an application of the finite theory of revision

An application of the finite theory of revision

Riccardo Bruni

Dept. of Literature and Philosophy

University of Florence

Institut für Informatik

University of Bern19 November 2015

Part 1The revision–theoretic approach to circularity

The revision–theoretic approach

circular predicate P(x) =def ϕ(x ,P)



hypothesis H = y | P(y)




revision step H 7→ H ′ = z | (M,H) |= ϕ(z ,P)




revision step H 7→ H ′ = z | (M,H) |= ϕ(z ,P)

regularities H 7→ H ′ 7→ H ′′ 7→ . . . 7→ H 7→ . . .

A

Quasi–inductive sequence of sets

Let ∆ : P(N) → P(N). Define the sequence 〈∆α | α ∈ ON〉 by

∆0 = ∅∆α+1 = ∆(∆α)

∆λ = lim infβ→λ∆β , λ limit

(where lim infβ→λ∆β := n | ∃α < λ∀β < λ(α ≤ β → n ∈ ∆β)).

B

Properties of QID

Properties of QID

the sequence admits a stability pair (∆+∞,∆−

∞) and(countably–many) countable ordinals representing it;

Properties of QID



high definitional complexity;

Properties of QID



high definitional complexity;

axioms QID(K) for any class K of operators (with unsharpproof–theoretic bounds);

C

Part 2The finite revision theory

The finite revision theory

Let δ : P(N) → P(N), H ⊆ N. Define 〈δn(H) | n ∈ N〉 by

δ0(H) = H

δn+1(H) = δ(δn(H))

Recurring hypotheses

Definition

H ⊆ N is m–reflexive, if δm(H) = H (m ∈ N+)

H ⊆ N is reflexive, if H is m–reflexive for some m ∈ N+

Sets definition

Sets definition

Definition

Z ⊆ N is definable iff Z =⋂

H | H is reflexive

Z ⊆ N is n–definable iff Z =⋂

H | H is n–reflexive

Sets definition

Theorem

1. Every n–definable set Z is Π11

2. Every definable set Z is Π11

Semantics

take LPA;

Semantics

take LPA;

define L+ := LPA + G, where G (x) := A(x ,G );

Semantics

Definition

1. A sentence of L+ is m–valid in a model M of LPA (m ∈ N+) if,

and only if A is true in (M,H) for every m–reflexive hypothesis H.2. A sentence of L+ is m–valid (m ∈ N) if, and only if A is m–valid

in M, for every model M of the base language.

Derivability

m–validity corresponds to derivability into system C0 by Guptaand Belnap (1993)

Derivability


this system has Hilbert–style, as well as well–behavingGentzen–style counterparts HC0 and GC0

Derivability



indexed formulas Ai for “A is true at stage i of a revisionsequence”

Derivability



indexed formulas Ai for “A is true at stage i of a revisionsequence”

an extra implication connective Ai → B j for “if Ai holds, thenB j holds”

D

Part 3Strategic rationality as a circular predicate

Rational choice

Rational choice

Rational agents are expected to maximize their utilities

Rational choice

Rational agents are expected to maximize their utilities

Agent A has to reason on what is rational for agent B , andviceversa

Finite games

Definition

A finite game is triple Γ = 〈P , (Si )i∈P , u〉 where

P is a finite set (the players of Γ)

(Si )i∈P is a family of finite lists (the strategies of each playerof Γ)

let S be the set of tuples s = si sj . . . (for every i , j , . . . ∈ P),u : P × S → Z is a function (the preferences of the players ofΓ)

Rational choice

Suppose Γ = 〈P , (Si )i∈P , u〉 where P = A,B and Si = i1, i2.

Rational choice


R(·)

Rational choice


R(·), uA/B(·)

Rational choice


R(·), uA/B(·), · = ·, · < ·

Rational choice


R(·), uA/B(·), · = ·, · < · ϕ1 :≡ (R(b1) ∧ uA(a2b1) < uA(a1b1)) ∨ (R(b2) ∧ uA(a2b2) < uA(a1b2))

Rational choice



R(x) =Def

(x = a1 ∧ϕ1)∨ (x = a2 ∧ϕ2)∨ (x = b1 ∧ψ1)∨ (x = b2 ∧ψ2)

Rational choice



R(x) =Def

(x = a1 ∧ϕ1)∨ (x = a2 ∧ϕ2)∨ (x = b1 ∧ψ1)∨ (x = b2 ∧ψ2)

Use the operator δΓ to revise hypotheses for “x is rational in Γ”

