Modal Logics of Strategic Interaction
Workshop LSB@NASSLLI 2016, Rutgers University
Delft University of Technology
Helle Hvid Hansen
14 July 2016
Helle Hvid Hansen (TU Delft)
Modal Logics of Strategic Interaction
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Outline
1 Logic for Social Software
2 Coalition Logic
3 Intermezzo: Neighbourhood Semantics
4 Game Logic
PDL
Game Logic (GL)
Helle Hvid Hansen (TU Delft)
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Outline
1 Logic for Social Software
2 Coalition Logic
3 Intermezzo: Neighbourhood Semantics
4 Game Logic
PDL
Game Logic (GL)
Helle Hvid Hansen (TU Delft)
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Logic for Social Software
Social Software — term coined by Parikh.
Logic in Computer Science
Specification and verification
of computer systems
Helle Hvid Hansen (TU Delft)
Logic for Social Software
Specification and verification
of social procedures
Modal Logics of Strategic Interaction
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Logic for Social Software
Social Software — term coined by Parikh.
Logic in Computer Science
Specification and verification
of computer systems
Logic for Social Software
Specification and verification
of social procedures
Algorithms, complexity:
• Algorithmic Game Theory
• Computational Social Choice
Helle Hvid Hansen (TU Delft)
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Logic for Social Software
Social Software — term coined by Parikh.
Logic in Computer Science
Specification and verification
of computer systems
Logic for Social Software
Specification and verification
of social procedures
Algorithms, complexity:
• Algorithmic Game Theory
• Computational Social Choice
Logic approach: Translate design problems into logic questions: e.g.
mechanism design via satisfiability.
Helle Hvid Hansen (TU Delft)
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Modal Logics of Strategic Interaction
• Coalition Logic (Pauly)
• Game Logic (Parikh)
• Alternating-time Temporal logic (Alur, Henzinger, Kupferman)
• Strategy Logic (Chatterjee et al.)
• ... (modal logics, AI literature)
Conferences: LOFT, TARK, WoLLiC, LORI, AAMAS, IJCAI, ECAI,
JELIA, ...
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Outline
1 Logic for Social Software
2 Coalition Logic
3 Intermezzo: Neighbourhood Semantics
4 Game Logic
PDL
Game Logic (GL)
Helle Hvid Hansen (TU Delft)
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Angelina, Edwin and the Judge
Example: Formalising constitutions (cf. Allan Gibbard, 1974)
Angelina has to decide whether she wants to marry Edwin, the (male)
judge, or stay single. Edwin and the judge each can similarly decide
whether they want to stay single or marry Angelina.
Constitution:
Everyone has the right to choose who they marry and to choose to
stay single.
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Angelina, Edwin and the Judge
Specify the rights (or abilities) of players as sets of outcomes that
they can “force”.
Players N = {a, e, j}
States S = {m0 , me , mj } representing
• m0 : Angelina stays single,
• me : Angelina marries Edwin,
• mj : Angelina marries the judge.
Players can force the outcome in some subset of states:
player(s)
a
e
j
can force
{m0 }, {m0 , me }, {m0 , mj }
{m0 , mj }
{m0 , me }
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Modal Logics of Strategic Interaction
player(s)
{a, e}
{a, j}
{e, j}
can force
{me }
{mj }
{m0 }
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Game Theoretic Modelling
Angelina chooses the table.
Edwin chooses the row.
Judge chooses the column.
s
m
s
m
m0 m0
m0 m0
Stay single
s
m
s
m
m0 m0
me me
Marry Edwin
s
m
s
m
m0 mj
m0 mj
Marry Judge
Note: No preferences over outcomes are specified.
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Effectivity in Strategic Games
Def. A game form G = (N, {Σi | i ∈ N}, S, o : Σ → S) consists of
• a non-empty, finite set N of players,
• for each i ∈ N, a non-empty (finite) set of strategies Σi ,
• a set S of states,
• an outcome function o :
Helle Hvid Hansen (TU Delft)
Q
i
Σi → S
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Effectivity in Strategic Games
Def. A game form G = (N, {Σi | i ∈ N}, S, o : Σ → S) consists of
• a non-empty, finite set N of players,
• for each i ∈ N, a non-empty (finite) set of strategies Σi ,
• a set S of states,
• an outcome function o :
Q
i
Σi → S
Def. The effectivity function of a game form G is the function
EG : P(N) → P(P(S)) defined by:
X ∈ EG (C )
iff
∃σC ∀σC : o(σC , σC ) ∈ X .
