Logic of Joint Action Natasha Alechina (joint work with Thomas - - PowerPoint PPT Presentation

Logic of Joint Action

Natasha Alechina (joint work with Thomas ˚ Agotnes) November 2011, St Andrews Workshop in Honour of Roy Dyckhoff

˚ Agotnes & Alechina Logic of Joint Action November 2011 1 / 24

Motivation

Translate Coalition Logic (which has complicated semantics) into multimodal K with intersection of modalities Benefits: for example, a more standard tableaux procedure

˚ Agotnes & Alechina Logic of Joint Action November 2011 2 / 24

Outline

Outline

1

Syntax and Semantics of Coalition Logic

2

Syntax and Semantics of K ∩

n 3

Idea of the Embedding

4

Logic of Joint Actions

˚ Agotnes & Alechina Logic of Joint Action November 2011 3 / 24

Syntax and Semantics of Coalition Logic

Syntax

Syntax of CL is defined relative to a set of primitive propositions Θ and a set of agents N (assume |N| = g) φ ::= p | ¬φ | φ ∧ φ | [C]φ where p ∈ Θ and C ⊆ N [C]φ means: coalition C can enforce the outcome φ

˚ Agotnes & Alechina Logic of Joint Action November 2011 4 / 24

Syntax and Semantics of Coalition Logic

CL Models

(Also called Concurrent Game Structures, CGS) M = S, V, Act, d, δ where S is a set of states; V is a valuation function, assigning a set V(s) ⊆ Θ to each state s ∈ S; Act is a set of actions; For each i ∈ N and s ∈ S, di(s) ⊆ Act is a non-empty set of actions available to agent i in s. D(s) = d1(s) × · · · × dg(s) is the set of full joint actions in s. δ is a transition function, mapping each state s ∈ S and full joint action a ∈ D(s) to a state δ(s, a) ∈ S.

˚ Agotnes & Alechina Logic of Joint Action November 2011 5 / 24

Syntax and Semantics of Coalition Logic

Truth in CL Models

M, s | = p ⇔ p ∈ V(s) M, s | = ¬φ ⇔ M, s | = φ M, s | = (φ1 ∧ φ2) ⇔ (M, s | = φ1 and M, s | = φ2) M, s | = [C]ψ ⇔ ∃aC ∈ DC(s)∀aC ∈ DC(s), M, δ(s, (aC, aC)) | = ψ

˚ Agotnes & Alechina Logic of Joint Action November 2011 6 / 24

Syntax and Semantics of K ∩

n

Kn with intersection of modalities

Assume a set of primitive propositions Θ and actions A: φ ::= p ∈ Θ | ¬φ | φ ∧ φ | [π]φ π ::= a | π ∩ π where a ∈ A. As usual, πφ is defined as ¬[π]¬φ

˚ Agotnes & Alechina Logic of Joint Action November 2011 7 / 24

Syntax and Semantics of K ∩

n

K ∩

n models

M = S, V, {Rπ : π ∈ Π} where S is a set of states; V : S → 2Θ is a valuation function; For each π ∈ Π, Rπ ⊆ S × S Rπ1∩π2 = Rπ1 ∩ Rπ2 (INT) The modality truth definition clause: M, s | = [π]φ iff ∀(s, s′) ∈ Rπ, M, s′ | = φ

˚ Agotnes & Alechina Logic of Joint Action November 2011 8 / 24

Idea of the Embedding

Idea of embedding

A CGS like this:

<a,b> s t

˚ Agotnes & Alechina Logic of Joint Action November 2011 9 / 24

Idea of the Embedding

Idea of embedding 2

can be represented by a K ∩

n model like this:

(1,a)∩(2,b) s t (1,a) (2,b)

˚ Agotnes & Alechina Logic of Joint Action November 2011 10 / 24

Idea of the Embedding

Idea of embedding

and CL formulas translated as follows: ([{i1, . . . , ik}]φ)′ ≡

• a1,...,ak∈Actφ
• 1≤j≤k

(ij, aj)⊤ ∧ [(i1, a1) ∩ . . . ∩ (ik, ak)]φ′ where Actφ is a finite set of actions that is “read off” the formula

