Propositional modal logic with implicit modal quantification
Propositional term modal logic is interpreted over Kripke structures with unboundedly many accessibility relations and hence the syntax admits variables indexing modalities and quantification over them. This logic is undecidable, and we consider a variable-free propositional bi-modal logic with implicit quantification. Thus asserts necessity over all accessibility relations and is classical necessity over some accessibility relation. The logic is associated with a natural bisimulation relation over models and we show that the logic is exactly the bisimulation invariant fragment of a two sorted first order logic. The logic is easily seen to be decidable and admits a complete axiomatization of valid formulas. Moreover the decision procedure extends naturally to the ‘bundled fragment’ of full term modal logic.
Keywords:Term modal logic Implicitly quantified modal logic Bisimulation invariance Bundled fragment
Propositional multi-modal logics[4, 14] are used extensively in the context of multi-agent systems, or to reason about labelled transition systems. In the former case, might refer to knowledge or belief of agent that holds. In the latter case, may assert the existence of an -labelled transition from the current state to one in which holds. Such applications include epistemic reasoning [7, 6], games, system verification[5, 1] and more.
In either of the settings, the indices of modalities come from a fixed finite set. However, the applications themselves admit systems of unboundedly many agents, or infinite alphabets of actions. The former is the case in dynamic networks of processes, and the latter in the case of systems handling unbounded data. In fact, the set of agents relevant for consideration may itself be dynamic, changing with state.
Such motivations naturally lead to modal logics with unboundedly many modalities, and indeed quantification over modal indices. Grove and Halpern [12, 11] discuss epistemic logics where the agent set is not fixed and the agent names are not common knowledge. Khan et al.  use unboundedly many modalities and allow quantification over them to model information systems in approximation spaces. Other works on indexed modalities include Passy and Tinchev , Gargov and Goranko , Blackburn .
Term Modal logic(), introduced by Fitting, Voronkov and Thalmann offers a natural solution to these requirements. It extends first order logic with modalities of the form where is a variable (and hence can be quantified over). Thus we can write a formula of the form: . Kooi () considers the expressivity of in epistemic setting. Wang and Seligman () introduce a restricted version of where we have assignments in place of quantifiers (formulas of the form where is a constant, whose interpretation as an agent will be assigned to ).
Note that extends first order logic, and hence its satisfiability problem is undecidable. In  we prove that the problem is undecidable even when the atoms are restricted to boolean propositions (). Hence the question of finding decidable fragments of is well motivated.
In  we prove that the monodic fragment of is decidable. The monodic fragment is a restriction allowing at most one free variable within the scope of a modality. i.e, every subformula of the form has .
Orlandelli and Crosi  consider two decidable fragments: When quantifier occurrence is restricted to the form: (denoted by ); Quantifiers appear in a guarded form: and (and their duals). The first of these fragments is inspired by  which cosniders a similar fragment of first order epistemic logic.
Shtakser () considers a more general guarded fragment (with propositional atoms) of the form and where is quantified over subsets of indices and is interpreted appropriately.
These fragments are semantically motivated from their interest in the epistemic logic to model the notion of ‘everyone knows’ and ‘someone knows’ and community knowledge(ex: All eye-witnesses know who killed Mary).
Note that when modalities and quantifiers are ‘bundled’ together and atomic formulas are propositional, can be replaced by a variable free modality , and similarly by . In some sense this is the most natural variable free fragment of with modalities being implicitly quantified. This is the logic studied in this paper.
Just as propositional modal logic is the bisimulation-invariant fragment of first order logic, we show that is the bisimulation-invariant fragment of an appropriate two -sorted first order logic. The notion of bisimulation needs to be carefully re-defined to account for quantification over edge labels. Other natural questions on such as decidability of satisfiablity and complete axiomatization of valid formulas are answered easily. Interestingly, the natural tableau procedure for the logic can be extended to the ‘bundled fragment’ of with predicates of arbitrary arity, by an argument similar to the one developed in  (for a ‘bundled fragment’ of first order modal logic).
2 The logic
We start with , the propositional fragment of Term-Modal logic. Since we will only study its variable free fragment later, we consider here only the pure vocabulary (no constant and function symbols) with only variables as terms and without equality.
Definition 1 ( syntax)
Let be a countable set of variables and be a countable set of propositions. The syntax of is given by:
where and .
The boolean operators are defined in the standard way. The dual operators for quantifiers and modalities are given by and . The notion of free variables and modal depth are standard.
In the semantics, unlike classical modal logics, the agent set is not fixed, but specified along with the structure. Thus the Kripke frame for is given by where is a set of worlds, is a potential set of agents and . The agent dynamics is captured by a function ( below) that specifies, at any world , the set of agents alive (or meaningful) at . Then coherence demands that whenever , we have that : only an agent alive at can consider accessible.
