A modal logic for games with lies

A modal logic for games with lies

Bruno Teheux

University of Luxembourg

The R^ÉNYI – U^LAM game

A searching game with lies

1. ALICEchooses an element in{1, . . . ,M}.

2. BOBtries to guess this number by asking Yes/No questions.

3. ALICEis allowed to lien−1 times in her answers.

BOBtries to guess ALICE’s number as fast as possible.

The game is around for more than 50 years

Finding an optimal strategy :

Pelc, A. Searching games with errors - fifty years of coping with liars, Theoretical Computer Science 270 (1):71-109.

Using logic and algebras to model the states of the games: Mundici, D. The logic of Ulam’s game with lies. InKnow- ledge, belief, and strategic interaction. Cambridge Uni- versity Press, 1992.

Tools for the algebraic/logical approach

Static model

Łukasiewicz logic MV-algebras

Dynamic model

Modal logic Kripke semantics

Łukasiewicz (n + 1)-valued logic and MV

-algebras

L={¬,→,1}

¬x :=1−x, x →y :=min(1,1−x +y)

2

•0

•1

φ∈CPL

⇐⇒

2|=φ

Ais a Boolean algebra

⇐⇒

A|=CPL

Ł_n

t0 u1

•¹_n

•ⁿ⁻¹

n

φ∈PŁn

⇐⇒ Ł_n|=φ

Ais a MV_n-algebra

⇐⇒ A|=PŁ_n

Łukasiewicz (n + 1)-valued logic and MV

-algebras

L={¬,→,1}

¬x :=1−x, x →y :=min(1,1−x +y)

2

•0

•1

φ∈CPL

⇐⇒

2|=φ

Ais a Boolean algebra

⇐⇒

A|=CPL

Ł_n

t0 u1

•_n¹

•ⁿ⁻¹_n

φ∈PŁn

⇐⇒ Ł_n|=φ

Ais a MV_n-algebra

⇐⇒ A|=PŁ_n

Łukasiewicz (n + 1)-valued logic and MV

-algebras

L={¬,→,1}

¬x :=1−x, x →y :=min(1,1−x +y)

2

•0

•1

φ∈CPL

⇐⇒

2|=φ

Ais a Boolean algebra

⇐⇒

A|=CPL

Ł_n

t0 u1

•_n¹

•ⁿ⁻¹_n φ∈PŁn

⇐⇒

Ł_n|=φ

Ais a MVn-algebra

⇐⇒

A|=PŁ_n

Static model

Łukasiewicz logic MV-algebras

Dynamic model

Modal logic Kripke semantics

An algebraic encoding of the states of knowledge

1. Knowledge spaceK =Ł^M_n.

2. A state of knowledges ∈Ł^M_n :s(m)is the ’distance’

betweenmand the set of numbers that can be discarded.

3. A questionQis a subset of{1, . . . ,M}.

4. The positive answer toQis the mapf_Q:M → {ⁿ⁻¹_n ,1}

defined by

f_Q(m) =1 ⇐⇒ m∈Q The negative answer toQ is the mapf_M\Q.

x⊕y := (¬x →y)

=min(1,x +y) in Łn

xy :=¬(¬x⊕ ¬y)

=max(0,x+y−1) in Ł_n

Proposition. If ALICEanswers positively toQat state of knowledgesthen the new state of knowledge is

s⁰ :=sf_Q.

Static model

Łukasiewicz logic MV-algebras

Dynamic model

Modal logic Kripke semantics

A dynamic model for every instance of the game

The model only talks aboutstatesof an instance of the game.

s₀ s₁ s_k−1 s_k

We want a language to talk aboutwhole instancesof the game.

s₀ s₁ sk−1 s_k

Q Q⁰

We want a language to talk aboutall instances of any game.

Q₁ Q₄

Q₂ Q₃

A dynamic model for every instance of the game

The model only talks aboutstatesof an instance of the game.

s₀ s₁ s_k−1 s_k

We want a language to talk aboutwhole instancesof the game.

s₀ s₁ sk−1 s_k

Q Q⁰

We want a language to talk aboutall instances of any game.

Q₁ Q₄

Q₂ Q₃

A dynamic model for every instance of the game

The model only talks aboutstatesof an instance of the game.

s₀ s₁ s_k−1 s_k

We want a language to talk aboutwhole instancesof the game.

s₀ s₁ sk−1 s_k

Q Q⁰

We want a language to talk aboutall instances of any game.

