1 Introduction

‘Distributed knowledge’ is one of the notions of group knowledge studied in multi-agent epistemic logic [⁶, ¹⁸]. A typical example of distributed knowledge is the following: a group consisting of a and b has distributed knowledge of a fact q when a knows that p→q and b knows that p. According to [¹, Section 1], “distributed knowledge is the knowledge of a third party, someone ‘outside the system’ who somehow has access to the epistemic states of all the group members”. Fagin et al. [⁶, p. 3] stated as an intuitive description for distributed knowledge “a group has distributed knowledge of a fact φ if the knowledge of φ is distributed among its members, so that by pooling their knowledge together the members of the group can deduce φ”. At first sight, the latter description seems clearer than the former. Ågotnes and Wang [¹] state, however, that the above intuitive description by Fágin et al. [⁶, p. 3] is inappropriate by an illustrative example given in [¹, Section 1].

Formally, distributed knowledge is expressed as a modal operator DG, parameterized by a finite group G of agents and the satisfaction of DGφ at a state w is defined as: φ holds at all states v such that v can be reached in a single step from w for all agents in G, i.e., wRav for all agents a∈G, where Ra is a binary relation on the states. As for the model-theoretic study of distributed knowledge, we can cite [¹, ²⁵, ¹⁰, ²⁸]. Proof-theoretic study is relatively less active, but there have been proposed several sequent calculi [¹², ²³, ¹¹, ¹⁹]. However, those cited here are all on the basis of classical logic.

Not to mention distributed knowledge, epistemic logic as a whole has been studied mainly in the classical setting. However, several kinds of intuitionistic epistemic logics have been proposed from different perspectives. Several philosophical logicians have proposed intuitionistic epistemic logics [³¹, ²⁴, ³] for the sake of analysis of Fitch’s knowability paradox [⁷], from the verificationist point of view.

Another kind of intuitionistic epistemic logic [¹⁴] is proposed for the analysis of distributed computing in the sense of [¹³, ²⁶]. Also, [²⁷] develops an intuitionistic epistemic logic from the game-theoretical point of view. The intuitionistic aspect of the logic is required for describing the property of asynchronous communication among agents in distributed computing.

Jäger and Marti [¹⁵] formulate intuitionistic epistemic logic with distributed knowledge for the first time, as far as the authors know, and prove semantic completeness of Hilbert systems of intuitionistic K and KT with distributed knowledge. Logics we investigate in the present paper is basically based on theirs, but differs in the following respects: firstly, in our logics, distributed knowledge operator is parameterized by a group, i.e., a subset of whole agents, while [¹⁵] deals with only distributed knowledge for the whole agents. Secondly, we handle more axioms than [¹⁵], that is, we propose intuitionistic K, KT, KD, K4, K4D, and S4 with distributed knowledge. One point to note here. Axioms (K), (T), and (4) in our logics are simply a DG-version of the respective axioms in the basic modal logic.

However, our axiom (D) is restricted to a single agent (i.e., ¬D{a}⊥). This is because seriality for each Ra is generally not preserved under taking intersection among a group (refer to [²]), while reflexivity and transitivity are always preserved. As for proof of the semantic completeness, we adopt a more standard method via the concept of “pseudo-model” than [¹⁵]. We also propose cut-free sequent calculi for our logics, based on the idea introduced in [¹⁹] and prove Craig interpolation theorem by Maehara’s method [¹⁶, ²¹]. Also, we establish decidability of the sequent calculi by the standard argument [⁸, ⁹] on a cut-free derivation of a sequent, while [¹⁵] does not show it for their Hilbert systems.

The paper is organized as follows. In Section 2, we introduce syntax and semantics for intuitionistic epistemic logic with distributed knowledge. Section 3 defines Hilbert systems of the logics, and state soundness results. In Section 4, strong completeness of the Hilbert systems of the logics is shown, via a notion of “pseudo-model”.

In Section 5, we introduce sequent calculi for the logics and prove the cut-elimination theorem, Craig interpolation theorem, and decidability.

2 Syntax and Semantics of Intuitionistic Epistemic Logics with Distributed Knowledge Operators

We denote a finite set of agents by Agt. We call a nonempty subset of Agt “group” and denote it by G, H, etc. We denote by Grp the set of all groups, i.e., the set (Agt)\{∅} of all non-empty subset of Agt. Let Prop be a countable set of propositional variables and Form be the set of formulas defined inductively by the following clauses:

We read DGφ as “φ is distributed knowledge among a group G”. We define ¬φ as φ→ ⊥ and the epistemic operator Kaφ (read “agent a knows that φ”) as D{a}φ. As noted above, an expression of the form D∅φ is not a well-formed formula, since we have excluded ∅ from our definition of groups.