Strict games

Strict games

2,2

3,1

0,1

1,0

b1 b2

a2

a1

Strict games

2,2

3,1

0,1

1,0

b1 b2

a2

a1

a1b1 7→ a2b1

Strict games

2,2

3,1

0,1

1,0

b1 b2

a2

a1

a1b1 7→ a2b1

a2b1 7→ a2b1

a1b2 7→ a2b1

a2b2 7→ a2b1

Strict games

The previous approach generalizes to any strict game Γ owing tothe fact, that

Fact

1. For every strategy profile s in Γ, there exists k(s) ∈ N such

that δk(s)Γ (s) is reflexive

2. There exists k ∈ N such that for every strategy profile s in Γ,

δkΓ(s) is reflexive

Strict games

Definition

For any two strategy profiles s, s ′ and player i of Γ, say that s 6=i s ′

holds in case s and s ′ differs only in the action for i they feature.

Strict games

Definition

For any two strategy profiles s, s ′ and player i of Γ, say that s 6=i s ′

holds in case s and s ′ differs only in the action for i they feature.

Definition

A strategy profile s is Nash equilibrium in a strict game Γ if for noi ∈ P and no s ′ such that s 6=i s ′, ui (s) < ui (s

′) holds.

Strict games

Theorem

If Γ is strict, then a strategy profile s is fixed point of the revision

operator if and only if it is Nash equilibrium

Part 3Dropping strictness

(j.w. with G. Sillari - LUISS Rome)

Quasi–strict games

3,1

0,1

1,0

1,0

2,22,2

b1 b2

a2

a1

a3


3,1

0,1

1,0

1,0

2,22,2

b1 b2

a2

a1

a3

if A plays a1, b1 and b2 yields one and the same payoff;


3,1

0,1

1,0

1,0

2,22,2

b1 b2

a2

a1

a3

if A plays a1, b1 and b2 yields one and the same payoff;

B notices that, in case A played a2 or a3, b1 would be better;so, she drops b2;


ψ∗

1 :≡ . . . ∨ (R(a1) ∧ uB(a1b1) = uA(a1b2) ∧ uB(a2b2) <uB(a2b1) ∧ uB(a3b2) ≤ uB(a3b1)) ∨ (R(a1) ∧ uB(a1b1) =uA(a1b2) ∧ uB(a2b2) ≤ uB(a2b1) ∧ uB(a3b2) < uB(a3b1)) ∨ . . .


ψ∗

1 :≡ . . . ∨ (R(a1) ∧ uB(a1b1) = uA(a1b2) ∧ uB(a2b2) <uB(a2b1) ∧ uB(a3b2) ≤ uB(a3b1)) ∨ (R(a1) ∧ uB(a1b1) =uA(a1b2) ∧ uB(a2b2) ≤ uB(a2b1) ∧ uB(a3b2) < uB(a3b1)) ∨ . . .

R∗(x) =

Def(x = a1 ∧ϕ

∗

1)∨ (x = a2 ∧ϕ∗

2)∨ (x = b1 ∧ψ∗

1 )∨ (x = b2 ∧ψ∗

2 )


3,1

0,1

1,0

1,0

2,22,2

b1 b2

a2

a1

a3


Definition

For any two strategy profiles s, s ′ and player i of Γ, say that s =i s ′

holds in case s and s ′ are equal only in the action for i they feature.

Definition

A finite game Γ is quasi–strict if for every strategy profiles s, s ′ andplayer i ∈ P for which both s 6=i s ′ and ui (s1) = ui (s2) hold, theneither for every s∗, s† such that s∗ =i s and s† =i s ′ bothui (s

∗) ≤ ui (s†) and there exists s‡ such that s‡ = s ′ and

ui (s∗) < ui (s

‡) are the cases, or for every s∗, s† such that s∗ =i s

and s† =i s ′ both ui (s†) ≤ ui (s

∗) and there exists s‡ such thats‡ = s ′ and ui (s

‡) < ui (s∗) are the cases.


Definition

A strategy profile s is “trembling–hand” equilibrium in a finite gameΓ if for no i ∈ P and s ′ such that s 6=i s ′, ui (s) < ui (s

′) holds, andfor no s∗, s† such that s∗ =i s and s∗ 6=i s†, ui (s

∗) < ui (s†).


Theorem

If Γ is quasi–strict, then a strategy profile s is fixed point of the

revision operator if and only if it is “trembling–hand” equilibrium.