A monotonicity condition is imposed on the accessibility relation as well: whenever , we have that . This is required to handle interpretations of free variables. Hence the models are called ‘increasing agents’ models. For more details on this restriction, refer [8, 9].
Definition 2 ( structure)
An (increasing agent) model for is a tuple where, is a non-empty set of worlds, is a non-empty set of agents, , assigns to each a non-empty local agent set s.t. implies for any , and .
To interpret free variables, we need a variable assignment function (interpretation) . Call relevant at for a formula if for all . The increasing agent condition ensures that whenever is relevant at for and we have , then is relevant at for all subformulas of .
Definition 3 ( semantics)
Given a model , a formula , , and an interpretation that is relevant at for , define inductively as follows:
where denotes an interpretation that is the same as except for mapping to .
Note that is inductively defined only when is relevant at . A formula is satisfiable, if there is some and some and an interpretation which is relevant at for such that . Also, is valid if is not satisfiable. In , we prove that the satisfiability problem for is undecidable.
As observed in the previous section, we consider the variable free fragment of , with implicit modal quantification ().
Definition 4 ( syntax)
Let be a countable set of propositions. The syntax of is given by:
Since there are no variables in , the models are simpler, and we can fix an implicit set of agents for the model with live subsets at states.
Definition 5 ( structure)
An structure is given by where is a non-empty set of worlds, is a non-empty countable index set and where each and is the valuation function.
Note that could be finite or countably infinite. Hence we assume to be some initial segment of or itself. Hence, we often denote the model as when is clear from the context. The semantics is defined naturally as follows:
Definition 6 ( semantics)
Given a model , a formula , , define inductively as follows:
The formula is satisfiable if there is some model and such that . A formula is said to be valid if is not satisfiable. Table 1 gives a complete axiom system for the valid formulas of .
In the sequel we adopt the following covention. Given any model , and a formula of the form , if and is the corresponding witness then we write (similarly we have ).
|All instances of propositional validities.|
The axiom include all instances of propositional validities. is -like’ axiom for operator. The axiom describes the interactions between and operators. The inference rules () is standard and the rules (Nec), (Nec) mimic the necessitation rule of classical modal logics . The (Nec) is sound since is non-empty. Note that the axioms are similar to that in , except for (Nec) and (Nec). This is due to the fact that there is no notion of names in as supposed to the logic considered in .
is sound and complete for .
The proof of completeness is by construction of canonical model for any consistent formula. The details are presented in Appendix A.
3 bisimulation and elementary equivalence
Modal logics are naturally associated with bisimulations. If two pointed models are bisimilar, the related worlds agree on propositions and satisfy the so-called “back and forth” property (). However, when we come to , since the agent set is not fixed, we need to have the notion of both ‘world bisimilarity’ and ‘agent bisimilarity’. Towards this, in , we introduce a notion of bisimulation for propositional term modal logic and show that it preserves formulas. Similar definitions of bisimulations for first order modal logics can be found in [2, 23].
Now we introduce the notion of bisimulation for . Here the idea is that two worlds are bisimilar if they agree on all propositions and every index in one structure has a corresponding index in the other. The following definition of bisimulation formalizes the notion of ‘corresponding index’.
Given two models and , an -bisimulation on them is a non-empty relation such that for all the following conditions hold:
For all there is some such that for all there is some such that .
For all there is some such that for all there is some such that .
For all and for all there is some and some such that .
For all and for all there is some and some such that .
Given two models and we say that are bisimilar if there is some bisimulation on the models such that and denote it . Also, we say if they agree on all formulas i.e, for all , iff .
Theorem 3.1 (Bisimulation preserves formula equivalence)
For any two models and and any and ,
if then .
Let which means there is some bisimulation such that . We need to show that for any we have iff .
We prove this for all by induction on structure of . The base case and boolean cases are routine.
For the case : Suppose , we need to prove that . Since , there is some such that . Now let be the witness for for condition (forth). We claim that . Suppose not; then and hence there is some such that . Since was the witness for for (forth) condition, there is some such that . By induction hypothesis, which contradicts . The other direction is proved symmetrically using (back) condition.
For the case : Suppose then there is some and some such that and . By condition (forth) there is some and some such that . By induction hypothesis and hence . The other direction is symmetrically argued using back) condition.
Now we prove that the converse holds over image finite models with finite index set . is said to be (index, image) finite if is finite and is finite for all and .
Theorem 3.2 (Formula equivalence corresponds to bisimulation over image finite models)
Suppose and are (index,image) finite models then
follows from Theorem 3.1.
