Recently [BeEndG20, Tu16, vBBeEn19a, vBBeEn19b, vBBeEnYu17, Yu18, Yu20], a variant of neighbourhood semantics for modal logics is given, under the name of instantial neighbourhood logic (INL), where existential information about the neighbourhoods of a given world can be added. This semantics is a mixture between relational semantics and neighbourhood semantics, and its expressive power is strictly stronger than neighbourhood semantics. In this semantics, the n+1-ary modality is true at a world if and only if there exists a neighbourhood such that is true everywhere in , and each is true at for some .
Instantial neighbourhood logic is first introduced in [vBBeEnYu17], where the authors defines the notion of bisimulation for instantial neighbourhood logic, gives a complete axiomatic system, and determines its precise SAT complexity; in [Tu16], the canonical rules are defined for instantial neighbourhood logic; in [vBBeEn19a], the game-theoretic aspects of instantial neighbourhood logic is studied; in [vBBeEn19b], a propositional dynamic logic IPDL is obtained by combining instantial neighbourhood logic with propositional dynamic logic (PDL), its sound and complete axiomatic system is given as well as its finite model property and decidability; in [BeEndG20], the duality theory for instantial neighbourhood logic is developed via coalgebraic method; in [Yu18], a tableau system for instantial neighbourhood logic is given which can be used for mechanical proof and countermodel search; in [Yu20]
, a cut-free sequent calculus and a constructive proof of its Lyndon interpolation theorem is given. However, the Sahlqvist-type correspondence theory is still unexplored, which is the theme of this paper.
In this paper, we define the Sahlqvist formulas in the instantial neighbourhood modal language, and give two different proofs of correspondence results. The first proof is given by standard translation and minimal valuation techniques as in [BRV01, Section 3.6], while the second proof uses bimodal translation method in monotone modal logic and neighbourhood semantics [Ha03, KrWo99, Pa85, vB03] to show that every Sahlqvist formula in the instantial neighbourhood modal language can be translated into a bimodal Sahlqvist formula in Kripke semantics, and hence has a first-order correspondent.
The structure of the paper is as follows: in Section 2, we give a brief sketch on the preliminaries of instantial neighbourhood logic, including its syntax and neighbourhood semantics. In Section 3, we define the standard translation of instantial neighbourhood logic into a two-sorted first-order language. In Section 4, we define Sahlqvist formulas in instantial neighbourhood logic, and prove the Sahlqvist correspondence theorem via standard translation and minimal valuation. In Section 5, we discuss the translation of instantial neighbourhood logic into normal bimodal logic, and prove Sahlqvist correspondence theorem via this bimodal translation. We give some remarks and further directions in Section 6.
2 Preliminaries on instantial neighbourhood logic
In this section, we collect some preliminaries on instantial neighbourhood logic, which can be found in [vBBeEnYu17].
The formulas of instantial neighbourhood logic are defined as follows:
where is a propositional variable, is an +1-ary modality for each . can be defined in the standard way. An occurence of is said to be positive (resp. negative) in if
is under the scope of an even (resp. odd) number of negations. A formulais positive (resp. negative) if all occurences of propositional variables in are positive (resp. negative).
For the semantics of instantial neighbourhood logic, we use neighbourhood frames to interpret the instantial neighbourhood modality, one and the same neighbourhood function for all the +1-ary modalities for all .
(Neighbourhood frames and models) A neighbourhood frame is a pair where is the set of worlds, is a map called a neighbourhood function, where is the powerset of . A valuation on is a map . A triple is called a neighbourhood model or a neighbourhood model based on if is a neighbourhood frame and is a valuation on it.
The semantic clauses for the Boolean part is standard. For the instantial neighbourhood modality , the satisfaction relation is defined as follows:
iff there is such that for all we have and for all there is an such that .
Semantic properties of instantial neighbourhood modalities
It is easy to see the following lemma, which states that the +1-ary instantial neighbourhood modality is monotone in every coordinate, and is completely additive (and hence monotone) in the first coordinates. This observation is useful in the algebraic correspondence analysis in instantial neighbourhood logic.
For any , any and any valuations such that , for all ,
if , then ;
For any , any and any valuation , fix an and a , and define such that for , and . Then the following holds:
iff there exists a such that .
Algebraically, if we view the +1-ary modality as an +1-ary function , then is completely additive (i.e. preserve arbitrary joins) in the first coordinate, and monotone in the last coordinate. This observation is useful in the algebraic correspondence analysis (see Section 6).