Non–strict games

3,1

0,-1

1,0

1,0

2,22,2

b1 b2

a2

a1

a3

Non–strict games

3,1

0,-1

1,0

1,0

2,22,2

b1 b2

a2

a1

a3

B sticks to b1 as it carries the lowest index

Non–strict games

3,1

0,-1

1,0

1,0

2,22,2

b1 b2

a2

a1

a3

Non–strict games

3,1

0,-1

1,0

1,0

2,22,2

b1 b2

a2

a1

a3

1,0

1,0

3,1

0,-1

2,22,2

b1 b2

a2

a1

a3

Non–strict games

3,1

0,-1

1,0

1,0

2,22,2

b1 b2

a2

a1

a3

1,0

1,0

3,1

0,-1

2,22,2

b1 b2

a2

a1

a3

3,1

0,-1

1,0

1,0

2,22,2

b1 b2

a2

a1

a3

Non–strict games: the canonical ordering

Definition

A strategy profile s is a weak equilibrium in a non–strict game Γ ifffor no i ∈ P and for no s ′ such that s 6=i s ′ we have ui (s) < ui (s

′).

Definition

A weak equilibrium s is more efficient than a weak equilibrium s ′ inΓ if for no i ∈ P it is ui (s) < ui (s

′).


Definition

A finite game Γ′ = 〈P ′, (S ′i )i∈P′ , u′〉 is a reordering of

Γ = 〈P , (Si )i∈P , u〉 iff P = P ′ and Si = S ′i for all i ∈ P and there

are i , j ∈ P such that, for every strategy profile s, ui (s) = u′j(s) and

uj(s) = u′i (s).

Definition

Γc = 〈P , (Si )i∈P , uc〉 is the canonical version of Γ = 〈P , (Si )i∈P , u〉

iff Γc is the reordering of Γ that features the least number ofchanges in the order such that every most efficient weak equilibriumis fixed point of the revision operator.


3,1

0,-1

1,0

1,0

2,22,2

b1 b2

a2

a1

a3


3,1

0,-1

1,0

1,0

2,22,2

b1 b2

a2

a1

a3

1,0

1,0

3,1

0,-1

2,22,2

b1 b2

a2

a1

a3

Limiting the lexicographic choice: a proposal

3,9

0,0

1,0

1,1

2,22,2

b1 b2

a2

a1

a3


3,9

0,0

1,0

1,1

2,22,2

b1 b2

a2

a1

a3

B observes that playing b1 is more risky than playing b2


3,9

0,0

1,0

1,1

2,22,2

b1 b2

a2

a1

a3

B observes that playing b1 is more risky than playing b2

B also observes that playing b1 may better compensate the risk


3,9

0,0

1,0

1,1

2,22,2

b1 b2

a2

a1

a3


the risk factor of an action b of B given A’s choice of a is

r(b)a := |maxuB(a′b) : a′ 6= a − minuB(a

′b) : a′ 6= a|


the risk factor of an action b of B given A’s choice of a is

r(b)a := |maxuB(a′b) : a′ 6= a − minuB(a

′b) : a′ 6= a|

the compensation by b of its own risk factor is

c(b)a := MeanuB(a′b) : a′ 6= a − r(b)a


Theorem

The definition of strategic rationality that contains the “risk and

compensation” clause is more robust than the previous one under

reordering of a game.

Some final remarks

Some final remarks

flexible

Some final remarks

flexible

computational

Some final remarks

flexible

computational

robust

Some questions to fix

what are fixed–points in the most general case?

Some questions to fix

what are fixed–points in the most general case?

what about other regular hypotheses?

Further work

Further work

extensions a place for logic

Thank you!

The limit rule and why we need it

Let P be T , and set

T (x) := (x = ps = tq ∧ val(s) = val(t)) ∨∨ (x = pT (s)q ∧ T (pT (s)q)) ∨∨ (x = p¬ϕq ∧ ¬T (pϕq)) ∨

. . .

ϕ(x ,T )

Namely: ϕ(x ,T ) is a formula defining the Tarskian truth predicate.


Let F be the set:

F := ϕ0, ϕ1, . . . , ϕn, . . .

with:

ϕ0 := ⊤

ϕn+1 := Tϕn


Start from H0 = ∅, then you have:

Input Output

∅ ⇒ ϕ0ϕ0 ⇒ ϕ0, ϕ1

ϕ0, ϕ1 ⇒ ϕ0, ϕ1, ϕ2ϕ0, ϕ1, ϕ2 ⇒ ϕ0, ϕ1, ϕ2, ϕ3

......