For suppose , then define . Note that . Hence it suffices to show that is indeed an bisimulation. For this, choose any . Clearly holds since agree on all propositions. Now we verify the other conditions:
Now suppose that the (forth) condition does not hold. Then there is some such that for all there is some (*) such that and for all we have . Let and let be the corresponding (*) for every .
Also let -successors of be . By above argument, we have for all and . Hence for every and every there is a formula such that but . Now consider the formula . Note that for all and for all -successors we have and hence which implies . On the other hand for every at we have and hence which contradicts .
The (back) condition is argued symmetrically.
Suppose that the (back) condition does not hold. Then there
is some and some such that
for all and for all we have
. Let and let
be the set of all successors of .
Since is (index, image) finite, let . By
above argument, for every there is a formula such that
and . Hence
consider . Clearly
(with and as witnesses). On
the other hand, for any successor of since
we have which contradicts
our assumption that and satisfy the same formulas.
The (forth) is argued symmetrically.
An important consequence of the theorem above is that we can confine ourselves to tree models for formulas, since it is easily seen that an model is bisimilar to its tree unravelling.
Given a tree model we define its restriction to level in the obvious manner: is simply the same as upto level and the remaining nodes in are ‘thrown away’.
We can now sharpen the result above: we can define a notion of -bisimilarity and show that it preserves formulas with modal depth at most .
Given two tree models and , and in , in , we say and are -bisimilar if .
For , we say and are -bisimilar if the following conditions hold:
For all there is some such that for all there is some such that and are -bisimilar.
For all there is some such that for all there is some such that and are -bisimilar.
For all and for all there is some and some such that and are -bisimilar.
For all and for all there is some and some such that and are -bisimilar.
We can now speak of an -bisimulation relation between models and speak of models being -bisimilar, and employ the notation . Clearly, for tree models iff .
A routine re-working of the proof above shows that when two tree models are -bisimilar, they satisfy the same formulas of modal depth at most . That is, we have . We can go further and show that every -bisimulation class is represented by a single formula of modal depth at most . For this, we assume (as is customary in modal logic), that we have only finitely many atomic propositions.
Suppose is a finite set, then for any and for any there is a formula of modal depth such that for any iff .
Note that follows from Theorem 3.1 specialized to -bisimulation. For the other direction, the proof is by induction on . For , since is finite, is the required formula.
Let . For the induction step, the characteristic formula is given by:
Note that the formula remains finite even if is infinite or the number of successors of is infinite since inductively there are only finitely many characteristic formulas of depth . Showing that these formulas capture -bisimulation classes is routine, working through the conditions for bisimulation carefully.
4 Bisimulation games and invariance theorem
Like every propositional modal logic, is also a fragment of first order logic. However, implicit quantification over domain elements in needs to be made explicit as well as quantification over worlds. Since these serve different purposes in the semantics, we use a two sorted first order logic.
Definition 9 ( syntax)
Let and be two countable and disjoint sorts of variables and a ternary predicate. The two sorted , corresponding to is given by:
where is the corresponding monadic predicate for every and and .
A structure is given by where is the two sorted domain and are interpretations with and where . The semantics is defined for in the standard way where the variables in range over the first sort () and variables of range over second ().
Given a structure the corresponding structure is given by where iff and iff . Similarly given any structure, it can be interpreted as an structure. Thus there is a natural correspondence between structures and structures. For any structure let the corresponding structure be denoted by .
Definition 10 ( to translation)
The translation of into a parametrized by is given by:
For any formula and any structure
Hence can be translated into with variables of sort and one variable of sort. Given two models and , the notion of bisimulation naturally translates to bisimulation over the corresponding models and .
Now we state the van Benthem type characterization theorem: bisimulation invariant formulas can be translated back into . We say that is bisimulation invariant if for all we have iff . We can similarly speak of being -bisimulation invariant as well. Also, is equivalent to some formula if there is some formula such that for all we have iff .
Let with one free variable . Then is bisimulation invariant iff is equivalent to some formula.
Note that follows from Theorem 3.1. To prove it suffices to show that if is bisimulation invariant then, for some it is -bisimulation invariant, since we have already shown in the last section that -bisimulation classes are defined by formulas.
Towards proving this, we introduce a notion of locality for formulas. For any tree model and let be the corresponding model of restricted to depth.
We say that a formula is -local if for any tree model , iff .
For any formula which is bisimulation invariant with then is -local for where where is the quantifier rank of sort in and is the quantifier rank of in .
Assuming this lemma, consider an formula which is bisimulation invariant. It is -local for a syntactically determined . We now claim that is -bisimulation invariant. To prove this, consider . We need to show that iff .
Suppose that . By locality, . Now observe that . By bisimulation invariance of , . But then again by locality, , and we are done.