(α-effectivity, as opposed to β-effecivity ∀...∃...)
EG (C ) is the collection of subsets for which C is effective.
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Properties of Effectivity Functions
For any game form G , the effectivity function EG : P(N) → P(P(S))
is
1
outcome-monotonic: for all C ⊆ N,
if X1 ⊆ X2 ⊆ S and X1 ∈ E (C ) then X2 ∈ E (C ).
2
coalition-monotonic: if C1 ⊆ C2 then E (C1 ) ⊆ E (C2 ).
3
regular: for all C ⊆ N, if X ∈ E (C ) then X ∈
/ E (C ).
4
maximal: for all C ⊆ N, if X ∈
/ E (C ) then X ∈ E (C ).
5
superadditive: for all C1 , C2 ⊆ N such that C1 ∩ C2 = ∅,
if X1 ∈ E (C1 ) and X2 ∈ E (C2 ) then X1 ∩ X2 ∈ E (C1 ∪ C2 ).
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Playable Effectivity Functions
When is a function E : P(N) → P(P(S)) the effectivity function of
some game form G ?
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Playable Effectivity Functions
When is a function E : P(N) → P(P(S)) the effectivity function of
some game form G ?
Def. A function E : P(N) → P(P(S)) is playable if
1
for all C ⊆ N: ∅ ∈
/ E (C )
2
for all C ⊆ N: S ∈ E (C ) (non-empty)
3
E is N-maximal: for all X ⊆ S, if X ∈
/ E (∅) then X ∈ E (N)
4
E is superadditive.
Theorem (Pauly): E is playable iff there is a G such that E = EG .
(In particular, playability implies monotonicity and regularity.)
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Dynamic Models of Coalitional Effectivity
• Extensive game with simultaneous moves is γ : S → Γ(N, S)
where Γ(N, S) are all game forms for N and S.
At each state, a strategic game is played that determines the
next state.
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Dynamic Models of Coalitional Effectivity
• Extensive game with simultaneous moves is γ : S → Γ(N, S)
where Γ(N, S) are all game forms for N and S.
At each state, a strategic game is played that determines the
next state.
• Def. A (coalitional) effectivity frame F = (N, S, E ) consists of a
set S of states and a function
E : S → P(N) → P(P(S))
such that for all s ∈ S, E (s) : P(N) → P(P(S)) is playable.
Notation:
Helle Hvid Hansen (TU Delft)
sEC X means X ∈ E (s)(C )
(“at state s, coalition C is effective for X ”).
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Coalition Logic: Syntax & Semantics
• Def. Let Φ0 be a set of atomic propositions.
An effectivity model M = (N, S, E , V ) is an effectivity frame
(N, S, E ) together with a valuation V : Φ0 → P(S).
• Coalition Logic syntax L(N, Φ0 ):
ϕ
Helle Hvid Hansen (TU Delft)
::=
p ∈ Φ0 | ¬ϕ | ϕ ∧ ϕ | [C ]ϕ, C ⊆ N
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Coalition Logic: Syntax & Semantics
• Def. Let Φ0 be a set of atomic propositions.
An effectivity model M = (N, S, E , V ) is an effectivity frame
(N, S, E ) together with a valuation V : Φ0 → P(S).
• Coalition Logic syntax L(N, Φ0 ):
ϕ
::=
p ∈ Φ0 | ¬ϕ | ϕ ∧ ϕ | [C ]ϕ, C ⊆ N
• Coalition Logic semantics: Let M = (N, S, E , V ) be an effectivity
model, and s ∈ S.
M, s
M, s
M, s
M, s
p
¬ϕ
ϕ∧ψ
[C ]ϕ
⇐⇒
⇐⇒
⇐⇒
⇐⇒
s ∈ V (p)
M, s 6 ϕ
M, s ϕ and M, s ψ
[[ϕ]]M ∈ EC (s)
where [[ϕ]]M = {t ∈ S | M, t ϕ}.
