# Bundled fragments of first-order modal logic: (un)decidability

Quantified modal logic provides a natural logical language for reasoning about modal attitudes even while retaining the richness of quantification for referring to predicates over domains. But then most fragments of the logic are undecidable, over many model classes. Over the years, only a few fragments (such as the monodic) have been shown to be decidable. In this paper, we study fragments that bundle quantifiers and modalities together, inspired by earlier work on epistemic logics of know-how/why/what. As always with quantified modal logics, it makes a significant difference whether the domain stays the same across worlds, or not. In particular, we show that the bundle ∀ is undecidable over constant domain interpretations, even with only monadic predicates, whereas ∃ bundle is decidable. On the other hand, over increasing domain interpretations, we get decidability with both ∀ and ∃ bundles with unrestricted predicates. In these cases, we also obtain tableau based procedures that run in . We further show that the ∃ bundle cannot distinguish between constant domain and increasing domain interpretations.

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## 1 Introduction

In Meaning and Necessity [1], Carnap remarked:

Any system of modal logic without quantification is of interest only as a basis for a wider system including quantification. If such a wider system were found to be impossible, logicians would probably abandon modal logic entirely.

However, it seems that history went exactly the other way around. Compared to the flourishing developments of propositional modal logic in the past decades with successful applications in various other fields, first-order modal logic () is much less studied. In addition to numerous philosophical controversies, is also infamously hard to handle technically: e.g., you often loose good properties of first-order logic and modal propositional logic when putting them together.

Among those technical hurdles, finding useful decidable fragments of has been a major one preventing the use of in computational applications. On the one hand, the decidable fragments of first-order logic have been well mapped out during the last few decades. On the other hand, we have a thorough understanding of the robust decidability of propositional modal logics. However, when it comes to finding decidable fragments of , the situation seems quite hopeless: even the two-variable fragment with one single monadic predicate is (robustly) undecidable over almost all useful model classes [7]. On the positive side, besides the severely restricted one-variable fragment, the only promising approach has so far come from the study of the so-called monodic fragment, which requires that there be at most one free variable in the scope of any modal subformula. Combining the monodic restriction with a decidable fragment of we often obtain decidable fragments of , as Table 1 shows (results mostly hold for the usual frame classes e.g., T, S4, ).

The reason behind this sad tale is not far to seek: the addition of gives implicitly an extra quantifier, over a fresh variable. Thus if we consider the two-variable fragment of , with only unary predicates in the syntax, we can use to code up binary relations and we ride out of the two-variable fragment as well as the monadic fragment of FOL. The monodic restriction confines the use of significantly so that it cannot introduce a fresh variable implicitly.

It is then natural to ask: apart from variable restrictions, is there some other way to obtain syntactic fragments of that are yet decidable ?

One answer came, perhaps surprisingly, from epistemic logic. In recent years, interest has grown in studying epistemic logics of knowing-how, knowing-why, knowing-what, and so on (see [11] for a survey). As observed in [11], most of the new epistemic operators essentially share a unified de re semantic schema of where is an epistemic modality.111Note that the quantifier is not necessarily first-order. For instance, knowing how to achieve means that there exists a mechanism which you know such that executing it will make sure that you end in a state [9]. Here the distinction between and is crucial. It is also observed that such logics are often decidable. This leads to the proposal in [10] of a new fragment of by packing and into a bundle modality, but without any restriction on predicates or the occurrences of variables.

 ϕ::=P¯¯¯x∣¬ϕ∣(ϕ∧ϕ)∣∃x□ϕ

Note that in this language, quantifiers have to always come with modalities. If does not appear in then is simply equivalent to the usual . Such a language suffices to say many interesting things besides knowing-how such as:

• I do not know who killed Mary:

• I know a theorem such that I do not know any proof:

• For each person I think it is possible that I know someone who is a friend of him or her: (note that is )

• I know a public key whose corresponding private key is known to agent but not to me: .

