[PPT] - Reasoning about Knowledge and Strategies: Epistemic Strategy Logic PowerPoint Presentation

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Reasoning about Knowledge and Strategies: Epistemic Strategy Logic

Francesco Belardinelli

Laboratoire IBISC, Universit´ e d’Evry

Strategic Reasoning – 5 April 2014

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Overview

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Motivation and Background

◮ logics for reasoning about strategies and knowledge 2

Epistemic Strategy Logic

◮ semantics ◮ syntax 3

Main Contribution

◮ model checking ESL is no harder than SL 4

Imperfect Information

◮ benefits of combining epistemic and strategy modalities 5

Conclusions and Future Work

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Motivation and Background

Logics of strategic abilities

Logics for strategic reasoning are a thriving area of research in AI and MAS.

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Motivation and Background

Logics of strategic abilities

Logics for strategic reasoning are a thriving area of research in AI and MAS.
Two lines of research:

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multi-modal logics to formalise strategic abilities and behaviours of individual agents and groups:

⋆ Alternating-time Temporal Logic [AHK02] ⋆ Coalition Logic [Pau02] ⋆ Strategy Logic [CHP10, MMV10] 2

extensions of logics for reactive systems with epistemic operators to reason about the knowledge agents have of the system’s evolution:

⋆ combinations of CTL and LTL with multi-modal epistemic logic S5n [HV86, HV89, FHMV95] ⋆ successfully applied to MAS specification and verification [GvdM04, KNN+08, LQR09] 3

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Motivation and Background

Logics of strategic abilities

Logics for strategic reasoning are a thriving area of research in AI and MAS.
Two lines of research:

1

multi-modal logics to formalise strategic abilities and behaviours of individual agents and groups:

⋆ Alternating-time Temporal Logic [AHK02] ⋆ Coalition Logic [Pau02] ⋆ Strategy Logic [CHP10, MMV10] 2

extensions of logics for reactive systems with epistemic operators to reason about the knowledge agents have of the system’s evolution:

⋆ combinations of CTL and LTL with multi-modal epistemic logic S5n [HV86, HV89, FHMV95] ⋆ successfully applied to MAS specification and verification [GvdM04, KNN+08, LQR09]

Along these lines, [vdHW03] introduced ATEL.

◮ spawned a wealth of contributions: ⋆ imperfect information/uniform strategies [Sch04, JvdH04] ⋆ constructive knowledge [J˚

A07]

⋆ irrevocable/feasible strategies [AGJ07, Jon03] 3

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Motivation and Background

Logics of strategic abilities

Logics for strategic reasoning are a thriving area of research in AI and MAS.
Two lines of research:

1

multi-modal logics to formalise strategic abilities and behaviours of individual agents and groups:

⋆ Alternating-time Temporal Logic [AHK02] ⋆ Coalition Logic [Pau02] ⋆ Strategy Logic [CHP10, MMV10] 2

extensions of logics for reactive systems with epistemic operators to reason about the knowledge agents have of the system’s evolution:

⋆ combinations of CTL and LTL with multi-modal epistemic logic S5n [HV86, HV89, FHMV95] ⋆ successfully applied to MAS specification and verification [GvdM04, KNN+08, LQR09]

Along these lines, [vdHW03] introduced ATEL.

◮ spawned a wealth of contributions: ⋆ imperfect information/uniform strategies [Sch04, JvdH04] ⋆ constructive knowledge [J˚

A07]

⋆ irrevocable/feasible strategies [AGJ07, Jon03]

Epistemic Strategy Logic = strategies + knowledge

◮ topic of interest [HvdM14b, HvdM14a] 3

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The Prisoner’s Dilemma

Games in Normal Form

Anne and Bob can either Cooperate or Defect
payoff ordering: a > b > c > d

Bob Cooperate Defect Anne Cooperate b, b d, a Defect a, d c, c

can Anne achieve payoff a?
does Anne know whether she can achieve payoff a?
does Bob know whether Anne has a strategy to achieve payoff a?
do Anne and Bob know (de dicto) whether they can reach a Nash equilibrium?
are there strategies such that Anne and Bob know (de re) that they can reach a Nash

equilibrium?

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Epistemic Concurrent Game Models

Agents

We adopt an agent-oriented perspective.

Definition (Agent)

An agent i is

situated in some local state li ∈ Li and . . .
performs the actions in Acti
. . . according to her protocol function Pri : Li → 2Acti

The setting is reminiscent of the interpreted systems semantics for MAS [FHMV95].

Example (Prisoner’s Dilemma)

Agent Anne = LA, ActA, PrA is defined as

LA = {ǫA, a, b, c, d}
ActA = {C, D, ∗}, where ∗ is the skip action
PrA(ǫA) = {C, D} and PrA(a) = PrA(b) = PrA(c) = PrA(d) = {∗}

The definition of agent Bob is symmetric.

