Game-Theoretic Semantics
for Alternating-Time Temporal Logic
Valentin Goranko
Stockholm University
Joint work with Antti Kuusisto and Raine Rönnholm
Highlights of Logic, Games and Automata 2016
Brussels, September 8, 2016
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Logic and games
Two approaches for relating logic and games:
1. Analyzing games with logic ATL
2. Analyzing logic with games GTS
We bring these two approaches together.
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Introduction: ATL and GTS
Alternating-time temporal logic (ATL): a logical formalism for reasoning
about strategic abilities of agents and coalitions in multi-agent systems.
The formula hhAii ϕ of ATL says, intuitively, that
the coalition of agents A have a strategy to co-operatively ensure
that ϕ holds, regardless of the behaviour of the other agents.
In game-theoretic semantics (GTS), truth of a formula ϕ is determined
in a formal dispute, called evaluation game, between two players:
Eloise, who is trying to verify ϕ, and Abelard, who is trying to falsify it.
GTS equates truth of ϕ with the existence of a winning strategy for
Eloise in the evaluation game for ϕ.
We develop GTS for ATL. A 2-way interaction of logic and games.
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Preliminaries: concurrent game models and ATL
A concurrent game model (CGM) consists of a set of agents Agt, a set
of states St, together mappings act, v , and o, defining for every s ∈ St:
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a set v (s) of atomic propositions pi that are true at s,
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a set act(s, a) of actions available at s, for each agent a ∈ Agt,
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a unique outcome / successor state o(s, α
~ ) of s for every admissible tuple
of actions α
~ chosen by all the agents in Agt.
For simplicity, I will only consider the following subset of ATL-formulae:
ϕ ::= p | ¬ϕ | (ϕ ∨ ϕ) | hhAii F ϕ | hhAii G ϕ
The semantics of ATL is based on truth at a state in a CGM, via the clauses:
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hhAii X ϕ: ‘The coalition A has a collective action that ensures that every
possible outcome (state) satisfies ϕ’,
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hhAii F ϕ: ‘The coalition A has a collective strategy to ensure that every
possible outcome play eventually satisfies ϕ’,
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hhAii G ϕ: ‘The coalition A has a collective strategy to ensure that every
possible outcome play forever satisfies ϕ’,
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Example: 2-round ‘Rock-paper-scissors’
s1
M:
s3
RS,PR,SP
WIN1
SP
RS
S
,S
PP
,
RR
RR,PP,SS
SR
RR
,P
,R
P,
PS
P,S
S
SR,RP,PS
Agt = {a1 , a2 }
act(si ,ai ) = {R,P,S}
SR,RP,PS
,
s0
RS,PR,SP
,
PR
WIN2
s2
s4
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The unbounded evaluation game for ATL
Consider a CGM M, a state s0 in M and an ATL-formula ϕ.
We introduce the unbounded evaluation game G = G(M, s0 , ϕ).
A position in G is a tuple (P, s, ψ), where P ∈ {Eloise, Abelard},
s is a state in M and ψ a subformula of ϕ.
The opposing player of P is denoted by P.
The game G begins from the initial position (Eloise, s0 , ϕ)
and proceeds according to the following rules.
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The unbounded evaluation game for ATL: rules
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In a position (P, s, p) for an atomic proposition p, the game ends.
P wins the game if p ∈ v (s). Otherwise P wins the game.
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In a position (P, s, ψ ∨ θ), the player P chooses a disjunct
α ∈ {ψ, θ}, and then the game continues from (P, s, α).
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In a position (P, s, ¬ψ), the game moves to the position (P, s, ψ).
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In a position (P, s, hhAii X ψ), the following one-step game gX
is played with starting position (P, A, s), as follows:
1. P chooses actions for the agents in A.
2. P chooses actions for the agents in Agt \A.
Thus, an action profile α
~ is selected.
3. The one-step game gX ends in position (P, s0 , ψ),
where s0 = o(s, α
~ ) is the resulting state.
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The unbounded evaluation game for ATL: rules (cont.)
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In a position (P, s, hhAii F ψ), the game enters an embedded
subgame gF . It begins from the state s and proceeds by playing
repeatedly the one-step game gX , starting from (P, A, s) and each
next round starting at the ending position of the previous one.
In the embedded game gF , the player P is the controlling player
who may decide to end gF at any state s0 that is reached.
When (if) the embedded game gF ends, in a position (P, s0 , ψ),
the evaluation game then continues from that position.
If the subgame gF goes on forever, the controlling player P loses
the entire evaluation game.
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In a position (P, s, hhAii G ψ), the players enter a dual embedded
subgame gG , just like gF , but now the controlling player is P.
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Unbounded game-theoretic semantics for ATL
Unbounded game-theoretic semantics for ATL :
M, s |=GTS ϕ iff Eloise has a winning strategy in G(M, s, ϕ).
Theorem
The unbounded GTS for ATL is equivalent to the standard
(compositional) semantics of ATL.
The unbounded evaluation games are determined, but possibly infinite.
A drawback.
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Finitely bounded evaluation games for ATL
We now modify the evaluation game G by associating with every
embedded subgame (gF or gG ) a (finite) time limit n ∈ N.
The time limit is selected and announced in the beginning of the
subgame by the controlling player.
The controlling player must end the subgame before time runs out.
Evaluation games with time limit define finitely bounded GTS for ATL.
. NB: an analogy with FOR-loops and WHILE-loops.
Theorem
The finitely bounded and unbounded GTS are generally non-equivalent.
However, they are equivalent on image finite models.
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Example
M:
1
p
0
¬p
p
0
p
0
¬p
p
0
p
0
p
0
¬p
p
0
p
0
p
0
p
Agt = {a}
act(s0 , a) = N+
2
s0
p
3
4
0
¬p
Here Eloise can win G(M, s0 , hhaii G p) with time limits, but not with
unbounded GTS.
Therefore, M, s0 |=fin
GTS ϕ but M, s0 6|=GTS ϕ.
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Ordinal bounded evaluation games for ATL
We now generalize the notion of time limit, by allowing time limits for
the embedded subgames to be any ordinals.
The initial time limit of the subgame is chosen by the controlling player.
Every round, the time limit decreases:
– if a successor ordinal, by 1;
– if a limit ordinal, the controlling player must choose a strictly smaller one.
Ordinal bounded GTS: using evaluation games with ordinal time limits.
NB. Since ordinals are well-founded, the ordinal bounded semantics
guarantees that evaluation games end in finite number of rounds.
Theorem
The ordinal bounded and the unbounded GTS are equivalent.
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Example
M:
1
p
0
¬p
p
0
p
0
¬p
p
0
p
0
p
0
¬p
p
0
p
0
p
0
p
Agt = {a}
act(s0 , a) = N+
2
s0
p
3
4
0
¬p
Here Abelard can win G(M, s0 , hhaii G p) with ordinal
bounded GTS by choosing the ordinal ω as the initial time limit.
After the 1st round, where Eloise chooses an action,
Abelard can always reduce the time limit appropriately to win.
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Concluding remarks
We argue that:
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the game-theoretic semantics is a natural framework for reasoning
about games by using games.
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limitations on time resources in evaluation games lead to
interesting semantic variants for ATL.
Ongoing work:
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develop game-theoretic semantics for ATL∗
and other extensions of ATL.
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consider other evaluation games with limiting time resources.
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Thank you for your attention!
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