It is shown that this fragment with arbitrary predicates is in fact -complete over increasing domain models. Essentially, the idea is similar to the “secret of success” of modal logic: guard the quantifiers, now with a modality. On the other hand, the same fragment is undecidable over S5 models, and this can be shown by coding first-order sentences in this language using S5 properties.

There are curious features to observe in this tale of (partial) success. The fragment in [10] includes the bundle but not its companion bundle, and considers only increasing domain models. The latter observation is particularly interesting when we notice that S5 models, where the fragment becomes undecidable, force constant domain semantics.

The last distinction is familiar to first-order modal logicians, but might come across as a big fuss to others. Briefly, it is the distinction between a possibilist approach and an actualist approach. In the former, the model has one fixed domain for all possible worlds, and quantification extends over the domain (rather than only those objects that exist in the current world). This is the constant domain semantics. In the latter approach, each possible world has its own domain, and quantification extends only over objects that exist in the current world. In increasing domain semantics, once an object exists in a world , it exists in worlds accessible from .

Given such subtlety, it is instructive to consider more general bundled fragments of , including both and as the natural first step, and study them over constant domain as well as increasing domain models. This is precisely the project undertaken in this paper, and the results are summarized in Table 2.

As we can see, the bundle behaves better computationally than the bundle. For , even the monadic fragment is undecidable over constant domain models: we can encode in this language, qua satisfiability, any first-order logic sentence with binary predicates by exploiting the power of . A straightforward consequence is that the fragment is also undecidable over constant domain models.

On the other hand, we can actually give a tableau method for the and fragment together, similarly as the tableau in [10], for increasing domain models. The crucial observation is that such models allow us to manufacture new witnesses for and formulas, giving considerable freedom in model construction, which is not available in constant domain models.

Indeed, the well-behavedness of the bundle is further attested to by the fact that it is decidable over constant domain models as well. So constant domain is not the culprit for undecidability of this fragment over S5 models. Actually, we can show the bundle does not distinguish increasing domain models and constant domain models.

The paper is structured as follows. After formal definitions of bundled fragments, we present undecidability results and then move on to tableaux procedures for the decidable fragments. We then show that the validities of over increasing domain are exactly the same as its validities over constant domain models, and end the paper with a re-look at mapping the terrain of these fragments.

## 2 The bundled fragment of FOML

Let Var be a countable set of variables, and P be a fixed set of predicate symbols, with denoting the set of all predicate symbols of arity . We use to denote a finite sequence of (distinct) variables in Var. We only consider the ‘pure’ first order unimodal logic: that is, the vocabulary is restricted to Var (no equality and no constants and no function symbols).

Given Var and P, the bundled fragment of denoted by is defined as follows:

 ϕ::=P¯¯¯x∣¬ϕ∣(ϕ∧ϕ)∣∃x□ϕ∣∃x◊ϕ

where , . We denote the fragment to be the formulas which contains only formulas and which contains only formulas.

(True, False, Or and Implies) are defined in the standard way. as is the dual of , and defined by is the dual of . With both bundles we can say, in an epistemic setting, that for each country I know its capital

The free and bound occurrences of variables are defined as in first-order logic, by viewing and as quantifiers. We denote as the set of free variables of . We write if all the free variables in are included in . Given a formula and , we write for the formula obtained by replacing every free occurrence of by . A formula is said to be a sentence if it contains no free variables.

The semantics is the standard increasing domain semantics of . The relation is specialized to the fragment.

An (increasing domain) model for is a tuple where, is a non-empty set of worlds, is a non-empty domain, , assigns to each a non-empty local domain s.t. implies for any , and such that assigns to each -ary predicate on each world an -ary relation on .

We often write for . A constant domain model is one where for any . A finite model is one with both finite and finite.

Consider a model , . To interpret free variables, we also need a variable assignment . Call relevant at if for all . The increasing domain condition ensures that whenever is relevant at and we have , then is relevant at as well. (In a constant domain model, every assignment is relevant at all the worlds.)

