• Modal logics and : For finite, elements of are represented as lists of elements of .

• Conditional logics and : For finite, elements of are represented as partial maps ; such an represents the total map that extends by when is undefined. (The use of partial maps avoids exponential blowup.)

• Conditional logic : For finite, a pair consisting of a partial map and represents the pair , where extends by in case is undefined.

• Modal logics of quantitative uncertainty: Suitable compact representations are described in [9, 14, 15].

3 Polynomially branching shallow models

We now turn to the announced construction of polynomially branching shallow models for modal logics whose one-step logic has a small model property; this construction leads to a PSPACE decision procedure.

Definition 3.1.

We say that has the one-step polysize model property (OSPMP) if there exist polynomials and such that, whenever a one-step pair over has a one-step model , then it has a one-step model such that , is representable with , and iff for all .

In analogy to the transition between rules and axioms described in [28], one-step pairs are interchangeable with one-step formulas. In particular, we have

Proposition 3.2.

The -structure has the OSPMP iff there exist polynomials , such that, whenever a one-step formula over has a one-step model , then it has a one-step model such that iff for all , , and is representable with , where with and a -substitution.

Proof.

Only if: Let be a one-step model of a one-step formula over . By Lemma 2.9, is equivalent to a one-step pair of the form , with as in the statement. By the OSPMP, has a one-step model such that , , and iff for all ; by Lemma 2.9, this model gives rise to a one-step model of with the components , , unchanged.

If: Let be a one-step model of a one-step pair over . By Lemma 2.9, is equivalent to a one-step formula of the form , where is a -substitution. By assumption, has a one-step model such that , , and iff for all . By Lemma 2.9, this model gives rise to a one-step model of with the components , , unchanged. ∎

Both formulations of the OSPMP easily reduce to the case that is a conjunctive clause.

Remark 3.3.

It is shown in [28] that the one-step logic always has an exponential-size model property: a one-step formula over has a one-step model iff it has a one-step model with carrier set .

We are now ready to prove the shallow model theorem.

Definition 3.4.

A supporting Kripke frame of an -coalgebra is a Kripke frame such that for each ,

 ξ(x)∈S0{y∣xRy}⊆S0X

(equivalently ). A state is a loop if .

Theorem 3.5 (Shallow model property).

If has the OSPMP, then has the polynomially branching shallow model property: There exist polynomials , such that every satisfiable -formula is satisfiable at the root of an -coalgebra which has a supporting Kripke frame such that removing all loops from yields a tree of depth at most and branching degree at most , and is representable with .

Definition 3.6.

For and a -valuation , we put

Proof of Theorem 3.5.

Induction over the rank of . If , then evaluates to and hence is satisfied in a singleton -coalgebra , which exists by Assumption 2.3.

Now let . Let be a state in an -coalgebra such that . Let denote the set of subformulas of occuring in within the scope of a modal operator, let be the set of variables , indexed over , and let denote the substitution taking to for all . Let be the conjunction of all literals such that is a subformula of and . (Recall that denotes either nothing or negation.) Moreover, let denote the propositional theory of , i.e. the conjunction of all clauses over such that is -valid.

Then is a one-step model of , where . By the OSPMP, it follows that has a one-step model of polynomial size in such that for all , iff , which in turn is equivalent to .

From this model, we now construct a shallow model for . To begin, note that is -satisfiable for every . For suppose not; then is -valid, hence is a conjunct of . Thus, , in contradiction to the fact that by construction. By induction, we thus have, for every , a shallow model of , where we may assume and , with depth at most . We take as the disjoint union of the over , extended by the state , for which we put .

We have to verify that . We will prove the stronger statement , i.e.

 t⊨1X,θ¯ψ, (1)

where for .

By induction over and naturality of predicate liftings, iff for and for every formula . In particular, for all , i.e.

 y⊨(X,ξ)ρ⟺y∈τ(aρ) (2)

for all . We prove by induction over that

 x0⊨(X,ξ)ρ⟺x0∈τ(aρ), (3)

which in connection with (2) yields

 [[ρ]](X,ξ)∩Y=τ(aρ). (4)

The steps for boolean operations are straightforward. For , we have

 x0⊨(X,ξ)L(ρ1,…,ρn) ⟺ t∈[[L]]Y([[ρi]](X,ξ)∩Y)i=1,…,n=[[L]]Y(τ(aρ1),…,τ(aρn)) ⟺ t⊨1(Y,τ)L(aρ1,…,aρn),

using naturality of in the first step and the inductive hypothesis in the shape of (4) in the subsequent equality. Since , the last statement is equivalent to . By the definition of , this is equivalent to , which in turn is equivalent to by construction of .

By (4) and naturality of predicate liftings, our remaining goal (1) reduces to , which holds by construction.

Finally, we have to establish that the overall branching degree of the model is polynomial in . The model is recursively constructed from polynomial-size one-step models for pairs whose second components are conjunctive clauses over atoms , where is a subformula of . Such conjunctive clauses are of at most quadratic size in (even if subformulas of are represented by pointers into ); this proves the claim. ∎

Remark 3.7.