˚ Agotnes & Alechina Logic of Joint Action November 2011 11 / 24

Idea of the Embedding

Problem

t1 s t2 <a1, b1> <a2, b2> <a1, b2> <a2, b1>

˚ Agotnes & Alechina Logic of Joint Action November 2011 12 / 24

Idea of the Embedding

Problem

t1 s t2 (1,a1), (2,b1), (1,a2), (2,b2) (1,a1)(2,b1), (1,a2)(2,b1), (1,a1)(2,b2), (1,a2)(2,b2) (1,a1), (2,b1), (1,a2), (2,b2) (1,a1)(2,b1), (1,a2)(2,b1), (1,a1)(2,b2), (1,a2)(2,b2)

˚ Agotnes & Alechina Logic of Joint Action November 2011 13 / 24

Idea of the Embedding

Injective CGS

CGS without two or more different full joint actions between the same two states are injective. Injective CGSs do not suffer this problem. Theorem (Goranko 2007) For every CGS M = S, V, Act, d, δ there is an injective CGS M′ with states S′ such that S ⊆ S′ and for all CL formulae φ and states s ∈ S, M, s | = φ iff M′, s | = φ. Moreover, if M is finite, then |S′| ≤ |S| + |δ|.

˚ Agotnes & Alechina Logic of Joint Action November 2011 14 / 24

Logic of Joint Actions

Logic of Joint Actions

Let Act be a finite set of actions and N a set of g agents. Define a set of atomic modalities as follows: A = N × Act an atomic modality in A is an individual action a composite modality π = π1 ∩ π2 is a joint action. joint actions of the form (1, a1) ∩ . . . ∩ (g, an) with one individual action for every agent in N will be called complete (joint) actions.

˚ Agotnes & Alechina Logic of Joint Action November 2011 15 / 24

Logic of Joint Actions

Models for Logic of JA

A K ∩

n model over A (where Act is finite) is a joint action model if it

satisfies: Seriality (SER) For any state s and agent i, at least one action is enabled in s for i. Independent Choice (IC) For any state s, agents C = {i1, . . . , ik} and actions a1, . . . , ak ∈ Act, if for every j aj is enabled for ij in s, then there is a state s′ such that (s, s′) ∈ R(i1,a1)∩···∩(ik,ak). Deterministic Joint Actions (DJA) For any complete joint action α and states s, s1, s2, (s, s1), (s, s2) ∈ Rα implies that s1 = s2. Unique Joint Actions (UJA) For any complete joint actions α and β and states s, t, if (s, t) ∈ Rα ∩ Rβ then α = β.

˚ Agotnes & Alechina Logic of Joint Action November 2011 16 / 24

Logic of Joint Actions

Translation

Given an injective CGS M = S, V, Act, d, δ where Act is finite, the corresponding joint action model ˆ M = S, V, {Rπ : π ∈ Π} over Θ and A is defined as follows: R(i,a) = {(s, s′) : ∃a ∈ D(s) s.t. ai = a and s′ = δ(s, a)}, when (i, a) ∈ A Rπ1∩π2 = Rπ1 ∩ Rπ2

˚ Agotnes & Alechina Logic of Joint Action November 2011 17 / 24

Logic of Joint Actions

Translation

p′ ≡ p (¬φ)′ ≡ ¬φ′ (φ1 ∧ φ2)′ ≡ φ′

1 ∧ φ′ 2

([{i1, . . . , ik}]φ)′ ≡

a1,...,ak∈Act

• 1≤j≤k(ij, aj)⊤

∧[(i1, a1) ∩ . . . ∩ (ik, ak)]φ′

˚ Agotnes & Alechina Logic of Joint Action November 2011 18 / 24

Logic of Joint Actions

Axiomatisation

K [π](φ → ψ) → ([π]φ → [π]ψ) A1

a∈Act(i, a)⊤

A2 πφ →

a∈Actπ ∩ (i, a)φ

A3

i∈N(i, ai)⊤ → (1, a1) ∩ . . . ∩ (g, ag)⊤

A4 (1, a1) ∩ · · · ∩ (g, ag)φ → [(1, a1) ∩ . . . ∩ (g, ag)]φ A5 [π]φ → [π ∩ π′]φ A6 [(i, a) ∩ (i, b)]⊥ when a = b MP From φ → ψ and φ infer ψ G From φ infer [π]φ