Helle Hvid Hansen (TU Delft)
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Coalition Logic: Axiomatisation
Def. Let CLN denote the smallest set of L(N, Φ0 )-formulas that
• contains all propositional tautologies,
• contains the following axiom (schemas):
(⊥)
(>)
(N)
(M)
(S)
¬[C ]⊥
[C ]>
¬[∅]¬ϕ → [N]ϕ
[C ](ϕ ∧ ψ) → [C ]ϕ
([C1 ]ϕ1 ∧ [C2 ]ϕ2 ) → [C1 ∪ C2 ](ϕ1 ∧ ϕ2 )
where C1 ∩ C2 = ∅.
• is closed under modus ponens and the congruence rule for all
C ⊆ N:
ϕ↔ψ
ϕ ϕ→ψ
ψ
[C ]ϕ ↔ [C ]ψ
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Completeness and Decidability
Theorem (Pauly): CLN is sound and complete w.r.t. the class of all
coalitional effectivity models.
Theorem (Pauly): The satisfiability problem (i.e. given formula ϕ, is
there a model in which ϕ is true?) is decidable in PSPACE.
Method: Use satisfiability to find implementation of a constitution
specified by ϕ.
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Extended of Coalition Logic
(cf. Pauly, 2001):
Add fixpoint (temporal) operators:
[C ∗ ]ϕ
[C × ]ϕ
“C can eventually bring about ϕ”.
“C can maintain ϕ in the long run”.
Coalition Logic and Extended Coalition Logic can be translated into
Alternating-time Temporal Logic (Goranko).
Helle Hvid Hansen (TU Delft)
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Nash-consistent Coalition Logic
(cf. Pauly & Hansen, 2004):
Add to CLN the rule:
W
W
ϕi
NC
i∈N [N \ {i}]ϕi
i∈N
Nash-consistent Coalition Logic is sound and complete w.r.t. the class
of Nash-consistent models, i.e., if all (implicit) game forms in the
model have a Nash equilibrium for any set of preferences the players
may have over outcome states.
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Application: Gibbard’s Paradox
• Two players N = {a, b} (Ann and Bob)
• Each player can choose which colour shirt to wear: white or blue.
• Φ0 = {pa (Ann wears white), pb (Bob wears white)}.
• Constitution:
Everyone has the right to decide on the colour of his/her own
shirt:
ϕ+ = [a]pa ∧ [a]¬pa ∧ [b]pb ∧ [b]¬pb
No single agent is effective for coordination or anti-coordination:
V
ϕ− =
¬[i]((pa ∧ pb ) ∨ (¬pa ∧ ¬pb )) ∧
Vi∈{a,b}
i∈{a,b} ¬[i]((pa ∧ ¬pb ) ∨ (¬pa ∧ pb ))
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Application: Gibbard’s Paradox (II)
Can the constitution ϕ = ϕ+ ∧ ϕ− be implemented?
That is, is ϕ satsifiable?
The situation can be described by the game form G below on the left:
w
b
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w
b
(w , w ) (w , b)
(b, w ) (b, b)
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Application: Gibbard’s Paradox (II)
Can the constitution ϕ = ϕ+ ∧ ϕ− be implemented?
That is, is ϕ satsifiable?
The situation can be described by the game form G below on the left:
w
b
w
b
(w , w ) (w , b)
(b, w ) (b, b)
w
b
w
b
(2, 4) (4, 1)
(3, 2) (1, 3)
G is not Nash-consistent. Consider scenario where:
Bob is conformist, and besides that he prefers white to blue.
Ann is non-conformist, and besides that she prefers white to blue.
This is modelled by the payoff matrix on the right. No
Nash-equilibrium.
Helle Hvid Hansen (TU Delft)
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Application: Gibbard’s Paradox (III)
We can prove that ϕ is not satisfiable in a Nash-consistent model,
since it is inconsistent:
From tautology
((pa ∧ pb ) ∨ (¬pa ∧ ¬pb )) ∨ ((pa ∧ ¬pb ) ∨ (¬pa ∧ pb ))
infer using NC rule
[b]((pa ∧ pb ) ∨ (¬pa ∧ ¬pb )) ∨ [a]((pa ∧ ¬pb ) ∨ (¬pa ∧ pb ))
which contradicts ϕ− .