While it is to be expected that the construction of polynomially branching models depends on a condition like the OSPMP, it does not seem to be the case that the precise formulation of this condition is implicit in the literature (not even for the trivially copointed case). Note in particular that the polynomial bound depends only on the second component of a one-step pair. This is crucial, as the first component of the one-step pair constructed in the above proof may be of exponential size. When we say in the introduction that the OSPMP can be obtained from off-the-shelf results (e.g. [9, 14, 15]), we refer to polynomial-size model theorems in which the polynomial bound depends, in the notation of Proposition 3.2, on , which may be exponentially larger than ; typically, only an inspection of the given proofs shows that the bound can be sharpened to be polynomial in .

The proof of Theorem 3.5 leads to the following nondeterministic decision procedure.

Algorithm 3.8.

(Decide satisfiability of an -formula ) Let have the OSPMP, and let , be polynomial bounds as in Definition 3.1.

• If , terminate successfully if evaluates to , else unsuccessfully. Otherwise:

• Take and as in the proof of Theorem 3.5, and guess a conjunctive clause over containing for each subformula of either or such that propositionally entails .

• Guess a -valuation for and with , where , such that .

• For each , check recursively that is satisfiable.

Since the rank decreases with each recursive call, the above algorithm can be implemented in polynomial space, provided that Step 3 can be performed in polynomial space.

Definition 3.9.

The one-step model checking problem is to check, given a string , a finite set , , and -ary, whether represents some and whether .

This property and the above algorithm lead to a PSPACE bound for the modal logic. Moreover, for bounded-rank fragments, the polynomially branching shallow model property becomes a polynomial size model property, thus leading to an NP upper bound:

Corollary 3.10.

Let have the OSPMP.

• If one-step model checking is in PSPACE, then the satisfiability problem of is in PSPACE.

• If one-step model checking is in , then the satisfiability problem of is in NP for every .

Proof.

1: By Algorithm 3.8.

2: Let and be polynomial bounds on the branching degree of supporting Kripke frames and on the size of successor structures as guaranteed by Theorem 3.5. Let . Then by Theorem 3.5, every satisfiable formula is satisfiable in a model such that and for all , where the second inequality relies also on Convention 2.11. Thus, the entire representation size of the model is bounded by , which is polynomial in . Thus, the following non-deterministic algorithm decides satisfiability of in polynomial time:

• Guess a model of size at most

• Check that is an -coalgebra.

• Check that .

The second step can be performed in polynomial time because one-step model checking is in . The third step can be performed in polynomial time by recursively computing extensions , again because one-step model checking is in . ∎

This generalises results for the modal logics and established in [13].

Example 3.11.
• Modal logics and : One-step model checking for and amounts to checking a subset inclusion and, in the case of , additionally an elementhood; this is clearly in . To verify the OSPMP for , let be a one-step model of a one-step pair over ; w.l.o.g.  is a conjunctive clause over atoms , where . For in , there exists such that . Taking to be the set of these , we obtain a polynomial-size one-step model of , where for all . The construction for is the same, except that the point of the original one-step model is retained in the carrier set , and becomes the point of the small model. By Corollary 3.10, this reproves Ladner’s PSPACE upper bounds for and  [20], as well as Halpern’s NP upper bounds for bounded-rank fragments [13].

• Conditional logic: It is easy to see that one-step model checking for , , and is in . (In particular, deciding whether a given string represents an element of just amounts to checking subset inclusions. Moreover, deciding whether , i.e. whether implies , can be done in polynomial time thanks to the choice of default value for ; cf. Example 2.12.2.)

To prove that has the OSPMP, let be a one-step model of a one-step pair , where w.l.o.g.  is a conjunctive clause . If , fix an element in the symmetric difference of and . Moreover, if is negation, fix . Let be the set of all and all . Let be the -valuation defined by , and let be represented by the partial map taking to for all (this is well-defined by construction of ). Then is a one-step model of . The cardinality of is quadradic in , and the representation size of is polynomial.

Thanks to the choice of default value, this construction of polynomial-size one-step models works also for . The construction for is almost identical, except that the point of is retained in the small one-step model ; here, due to the different choice of default value.

We thus obtain that , , and are in PSPACE (hence PSPACE-complete, as these logics contain  and — in the case of , respectively, as sublogics). This has previously been proved using a detailed analysis of a labelled sequent calculus [22] (the method of [22] yields an explicit polynomial bound on space usage which is not matched by the generic algorithm). The NP upper bound for bounded-rank fragments of , , and arising from Corollary 3.10.2 is, to our knowledge, new.

• Modal logics of quantitative uncertainty: Polynomial size model properties for one-step logics have been proved for the logic of probability [9], the logic of upper probability [14], and various logics of expectation [15]. As indicated in Remark 3.7, the polynomial bounds are stated in the cited work as depending on the size of the entire one-step formula ; however, inspection of the given proofs shows that the polynomial bound in fact depends only on the number of likelihoods or expectations in , respectively, and on the representation size of the largest coefficient. By Proposition 3.2, it follows that the respective logics have the OSPMP. Suitable complexity estimates for one-step model checking are also found in the cited work.