3 Standard translation of instantial neighbourhood logic
3.1 Two-sorted first-order language and standard translation
Given the INL language, we consider the corresponding two sorted first-order language , which is going to be interpreted in a two-sorted domain . It has the following ingredients:
world variables , to be interpreted as possible worlds in the world domain ;
subset variables , to be interpreted as objects in the subset domain ;111Notice that here the subset variables are treated as first-order variables in the subset domain , rather than second-order variables in the world domain .
a binary relation symbol , to be interpreted as the reverse membership relation such that iff ;
a binary relation symbol , to be interpreted as the neighbourhood relation such that iff ;
unary predicate symbols , ,…, to be interpreted as subsets of the world domain .
We also consider the following second-order language which is obtained by adding second-order quantifiers ,…over the world domain . Existential second-order quantifiers are interpreted in the standard way. Notice that here the second-order variables ,…are different from the subset variables , since the former are interpreted as subsets of , and the latter are interpreted as objects in .
Now we define the standard translation as follows:
(Standard translation) For any INL formula and any world symbol , the standard translation of at is defined as follows:
For any neighbourhood frame , it is natural to define the following corresponding two-sorted Kripke frame , where
such that for any and , iff ;
such that for any and , iff .
Given a two-sorted Kripke frame , a valuation is defined as a map . Notice that here the in the definition of is understood as the powerset of the first domain, rather than the second domain itself.
For this standard translation, it is easy to see the following correctness result:
For any neighbourhood frame , any valuation on , any , any INL formula ,
4 Sahlqvist correspondence theorem in instantial neighbourhood logic via standard translation
In this section, we will define the Sahlqvist formulas in instantial neighbourhood logic and prove the correspondence theorem via standard translation and minimal valuation method. First we recall the definition of Sahlqvist formulas in normal modal logic. Then we identify the special situations where the instantial neighourhood modalities “behave well”, i.e. have good quantifier patterns in the standard translation. Finally, we define INL-Sahlqvist formulas step by step in the style of [BRV01, Section 3.6], and prove the correspondence theorem.
4.1 Sahlqvist formulas in normal modal logic
In this subsection we recall the syntactic definition of Sahlqvist formulas in normal modal logic (see [BRV01, Section 3.6]).
(Sahlqvist formulas222Here what we call Sahlqvist formulas are called Sahlqvist implications in [BRV01, Section 3.6]. in normal modal logic) A boxed atom is a formula of the , where are (not necessarily distinct) boxes. In the case where =0, the boxed atom is just .
A Sahlqvist antecedent is a formula built up from , boxed atoms, and negative formulas, using and existential modal operators ( and ). A Sahlqvist formula is an implication in which is positive and is a Sahlqvist antecedent.
As we can see from the definition above, the Sahlqvist antecedents are built up by and negative formulas using . If we consider the standard translations of Sahlqvist antecedents, the inner part are translated into universal quantifiers, and the outer part are translated into existential quantifiers.
4.2 Special cases where the instantial neighbourhood modalities become “normal”
As is mentioned in [vBBeEnYu17, Section 7] and as we can see in the definition of the standard translation, the quantifier pattern of is similar to the case of monotone modal logic [Ha03] which has an pattern. As a result, even with two layers of INL modalities the complexity goes beyond the Sahlqvist fragment. However, we can still consider some special situations where we can reduce the modality to an -ary normal diamond or a unary normal box.
-ary normal diamond.
We first consider the case where is a pure formula without any propositional variables, i.e., all propositional variables are substituted by or . In this case is a first-order formula without any unary predicate symbols …. Therefore, in the shape of the standard translation of , the universal quantifier is not touched during the computation of minimal valuation, since there is no unary predicate symbol there. Indeed, we can consider the following equivalent form of :
Now is essentially in a form similar to in the normal modal logic case; indeed, when we compute the minimal valuation here, can be recognized as an integrity and stay untouched during the process.
From now onwards we can denote by where is pure.
As we can see from above, in , we can replace propositional variables in by and to obtain -ary normal diamond modalities. By using the composition with negations, we can get the unary box modality, i.e. we can have a modality
Now we can consider the standard translation of :
where does not contain unary predicate symbols . Now we can see that has a form similar to where is a normal unary box, by taking as an integrity.
4.3 The definition of INL-Sahlqvist formulas in instantial neighbourhood logic
Now we can define the INL-Sahlqvist formulas in instantial neighbourhood logic step by step in the style of [BRV01, Section 3.6].
4.3.1 Very simple INL-Sahlqvist formulas
Definition 5 (Very simple INL-Sahlqvist formulas).
A very simple INL-Sahlqvist antecedent is defined as follows:
where is a propositional variable, is a pure formula without propositional variables. A very simple INL-Sahlqvist formula is an implication where is positive (see page 2), and is a very simple INL-Sahlqvist antecedent.
For very simple INL-Sahlqvist formulas, we allow -ary normal diamonds in the construction of , while for the +1-ary modality , we only allow propositional variables to occur in the +1-th coordinate.