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Coalition Logic References
• M. Pauly. A modal logic for coalitional power in games. Journal
of Logic and Computation, 12(1):149166, 2002.
• M. Pauly. Logic for Social Software. PhD Thesis, University of
Amsterdam, 2001.
• H.H. Hansen & M. Pauly, Axiomatising Nash-Consistent
Coalition Logic. JELIA 2002, LNAI 2424, Springer, 2002.
• Several more recent extensions: epistemic, announcements,
quantification over coalitions, resource-bounded,... (cf. Ågotnes,
Alechina, Van Ditmarsch, Van der Hoek, Wooldridge,...)
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Outline
1 Logic for Social Software
2 Coalition Logic
3 Intermezzo: Neighbourhood Semantics
4 Game Logic
PDL
Game Logic (GL)
Helle Hvid Hansen (TU Delft)
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From Effectivity to Neighbourhoods
• For every state s,
E (s) : P(N) → P(P(S))
is an effectivity function
• For every coalition C ,
EC : S → P(P(S))
is a neighbourhood function.
• A coalitional effectivity frame (N, S, E ) can be viewed as a
coalition-indexed neighbourhood frame:
(S, {EC : S → P(P(S)) | C ⊆ N})
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Neighbourhood Semantics of Modal Logic
• A neighbourhood model M = (S, ν, V ) consists of:
• state space S,
• neighbourhood function ν : S → P(P(S)),
• valuation V : Φ0 → P(S)
• Basic Modal Language L(Φ0 ):
ϕ
::=
p ∈ Φ0 | ¬ϕ | ϕ ∧ ϕ | 2ϕ
• Neighbourhood semantics of modal logic:
M, s
M, s
M, s
M, s
p
¬ϕ
ϕ∧ψ
2ϕ
⇐⇒
⇐⇒
⇐⇒
⇐⇒
s ∈ V (p)
M, s 6 ϕ
M, s ϕ and M, s ψ
[[ϕ]]M ∈ ν(s)
where [[ϕ]]M = {t ∈ S | M, t ϕ}.
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Why can’t we use good old Kripke models?
• The principle
[C ]ϕ ∧ [C ]ψ → [C ](ϕ ∧ ψ)
is generally NOT valid, because ϕ-strategy and ψ-strategy may
be incompatible.
• But formula 2p ∧ 2q ↔ 2(p ∧ q) is valid in all Kripke models
(i.e. true at all states in all models).
• Not so in neighbourhood models:
[[p]] ∈ ν(s), [[q]] ∈ ν(s) 6⇒ [[p ∧ q]] = [[p]] ∩ [[q]] ∈ ν(s)
(since ν(s) need not be closed under intersection).
• If for all s, ν(s) is upward closed, i.e., X ∈ ν(s), X ⊆ Y implies
Y ∈ ν(s), then 2 is monotonic:
2(p ∧ q) → 2p ∧ 2q is valid,
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or equiv.
Modal Logics of Strategic Interaction
ϕ→ψ
is valid
2ϕ → 2ψ
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Non-normal Modal Logic
• Logics that are weaker than K (modal logic of all Kripke frames)
are called non-normal.
• Non-normal modal logics were studied early on (cf. Segerberg,
Chellas), mainly from a proof-theoretic perspective.
• Classical modal logic is the logic of all neighbourhood models:
• All propositional tautologies
• Closed under modus ponens and congruence rule:
ϕ↔ψ
2ϕ ↔ 2ψ
• Monotonic modal logic is the logic of all monotonic
neighbourhood models.
• All propositional tautologies
• Closed under modus ponens and monotonicity rule:
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Modal Logics of Strategic Interaction
ϕ→ψ
2ϕ → 2ψ
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Theory of Neighbourhood Structures
• Bounded morphisms X
• Bisimulation X
• Van Benthem Characterisation Theorem X
• Algebraic duality X
• ...
(cf. Chellas 1980, Segerberg 1971, Hansen, Kupke, Pacuit, ...)