...


Start from H0 = ∅, then you have:

Input Output

∅ ⇒ ϕ0ϕ0 ⇒ ϕ0, ϕ1

ϕ0, ϕ1 ⇒ ϕ0, ϕ1, ϕ2ϕ0, ϕ1, ϕ2 ⇒ ϕ0, ϕ1, ϕ2, ϕ3

......

...

PROBLEM: we cannot conclude F ⊆ T by finitary means.

QID: Alternative views

Herzberger’s

(the ‘pure’ theory)

∅

arithmetical

liminf

Gupta’s

(the ‘corrected’ th.)

Z ⊆ N

=

liminf ∪ h0

Belnap’s

(the ‘maximal’ th.)

Z ⊆ N

=

X |X coherent†

†X ⊆ N s. t., for λ limit

h+<λ ⊆ X X ∩ h−

<λ = ∅

with h+/−<λ sets of positive/negative stable elements (below λ).

Axioms: logical, arithmetical, ordinal–theoretical

For a fixed K = ∆n,Πn,Σn, the axioms of QID(K) amount at:

a complete axiomatization of first–order classical logic with

equality

the axioms of arithmetic (with CI)

standard assumptions on the ordering <Ω, on ordinal individual

constants, the defining equations of the stock of primitiveordinal functions as well as axioms on their basic properties

(monotonicity, inverses), plus a schema of transfinite induction

Ordinal axioms

(Ω.1) ∀αβ(α = β ∨ α < β ∨ β ∨ α)

(Ω.2) ∀α(¬α < α)

(Ω.3) ∀αβγ(α < β ∧ β < γ → α < γ)

(Ω.4) ∀α(0Ω ≤ α) [where α ≤ β := (α < β ∨ α = β)]

(Ω.5) ∀α(α < α′) [with α′ = succΩ(α)]

(Ω.6) ∀αβ(α < β → α′ ≤ β)

(Ω.7) 0Ω < ω ∧ ∀α < ω(α′ < ω)

(Ω.8) ∀λ(Lim(λ) → ω ≤ λ)

[where Lim(α) := (0 < α ∧ ∀β < α(β′ < α))]

Ordinal axioms

(Ω.9) ∀α(α+ 0Ω = α)

(Ω.10) ∀αβ(α+ β′ = (α+ β)′)

(Ω.11) ∀αβγ(α < β → γ + α < γ + β)

(Ω.12) ∀αβγ(α ≤ β → α+ γ ≤ β + γ)

(Ω.13) ∀α(α0Ω = 0Ωα = 0Ω)

(Ω.14) ∀αβ(αβ′ = αβ + α)

(Ω.15) ∀αβγ(0Ω < γ ∧ α < β → γα < γβ)

(Ω.16) ∀αβγ(α ≤ β → αγ ≤ βγ)

(Ω.17) ∀αβ(α < β → ∃γ ≤ β(α+ γ = β))

(Ω.18) ∀αβ(0Ω < β → ∃γ ≤ α∃δ < β(α = βγ + δ))

Ordinal axioms

(L(K)− IN) A(0) ∧ ∀x(A(x) → A(x ′)) → ∀xA(x)

(L(K)− IΩ) ∀α((∀β < α)A(β) → A(α)) → ∀αA(α)

Axioms: QID

(QID.1) x ∈ H0

A → x 6= x

(QID.2) x ∈ Hα+1

A ↔ A(x ,HαA)

(QID.3) Lim(λ) → [x ∈ HλA ↔ (∃α < λ)(∀β < λ)(α ≤ β → x ∈ Hβ

A)]

Axioms: QID

(QID.1) x ∈ H0

A → x 6= x

(QID.2) x ∈ Hα+1

A ↔ A(x ,HαA)

(QID.3) Lim(λ) → [x ∈ HλA ↔ (∃α < λ)(∀β < λ)(α ≤ β → x ∈ Hβ

A)]

(QID.4) ∀α∃λ(Lim(λ)∧α < λ∧(H+A (λ) ≡ H+

A (∞))∧(H−A (λ) ≡ H−

A (∞)))

Alphabet and syntax

Let L+ = L∪ G (·). We introduce a language L(I ) extending L+

by means of indices for formulas, and a new implication connective→.