We introduce Kripke semantics for intuitionistic multi-agent epistemic logic with distributed knowledge, along the lines of [¹⁵].

Definition 2.1 (Frame, Model). A tuple F=(W,≤,(Ra)a∈Agt) is a frame if: W is a set of states; ≤ is a preorder on W; (Ra)a∈Agt is a family of binary relations on W , indexed by agents; and ≤;Ra⊆Ra (for all a∈Agt), where R1;R2:={(x,z)| there exists y such that xR1y and yR2z}.

A pair M=(F,V) is a model if F is a frame, and a valuation function V: Prop→P(W) satisfies the heredity condition, i.e., if w∈V(p) and w≤v, then v∈V(p). We denote an underlying set of states of a frame F or a model M by |F| or |M|.

For a model M=(W,≤,(Ra)a∈Agt,V) and a state w∈W, a pair (M,w) is called a pointed model.

Satisfaction relation M,w⊨φ on pointed models and formulas is defined recursively as follows:

It is noted from our definition of Kaφ:=D{a}φ that the satisfaction of Kaφ at a state w of a model M is given as follows:

As is the case with ordinary intuitionistic logic, we have the following heredity property for a formula.

Proposition 2.2 (Heredity). If M,w⊨φ and w≤v, then M,v⊨φ.

Proof. By induction on φ. For the case where φ≡DGψ, it is noted that the condition ≤;Ra⊆Ra of a frame implies that ≤;∩a∈GRa⊆∩a∈GRa.

Fig. 1 is an example of a frame. The preorder is depicted by a dotted arrow. Note that we omit reflexive arrows for the preorder. If a valuation is defined by, for example, V(p)={v} for any p∈Prop, V satisfies the heredity condition. In this model, it can be seen that different groups have different distributed knowledge even at the same state. Indeed, D{a,b}p is true at w, but D{a,c}p is false at w. Further, we can also see that seriality for each agent’s relation is not always preserved under taking intersection among a group. Namely, Rb and Rc are serial but Rb∩Rc is not in the example. This is why we should restrict (D) axiom to ¬D{a}⊥, as defined in Table 1. Given a frame F=(W,≤,(Ra)a∈Agt), we say that a formula φ is valid in F (notation: F⊩φ) if (F,V),w⊩φ for every valuation function V and every w∈W. Moreover, a formula φ is valid in a class F of frames (notation: F⊩φ) if F⊩φ for every F∈F.

Fig. 1 Example of a frame 

Definition 2.3. A formula φ is a semantic consequence of Γ in a frame class F if for all frame F∈F, a valuation V on F, a state w∈|F|, if (F,V),w⊩Γ, then (F,V),w⊩φ. We write it as “Γ⊩Fφ”.

3 Hilbert Systems

Hilbert systems for intuitionistic epistemic logics with DG operators are constructed from axioms and rules shown in Table 1.

A Hilbert system H(IK) consists of axioms and rules for intuitionistic logic, axioms (Incl) and (K), and a rule (Nec). Hilbert systems H(IKT), H(IKD), H(IK4), H(IK4D), and H(IS4) are defined as axiomatic expansions of H(IK) with (T), (D), (4), (4) and (D), and (T) and (4), respectively. Let X be any of IK, IKT, IKD, IK4, IK4D, and IS4 in what follows. The notion of provability in each system is defined as usual, and the fact that a formula φ is provable in H(X) is denoted by “⊢H(X)φ”. We also define derivability relation between a set Γ of formulas and a formula φ as below.

Definition 3.1. A formula φ is derivable from Γ in a logic X if ⊢H(X)∧Γ′→φ for some finite set Γ′ which is a subset of Γ. We write it as “Γ⊢H(X)φ”.

We introduce a class of frames corresponding to each logic, in order to state soundness of our axiomatization.

Definition 3.2. A class of frames F(X) is defined as follows:

Here, reflexivity, seriality, and transitivity are defined ordinarily.

We can prove the following soundness theorem by induction on φ. Note that axioms (T) and (4) are valid in reflexive and transitive frames, respectively, because if Ra is reflexive or transitive for any a∈G,∩a∈GRa is also reflexive or transitive, respectively.