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Epistemic Concurrent Game Models

ECGM

The interactions amongst agents generate ECGM.

related to CGS [AHK02, MMV10] and AETS [vdHW03]
global states are not primitive: s = l0, . . . , lℓ ∈ G = Πi∈AgLi
joint actions are tuples σ = σ0, . . . , σℓ ∈ Act = Πi∈AgActi

Definition (ECGM)

Given

◮ a set Ag = {i0, . . . , iℓ} of agents ◮ a set AP of atomic propositions

an ECGM P includes

◮ a finite set I ⊆ G of initial global states ◮ a transition function τ : G × Act → G ◮ an interpretation π : AP → 2G of atomic propositions

the epistemic indistinguishability relation is not primitive: s ∼i s′ iff li = l′

i

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The Prisoner’s Dilemma as an ECGM

Let AP = {ai, bi, ci, di} for i ∈ {A, B}. ǫA, ǫB s0 b, b d, a a, d c, c (C, C) (C, D) (D, C) (D, D) (∗, ∗) (∗, ∗) (∗, ∗) (∗, ∗)

Example (ECGM Ppd)

For the set Ag = {A, B} of agents, the prisoner’s dilemma ECGM Ppd includes

the set I = {s0} of initial states, with s0 = (ǫA, ǫB)
the transition function τ, given as

◮ τ(s0, (C, C)) = (b, b) ◮ τ(s0, (C, D)) = (d, a) ◮ τ(s0, (D, C)) = (a, d) ◮ τ(s0, (D, D)) = (c, c) ◮ τ(s, (∗, ∗)) = s, for every state s different from s0

the interpretation π s.t. a state (lA, lB) belongs to π(pi) iff li = p.

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Epistemic Strategy Logic

ESL

ESL extends SL with epistemic operators Ki for individual knowledge.

we introduce a set Vari of strategy variables for each agent i ∈ Ag

Definition (ESL)

ESL formulas are defined in BNF as follows: φ ::= p | ¬φ | φ → φ | Xφ | φUφ | ∃xiφ | Kiφ

we consider a multi-agent setting (= [CHP10])
the language does not include the binding operator (α, x) (= [MMV10])

The questions above can be recast as model checking problems: Ppd

?

| = ∃xAFaA Ppd

?

| = KA(∃xAFaA ∨ ¬∃xAFaA) Ppd

?

| = KB(∃xAFaA ∨ ¬∃xAFaA)

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Epistemic Concurrent Game Models

Strategies

Definition (Strategy)

An A-strategy is a mapping fA : G + → ActA from finite sequences of states to enabled A-actions.

a run λ is a sequence s0 → s1 → . . . of global states
a run λ belongs to outcome out(s, fA) iff λ(i + 1) ∈ ˆ

τ(λ(i), fA(λ[. . . , i])) ⇒ a group strategy is really the composition of its members’ strategies

an assignment χ maps each agent i ∈ Ag to an i-strategy fi

◮ f χ is the Ag-strategy χ(i0) × . . . × χ(iℓ)

Definition (Satisfaction)

An ECGM P satisfies an ESL formula ϕ in a state s for an assignment χ, iff (P, s, χ) | = p iff s ∈ π(p) (P, s, χ) | = Xψ iff for λ = out(s, f χ), (P, λ(1), χ) | = ψ (P, s, χ) | = ψUψ′ iff for λ = out(s, f χ) there is k ≥ 0 s.t. (P, λ(k), χ) | = ψ′ and 0 ≤ j < k implies (P, λ(j), χ) | = ψ (P, s, χ) | = ∃xiψ iff there exists an i-strategy fi s.t. (P, s, χi

fi ) |

= ψ (P, s, χ) | = Kiψ iff for every s ∈ S, s ∼i s′ implies (P, s′, χ) | = ψ

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Expressiveness

Knowledge of Nash Equilibria

given an n-player game in normal form with payoff ordering a1 > . . . > ak, define

ψNE ::=

n

i=1

k

i=1

   

i−1

j=1

¬∃yjXaj   ∧ ∃yiXai → Xai  

Proposition

(Ppd, s0, χ) | = ψNE iff (χ(1)(s0), . . . , χ(n)(s0)) is a Nash equilibrium

for the prisoner’s dilemma,

(Ppd, s0, χ) | = ψNE iff (χ(1)(s0), χ(2)(s0)) is a Nash equilibrium iff χ(1)(s0) = χ(2)(s0) = D The questions above can be recast as model checking problems: Ppd

?

| = KA∃xA, xBψNE ∧ KB∃xA, xBψNE Ppd

?

| = ∃xA, xB(KAψNE ∧ KBψNE )

