# When Do Introspection Axioms Matter for Multi-Agent Epistemic Reasoning?

The early literature on epistemic logic in philosophy focused on reasoning about the knowledge or belief of a single agent, especially on controversies about "introspection axioms" such as the 4 and 5 axioms. By contrast, the later literature on epistemic logic in computer science and game theory has focused on multi-agent epistemic reasoning, with the single-agent 4 and 5 axioms largely taken for granted. In the relevant multi-agent scenarios, it is often important to reason about what agent A believes about what agent B believes about what agent A believes; but it is rarely important to reason just about what agent A believes about what agent A believes. This raises the question of the extent to which single-agent introspection axioms actually matter for multi-agent epistemic reasoning. In this paper, we formalize and answer this question. To formalize the question, we first define a set of multi-agent formulas that we call agent-alternating formulas, including formulas like Box_a Box_b Box_a p but not formulas like Box_a Box_a p. We then prove, for the case of belief, that if one starts with multi-agent K or KD, then adding both the 4 and 5 axioms (or adding the B axiom) does not allow the derivation of any new agent-alternating formulas – in this sense, introspection axioms do not matter. By contrast, we show that such conservativity results fail for knowledge and multi-agent KT, though they hold with respect to a smaller class of agent-nonrepeating formulas.

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## 1 Introduction

The classic early works on epistemic logic in philosophy by Hintikka [14] and Lenzen [20] focused on the logic of knowledge and belief for a single agent,111Only §§ 4.1-4.6 and § 4.13 of [14] and pp. 59, 66, and 70 of [20] contain discussion of multi-agent formulas. especially on controversies about “introspection axioms”: for example, if an agent knows , does she know that she knows (formalized by the axiom of modal logic, )? If an agent does not know , does she know that she does not know (formalized by the axiom of modal logic, )? By contrast, the later literature on epistemic logic in computer science (e.g., [22, 6]) and game theory (e.g., [3]) focused on multi-agent epistemic reasoning, especially as required for coordination between agents or strategic reasoning against opponents. In this literature, the single-agent introspection principles formalized by the and axioms are largely taken for granted (for exceptions, see, e.g., [26, 18, 16]). In the relevant multi-agent scenarios, it is often important to reason about what agent A believes about what agent B believes about what agent A believes (); but it is rarely important to reason just about what agent A believes about what agent A believes (). Consider the following famous examples of multi-agent epistemic reasoning.

### Muddy children

We assume familiarity with the 3-agent Muddy Children puzzle where two children have mud on their foreheads (see, e.g., § 1.1 of [6]). The following is a derivation in the bimodal version of the minimal normal modal logic K showing how one of the muddy children comes to realize that she is muddy.222‘PL’ stands for propositional logic, ‘Nec’ stands for the necessitation rule, and ‘RM’ stands for the monotonicity rule that if is a theorem, then so is . Note that in the derivation, RM is only applied to theorems of the logic. For example, to obtain , RM is applied to the theorem where and . Note that (i) no introspection axioms are used, and in fact (ii) modalities occur only “alternatingly,” in the sense that no occurrence of a modality for an agent has scope over another occurrence of a modality for without an intervening occurrence of some modality for an agent .

1. [label=()]

2. (assumption: 1 knows that 2 knows that at least one child is muddy)

3. (assumption: 1 knows that 2 can see 3, who is not muddy)

4. (assumption: 1 knows that 2 can see 1)

5. (assumption: 1 knows that 2 did not step forward after the parent’s first question)

1. (K axiom)

2. (from (1) by RM)

3. (from (a) and (2) by PL)

4. (from (3) using PL and RM)

5. (from (d) and (4) by K and PL)

6. (theorem of K)

7. (from (6) by RM)

8. (from (5) and (7) by PL)

9. (from (b) and (8) using PL, Nec, and K)

10. (from (c) and (9) using PL, Nec, and K)

### Backward induction

We assume familiarity with the classic backward induction reasoning in extensive form games (see, e.g., [23, § 6.2]). In [27], Vilks provides a syntactical derivation of backwards induction in the bimodal version of the modal logic KT, which we reproduce below. Again note that (i) no introspection axioms are used, and in fact (ii) modalities occur only “alternatingly” as above.