Given , , and an assignment relevant at , define inductively as follows:

 M,w,σ⊨P(x1⋯xn)⇔(σ(x1),⋯,σ(xn))∈ρ(P,w)M,w,σ⊨¬ϕ⇔M,w,σ⊭ϕM,w,σ⊨(ϕ∧ψ)⇔M,w,σ⊨ϕ and% M,w,σ⊨ψM,w,σ⊨∃x□ϕ⇔there is some d∈δ(w) such that M,v,σ[x↦d]⊨ϕ for all v s.t.\ wRvM,w,σ⊨∃x◊ϕ⇔there is some d∈δ(w) and some v∈W  such that wRv and M,v,σ[x↦d]⊨ϕ

where denotes another assignment that is the same as except for mapping to .

It is easily verified that is defined only when is relevant at . In general, when considering the truth of in a model, it suffices to consider , assignment restricted to the free variables occurring free in . When and , We write to denote for any such that and for all we have . Hence when is a sentence, we can simply write .

We say is valid, if is true on any w.r.t. any relevant at . is satisfiable if is not valid.

## 3 Undecidability results

In this section we prove that the satisfiability problem for the fragment with constant domain semantics is undecidable even when the atomic predicates are restricted to be unary. We prove this by reduction from the satisfiability problem for with one arbitrary binary predicate, which is known to be undecidable (from [4]).

That full with constant domain semantics is undecidable even when the atomic predicates are only unary is well known; it was shown by Kripke[6]. That we need only 2 variables along with propositions to make Monadic undecidable was shown by Gabbay and Shehtman [3]. That propositions can be eliminated was observed by Kontchakov, Kurucz and Zakharyaschev [5].

Consider , the first order logic with only variables as terms and no equality, and the single binary predicate . To translate sentences to formulas, we use two unary predicate symbols in the latter. The main idea is that the atomic formula is coded up as the formula , where is a new variable, distinct from and .222This is similar to the approach used by Kripke [6], specialized for the fragment. In the model constructed, it will turn out that . But in which world is this to be enforced ? We will enforce that all worlds at a specific modal depth interpret in the same way, thus ruling out any ambiguity, crucially using the bundle and constant domain semantics.

For any quantifier free formula , we define the translation of to formula inductively as follows.

• , where is distinct from and .

• and .

Now consider an sentence (having no free variables) and presented in prenex form: where is quantifier free. We define to be the conjunction of the following three sentences:

• where if and if .

• .

• .

Of these, ensures that the formulas are interpreted over the same domain, and that the meaning of is given as in a world at depth . ensures that all worlds at depth agree on and and hence on . asserts that every path can be extended until depth , one never gets stuck earlier.

The role of dummy variables

and in and depth in may need an explanation. First note that the interpretation for is collected at depth , and the extra depth is because the extra quantification in the coding of added successor worlds. Now we need these two variables to refer to elements of and at depth , but in the bundled fragment, any variable comes packed with a modality. Thus we get depth . Further we could not use variables from (which might be quantified existentially), so fresh variables are needed.

The sentence is satisfiable iff the sentence is constant domain satisfiable.

We sketch the proof here, the details are given in Appendix A. Fix , where is quantifier free. To prove , assume that is satisfiable. Let be some domain such that where is the interpretation for .

Define where:

• .

• .

• for all .

• For all and and define and for all and .

By construction, is a model that is a path of length originating from until at which point we have a tree of depth , with children , one for each . Therefore, it is easy to see that .

Note that is a constant domain model. Further, it can be easily checked that iff . Thus iff . Hence a routine induction shows that for any quantifier free formula , iff . Further, since is a path model until and there is a path of length starting from , we see that . We then show that , which would complete the forward direction of the proof. This is proved by reverse induction on . The base case, when , follows from our assertion above on the interpretation of at . For the induction step, we crucially use the fact the model constructed is a path and hence and coincide along the path.