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Simulating Non-normal Modal Logics
We can translate non-normal modal logic into normal multi-modal
logic [Kracht & Wolter]. Basic idea:
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Simulating Non-normal Modal Logics
We can translate non-normal modal logic into normal multi-modal
logic [Kracht & Wolter]. Basic idea:
Monotonic modal logic:
Syntax :
2p
Semantics : (S, ν)
hνi[3]p
(S ∪ P(S), Rν , R3 , PS )
where Rν ⊆ S × P(S) : (s, U) ∈ Rν iff U ∈ ν(s)
R3 ⊆ P(S) × S : (U, s) ∈ R3 iff U 3 s
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Simulating Non-normal Modal Logics
We can translate non-normal modal logic into normal multi-modal
logic [Kracht & Wolter]. Basic idea:
Monotonic modal logic:
Syntax :
2p
hνi[3]p
(S ∪ P(S), Rν , R3 , PS )
Semantics : (S, ν)
where Rν ⊆ S × P(S) : (s, U) ∈ Rν iff U ∈ ν(s)
R3 ⊆ P(S) × S : (U, s) ∈ R3 iff U 3 s
Classical modal logic:
Syntax :
2p
Semantics : (S, ν)
hνi([3]p ∧ [63]¬p)
(S ∪ P(S), Rν , R3 , R63 , PS )
(Interesting for correspondence theory, but not for completeness etc.)
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Neighbourhood Semantics References
• B. F. Chellas. Modal logic, an intoduction. Cambridge University
Press, 1980.
• K. Segerberg. An Esssay in Classical Modal Logic. Number 13 in
Filosofiska Studier. Uppsala Universitet, 1971.
• M. Kracht and F. Wolter. Normal monomodal logics can simulate
all others. Journal of Symbolic Logic, 64(1):99138, 1999.
• H. H. Hansen. Monotonic Modal Logics. MSc Thesis
Mathematics. Universiteit van Amsterdam, 2003. Available as
ILLC technical report PP-2003-24.
• Helle Hvid Hansen, Clemens Kupke and Eric Pacuit.
Neighbourhood Structures: Bisimilarity and Basic Model Theory.
Logical Methods in Computer Science, vol. 5, issue 2, paper 2,
2009.
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Outline
1 Logic for Social Software
2 Coalition Logic
3 Intermezzo: Neighbourhood Semantics
4 Game Logic
PDL
Game Logic (GL)
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An External View: Dynamic Logics
• Logic for Social Software: From program logics to game logics.
• External (compositional) view: games are an explicit part of the
syntax.
• Internal (monolithic) view: LTL, CTL.
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Reasoning About Programs
Consider the following program which takes as input a pair of integers
in the variables x and y,
while y 6= 0 do
begin
z := x mod y;
x := y;
y := z
end
What is the value of x when the program halts?
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Reasoning About Programs
Consider the following program which takes as input a pair of integers
in the variables x and y,
while y 6= 0 do
begin
z := x mod y;
x := y;
y := z
end
What is the value of x when the program halts?
Euclidean algorithm for computing the greatest common divisor
(GCD) of x and y.
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Example Execution
Initial state (x,y,z) = (15,27,0).
command
state
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y=
6 0; z := x mod y ;
x := y ;
y := z;
true
(15, 27, 15)
(27, 27, 15) (27, 15, 15)
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Example Execution
Initial state (x,y,z) = (15,27,0).
command
state
command
state
command
state
command
state
command
state
y=
6 0;
true
y 6= 0;
true
y 6= 0;
true
y 6= 0;
true
y 6= 0;
false
z := x mod y ;
x := y ;
y := z;
(15, 27, 15)
(27, 27, 15) (27, 15, 15)
z := x mod y ;
x := y ;
y := z;
(27, 15, 12)
(15, 15, 12) (15, 12, 12)
z := x mod y ;
x := y ;
y := z;
(15, 12, 3)
(12, 12, 3)
(12, 3, 3)
z := x mod y ;
x := y ;
y := z;
(12, 3, 0)
(3, 3, 0)
(3, 0, 0)
Return 3.