• (both play left)

• (1 play left, 2 play right)

• (1 play right, 2 play left)

• (both play right)

• (players’ preferences)

• (description of the game)

1. [label=()]

2. (assumption: 1 knows the game)

3. (assumption: 1 knows that if 2 is at then 2 considers the move possible)

4. (assumption: similar to (b))

5. (assumption: similar to (b))

6. (assumption: follows from assuming 1 knows that 2 is rational)

7. (assumption: similar to (e))

8. (assumption: follows from 1 being rational)

1. (from (a), (b), and (e) using PL, Nec, and K)

2. (from (a), (c), and (f) using PL, Nec, and K)

3. (from (d) by T)

4. (from (a) by T)

5. (from (4) by PL)

6. (from (3) and (5) by PL)

7. (from (g), (1), (2), (4), and (6) by PL)

8. (from (5) and (7) by PL)

9. (from (2) by T)

10. (from (8) and (9) by PL)

In general, in typical strategic form games a player needs to reason about the beliefs of her opponents, as which action is best for her depends on her opponents’ actions, which in turn depend on their beliefs. On the other hand, reasoning about one’s own beliefs seems unnecessary, as the dependencies just mentioned seem to be tight: which action is the best for a player depends on what her opponents’ actions are alone, which in turn depend on their beliefs over what their opponents’ actions are alone. We can then iterate this reasoning, and it seems there is no place for reasoning about one’s own beliefs. In Appendix A we provide a formalization of this idea using Kripke models of games in the style of [25] and [5], where only formulas with no modality scoping immediately over a modality of the same agent are used to ensure that rationalizable strategies are played.333We are not arguing that introspection assumptions never matter in multi-agent epistemic reasoning. For example, it is shown in [9, 19] that Aumann’s [2] theorem on agreeing to disagree fails without the assumption of positive introspection.

These considerations raise the question of the extent to which single-agent introspection axioms actually matter for multi-agent epistemic reasoning. In particular, as motivated by the above examples, we can ask: in situations where the agents and also the analyst only need to reason about formulas where modalities occur only alternatingly, would the commonly debated introspection axioms still matter, in the sense that assuming them allows us to derive more conclusions?

This question has indeed been partially investigated previously, though motivated not by the question of whether introspection axioms may in practice be “irrelevant” but rather by the goal of devising efficient reasoning algorithms for the system . In [17], it is explicitly stated (Lemma 5) that when restricted to the fragment of the multi-agent language in which modalities occur only in the agent-alternating way, and derive the same set of theorems.444The authors refer to [12] for the proof of this lemma, though we are unable to locate an explicit proof there. This facilitates reasoning in since it is also known that every formula is provably equivalent in to an agent-alternating formula,555In Appendix B, we show the semantic counterpart of this proposition and further show that and are in a sense necessary. See also Theorem 1 of [24] for an early precursor of this result. which is then derivable in iff it is derivable , making the efficient methods of deciding theoremhood in applicable to . Subsequently, the idea of agent-alternating formulas was also used in the axiomatization of refinement quantification logics [11, 10] and in epistemic planning [15, 21, 7].

In this paper, we study the question more systematically. In § 2, we provide multiple ways to define the agent-alternating formulas, which include formulas like but not . In § 3, we first provide a bisimulation notion for the fragment of agent-alternating formulas and then use it to completely chart the relationships of the modal logics in the well-known “Modal Logic Cube” when restricted to the fragment of agent-alternating formulas. We prove that if one starts with multi-agent K or KD, then adding both the and axioms (or adding the axiom) does not allow the derivation of any new agent-alternating formula—in this sense, introspection axioms do not matter. By contrast, we show that such conservativity results fail for knowledge and multi-agent KT, though they hold with respect to a smaller class of agent-nonrepeating formulas introduced in § 4. In § 5, we report on preliminary investigations of how these results are affected in the presence of a common belief operator in the language. Finally, we conclude in § 6 with some directions for future research.

## 2 Agent-Alternating Formulas

Fix a set of agents with and a countably infinite set of proposition letters.

###### Definition 2.1.

The language of multi-agent epistemic logic is defined inductively by

 L∋φ::=p∣¬φ∣(φ∧φ)∣□_aφ

where and . Connectives , , and are abbreviations as usual.

We adopt the standard definition of when one formula is a subformula of another.

###### Notation 2.2.

For , let indicate that is a subformula of and that is a proper subformula of .

Intuitively, agent-alternating formulas are those formulas in which an operator does not immediately scope over another operator of the same agent . We now offer two ways to precisely capture this intuition, one using immediate subformulas and occurrences, and one using simultaneous induction.

###### Definition 2.3.

For , we say is an immediate subformula of , and write , if is either , or for some , or for some , or for some . Note that the reflexive and transitive closure of is precisely .

For any , an occurrence type of is a finite sequence of formulas in such that and for each between and , . Let be the set of occurrence types of and the prefix-extension relation: iff is a suffix of . It is then easy to see that is a (downward-growing) tree.