To prove , suppose that is satisfiable, and let be a constant domain model such that . Without loss of generality, we can assume to be a tree rooted at , and ensures that every path in it has length at least . Let be any world at height . Define . For world at height , define . Since , we see that , for all at height . Hence we unambiguously define , thus defining the first order model . We now claim that the formula is satisfied in this model. The definition of ensures that the atomic formulas are correctly satisfied. We proceed by subtree induction noting that all children of a node satisfy subformulas equivalently (which is needed for formulas).

## 4 Decidability results

Having seen that the fragment is undecidable over constant domain models, and noted that the bundle is decidable over increasing domain models ([10]), it is natural to wonder whether the problem is decidable with the bundle or constant domain semantics, or both. In this section, we show that it is indeed the combination that is the culprit, by showing that relaxing either of the conditions leads to decidability. First, we show that the full fragment is decidable over increasing domain models, and then show that the bundle is decidable over constant domain models.

### 4.1 Increasing domain models

We consider formulas given in negation normal form (NNF):

 ϕ::=P¯¯¯x∣¬P¯¯¯x∣(ϕ∧ϕ)∣(ϕ∨ϕ)∣∃x□ϕ∣∃x◊ϕ∣∀x□ϕ∣∀x◊ϕ

Formulas of the form and are literals. Clearly, every -formula can be rewritten into an equivalent formula in NNF.

We call a formula clean if no variable occurs both bound and free in it and every use of a quantifier quantifies a distinct variable. Note that every -formula can be rewritten into an equivalent clean formula. (For instance, and are unclean formulas, whereas and are their clean equivalents.)

We define the following tableau rules for all formulas in NNF. The tableau is a tree structure where is a finite set, is a rooted tree and is a labelling map. Each element in is of the form , where , is a finite set of formulas and is a finite set. The intended meaning of the label is that the node constitutes a world that satisfies the formulas in with the ‘assignment’ , with each variable in denoting one that occurs free in and as we will see, the interpretation will be the identity.

A rule specifies that if a node labelled by the premise of the rule exists, it can cause one or more new nodes to be created as children with the labels as given by the completion of the rule.

Tableau rules

 w:ϕ1∨ϕ2,Γ,Fw:ϕ1,Γ,σ∣w:ϕ2,Γ,F (∨)  w:ϕ1∧ϕ2,Γ,Fw:ϕ1,ϕ2,Γ,F(∧) Given n1,m1≥1,n2,m2,s≥0: (BR) where i∈[1,n1],k∈[1,m1],y∈F′ Given n2,m2≥1; s≥0: w:∃y1□β1,⋯,∃yn2□βn2,∀z′1□ψ1,⋯,∀z′m2□ψm2,r1…rs,Fw:r1…rs,F (END)

where and (the literals).

The rules are standard. The rule says that in the absence of any formulas, with , the branch does not need to be explored further, only the literals remain. Further, note that there is an implicit ordering on how rules are applied: insists on the label containing no top level conjuncts or disjuncts, and hence may be applied only after the and rules have been applied as many times as necessary.

The rule looks complicated but asserts standard modal validities, but with multiplicity. To see how it works, consider a model , a world and assignment such that . Then for some domain element , we have a successor world such that , where and . Further if then for every domain element , we have a successor world such that , where and . When the domain elements we use are themselves variables, they can be substituted into formulas so we could well write . The rule achieves just this, but has to do all this simultaneously for all the quantified formulas at the node “in one shot”, and has to keep the formulas clean too.

We need to check that the rule is well-defined. Specifically, if the label in the premise contains only clean formulas, we need to check that the label in the conclusion does the same. To see this, observe the following, with being the set of clean formulas in the premise. Let stand for any modality.

• Note that if and are both in , with any quantifier, then and neither occurs free in nor occurs free in , also or do not contain any subformula that quantifies over or .

• Hence, in the conclusion of , every substitution of the form or results in a clean formula, since occurs free in and does not occur at all in and similarly for .