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Expressiveness

Knowledge de re v. Knowledge de dicto

knowledge de re ⇒ knowledge de dicto:

| = ∃xiKjφ → Kj∃xiφ also, knowledge de dicto ⇒ knowledge de re: | = Kj∃xiφ → ∃xiKjφ indeed, agents have perfect information of the game

individual strategies depend on global states [JvdH04]

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Model Checking ESL

Theorem (Hardness)

The model checking problem for ESL is Non-ElementarySpace-hard.

reduction to satisfiability for quantified propositional temporal logic (QPTL)
differently from [MMV10] the syntax does not include the binding operator

Theorem (Completeness)

The model checking problem for ESL is PTime-complete w.r.t. the size of the model and Non-Elementary w.r.t. the size of the formula.

reduction to non-emptyness for alternating tree automata [MMV10]

⇒ The model checking problem is no harder for ESL than for SL.

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Imperfect Information

ǫA, ǫB λ, 0 λ, 1 0, 0 1, 0 0, 1 1, 1 (∗, 0) (∗, 1) (0, ∗) (1, ∗) (0, ∗) (1, ∗) A (∗, ∗) (∗, ∗) (∗, ∗) (∗, ∗)

Bob chooses secretly between 0 and 1
at the next step Anne also chooses between 0 and 1
Anne wins the game iff the values provided by the two players coincide
the dotted line indicates epistemic indistinguishability
Anne knows that there exists a strategy to win the game . . .

. . . however, she is not able to point this strategy out ⇒ Anne has imperfect information of the game

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Imperfect Information

Uniform Strategies

Under imperfect information, strategies depend on the local state of agents only.

Definition (Uniform Strategies [JvdH04])

A (positional) i-strategy is uniform iff for all states s, s′, s ∼i s′ implies fi(s) = fi(s′).

Anne knows that there exists a strategy to win the game . . .

(Q, sλ0) | =ii KA ∃xA X win . . . however, there is no strategy that she knows to be winning: (Q, sλ0) | =ii ∃xA KA X win

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Imperfect Information

Fixed-point Characterisations

Epistemic modalities allow us to recover some fixed-point characterisations of ATL operators.

for A = {i0, . . . , iℓ},

xA = xi0, . . . , xiℓ and ¯ A = Ag \ A, define

A

φ ::= ∃ xA∀ x¯

Aφ

under imperfect information we have that
A

Gφ ⇔ φ ∧ A X A Gφ

A

Fφ ⇔ φ ∨ A X A Fφ

A

φUφ′ ⇔ φ′ ∨ (φ ∧ A X A φUφ′)

by using epistemic modalities we can recover fixed-point characterisations:
i

Gφ ⇔ φ ∧ i X i (Gφ ∧ Ki( i Gφ → Gφ))

i

Fφ ⇔ φ ∨ i X i (Fφ ∧ Ki( i Fφ → Fφ))

i

φUφ′ ⇔ φ′ ∨ (φ ∧ i X i (φUφ′ ∧ Ki( i φUφ′ → φUφ′)))

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Conclusions

Results:

ESL: a logic for reasoning about knowledge and strategies in a multi-agent setting
the model checking problem is no harder than for SL
under imperfect information ESL allows us to recover the fixed-point characterisation of ATL
perators

and Future Work:

fragments of ESL: better computational complexity?
epistemic operators for group knowledge (distributed, common knowledge, etc.)
imperfect information

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References

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J. ACM, 49(5):672–713, 2002.

Krishnendu Chatterjee, Thomas A. Henzinger, and Nir Piterman. Strategy logic.

Inf. Comput., 208(6):677–693, 2010.

Ronald Fagin, Joseph Y. Halpern, Yoram Moses, and Moshe Y. Vardi. Reasoning About Knowledge. The MIT Press, 1995. Peter Gammie and Ron van der Meyden. Mck: Model checking the logic of knowledge. In Rajeev Alur and Doron Peled, editors, CAV, volume 3114 of Lecture Notes in Computer Science, pages 479–483. Springer, 2004. Joseph Y. Halpern and Moshe Y. Vardi. The complexity of reasoning about knowledge and time: Extended abstract. In Juris Hartmanis, editor, STOC, pages 304–315. ACM, 1986. Joseph Y. Halpern and Moshe Y. Vardi. The complexity of reasoning about knowledge and time. i. lower bounds.

J. Comput. Syst. Sci., 38(1):195–237, 1989.
X. Huang and R. van der Meyden.

An epistemic strategy logic (extended abstract). In Proceedings of 2nd International Workshop on Strategic Reasoning, 2014.

X. Huang and R. van der Meyden.

A temporal logic of strategic knowledge. In Proceedings of the 14th International Conference on Principles of Knowledge Representation and Reasoning, 2014. Wojciech Jamroga and Thomas ˚ Agotnes. 17