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Basic (Imperative) Programming Constructs
While-language:
• x := t, where t is a term
• α; β
(simple assignments)
(sequential composition)
• if ϕ then α else β
• while ϕ do α
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(conditional)
(loop)
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PDL Programs and Formulas
Assume given:
• a set A0 of atomic programs, and
• a set Φ0 of atomic propositions
The set of PDL programs and formulas (tests) is defined by mutual
induction:
formulas Φ 3 ϕ ::= p ∈ Φ0 | ¬ϕ | ϕ ∨ ϕ | [α]ϕ
programs A 3 α ::= a ∈ A0 | α; α | α ∪ α | α∗ | ϕ?
PDL program constructs:
• α; β is sequential composition (do α and then β)
• α ∪ β is nondeterministic choice between α and β.
• α∗ is iteration of α (execute α a nondeterministically chosen
finite number of times)
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PDL Programs and Formulas (II)
• PDL programs subsume the While-language:
if ϕ then α else β
while ϕ do α
def
=
ϕ?; α ∪ ¬ϕ?; β
def
(ϕ?; α)∗ ; ¬ϕ?
=
• PDL subsumes Hoare logic (weakest precondition calculus)
{ϕ}α{ψ}
!
ϕ → [α]ψ
if ϕ holds, then after executing α, ψ holds
([α]ψ is the weakest precondition for ψ.)
• PDL subsumes Kleene Algebra with Tests:
α≡β
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!
[α]p ↔ [β]p
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PDL Models
PDL was introduced by Fischer & Ladner, 1977.
[α]ϕ
“after all successful executions of program α, ϕ holds”
Nondeterministic programs are modelled as relations.
A PDL model M = (S, {Rα | α ∈ A}, V ) consists of
• a set S (state space),
• a relation Rα ⊆ S × S for all α ∈ A0 .
• a valuation V : Φ0 → P(S)
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PDL Semantics
• Interpretation of formulas (as usual):
M, s |= [α]ϕ
iff
for all t ∈ S s.t. (s, t) ∈ Rα : M, t |= ϕ.
• Interpretation of programs via relation algebra:
Rα;β
Rα∪β
Rα∗
Rϕ?
=
=
=
=
Rα ◦ Rβ (relation composition)
Rα ∪ Rβ
Rα∗ (reflexive, transitive closure)
{(s, s) | s ∈ [[ϕ]]}
A PDL model is a special kind of multi-modal Kripke model.
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PDL Axiomatisation
Def. Let PDL be the smallest set of formulas which contains
• normal modal logic K([α]) for all α ∈ A
• the following axiom schemas:
[α; β]ϕ ↔ [α][β]ϕ
[α ∪ β]ϕ ↔ [α]ϕ ∧ [β]ϕ
[ψ?]ϕ ↔ (ψ → ϕ)
ϕ ∧ [α][α∗ ]ϕ ↔ [α∗ ]ϕ
ϕ ∧ [α∗ ](ϕ → [α]ϕ) → [α∗ ]ϕ
• and is closed under modus ponens.
Theorem (Kozen & Parikh, 1981): PDL is sound and complete with
respect to the class of PDL models.
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Game Logic (GL)
• Parikh, 1985. Strategic ability in determined 2-player games.
hγiϕ “player 1 has strategy in γ to ensure outcome satisfies ϕ”
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Game Logic (GL)
• Parikh, 1985. Strategic ability in determined 2-player games.
hγiϕ “player 1 has strategy in γ to ensure outcome satisfies ϕ”
• Determined 2-player game γ is modelled as a monotone
neighbourhood function Eγ : S → P(P(S)) where
U ∈ Eγ (s) means “player 1 is effective for U in γ at state s”.
• Determinacy means that
U 6∈ Eγ (s) iff “player 2 is effective for U in game γ at state s”.
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Models for Game Logic
Assume given:
• a set Γ0 of atomic games, and
• a set Φ0 of atomic propositions
Def. A GL model M = (S, {Eγ | γ ∈ Γ0 }, V ) consists of
• a set S of states,
• for each atomic game γ ∈ Γ0 , Eγ : S → PP(S) a monotonic
neighbourhood function,
if U ∈ Eγ (s) and U ⊆ U 0 then U 0 ∈ Eγ (s).
• a valuation V : Φ0 → P(S).