Thus, maintaining ‘cleanliness’ allows us to treat existential quantifiers as giving their own witnesses. The ‘increase’ in the domain is given by the added elements in in the conclusion. Note that with each node creation either the number of boolean connectives or the maximum quantifier rank of formulas in the label goes down, and hence repeated applications of the tableau rules must terminate, thus guaranteeing that the tableau generated is always finite.

A tableau is said to be open if it does not contain any node such that its label contains a literal as well as its negation. Given a tableau , we say a node is a branching node if it is branching due to the application of . We call the last node of , if it is a leaf node or a branching node. Clearly, given any label appearing in any node of a tableau , the last node of uniquely exists. If it is a non-leaf node, every child of is labelled for some .

Let denote the last node of in tableau and let . If it is a non-leaf node, then it is a branching node with rule applying to it with as its conclusion. We let denote the set in this case and otherwise.

For any clean -formula in NNF, there is an open tableau from where , where does not appear in , iff is satisfiable in an increasing domain model.

Let be any tableau starting from where is clean. We observe that for any node in , we have the following. If and occurs in a formula in then every occurrence is free. Further, every variable occurring free in a formula in is in . These are proved by induction on the structure of using the fact that the rule , when applied to clean formulas, results in clean formulas.

To prove the theorem, given an open tableau with root node labelled by , we define where: ; iff for some ; ; iff , where . Clearly, if then , and hence is indeed an increasing domain model.

Moreover is well-defined due to openness of . We now show that is indeed a model of , and this is proved by the following claim.

#### Claim.

For any tree node in if and if then . (Below, we abuse notation and write for where .)

The proof proceeds by subtree induction on the structure of . The base case is when the node considered is a leaf node and hence it is also the last node with that label. The definition of ensures that the literals are evaluated correctly in the model.

For the induction step, the cases for the conjunction and disjunction rules are standard. Now consider the application of rule at a branching node with label . Let

 Γ={∃xi◊αi∣i∈[1,n1]}∪{∃yj□βj∣j∈[1,n2]}∪{∀zk◊ϕk∣k∈[1,m1]}∪{∀z′l□ψl∣l∈[1,m2]}∪{r1…rs}.

By induction hypothesis, we have that for every , and for every and , , where .

Note that . We need to show that for each . Every such is either a literal or a bundle formula. The assertion for literals follows from the definition of . For we have the successor where is true. Similarly for every and we have the successor where is true.

Now for the case : by induction hypothesis, for all successors of where is either empty or we have . By cleanliness of , for all and for all we have that are not free in . Hence for each . Since we have .

The case is similar. By induction hypothesis, we have for every and again by cleanliness of , for all and for all we note that are not free in . Thus for all . Hence .

Thus the claim is proved and hence it follows that .

Completeness of tableau construction:
We only need to show that all rule applications preserve the satisfiability of the formula sets in the labels. This would ensure that there is an open tableau since satisfiability of formula sets ensures lack of contradiction among literals. It is easy to see that the rules and preserve satisfiability. If one of the conclusions of the rule is satisfiable then so is the premise. It remains only to show that preserves satisfiability. Consider a label set of clean formulas at a branching node. Let be satisfiable at a model , and an assignment such that for all and .

By the semantics, we have the following: (A): There exist and successors of such that . (B): There exist such that for all if then . (C): For all there exist , successors of such that . (D): For all and for all if then .

By cleanliness of , each is free only in and is not free in any for and for all . Similarly is free only in and is not free in any for and for all . Thus, due to (B) and (D), we can rewrite (A) and (C) as: (A’): There exists some and for all such that there exist and , successor of such that
.

(C’): there exists and for all such that for all there exist , successors of such that .

Now all the nodes in the conclusion of the rule have formulas as described in type A’ or C’ and are hence satisfiable.