Theorem 3.3. If ⊢H(X)φ, then F(X)⊩φ.

4 Completeness

In the present section, we explain a proof of the strong completeness theorem of our logic. Let Γ be a set of formulas and φ be a formula. The strong completeness theorem is stated as follows.

Theorem 4.1. Let X be any of IK, IKT, IKD, IK4, IK4D, and IS4. Then, if Γ⊩F(X)φ, then Γ⊢H(X)φ.

As in [⁵], we show the theorem in two steps via the notion of “pseudo-model”, that is, we first construct a canonical pseudo-model satisfying truth lemma, and then transform it into an equivalent pseudo-model which can be regarded as a model in the sense of Definition 2.1.

Definition 4.2 (Pseudo-frame, Pseudo-model). A tuple F=(W,≤,(RG)G∈Grp) is a pseudo-frame if: ≤;RG⊆RG for any G∈Grp and RH⊆RG if G⊆H.

A pair M=(F,V) is a pseudo-model if F is a pseudo-frame, and a valuation function V:Prop→P(W) satisfies the heredity condition, i.e., if w∈V(p) and w≤v, then v∈V(p).

Example 4.3. Fig. 2 is an example of a pseudo-frame. We name it Fex. Note that {a} is written as “a” and R{a,b} is defined as 0 here. Since R{a,b}=0, the condition of “RH⊆RG if G⊆H” is self-evidently satisfied, i.e., R{a,b}⊆R{a} and R{a,b}⊆R{b}. Note that R{a}∩R{b}⊆R{a,b} in Fex, while the contrary is guaranteed by the condition of “RH⊆RG if G⊆H”. Any frame can be regarded as a pseudo-frame with only relations for singleton groups, as in Fex.

Fig. 2 Example of a pseudo-frame 

Definition 4.4 (Pseudo-satisfaction Relation). For a pseudo-model M, a state w∈|M|, and a formula φ, a pseudo-satisfaction relation M,w⊩psφ is defined the same as the satisfaction relation ⊩, except for the clause for DGφ: that is:

Namely, in a pseudo-model, an operator DG is treated like a primitive box operator, parameterized by a group.

Considering the definition of satisfaction relation for DGφ, a pseudo-frame can be seen as a frame if the condition RG=∩a∈GR{a} is satisfied for any group G.

So, we can prove the strong completeness theorem by transforming a canonical pseudo-model into a pseudo-model enjoying the condition above without changing satisfaction. We do this by a method of “tree unraveling”.

4.2 Tree Unraveling

We introduce a method called “tree unraveling”, which transforms a pseudo-model into another pseudo-model satisfying ∩a∈GR{a}=RG (i.e., a model in the sense of Definition 2.1). Our definitions below are intuitionistic generalizations of definitions proposed in [⁵] over classical logic.

Definition 4.13. Let M=(W,≤,(RG)G∈Grp,V) be a pseudo-model. A pseudo-model M′=(W′,≤∩(W′×W′),(RG∩(W′×W′))G∈Grp,V′) is a generated submodel of M if: W′⊆W; If w∈W′ and w≤w′ then w′∈W′; If w∈W′ and wRGw′ then w′∈W′; and V′(p)=V(p)∩W′ for any p∈Prop.

For X⊆|M|, we define MX as the smallest generated submodel containing X. If M=MX, we say

that M is generated by X.

Definition 4.14. Let M=(F,V) be a pseudo-model generated by w∈W, where F=(W,≤,(RG)G∈Grp):

• — We put w0:=w and define Finpath(F,w) as {〈w0,L1,w1,L2,⋯,Ln,wn〉|n≥0,Li∈{≤,RG}G∈Grp,wi−1Liwi for all 1≤i≤n}. We call an element of Finpath(F,w) “a path (from a state w)” and denote it by u→, v→, etc.

• — For u→=〈w0,L1,w1,L2,⋯,Ln−1,wn−1,Ln,wn〉∈Finpath(F,w),tail(u→) is defined as wn.

• — We say that paths u→,v→∈Findpath(F,w) satisfy a relation u→≼v→ if and only if v→≡u→⌢〈≤,w′〉, where ⌢ means concatenation of two tuples.

• — We say that paths u→,v→∈Findpath(F,w) satisfy a relation u→ℛGv→ if and only if v→≡u→⌢〈RH,w′〉 and G⊆H.

• — A valuation V:Prop→P(Finpath(F,w)) is defined by:

V(p)={u→∈Finpath(W,w)|tail(u→)∈V(p)}

Take Fex in Fig. 2 as an example. The set Finpath(Fex,w) of paths on Fex and ≼ and ℛG on this set are drawn in Fig. 3. The point is that the a-arrow and b-arrow on w in Fex are transformed into two arrows with different destinations, so that the condition “R{a}∩R{b}=R{a,b}” is not satisfied in Fex but becomes satisfied in Finpath(Fex,w). However, as it is, (Finpath(Fex,w)≼,(ℛG)G∈Grp) is not a pseudo-frame, since ≼ itself is not a preorder and the condition “≤;RG⊆RG” is not satisfied because, for example, there is no a-arrow from 〈w〉 to (w,≤,w,a,w). Therefore, a preorder and relations for DG on Finpath(F,w) in general should be defined as follows.

Fig. 3 Tree unraveling 

Definition 4.15 (Tree Unraveling). Let M=(F,V) be a pseudo-model generated by w∈W, where F=(W,≤,(RG)G∈Grp). A tree unraveling pseudo-model Tree(M,w) of a pointed pseudo-model (M,w) is defined as a tuple:

(Finpath(F,w),≼*,(≼*;ℛG)G∈Grp,V),

where R* is defined as the reflexive and transitive closure of a relation R.

We can easily show that Tree(M,w) is indeed a pseudo-model. Moreover, as explained above with Fig. 3, ∩a∈Gℛ{a}=ℛG, from which it is also shown that ∩a∈G≼*;ℛ{a}= ≼*;ℛG by a simple argument using property of a tree unraveling pseudo-model. Therefore, Tree(M,w) can be seen as a model in the sense of Definition 2.1. The following is a key property of tree unraveling.

Lemma 4.16. Let M=(F,V) be a pseudo-model generated by w∈W, where F=(W,≤,(RG)G∈Grp). Then, M,w⊩psφ iff Tree(M,w),〈w〉⊩psφ for any formula φ.

Proof. The function u→↦tail(u→) is a bounded morphism (which takes not only relations for DG but also a preorder into account) from Tree(M,w) to M.

We end the present section by proving Theorem 4.1.

Proof. (Outline) First, we show the case of IK. We show the contraposition. Assume Γ⊢H(X)φ. By Lemma 4.8, We can find an X-prime and X-consistent theory Γ+ such that Γ⊆Γ+ and Γ+⊢H(X)φ. Since Γ⊆Γ+, MX,Γ+⊩psΓ by the left-to-right of Lemma 4.11. On the other hand, MX,Γ+⊩psφ by the right-to-left of Lemma 4.11 and item 1 of Lemma 4.7. We can take Tree(MΓ+X,Γ+), because, by Proposition 4.10, MΓ+X is a pseudo-model generated by Γ+. Since any tree unraveling pseudo-model can be seen as a model in the sense of Definition 2.1, it suffices to show that (M+,Γ+) satisfies exactly the same formulas as (Tree(MΓ+X,Γ+),〈Γ+〉). First, (MX,Γ+) satisfies exactly the same formulas as (MΓ+X,Γ+). Then, by Lemma 4.16, (MΓ+X,Γ+) satisfies exactly the same formulas as (Tree(MΓ+X,Γ+),〈Γ+〉).

For the remaining logics, basically, a similar argument can be applied, but definitions and proofs become more involved. In order to make relations for DG have the desired property, such as reflexivity or transitivity, the relation ≼*;ℛG should be replaced by ≼*;ℛG∘, (≼*;ℛG+)+, and (≼*;ℛG∗) for IKT, IK4 and IK4D, and IS4, respectively, in the definition of tree unraveling.

Here, R° and R+ are defined as the reflexive closure and transitive closure of a relation R, respectively. Also, note that ≼*;ℛG and (≼*;ℛG+)+ are serial if RG is serial and that R{a}X is serial if X has the axiom (D) (by Proposition 4.12). The resulting tree unravelings are also easily shown to be pseudo-models. The condition “∩a∈GR{a}=RG in the tree unraveling pseudo-models also can be shown to be satisfied, by using the property of a tree unraveling pseudo-model. Therefore, from the above argument, Tee(MΓ+X,Γ+) can be seen as a model, whose underlying frame is an element of F(X). The fact that the function u→↦tail(u→) is a bounded morphism also in the respective tree unraveling pseudo-models is needed, and can be shown straightforwardly.

5 Sequent Calculi of Intuitionistic Epistemic Logics with Distributed Knowledge