We call an occurrence type of with an -occurrence of . If this is for some and , then we also call a -occurrence. We typically denote an -occurrence by .

###### Definition 2.4.

A formula is an agent-alternating formula iff for any and any two different occurrences and such that , there is a and a -occurrence of such that . In other words, is agent alternating iff in the tree , between any two -occurrences, there is a -occurrence for some .

###### Example 2.5.

Assuming are different elements in , examples of agent-alternating formulas include:

 □_ap, □_a□_bp, □_a□_b□_ap, □_a□_b□_cp, □_a(p∧□_bq).

Non-examples include:

 □_a□_ap, □_a□_b□_a□_ap, □_a(□_b□_ap∧□_aq).

We now give an equivalent inductive definition of the set of agent-alternating formulas.

###### Definition 2.6.

Define a family of languages through the following simultaneous induction:

 L_−a∋φ::=p∣□_xψ∣¬φ∣(φ∧φ)

where and while . Then the language is defined inductively by

 L_alt∋φ::=p∣χ∣¬φ∣(φ∧φ)

where and .

Note that does not cover all of . For example, when with , is in but not in .

It is not hard to verify that the two definitions above are equivalent, suggesting that our formal definitions captures the intended intuition. Due to limited space, we omit the proof of this equivalence, but the idea is simply to examine the parsing trees of formulas.

###### Proposition 2.7.

For any , is agent alternating iff .

## 3 Collapsing logics by L_alt

We now investigate which logics are indistinguishable by formulas in . For any normal modal logic (defined as a set of formulas in satisfying the usual closure properties), let . Then the general question is: for which modal logics and are and the same?

More specifically, since we are mainly interested in the introspection axioms and , we focus on the logics appearing in the classic modal logic cube shown in Figure 2 below.666Figure 2 is reproduced from [8]. Our main result is that the two shaded areas in Figure 2 are collapsed in but no other logics are. To establish this result, we need to first develop bisimulation and unraveling concepts for agent-alternating formulas.

###### Notation 3.1.

For convenience, we consider as an object not in . Also for any set of formulas, means that for all , iff .

###### Definition 3.2 (Agent-alternating bisimulation relation).

An agent-alternating bisimulation family between two models and is a family of binary relations between and such that for every and every and such that :

• (Atom) for all , iff ;

• (Zig) for all and , there is such that ;

• (Zag) for all and , there is such that .

Then we say is agent-alternating bisimilar to if there is an agent-alternating bisimulation family between and such that .

###### Lemma 3.3.

For any models and , agent-alternating bisimulation family between and , and , if , then , and if , then .

###### Proof.

A simple induction on modal depth. ∎

###### Definition 3.4 (Agent-alternating unraveling).

Given a model , its agent-alternating unravelings are all models of the form satisfying the following conditions:

• is the set of all nonempty finite sequences of pairs in such that

• ,

• for all , and

• letting for all , and for all ;

• for all and such that , for all , (note that this is precisely ;

• for every and , iff .

Let denote the set of all agent alternating unraveling of . Then for every , we define a family of binary relations between and , which we denote as , by

 uPN_as⟺s_len(s)=⟨a,u⟩.
###### Lemma 3.5.

For any model and , is an agent-alternating bisimulation family between and . Consequently, by Lemma 3.3, for every , .

###### Proof.

Immediate from Definition 3.4 and the recursive structure of as defined in Definition 2.6. ∎

Now we can formally state our main result.

###### Theorem 3.6.

Among the systems displayed in Figure 2:

1. ;

2. ;

3. no other collapse happens when restricting to .

The results are summarized in Figure 2, where systems in the same shaded region in Figure 2 collapse.

###### Proof.

Combining Proposition 3.7, 3.8, and 3.9 below, we have all the collapsing and non-collapsing results in the three layers of Figure 2. To see that the three layers do not collapse, it is enough to observe that the axioms and are in . ∎

and

###### Proof.

The right-to-left direction of both equations is trivial. For the left-to-right direction, by completeness, we need only show that for every , if is satisfied by a pointed model, then it is also satisfied by a pointed model based on a transitive and Euclidean frame. Further, if the first model is based on a serial frame, then the frame of the second model is also serial. So it is enough to show the following: for every pointed model , there exists a pointed model such that:

1. if for every , is serial, then for every , is also serial;

2. for every , is transitive and Euclidean;

3. .

Now let be constructed by adding to the definition of being in as in Definition 3.4 the following:

• for all and such that , .