Alphabet and syntax

Let I be the set of index terms, which contains terms p forevery p ∈ Z;

Alphabet and syntax


Terms of L(I ) are the terms of L+ and index terms;

Alphabet and syntax



Formulas of L(I ) are obtained by indexing formulas of L+, andby composing them by means of the new implicationconnective, according to the clauses:

Alphabet and syntax




Ai is a formula of L(I ), where A is a formula of L+ and i ∈ I ;

Alphabet and syntax




Ai is a formula of L(I ), where A is a formula of L+ and i ∈ I ; B i → C j is a formula of L(I ), where B i,C j are formulas of

L(I ), with i = i1, . . . , ik , j = j1, . . . , jq(i1, . . . , ik , j1, . . . , jq ∈ I ).

Axioms: logical

Ai → (B j → Ai), (Ai → (Bh → C j)) → ((Ai → Bh) → (Ai → C j));

Ai → (A ∨ B)i , B i → (A ∨ B)i ;

(Ai → C i ) → ((B i → C i ) → ((A ∨ B)i → C i ));

(A ∧ B)i → Ai , (A ∧ B)i → B i , Ai → (B i → (A ∧ B)i );

(B ∧ ¬B)i → Ai, (¬¬A)i → Ai ;

(∀xA)i → A[x/t]i , A[x/t]i → (∃xA)i ;

Ai → B(x)i , x 6∈ FV (A)Ai → (∀xB)i

(GEN),A(x)i → B i, x 6∈ FV (B)

(∃xA)i → B i (PAR)

B i → C j B i

C j (MP)

(where A,B,C are formulas of L(I )n wherever they carry sequences ofindices, formulas of L+ elsewhere).

System HC0: special axioms

The system HC0 features axioms for G–definition and index shift:



(DEF) AG (t,G )i ↔ G (t)i+1



(DEF) AG (t,G )i ↔ G (t)i+1

(IS) B i → B j , B formula of L, i , j ∈ I

Systems HCn: special axioms

In addition, systems HCn (n > 0) feature the followinggeneralization of index shift:

(ISn) B i ↔ B i+n, B formula of L+, i ∈ I

Rules: the system GC0

Let Γ,∆, ... range over multisets of formulas of L(I )n, and A,Bover formulas of L+.


Let Γ,∆, ... range over multisets of formulas of L(I )n, and A,Bover formulas of L+.

A. AxiomsP i , Γ ⇒ ∆,P j (Ax)

[where P is an atomic formula of L and possibly i 6= j ]

G (t)i , Γ ⇒ ∆,G (t)i (Ax)


B. Logical Rules

Ai , Γ ⇒ ∆Γ ⇒ ∆, (¬A)i

(L¬) Γ ⇒ ∆,Ai

(¬A)i , Γ ⇒ ∆(R¬)

Aim, Γ ⇒ ∆

(A1 ∧ A2)i , Γ ⇒ ∆

(L∧m)Γ ⇒ ∆,Ai Γ ⇒ ∆,B i

Γ ⇒ ∆, (A ∧ B)i(R∧)

Γ ⇒ ∆,Ai B j , Γ ⇒ ∆Ai → B j , Γ ⇒ ∆

(L →)Ai , Γ ⇒ ∆,B j

Γ ⇒ ∆,Ai → B j (R →)

[where, in (L/R →), i 6= j is possible (NB: A,B are formulas ofL+)]

A[v := t]i , Γ ⇒ ∆(∀vA)i , Γ ⇒ ∆

(L∀)Γ ⇒ ∆,A[v := v ′]i

Γ ⇒ ∆, (∀vA)i(R∀)

[v ′ is not free in the conclusion of (R∀)]


C. Revision Rules

AG (t,G )i , Γ ⇒ ∆G (t)i+1, Γ ⇒ ∆

(L1)Γ ⇒ ∆,AG (t,G )i

Γ ⇒ ∆,G (t)i+1 (R1)


E. Structural Rules

Ai ,Ai , Γ ⇒ ∆Ai , Γ ⇒ ∆

(LC )Γ ⇒ ∆,Ai ,Ai

Γ ⇒ ∆,Ai (RC )

B j → Ch,B j → Ch, Γ ⇒ ∆B j → Ch, Γ ⇒ ∆

(LC→)Γ ⇒ ∆,B j → Ch,B j → Ch

Γ ⇒ ∆,B j → Ch (RC→)

Systems GCn, n > 0

As before, plus:

D. Reflexivity Rules

Ai , Γ ⇒ ∆Ai+n, Γ ⇒ ∆

(Ln)Γ ⇒ ∆,Ai

Γ ⇒ ∆,Ai+n (Rn)

[where A is any formula of L+]

an application of the finite theory of revision

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