We can show that very simple INL-Sahlqvist formulas have first-order correspondents:
For any given very simple INL-Sahlqvist formula , there is a two-sorted first-order local correspondent such that for any neighbourhood frame , any ,
The proof strategy is similar to [BRV01, Theorem 3.42, Theorem 3.49], with some differences in treating .
We first start with the two-sorted second-order translation of , namely , where are the two-sorted first-order standard translations of .
For any very simple INL-Sahlqvist antecedent , we consider the shape of defined inductively,
Now we can denote as , and thus get
By using the equivalences
it is easy to see that the two-sorted first-order formula is equivalent to a formula of the form , where:
is a (possibly empty) conjunction of formulas of the form or ;
ATProp is a conjunction of formulas of the form or or or .
Therefore, by using the equivalences
it is immediate that is equivalent to
where and ATProp are given as above, and POS is the standard translation .
Now we can use similar strategy as in [BRV01, Theorem 3.42, Theorem 3.49]. To make it easier for later parts in the paper, we still mention how the minimal valuation and the resulting first-order correspondent formula look like. Without loss of generality we suppose that for any unary predicate that occurs in the POS also occurs in AT; otherwise we can substitute by for to eliminate .
Now consider a unary predicate symbol occuring in ATProp, and , , …, are all occurences of in ATProp. By taking to be
we get the minimal valuation. The resulting first-order correspondent formula is
From the proof above, we can see that the part corresponding to is essentially treated in the same way as an -ary diamond in the normal modal logic setting, and is treated as where is an +1-ary normal diamond, is a unary normal diamond and is a unary normal box, therefore we can guarantee the compositional structure of quantifiers in the antecedent to be as a whole.
4.3.2 Simple INL-Sahlqvist formulas
Similar to simple Sahlqvist formulas in basic modal logic, here we can define simple INL-Sahlqvist formulas:
Definition 6 (Simple INL-Sahlqvist formulas).
A pseudo-boxed atom is defined as follows:
where is a pure formula without propositional variables. Based on this, a simple INL-Sahlqvist antecedent is defined as follows:
where is a pure formula without propositional variables and is a pseudo-boxed atom. A simple INL-Sahlqvist formula is an implication where is positive, and is a simple INL-Sahlqvist antecedent.
For any given simple INL-Sahlqvist formula , there is a two-sorted first-order local correspondent such that for any neighbourhood frame , any ,
We use similar proof strategy as [BRV01, Theorem 3.49]. The part that we needs to take care of is the way to compute the minimal valuation. Now without loss of generality (by renaming quantified variables) we have the following Backus-Naur form of defined inductively for any pseudo-boxed atom :
The Backus-Naur form of is defined inductively for any simple Sahlqvist antecedent :
Now we can denote as and for (note that the only possible free variable in is ), then the Backus-Naur form of and can be given as follows:
Now we denote as , and we get the Backus-Naur form of pseudo-boxed atom as follows:
Now using the following equivalences:
(where does not occur in );
for any pseudo-boxed atom , the first-order formula is equivalent to a conjunction of two-sorted first-order formulas of the form , where:
is a (possibly empty) conjunction of formulas of the form ;
AT is a formula of the form or or where is bounded by (here we do not need to take the conjunction because of ).
It is easy to see that does not contain any unary predicate symbol . By the equivalence where does not contain , we can transform into , where is or or .
We can introduce a new binary relation symbol which is . Then is a conjunction of formulas of the form .
Now we somehow come back to the situation of the basic normal modal logic case, where is a real relation symbol. The Backus-Naur form of for simple INL-Sahlqvist antecedent can be recursively defined as follows:
since is a conjunction of formulas of the form , we have
Now the situation is similar to the very simple INL-Sahlqvist formula case. We can see how the minimal valuation is computed:
for the part, when is , its corresponding minimal valuation is ; when is or , we can replace by or , respectively;
for the part, it is equivalent to ;
for the part, it is equivalent to ;
for the part, its corresponding minimal valuation is ;
for the part, when is , its corresponding minimal valuation is ; when is or , we can replace by or , respectively.
Now for each propositional variable , we take the minimal valuation to be the disjunction of all the corresponding minimal valuations where the branch has an occurence of . By essentially the same argument as in [BRV01, Theorem 3.49], we get the first-order correspondent of . ∎
4.3.3 INL-Sahlqvist formulas
In the present section, we add negated formulas and disjunctions in the antecedent part, which is analogous to [BRV01, Definition 3.51].
Definition 7 (INL-Sahlqvist formulas).
For any given INL-Sahlqvist formula , there is a two-sorted first-order local correspondent such that for any neighbourhood frame , any ,
We use similar proof strategy as [BRV01, Theorem 3.54]. The part that we needs to take care of is the way to compute the minimal valuation. Now for each INL-Sahlqvist antecedent , we consider the Backus-Naur form of :
By denoting as , as , we can rewrite the Backus-Naur form of as follows: