Strategic Coalitions with Perfect Recall

07/13/2017 ∙ by Pavel Naumov, et al. ∙ Lafayette College Vassar College 0

The paper proposes a bimodal logic that describes an interplay between distributed knowledge modality and coalition know-how modality. Unlike other similar systems, the one proposed here assumes perfect recall by all agents. Perfect recall is captured in the system by a single axiom. The main technical results are the soundness and the completeness theorems for the proposed logical system.

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Introduction

Autonomous agents such as self-driving cars and robotic vacuum cleaners are facing the challenge of navigating without having complete information about the current situation. Such a setting could be formally captured by an epistemic transition system where an agent uses instructions to transition the system between states without being able to distinguish some of these states. In this paper we study properties of strategies in such systems. An example of such a system is the epistemic transition system , depicted in Figure 1. It has six states named and two instructions and that an agent can use to transition the system from one state to another. For instance, if an instruction is given in state , then the system transitions into state . The system is called epistemic because the agent cannot distinguish state from state for each . The indistinguishability relation is shown in the figure using dashed lines. Atomic proposition is true only in state .

Figure 1: Epistemic Transition System .

The logical system that we propose consists of two modalities. The first is the knowledge modality . Imagine that the system starts in state . Since agent cannot distinguish state from state where statement is not satisfied, the agent does not know if is true or not. We write this as . Next, suppose that the system started in state and the agent used instruction to transition the system into state . In this paper we assume that all agents have perfect recall, so in state the agent remembers history . However, such a history is indistinguishable from history because the agent cannot distinguish state from state and state from state . Thus, the agent does not know that proposition is true in the state even with history . We denote this by . Finally, assume that the system started in state and the agent first used instruction to transition it into state and later instruction to transition it to state . Thus, the history of the system is . The only history that the agent cannot distinguish from this one is history . Since both of these histories end in a state where proposition is satisfied, agent does know that proposition is true in state , given history . We write this as .

The other modality that we consider is the strategic power. In system , the agent can transition the system from state to state by using instruction 0. Similarly, the agent can transition the system from state to state by using instruction 1. In other words, given either history or history the agent can transition the system to a state in which atomic proposition is satisfied. We say that, given either history, agent has a strategy to achieve . Histories and are the only histories indistinguishable by agent from history . Since she has a strategy to achieve under all histories indistinguishable from history , we say that given history the agent knows that she has a strategy. Similarly, given history , she also knows that she has a strategy. However, since indistinguishable histories and require different strategies to achieve , given history she does not know what the strategy is. We say that she does not have a know-how strategy. We denote this by , where stands for know-ow. Of course, it is also true that .

The situation changes if the transition system starts in state instead of state and transitions to state under instruction . Now the history is and the histories that the agent cannot distinguish from this one are history and history itself. Given both of these two histories, agent can achieve using the same transition . Thus, .

Finally note that there are only two histories: and indistinguishable from . Given either history, agent can achieve using instruction . Thus, . That is, given history agent knows how to transition to a state in which formula is satisfied.

Multiagent Systems

Like many other autonomous agents, self-driving cars are expected to use vehicle-to-vehicle communication to share traffic information and to coordinate actions [Harding et al.2014]. Thus, it is natural to consider epistemic transition systems that have more than one agent. An example of such a system is depicted in Figure 2.

Figure 2: Epistemic Transition System .

This system has five epistemic states: , and and three agents: , , and . In each state the agents vote either 0 or 1 and the system transitions into the next state based on the majority vote. For example, since the directed edge from state to state is labelled with 1, if the majority of agents in state votes 1, then the system transitions into state . Since coalition forms a majority, this coalition has a strategy to transition the system from state to state and, thus, to achieve . Note that agent cannot distinguish state from state and thus agent does not know what she should vote for to achieve . Similarly, agent also does not know what she should vote for to achieve because she cannot distinguish state from state . In this paper, we assume that members of a coalition make the decisions based on combined (distributed) knowledge of the whole coalition. In our example, coalition can distinguish state from both state and state . Thus, given history the coalition knows how to achieve . We denote this by , or simply as .

Universal Principles

We have discussed a statement being true or false given a certain history. This paper focuses on the logical principles that are true for each history in each epistemic transition system. An example of such a principle is the strategic positive introspection: This principle says that if a coalition knows how to achieve , then the coalition knows that it knows how to achieve . Informally, this principle is true because in order for statement to be satisfied for a given history , coalition must have a strategy to achieve that works under any history indistinguishable from history by the coalition. Thus, the same strategy must work for any history indistinguishable from history by the coalition. In other words, it is also true that . Recall that is an arbitrary history indistinguishable from history by coalition . Hence, according to the standard semantics of the epistemic modality . A similar argument can be used to justify the strategic negative introspection:

Another universal principle is the empty coalition principle: Indeed, means that statement is true under any history indistinguishable from the given history by an empty coalition. Since an empty coalition cannot distinguish any two histories, the assumption means that statement is true under any history. In particular, this statement is true after the next transition no matter how agents vote. Hence, .

The epistemic modality also satisfies axioms of epistemic logic S5 for distributed knowledge. Know-how modality satisfies the unachievability of falsehood principle: , stating that no coalition can achieve . Know-how modality also satisfies a form of cooperation principle [Pauly2001, Pauly2002]:

Perfect Recall

A complete trimodal logical system describing the interplay between distributed knowledge modality , coalition know-how modality , and standard (not know-how) strategic power modality in the imperfect recall setting was proposed by [Naumov and Tao2017b]. We provide a complete axiomatization of the interplay between modalities and in the perfect recall setting. Surprisingly, the assumption of perfect recall by all agents is captured by a single principle that we call the perfect recall principle: where . This principle says that if a sub-coalition can achieve , then after the vote the whole coalition will know that is true. Informally, this principle is true because coalition is able to recall how sub-coalition voted and, thus, will deduce that formula is true after the transition. As an empty coalition has no memory even in the perfect recall setting, it is essential for coalition to be nonempty. However, the sub-coalition can be empty.

Literature Review

Non-epistemic logics of coalition power were developed by [Pauly2001, Pauly2002], who also proved the completeness of the basic logic of coalition power. His approach has been widely studied in the literature [Goranko2001, van der Hoek and Wooldridge2005, Borgo2007, Sauro et al.2006, Ågotnes et al.2010, Ågotnes, van der Hoek, and Wooldridge2009, Belardinelli2014, Goranko, Jamroga, and Turrini2013]. An alternative logical system for coalition power was proposed by [More and Naumov2012].

[Alur, Henzinger, and Kupferman2002] introduced Alternating-Time Temporal Logic (ATL) that combines temporal and coalition modalities. [van der Hoek and Wooldridge2003] proposed to combine ATL with epistemic modality to form Alternating-Time Temporal Epistemic Logic. They did not prove the completeness theorem for the proposed logical system. Aminof, Murano, Rubin and Zuleger [Aminof et al.2016] studied model-checking problems of an extension of ATL with epistemic and “prompt eventually” modal operators.

[Ågotnes and Alechina2012] proposed a complete logical system that combines the coalition power and epistemic modalities. Since their system does not have epistemic requirements on strategies, it does not contain any axioms describing the interplay of these modalities. In the extended version, [Ågotnes and Alechina2016] added a complete axiomatization of an interplay between single-agent knowledge and know-how modalities.

Know-how strategies were studied before under different names. While [Jamroga and Ågotnes2007] talked about “knowledge to identify and execute a strategy”, [Jamroga and van der Hoek2004] discussed “difference between an agent knowing that he has a suitable strategy and knowing the strategy itself”. [van Benthem2001] called such strategies “uniform”. [Broersen2008] investigated a related notion of “knowingly doing”, while [Broersen, Herzig, and Troquard2009] studied modality “know they can do”. [Wang2015, Wang2016] captured the “knowing how” as a binary modality in a complete logical system with a single agent and without the knowledge modality.

Coalition know-how strategies for enforcing a condition indefinitely were investigated by [Naumov and Tao2017a]. Such strategies are similar to [Pauly2001, p. 80] “goal maintenance” strategies in “extended coalition logic”. A similar complete logical system in a single-agent setting for know-how strategies to achieve a goal in multiple steps rather than to maintain a goal is developed by [Fervari et al.2017].

[Naumov and Tao2017b] also proposed a complete trimodal logical system describing an interplay between distributed knowledge modality , coalition know-how modality , and standard (not know-how) strategic power modality in the imperfect recall setting.

In this paper we provide a complete axiomatization of an interplay between modalities and in the perfect recall setting. The main challenge in proving the completeness, compared to [Ågotnes and Alechina2016, Fervari et al.2017, Naumov and Tao2017b, Naumov and Tao2017a], is the need to construct not only “possible worlds”, but the entire “possible histories”, see the proof of Lemma 22.

Outline

The rest of the paper is organized as follows. First, we introduce the syntax and semantics of our logical system. Next, we list the axioms and give examples of proofs in the system. Then, we prove the soundness and the completeness of this system.

Syntax and Semantics

Throughout the rest of the paper we assume a fixed set of agents . By we denote the set of all functions from set to set , or in other words, the set of all tuples of elements from set indexed by the elements of set . If is such a tuple and , then by we denote the -th component of tuple .

We now proceed to describe the formal syntax and semantics of our logical system starting with the definition of a transition system. Although our introductory examples used voting to decide on the next state of the system, in this paper we investigate universal properties of an arbitrary nondeterministic action aggregation mechanism.

Definition 1

A tuple is called an epistemic transition system, if

  1. is a set of epistemic states,

  2. is an indistinguishability equivalence relation on for each ,

  3. is a nonempty set called domain of choices,

  4. is an aggregation mechanism,

  5. is a function that maps propositional variables into subsets of .

For example, in the transition system depicted in Figure 1, the set of states is and relation is a transitive reflexive closure of , .

Informally, an epistemic transition system is regular if there is at least one next state for each outcome of the vote.

Definition 2

An epistemic transition system is regular if for each and each , there is such that .

A coalition is a subset of . A strategy profile of coalition is any tuple in the set .

Definition 3

For any states and any coalition , let if for each agent .

Lemma 1

For each coalition , relation is an equivalence relation on the set of epistemic states .

Definition 4

For all strategy profiles and of coalitions and respectively and any coalition , let if for each .

Lemma 2

For any coalition , relation is an equivalence relation on the set of all strategy profiles of coalitions containing coalition .

Definition 5

A history is an arbitrary sequence such that and

  1. for each ,

  2. for each ,

  3. for each .

In this paper we assume that votes of all agents are private. Thus, an individual agent only knows her own votes and the equivalence classes of the states that the system has been at. This is formally captured in the following definition of indistinguishability of histories by an agent.

Definition 6

For any history , any history , and any agent , let if and

  1. for each ,

  2. for each .

Definition 7

For any histories and any coalition , let if for each agent .

Lemma 3

For any coalition , relation is an equivalence relation on the set of histories.

The length of a history is the value of . By Definition 7, the empty coalition cannot distinguish any two histories, even of different lengths.

Lemma 4

for each histories and such that for some nonempty coalition .

For any sequence and an element , by sequence we mean . If sequence is nonempty, then by we mean element .

Lemma 5

If , then , , and .

Definition 8

Let be the language specified as follows where .

Boolean constants and are defined as usual.

Definition 9

For any history of an epistemic transition system and any formula , let satisfiability relation be defined as follows

  1. if and is a propositional variable,

  2. if ,

  3. if or ,

  4. if for each history s.t. ,

  5. if there is a strategy profile such that for any history , if and , then .

Axioms

In additional to propositional tautologies in language , our logical system consists of the following axioms:

  1. Truth: ,

  2. Negative Introspection: ,

  3. Distributivity: ,

  4. Monotonicity: , if ,

  5. Strategic Positive Introspection: ,

  6. Cooperation: , where ,

  7. Empty Coalition: ,

  8. Perfect Recall: , where ,

  9. Unachievability of Falsehood: .

We write if formula is provable from the axioms of our logical system using Necessitation, Strategic Necessitation, and Modus Ponens inference rules:

We write if formula is provable from the theorems of our logical system and a set of additional axioms using only Modus Ponens inference rule.

The next lemma follows from a well-known observation that the Positive Introspection axiom is provable from the other axioms of S5.

Lemma 6

.

Proof. Formula is an instance of the Negative Introspection axiom. Thus, by the law of contrapositive in the propositional logic. Hence, by the Necessitation inference rule. Thus, by the Distributivity axiom and the Modus Ponens inference rule,

(1)

At the same time, is an instance of the Truth axiom. Thus, by contraposition. Hence, taking into account the following instance of the Negative Introspection axiom , one can conclude that . The latter, together with statement (1), implies the statement of the lemma by the laws of propositional reasoning.

Derivation Examples

This section contains examples of formal proofs in our logical system. The results obtained here are used in the proof of completeness. The proof of Lemma 7 is based on the one proposed to us by Natasha Alechina.

Lemma 7 (Alechina)

.

Proof. By the Positive Strategic Introspection axiom, . Thus, by the contrapositive. Hence, by the Necessitation inference rule. Then, by the Distributivity axiom and the Modus Ponens inference rule . Thus, by the Negative Introspection axiom and the laws of propositional reasoning, . Note that is the contrapositive of the Truth axiom. Therefore, by the laws of propositional reasoning, .

Lemma 8

, where .

Proof. Note that is a propositional tautology. Thus, . Hence, by the Strategic Necessitation inference rule. At the same time, by the Cooperation axiom, due to the assumption . Therefore, by the Modus Ponens inference rule.

Lemma 9

If , then

  1. ,

  2. , where sets are pairwise disjoint.

Proof. To prove the second statement, apply deduction lemma for propositional logic time. Then, we have . Thus, by the Strategic Necessitation inference rule, Hence, by the Cooperation axiom and the Modus Ponens inference rule. Then, by the Modus Ponens inference rule. Thus, again by the Cooperation axiom and the Modus Ponens inference rule we have . Therefore, by repeating the last two steps times, . The proof of the first statement is similar, but it uses the Distributivity axiom instead of the Cooperation axiom.

Soundness

In this section we prove the following soundness theorem for our logical system.

Theorem 1

If , then for each history of each regular epistemic transition system.

The proof of the soundness of axioms of epistemic logic S5 with distributed knowledge is standard. Below we prove the soundness of each remaining axiom as a separate lemma.

Lemma 10

If , then .

Proof. Due to Definition 9, assumption implies that there is a strategy profile of coalition such that for any history , if and , then .

Consider any history such that . By Definition 9, it suffices to show that . Furthermore, by the same Definition 9, it suffices to prove that for any history , if and , then .

Suppose that and . By Lemma 3, statements and imply that . Therefore, by the choice of .

Lemma 11

If , , and , then .

Proof. By Definition 9, assumption implies that there is a strategy profile of coalition such that for any history , if and , then .

Similarly, assumption implies that there is a profile of coalition such that for any history , if and , then .

Consider strategy profile of coalition such that

Strategy profile is well-defined because coalitions and are disjoint. By Definition 9, it suffices to show that for any history , if and , then .

Suppose that and . Then, and . Hence, by the choice of strategy . Similarly, . Therefore, by Definition 9.

Lemma 12

If , then .

Proof. Let be the empty tuple. By Definition 9, it suffices to show that for any history we have . Definition 7 implies . Thus, by Definition 9 and .

Lemma 13

If , then , where and .

Proof. By Definition 9, assumption implies that there is a strategy profile of coalition such that for any history , if and , then .

Consider any history such that and . By Definition 9, it suffices to prove that . Let be any such history that . Again by Definition 9, it suffices to prove that .

By Lemma 4, assumptions and imply that . Thus, for some history , some complete strategy profile , and some epistemic state .

Then, . Hence, and by Lemma 5. Hence, and by the choice of history . Then, and by Lemma 3, Lemma 2, and because . Thus, by the choice of strategy profile . Therefore, , because .

Lemma 14

for any history of any regular epistemic transition system.

Proof. Suppose . By Definition 9, there is a strategy profile such that for any history , if and , then .

By Definition 1, set contains at least one element . Let be a complete strategy profile such that

(2)

By Definition 2, there is an epistemic state such that . Thus, is a history by Definition 5. Note that by Lemma 3 and due to equation (2). Thus, by the choice of strategy profile , which contradicts Definition 9 and the definition of . This concludes the proof of Theorem 1.

Completeness

In the rest of this paper we focus on the completeness theorem for our logical system with respect to the class of regular epistemic transition systems. We start the proof of completeness by fixing a maximal consistent set of formulae and defining a canonical epistemic transition system using the “unravelling” technique [Sahlqvist1975]. Note that the domain of choices in the canonical model is the set of all formulae .

Canonical Epistemic Transition System

Definition 10

The set of epistemic states consists of all sequences , such that and

  1. is a maximal consistent subset of for each ,

  2. for each ,

  3. , for each .

Definition 11

Suppose that and are epistemic states. For any agent , let if there is a non-negative integer such that

  1. for each such that ,

  2. for each such that ,

  3. for each such that ,

  4. for each such that .

Lemma 15

For any epistemic state and any integer , if and for each integer such that , then .

Proof. Suppose for some . Let be the maximal such . Note that by the assumption of the lemma. Thus, .

Assumption implies by the maximality of the set . Hence, by the Negative Introspection axiom. Thus, by the Monotonicity axiom and the assumption of the lemma (recall that ). Then, due to the maximality of the set . Hence, by Definition 10. Thus, due to the consistency of the set , which contradicts the choice of .

Lemma 16

For any epistemic state and any integer , if and for each integer such that , then .

Proof. We prove the lemma by induction on the distance between and . In the base case . Then the assumption implies by the Truth axiom. Therefore, due to the maximality of set .

Suppose that . Assumption implies by Lemma 6. Thus, by the Monotonicity axiom, the condition of the inductive step, and the assumption of the lemma. Then, by the maximality of set . Hence, by Definition 10. Therefore, by the induction hypothesis.

Lemma 17

For any epistemic states such that , if , then .

Proof. The statement of the lemma follows from Lemma 15 and Lemma 16 as well as Definition 11 because there is a unique path between any two nodes in a tree.

Next, we specify the aggregation mechanism of the canonical epistemic transition system. Informally, if a coalition has a know-how strategy to achieve and all members of the coalition vote for , then must be true after the transition.

Definition 12

For any states and any complete strategy profile , let if

Definition 13

.

This concludes the definition of the canonical epistemic transition system . We prove that this system is regular in Lemma 25.

-child Lemmas

The following technical results (Lemmas 1822) about the knowledge modality are used in the proof of completeness.

Lemma 18

For any epistemic state if , then there is an epistemic state such that and .

Proof. Consider the set . First, we show that set is consistent. Assume the opposite. Then, there exist formulae such that . Thus, by Lemma 9. Therefore, by the choice of formulae , which contradicts the consistency of the set due to the assumption .

Let be a maximal consistent extension of set and let be sequence . Note that by Definition 10 and the choice of set . Furthermore, by Definition 11. To finish the proof, note that by the choice of set .

History is a sequence whose last element is an epistemic state. Epistemic state , by Definition 10, is also a sequence. Expression denotes the last element of the sequence .

Lemma 19

For any history , if , then for each history such that .

Proof. Assumption by Definition 6 implies that . Therefore, by Lemma 17 and the assumption .

Figure 3: Illustration for Lemma 20.
Lemma 20

For any nonempty coalition , any states , and any complete strategy profile such that and , see Figure 3, there is a state and a complete strategy profile such that