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Syntactic Game Constructs
PDL constructs extended with dual operation on games:
• γ1 ; γ2 : play γ1 then γ2 ,
• γ1 ∪ γ2 : player 1 chooses to play γ1 or γ2 ,
• γ ∗ : γ is repeatedly played a finite number of times, player 1
chooses when to stop.
• γ d : players switch roles.
We can define
def
• α ∩ β = (αd ∪ β d )d (demonic choice, choice for player 2)
def
• α× = ((αd )∗ )d (demonic iteration, player 2 chooses when to
stop)
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Semantic Game Constructs
• There is 1-1 correspondence:
Eγ : S → P(P(S)) (upward-closed nbhd function)
Êγ : P(S) → P(S) (monotonic predicate transformer)
given by transpose: s ∈ Êγ (X ) ⇐⇒ X ∈ Eγ (s).
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Semantic Game Constructs
• There is 1-1 correspondence:
Eγ : S → P(P(S)) (upward-closed nbhd function)
Êγ : P(S) → P(S) (monotonic predicate transformer)
given by transpose: s ∈ Êγ (X ) ⇐⇒ X ∈ Eγ (s).
• Algebraic operations on monotonic neighbourhood functions
Eγ : S → P(P(S)) are conveniently described in terms of Êγ :
Êγ1 ;γ2 (X )
= Êγ1 (Êγ2 (X ))
Êγ1 ∪γ2 (X ) = Êγ1 (X ) ∪ Êγ2 (X )
Êγ ∗ (X )
= µY .X ∪ Êγ (Y ) (LFP of Y 7→ X ∪ Êγ (Y ))
Êγ d (X )
/ Eγ (s))
= Êγ (X ), (i.e. X ∈ Eγ d (s) iff X ∈
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Game Logic Semantics
Interpretation of formulas and games in a GL model is defined by
mutual induction:
• Interpretation of formulas (as usual):
M, x |= hγiϕ
iff
[[ϕ]]M ∈ Eγ (x)
• Interpretation of games (as on previous slide) and for tests:
Êϕ? (X ) = [[ϕ]]M ∩ X
GL models are special multi-modal monotonic neighbourhood models.
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Game Logic Axiomatisation
Let GL be the smallest set of formulas which contains
• monotonic modal logic M(hγi) for all γ ∈ Γ
• the following axiom schemas
hγ; δiϕ ↔ hγihδiϕ
hψ?iϕ ↔ (ψ ∧ ϕ)
hγ ∪ δiϕ ↔ hγiϕ ∨ hδiϕ
hγ d iϕ ↔ ¬hγi¬ϕ
ϕ ∨ hγihγ ∗ iϕ → hγ ∗ iϕ
• and is closed under modus ponens and ϕ ∨ hγiϕ → ψ
hγ ∗ iϕ → ψ
Theorem (Parikh, 1985): GL without dual is sound and complete
wr.t. dual-free GL models.
Theorem (Pauly, 2001): GL without iteration is sound and strongly
complete w.r.t iteration-free GL models.
Open question: Completeness of full GL.
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Complexity of Decision Procedures
Theorem (Pauly): Game Logic satisfiability is decidable in EXPTIME.
Theorem (Pauly): Model checking for Game Logic is polynomial for
bounded alternation depth.
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Game Logic References
• R. Parikh (1985): The logic of games and its applications. In:
Topics in the Theory of Computation, Annals of Discrete
Mathematics 14, Elsevier.
• M. Pauly & R. Parikh (2003): Game Logic: An Overview. Studia
Logica 75(2), pp. 165-182.
• Extensions/variarions: stochastic GL (Doberkat), differential GL
(Platzer), Multi-Agent Strategy Logic (Van Eijck),...
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Concluding Remarks
Current Research: A unifying (coalgebraic) framework for dynamic
logics like PDL, Game Logic and Coalition Logic.
• PDL-frame S → P(S)A where P is the powerset functor.
• GL-frame S → M(S)A where M is the monotone
neighbourhood functor
• Both P and M are monads:
sequential composition = Kleisli composition,
skip program = unit of monad.
• A general framework for developing dynamic logics for structure
S → T (S)A parametric in the type T .
• Challenges: reduction axioms, identify algebraic “program/game”
structure, how to prove completeness with fixpoint operators (cf.
modal µ-calculus).
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