Q₁ Q₄

Q₂ Q₃

Static model

Łukasiewicz logic MV-algebras

Dynamic model

Modal logic Kripke semantics

We need K

models with many-valued worlds

Π0={atomic programs}

Prop={propositional variables}

Definition.A(dynamic n+1-valued)KRIPKEmodel is M=hW,R,Vali

where

I W 6=∅,

I R maps anya∈Π₀toRa⊆W ×W,

I Val(u,p)∈Ł_nfor anyu∈W and anyp∈Prop.

The R

- U

game has a K

model

Language :

I Prop={p_m|m∈M}where

pm ≡howmis far from the set of rejected elements.

I Π₀={m|m∈M}.

Model :

I W =Ł^M_n is the knowledge space.

I (s,t)∈Rmift =sf_{m}

I Val(s,pm) =s(m).

A modal language for the Kripke models

Programsα∈Πand formulasφ∈Form are defined by Formulas φ::=p |0|φ→φ| ¬φ|[α]φ Programs α::=a|φ?|α;α|α∪α|α^∗ wherep∈Prop anda∈Π₀.

Word Reading

α;β αfollowed byβ

α∪β αorβ

α^∗ any number of execution ofα

φ? testφ

[α] after any execution ofα

Interpreting formulas in Kripke Models

Val(·,·)andRare extended by induction :

I In a truth functional way for¬and→,

I Val(u,[α]ψ) :=V{Val(v, ψ)|(u,v)∈R_α},

I R_α;β :=R_α◦R_β and R_α∪β :=R_α∪R_β,

I R_φ?={(u,u)|Val(u, φ) =1},

I Rα^∗ := (Rα)^∗ =S

k∈ωR_α^k.

Definition. M,u|=φifVal(u, φ) =1 andM |=φifM,u|=φ for everyu∈W.

Tn:={φ| M |=φfor every Kripke modelM} Find an axiomatization of Tn.

Interpreting formulas in Kripke Models

Val(·,·)andRare extended by induction :

I In a truth functional way for¬and→,

I Val(u,[α]ψ) :=V{Val(v, ψ)|(u,v)∈R_α},

I R_α;β :=R_α◦R_β and R_α∪β :=R_α∪R_β,

I R_φ?={(u,u)|Val(u, φ) =1},

I Rα^∗ := (Rα)^∗ =S

k∈ωR_α^k.

Definition. M,u|=φifVal(u, φ) =1 andM |=φifM,u|=φ for everyu∈W.

Tn:={φ| M |=φfor every Kripke modelM}

Find an axiomatization of Tn.

Static model

Łukasiewicz logic MV-algebras

Dynamic model

Modal logic Kripke semantics

Modal extensions of Łukasiewicz (n + 1)-valued logic

L={,¬,→,1}, M=hW,R,Vali

Theorem.The set K_n:={φ∈FormL |φis a tautology}is the smallest subset of FormL that

I contains tautologies of Łukasiewicz(n+1)-valued logic

I contains(K)(φ→ψ)→(φ→ψ)

I contains

(φ ? φ)↔(φ)?(φ) for?∈ {,⊕}

I is closed under MP, substitution and generalization.

Static model

Łukasiewicz logic MV-algebras

Dynamic model

Modal logic Kripke semantics

Tn:={φ| M |=φfor every Kripke modelM}

There are three ingredients in the axiomatization

Definition.PDL_nis the smallest set of formulas that contains formulas in Ax₁, Ax₂, Ax₃and closed for the rules in Ru₁, Ru₂.

Łukasiewiczn+1-valued logic Ax₁ Axiomatization Ru1 MP, uniform substitution

Crisp modaln+1-valued logic Ax2

[α](p→q)→([α]p→[α]q), [α](p⊕p)↔[α]p⊕[α]p, [α](pp)↔[α]p[α]p,

Ru₂ φ[α]φ

Program constructions

Ax₃

[α∪β]p↔[α]p∧[β]p [α;β]p↔[α][β]p, [q?]p↔(¬qⁿ∨p) [α^∗]p↔(p∧[α][α^∗]p), [α^∗]p→[α^∗][α^∗]p,

(p∧[α^∗](p →[α]p)ⁿ)→[α^∗]p.

Theorem(Completeness).

T_n = PDL_n

Remark.Ifn=1, it boils down to PDL (introduced by FISCHER

and LADNERin 1979).

Focus on the induction axiom

The formula

(p∧[α^∗](p →[α]p)ⁿ)→[α^∗]p means

‘if after an undetermined number of executions of α the truth value of pcannot decrease after a new execution of α,thenthe truth value ofpcannot de- crease after any undetermined number of execu- tions ofα’.