Thus it only remains to prove the locality lemma. For this, it is convenient to consider the Ehrenfeucht-Fraisse () game for . In this game we have two types of pebbles, one for and the other for .
The game is played between two players Spoiler() and Duplicator() on two structures. A configuration of the game is given by where is a finite string and similarly .
Suppose the current configuration is . In a round, places a pebble on some sort in one of the structure and responds by placing a pebble on a sort in the other structure. In a round, similarly picks one structure and places an pebble on some sort and responds by placing an pebble on some sort in the other structure. In both cases, the new configuration is updated to where and are the new elements(either or sort) picked in the corresponding structures.
A round game is one where many pebbles of type are used and many pebbles of type is used. Player wins after if after rounds, if is the mapping forms a partial isomorphism over and . Otherwise wins.
It can be easily shown that has a winning strategy in the round game over two structures iff they agree on all formulas with quantifier rank of sort and quantifier rank of sort .
Let be any tree structure. To prove lemma 2, we need to prove that iff .
Let and be disjoint copies of and . Note that inclusion relation over and forms a bisimulation. Also note that continues to be a bisimulation over the disjoint union of and . Moreover, notice that is bisimlar to and is bisimilar to .
Now since is bisimulation invariant, it is enough to show that has a winning strategy in the game starting from .
The winning strategy for is to ensure that at every round the critical distance is respected:
If places pebble on a sort which is within of an already pebbled pebble, plays according to a local isomorphism in the - neighbourhoods of previously pebbled elements (exists since and ); if places a pebble on which is beyond distance from all pebbles previously used, then, responds in a fresh isomorphic copy of type or correspondingly (again, it is guaranteed to exists since previously at most would have been used).
If decides to use an pebble and places it on some sort in one structure, then responds by placing an pebble on in the mirror copy in the other structure, where by mirror copy we mean : for or in then the mirror copy in the other structure is itself and the original and are mirror copies of each other.
5 Satisfiability problem
The satisfiability problem for can be solved by sharpening the completeness proof of the axiom system by showing that every consistent formula is satisfied in a model of bounded size. Indeed, a decision procedure can be given along the lines of Grove and Halpern . However, we give a tableau procedure for which is instructive, and as we will observe later, neatly generalizes to more expressive logics.
Given a formula , we set where , the set of subfomulas of is defined in the standard way. This forms the index set where and act as witnesses for the corresponding formulas.
We construct a tableau 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 . The intended meaning of the label is that the node constitutes a world that satisfies the formulas in and is the incoming label edge of .
The tableau rules for are inspired from the tableau procedure for the bundled fragment of first order modal logic introduced in . The and tableau rules are standard. For the modalities, the intuition for the corresponding tableau rule is the following: Suppose that we are in an intermediate step of tableau construction when we have formulas to be satisfied at a node . For this, first we need to add a new successor node where holds; this new node inherits not only but also . Also, we need a successor which inherits and . Finally for each we need a -successor which also inherits .
The tableau rule extends this idea when there are multiple occurrences of each kind of formulas above. In general if the set of formulas considered at node where and . Let . The rule is given as follows:
From an ‘open tableau’ we can construct a model for , along the lines of . Conversely it can be proved that every satisfiable formula has an open tableau.
This tableau construction can be extended to the ‘bundled fragment’ of full where we have predicates of arbitrary arity and the quantifiers and modalities occur (only) in the form and (and their duals). The proof follows the lines of .
We have studied the variable-free fragment of , with implicit modal quantification. We could also consider more forms of implicit quantification such as and modalities, though there is no obvious semantics to them. These logics are the obvious variable free versions of monadic ‘bundled’ fragments of . One could consider a similar exercise for ‘bundled’ fragments of first order modal logic (). As  shows, this is a decidable logic for increasing domain semantics.
Our study suggests that there are other forms of implicitly quantified modal logics. For instance, is there an implicit hybrid version of the logic studied by Wang and Seligman?
A natural question is the delimitation of expressiveness of these logics: which are the properties of models expressed only by or only by modalities ? How does nesting of these modalities increase expressive power ? We believe that the model theory of implicit modal quantification may offer interesting possibilities for abstract specifications of some infinite-state systems. However, for such study, we will need to consider transitive closures of accessibility relations, and this seems to be quite challenging.
We thank Yanjing Wang for his insightful and extensive discussions on the theme of this paper.
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The axiom system is sound for .
To see that is a validity, for any model and any world let and . Since for any and for any we have . Further since , for any there is some such that and . But then and hence . Thus by semantics, .
Similarly validity of which is the variant of standard axiom can be verified. Also notice that the inference rules and both () preserve validities. Hence is sound.
The notions of consistent set of formulas and maximally consistent set of formulas is defined in the standard way.
For any set of formulas , if is a maximal consistent set then
if then is consistent.