This proves the theorem, offering us a tableau construction procedure for every formula: we have an open tableau iff the formula is satisfiable. Now note that the tableau is not only of depth linear in the size of the formula, but also that subformulas are never repeated across siblings. Hence an algorithm can explore the tableau depthwise and reuse the same space when exploring other branches. The techniques are standard as in tableau procedures for modal logics. The extra space overhead for keeping track of domain elements is again only linear in the size of the formula. This way, we can get a -algorithm for checking satisfiability. On the other hand, the lower bound for propositional modal logic applies as well, thus giving us the following corollary.

Satisfiability of -formulas is -complete.

### 4.2 Constant domain models

We now take up the second task, to show that over constant domain models, the culprit is the bundle, by proving that the satisfiability problem for the is decidable over constant domain models. [10] already showed decidability of the over increasing domain models. Taken together, we see that the fragment is computationally robust.

The central idea behind the tableau procedure in the previous section was the use of existential quantifiers to offer their own witnesses, and cleanliness of formulas ensures that these are new every time they are encountered. This works well with increasing domain models, but in constant domain models, we need to fix the domain right at the start of the tableau construction and use only these elements as witnesses. Yet, a moment’s reflection assures us that we can give a precise bound on how many new elements need to be added for each subformula of the form

, and hence we can include as many elements as needed at the beginning of the tableau construction.

Let stand for the finite set of subformulas of . Given a clean formula in NNF, for every let . Now, cleanliness has its advantages: every subformula of a clean formula is clean as well. Hence, when and are both in , . Similarly, when and , again .

Fix a clean formula in NNF with modal depth . For every define to be the set of fresh variables , and let , be the set of new variables to be added. Note that when . Fix a variable not occurring in . Define does not occur in . Note that is non-empty.

The tableau rules are given by:

 w:ϕ1∨ϕ2,Γ,Cw:ϕ1,Γ,C∣w:ϕ2,Γ,C (∨)  w:ϕ1∧ϕ2,Γ,Cw:ϕ1,ϕ2,Γ,C(∧) Given n,s≥0; m≥1: w:∃x1□ϕ1,…,∃xn□ϕn,∀y1◊ψ1,…,∀ym◊ψm,r1…rs,C⟨(wvyyi:{ϕj[xkjj/xj]∣1≤j≤n},ψi[y/yi],C′)⟩where y∈Dθ,i∈[1,m] (BR) Given n≥1,s≥0: w:∃x1□ϕ1,⋯,∃xn□ϕn,r1,⋯rs,Cw:r1⋯rs,C (END)

where and where is the smallest number such that and .

Note that the rule starts off one branch for each , since the connective requires this over the fixed constant domain . keeps track of the variables used already along the path from the root till the current node. These are now fixed, so the witness for is picked from the remaining variables in . Note that the variables in are introduced only by applying . Since is the modal depth, we always have a fresh to choose.

The notion of open tableau is as before, and the following observation is very useful:

The rule preserves cleanliness of formulas: if a tableau node is labelled by , is clean, and a child node labelled is created by then is clean as well.

An important corollary of this proposition is that for all , at any tableau node all occurrences of in are free. Therefore, for any formula of the form in the conclusion of the rule, is free and does not occur at all in .

For any clean -formula in NNF, there is an open constant tableau from iff is satisfiable in a constant domain model.

The structure of the proof is very similar to the earlier one, but we need to be careful to check that sufficient witnesses exist as needed, since the domain is fixed at the beginning of tableau construction. The proposition above, that the rule preserves cleanliness of formulas, does the bulk of the work. The details are presented in Appendix B.

The complexity of the decision procedure does not change from before, since we add only polynomially many new variables.

The satisfiability problem for -formulas over constant domain models is -complete.

## 5 Between Constant Domain and Increasing Domain

We now show that the fragment cannot distinguish increasing domain models and constant domain models. Note that in this distinction is captured by the Barcan formula ; but this is not expressible in .333However, with equality added in the language we can distinguish the two by a formula. We can also accomplish this in the fragment:

The tableau construction for the fragment over increasing domain models is a restriction of the one in the last section, and was presented in [10].

Given :