By the above results, it follows that the respective modal logics of quantitative uncertainty are in PSPACE (hence PSPACE-complete, as one can embed  by mapping to ), and in NP when the modal nesting depth is bounded. For the modal logic of probability, a proof of the PSPACE upper bound is sketched in [8]. The PSPACE upper bounds for the remaining cases (e.g. the modal logic of upper probability and the various modal logics of expectation) seem to be new, if only for the reason that only the one-step versions of these logics appear in the literature. Similarly, all NP upper bounds for bounded-rank fragments are, to our knowledge, new. Moreover, the upper bounds extend easily to modal logics of uncertainty with non-iterative axioms, e.g. an axiom which states that the present state remains stationary with likelihood at least .

4 Extended Example: Elgesem’s modal logic of agency

There have been numerous approaches to capturing the notion of agents bringing about certain states of affairs, one of the most recent ones being Elgesem’s modal logic of agency ([7] and references therein, [12]). Modal logics of agency play a role e.g. in planning and task assignment in multi-agent systems (cf. e.g. [4, 18]).

Elgesem defines a logic with two modalities and (in general indexed over agents; all results below easily generalise to the multi-agent case), read ‘the agent brings about’ and ‘the agent is capable of realising’, respectively. The semantics is given by a class of conditional frames (Example 2.10.3), called selection function models in this context. The clauses for the modal operators are

 x⊨Eϕ iffx∈f(w)([[ϕ]])and x⊨Cϕ ifff(w)([[ϕ]])≠∅.

The relevant class of selection function models is defined by the conditions

 \tabularcell@hbox(E1)\tabularcell@hbox$f(x)(X)=∅$\tabularcell@hbox(E2)\tabularcell@hbox$f(x)(A)∩f(x)(B)⊆f(x)(A∩B)$\tabularcell@hbox(E3)\tabularcell@hbox$f(x)(A)⊆A$.

It is shown in [7, 12] that the logic of agency is completely axiomatised by , , , , and . Notably, the agent is incapable of realising what is logically necessary (), i.e. the notion of realising a state of affairs entails actual attributability (this axiom is weaker than previous formulations using avoidability; cf. the baby food example in [7]). Monotonicity is not imposed. The axiom is due to [12].

Most of the information in selection function models (motivated by philosophical considerations in [7]) is irrelevant for the semantics of and : one only needs to know whether is non-empty, and whether it contains . Moreover, the selection function semantics fails to be coalgebraic, as the naturality condition fails for the (generalised) predicate lifting implicit in the clause for . Both problems are easily remedied by moving to the following coalgebraic semantics: put (to represent the cases , , and , respectively), and take as signature functor the -valued neighborhood functor given by (with denoting contravariant powerset). We define the copointed functor as the subfunctor of such that iff for all ,

 \tabularcell@hbox(E1$′$)\tabularcell@hbox$f(X)=⊥$\tabularcell@hbox(E2$′$)\tabularcell@hbox$f(A)∧f(B)≤f(A∩B)$\tabularcell@hbox(E3a$′$)\tabularcell@hbox$f(∅)=⊥$\tabularcell@hbox(E3b$′$)\tabularcell@hbox$f(A)=⊤⟹x∈A$

where and refer to the ordering . We define a structure over for the modal logic of agency by

 [[E]]XA ={f:Q→3∣f(A)=⊤} [[C]]XA ={f:Q→3∣f(A)≠⊥}.
Proposition 4.1.

A formula of the modal logic of agency is satisfiable in a selection function model iff it is satisfiable over .

Proof.

‘Only if:’ Given a selection function model , define an -coalgebra by

 ~f(x)(A)=⎧⎪⎨⎪⎩⊤if x∈f(x)(A)∗if x∉f(x)(A)≠∅⊥if f(x)(A)=∅.

It is clear that is an -coalgebra and that satisfies the same formulas in as in .

’If’: Let be an -coalgebra. We can assume that for all formulas (otherwise, form the coproduct of with itself, so that each state has a twin satisfying the same formulas). We define a selection function model by

 ¯f(x)(A)=⎧⎪⎨⎪⎩Aif f(x)(A)=⊤A−{x}if f(x)(A)=∗∅if f(x)(A)=⊥.

It is clear that satisfies E1–E3. One shows by induction over the formula structure that satsifies the same formulas in as in , with the only non-trivial point being that in the step for the modal operator , one has to note that, by the above assumption, whenever . ∎

To avoid exponential explosion, we represent elements of , for finite, using partial maps . To enforce (E2), we let such an represent the map that maps to the maximum of taken over all sets such that and is defined for all ; when no such sets exist, the maximum is understood to be .

Lemma 4.2.

Let and be as above.

• Whenever is defined, then .

• Let . Then iff

• The pair satisfies (E1) iff is either undefined or equals .

• The pair satisfies (E2).

• The pair satisfies (E3a) iff

• The pair