This construction is possible because crucially the definition of being an agent-alternating unraveling of is silent on what should be when ends in for . Also, when ends in for some , and does not end in , which means that is defined in Definition 3.4.

Now we can show that satisfies all the requirements. It is not hard to see that if is serial, then so is . The key observation is that for any , letting , must include , and for any must be nonempty since is nonempty. Hence we are done with (1). To see that for every , is transitive and Euclidean, note that for any , letting , we have the following:

• If , then for every , ends in , and . This means that our construction above applies to and .

• If , then our construction above applies to : letting , does not end in , and by our definition. Then it is easy to see that for every , , and is also . This means that ends in , and our construction above also applies to . Hence .

Adding the above two points together, we have shown that for every and , . This is precisely transitivity plus Euclideanness.

By Lemma 3.5, since is an agent-alternating bisimulation family and . Thus, all three requirements are satisfied, so we are done. ∎

and

###### Proof.

Following the strategy of the proof of Proposition 3.7, we only need to show that for every pointed model , there exists an agent-alternating unraveling of such that for every , is symmetric.

Indeed, let be the agent-alternating unraveling of such that for every and such that , . Then it is easy to see that for every , is symmetric: for every , if , then we have the following.

• If , then must be by our construction. By the definition of unraveling, .

• If , then must be for some such that letting , . Then our construction applies to and .

Putting the above two points together, is symmetric, so we are done. ∎

###### Proposition 3.9 (Non-collapsing results).

, , , are all nonempty.

###### Proof.

Let be two different elements in . In (), we have the following theorems.

 ⊢_B□_b□_ap→□_ap (1)[T] ⊢_B◊_a□_b□_ap→◊_a□_ap (2)[RM,1] ⊢_B◊_a□_ap→p (3)[B] ⊢_B◊_a□_b□_ap→p (4)[MP,2,3].

Now the last formula, formula (4), is agent-alternating. However, . Using soundness, it is enough to find an model refuting (4). Consider the following model:

By focusing on the restriction of to and , respectively, it is easy to see that is based on an frame. Indeed, the accessibility relation for is even an equivalence relation. Now, since . Also we have . Hence , and thus . This shows that is nonempty.

In the same spirit, . The derivation of in is essentially the same as above: using we can eliminate the in between the two ’s. A symmetric countermodel of this formula is as follows.

In we do not have the axiom. So and are not in . However, we only need to add to the antecedents. Specifically, note that the formula is in . Hence:

• ;

• .

Their derivations in are in the same spirit as above, and and can be reused. ∎

## 4 Agent-nonrepeating formulas

The above non-collapsing results raise a natural question: is there a smaller fragment defined in the same spirit that also collapses to ? Recall that the non-collapsing results are witnessed by formulas like . When is factive, agent is ipso facto introspecting since we can eliminate by . In this section we identify a fragment of agent-nonrepeating formulas in which this cannot happen and does collapse to . The key idea is that we need to forbid to appear at all in the scope of . Again, to formalize this idea, we provide an occurrence-based definition and an inductive definition.

###### Definition 4.1.

A formula is an agent-nonrepeating formula iff for any and occurrence , there is no other occurrence such that

###### Definition 4.2.

Define a family of fragments of through the following simultaneous induction:

 L_X∋φ::=p∣□_xψ∣¬φ∣(φ∧φ)

where and while .

The following equivalence is easily verified.

###### Proposition 4.3.

For any , iff is agent nonrepeating.

As before, we need a notion of bisimulation appropriate for the fragment.

###### Definition 4.4.

An agent-nonrepeating bisimulation family between two models and is a family of binary relations between and such that for every and every and such that :

• (Atom) for all , iff ;

• (Zig) for all and , there is such that ;

• (Zag) for all and , there is such that .

Then we say is agent-nonrepeating bisimilar to if there is an agent-nonrepeating bisimulation family between and such that .

###### Lemma 4.5.

For any models and , agent-nonrepeating bisimulation family between and , and , if , then . Hence whenever is agent-nonrepeating bisimilar to , we have .

For any logic , we write for . We can now prove the desired collapse result.

###### Theorem 4.6.

For every reflexive pointed model , there is a partition model such that is agent-nonrepeating bisimilar to . Consequently, .

###### Proof.

Let be a reflexive model. We construct . Let be the set of all nonempty finite sequences of pairs in such that, letting , (1) , and (2) for all and , there is such that and .

To make the rest of the construction easier, we make a few auxiliary definitions. For each , define to be when and otherwise the with and when . Intuitively, denotes the last accessibility relation used in the sequence . It is easy to observe from the definition above that for any with