˚ Agotnes & Alechina Logic of Joint Action November 2011 19 / 24

Logic of Joint Actions

Tableaux procedure 1

Based on Lutz and Sattler 2001 for K ∩

n .

in what follows, assume that formulas do not contain diamond modalities and disjunctions notation: for two modalities π1 and π2 we say π1 ≤ π2 if the set of individual actions of π1 is a subset of that of π2. For example, (2, b) ≤ (1, a) ∩ (2, b) ∩ (3, c)

˚ Agotnes & Alechina Logic of Joint Action November 2011 20 / 24

Logic of Joint Actions

Tableaux procedure 2

Given a set of formulas X, we use Cl(X) to denote the smallest set containing all subformulas of formulas in X such that: (a) for each agent i and action a, [(i, a)]⊥ ∈ Cl(X) (b) for every complete joint action α ∈ JA, [α]⊥ ∈ Cl(X) (c) if ¬[(1, a1), . . . , (g, ag)]ψ ∈ Cl(X), then [(1, a1), . . . , (g, ag)] ∼ ψ ∈ Cl(X), where ∼ ψ = ¬ψ if ψ is not of the form ¬χ, and ∼ ψ = χ otherwise (d) for each i and a = b, [(i, a) ∩ (i, b)]⊥ ∈ Cl(X) (e) if ψ ∈ Cl(X), then ∼ ψ ∈ Cl(X)

˚ Agotnes & Alechina Logic of Joint Action November 2011 21 / 24

Logic of Joint Actions

Tableaux procedure 3

For sets of formulas ∆ and S where S is closed as above, Tab(∆, S) returns true iff (A) ∆ is a maximally propositionally consistent subset of S, that is, for each ¬ψ ∈ S, ψ ∈ ∆ iff ¬ψ ∈ ∆ and for each ψ1 ∧ ψ2 ∈ S, ψ1 ∧ ψ2 ∈ ∆ iff ψ1 ∈ ∆ and ψ2 ∈ ∆. (B) There is a partition of the set {¬[π]ψ : ¬[π]ψ ∈ ∆} into sets Wα (at most one for each α ∈ JA) such that if ¬[π]ψ ∈ Wα then π ≤ α and (i) ¬ψ ∈ ∆α (ii) for each π′ and ψ′, if [π′]ψ′ ∈ ∆ and π′ ≤ α, then ψ′ ∈ ∆α (iii) Tab(∆α, S′) returns true, where S′ = Cl({ψ′ : [π′]ψ′ ∈ ∆ and π′ ≤ α} ∪ {¬ψ : ¬[π]ψ ∈ Wα})

˚ Agotnes & Alechina Logic of Joint Action November 2011 22 / 24

Logic of Joint Actions

Tableaux procedure 4

(C) for each i ∈ N, ¬[(i, a)]⊥ ∈ ∆ for some a ∈ Act (D) if ¬[(1, a1)]⊥, . . . , ¬[(g, ag)]⊥ ∈ ∆, then ¬[(1, a1)∩ . . . ∩(g, ag)]⊥ ∈ ∆ (E) if ¬[α]ψ ∈ ∆, then [α] ∼ ψ ∈ ∆ (F) for every i ∈ N and a, b ∈ Act such that a = b, [(i, a) ∩ (i, b)]⊥ ∈ ∆

˚ Agotnes & Alechina Logic of Joint Action November 2011 23 / 24

Logic of Joint Actions

Conclusion

we proposed a different way of looking at Coalition Logic: as an extension of K ∩

n with a few natural properties

has implications for tools available for reasoning with CL, for example an alternative tableaux procedure.

˚ Agotnes & Alechina Logic of Joint Action November 2011 24 / 24