Focus on the homogeneity axioms

Proposition.The axioms

[α](φ ? φ)↔([α]φ ?[α]φ), ?∈ {,⊕},

can be replaced bynaxioms that state equivalence between

‘the truth value of[α]φis at least _nⁱ’ and

‘after any execution ofα, the truth-value ofφis at least _nⁱ’

About the expressive power of the many-valued modal language.

The ability to distinguish between frames

L ={,¬,→,1}

Frame :F=hW,Ri

Modn(Φ) :={F| M |= Φfor any(n+1)-valuedMbased onF}

Definition.A classCof frames is Łn-definable if there is Φ⊆FormL such that

C=Modn(Φ).

Proposition.IfCisŁ₁-definable then it isŁ_n-definable for every n≥1.

The ability to distinguish between frames

L ={,¬,→,1}

Frame :F=hW,Ri

Modn(Φ) :={F| M |= Φfor any(n+1)-valuedMbased onF}

Definition.A classCof frames is Łn-definable if there is Φ⊆FormL such that

C=Modn(Φ).

Proposition.IfCisŁ₁-definable then it isŁ_n-definable for every n≥1.

Enriching the signature of frames

Theorem.IfCcontains ultrapowers of its elements, then CisŁ_n-definable if and only if CisŁ₁-definable.

The many-valued modal language isnot adaptedfor the signature of frames.

Definition.An Ł_n-frame is a structure

F=hW,R,{r_m|mis a divisor ofn}i,

whererm ⊆W for anym. A modelM=hW,R,Valiisbased on F if

u∈r_m =⇒ Val(u, φ)∈Ł_m

Enriching the signature of frames

Theorem.IfCcontains ultrapowers of its elements, then CisŁ_n-definable if and only if CisŁ₁-definable.

The many-valued modal language isnot adaptedfor the signature of frames.

Definition.An Ł_n-frame is a structure

F=hW,R,{r_m|mis a divisor ofn}i,

whererm⊆W for anym. A modelM=hW,R,Valiisbased on F if

u∈r_m =⇒ Val(u, φ)∈Ł_m

Ł

-frames

F=hW,R,{r_m|mis a divisor ofn}i, u∈r_m =⇒ Val(u, φ)∈Ł_m

Example.(Forbidden situation)

r₂ r₃

u v

IfVal(u,p) =1 andVal(v,p) =1/3 thenVal(u,p) =1/36∈Ł₂

Additional conditions on Łn-frames :

I rm∩r_k =r_gcd(m,k)

I u ∈rm=⇒Ru⊆rm

Ł

-frames

F=hW,R,{r_m|mis a divisor ofn}i, u∈r_m =⇒ Val(u, φ)∈Ł_m

Example.(Forbidden situation)

r₂ r₃

u v

IfVal(u,p) =1 andVal(v,p) =1/3 thenVal(u,p) =1/36∈Ł₂ Additional conditions on Ł_n-frames :

I rm∩r_k =r_gcd(m,k)

I u ∈rm=⇒Ru⊆rm

Goldblatt - Thomason theorem

C ∪ {F}=class of Ł_n-frames

Mod(Φ) :={F| M |= Φfor any(n+1)-valuedMbased onF}

Definition.A classCof Ł_n-frames is definable if there is Φ⊆Form_L such that

C =Mod(Φ).

Theorem.IfC is closed under ultrapowers, then the following conditions are equivalent.

I Cis definable

I Cis closed under Łn-generated subframes, Łn-bounded morphisms, disjoint unions and reflects canonical extensions.

Goldblatt - Thomason theorem

C ∪ {F}=class of Ł_n-frames

Mod(Φ) :={F| M |= Φfor any(n+1)-valuedMbased onF}

Definition.A classCof Ł_n-frames is definable if there is Φ⊆Form_L such that

C =Mod(Φ).

Theorem.IfCis closed under ultrapowers, then the following conditions are equivalent.

I Cis definable

I Cis closed under Łn-generated subframes, Łn-bounded morphisms, disjoint unions and reflects canonical extensions.

What to do next ?

1. Shows that PDL_ncan actually help in stating many-valued program specifications.

2. There is an epistemic interpretation of PDL. Can it be generalized to then+1-valued realm ?

3. What happens if KRIPKEmodels are not crisp.

4. Can coalgebras explain why PDL and PDL_nare so related ?