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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Combining equilibrium logic and dynamic logic (an introduction and a very brief overview)

Luis Fariñas del Cerro, Andreas Herzig and Ezgi Iraz Su

University of Toulouse, IRIT-CNRS, France

Toulouse, July 5, 2013

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Outline

1

Introduction

2

HT logic and equilibrium logic

3

A dynamic extension of HT logic and of equilibrium logic

4

DL-PA: dynamic logic of propositional assignments

5

Relating D-HT and DL-PA

6

Conclusion

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Motivation

motivation in a broad sense: recent studies reveals:

Answer Set Programming (ASP): central to various approaches in non-monotonic reasoning equilibrium logic: semantical framework for ASP [Pearce, Lifschitz, . . . ] then, need for an extension of the language of ASP ... (with some supportive concepts like:)

the representations of modalities actions

• ntologies

updates

main goal: beyond updates, adding other modalities to equilibrium logic (and, via that, to ASP) motivation, in particular, for this work: the update of answer set programs aim (here): manage atomic change of equi. models

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Outline

1

Introduction

2

HT logic and equilibrium logic

3

A dynamic extension of HT logic and of equilibrium logic

4

DL-PA: dynamic logic of propositional assignments

5

Relating D-HT and DL-PA

6

Conclusion

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Here-and-there models

a bit history: (idea [Gödel]) strength of “→” between material implication “⊃” and intuitionistic implication “⇒” here-and-there model (HT model) =

• H, T
• :

H, T sets of propositional variables from P with H ⊆ T

{p, q}

‘there’

∅

• ‘here’

truth conditions: H, T |= p iff p ∈ H H, T |= ⊥ H, T |= ϕ ∧ ψ iff H, T |= ϕ and H, T |= ψ H, T |= ϕ ∨ ψ iff H, T |= ϕ or H, T |= ψ H, T |= ϕ → ψ iff H, T |= ϕ ⊃ ψ and T, T |= ϕ ⊃ ψ

(H, T) is a HT model of ϕ

iff H, T |= ϕ

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Equilibrium models

Definition

T ⊆ P is an equilibrium model of ϕ iff

1

(T, T) is a HT model of ϕ;

2

(minimality condition) no (H, T) with H ⊂ T is a HT model of ϕ. example: T = ∅ is an equilibrium model of ¬p = p → ⊥:

1

∅, ∅ |= p → ⊥

2

• min. cnd. always satisfied for T = ∅.

3

actually the only one: e.g. for T = {q} we have ∅, {q} |= p → ⊥

p ∨ q has only 2 equi. models, namely T = {p} and T = {q}.

¬¬p has no equilibrium model:

1

{p}, {p} |= ¬¬p

2

however, min. cnd. fails since ∅, {p} |= ¬¬p.

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Equilibrium models

Definition

T ⊆ P is an equilibrium model of ϕ iff

1

(T, T) is a HT model of ϕ;

2

(minimality condition) no (H, T) with H ⊂ T is a HT model of ϕ. example: T = ∅ is an equilibrium model of ¬p = p → ⊥:

1

∅, ∅ |= p → ⊥

2

• min. cnd. always satisfied for T = ∅.

3

actually the only one: e.g. for T = {q} we have ∅, {q} |= p → ⊥

p ∨ q has only 2 equi. models, namely T = {p} and T = {q}.

¬¬p has no equilibrium model:

1

{p}, {p} |= ¬¬p

2

however, min. cnd. fails since ∅, {p} |= ¬¬p.

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Equilibrium logic

χ |= ϕ: logical consequence in HT models χ | ≈ ϕ: logical consequence in equilibrium models Definition χ | ≈ ϕ iff for every equil. model T of χ, (T, T) is HT model of ϕ.

example: ⊤ |

≈ ¬p and ¬p → q | ≈ q

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Equilibrium logic

χ |= ϕ: logical consequence in HT models χ | ≈ ϕ: logical consequence in equilibrium models Definition χ | ≈ ϕ iff for every equil. model T of χ, (T, T) is HT model of ϕ.

example: ⊤ |

≈ ¬p and ¬p → q | ≈ q

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Outline

1

Introduction

2

HT logic and equilibrium logic

3

A dynamic extension of HT logic and of equilibrium logic

4

DL-PA: dynamic logic of propositional assignments

5

Relating D-HT and DL-PA

6

Conclusion

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

The language LD-HT

extension of LHT with dynamic modalities: (common to D-HT and dynamic equilibrium logic)

LD-HT : ϕ p | ⊥ | ϕ ∧ ϕ | ϕ ∨ ϕ | ϕ → ϕ | [π]ϕ | πϕ π +p | −p | π; π | π ∪ π | π∗ | ϕ?

where p ranges over P. atomic programs: +p and −p

minimally update an HT model (H, T)

abbreviations:

¬ϕ = ϕ → ⊥ ⊤ = ⊥ → ⊥

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Dynamic here-and-there logic (1)

notation:

HT = {(H, T) : H ⊆ T ⊆ P}: the set of all HT models

interpretation of formulas and programs in D-HT:

for a formula ϕ, ϕD-HT ⊆ HT. examples:

1

¬pD-HT = {(H, T) : p T} (and so p H by the heredity constraint in HT logic, i.e., H ⊆ T)

2

p ∨ ¬pD-HT = {(H, T) : p ∈ H or p T}

3

(trivial, but important) ¬¬pD-HT = {(H, T) : p ∈ T} (and therefore, upshot: pD-HT ⊂ ¬¬pD-HT )

for a program π, πD-HT is a relation on HT.

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Dynamic here-and-there logic (1)

notation:

HT = {(H, T) : H ⊆ T ⊆ P}: the set of all HT models

interpretation of formulas and programs in D-HT:

for a formula ϕ, ϕD-HT ⊆ HT. examples:

1

¬pD-HT = {(H, T) : p T} (and so p H by the heredity constraint in HT logic, i.e., H ⊆ T)

2

p ∨ ¬pD-HT = {(H, T) : p ∈ H or p T}

3

(trivial, but important) ¬¬pD-HT = {(H, T) : p ∈ T} (and therefore, upshot: pD-HT ⊂ ¬¬pD-HT )

for a program π, πD-HT is a relation on HT.

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Dynamic here-and-there logic (1)

notation:

HT = {(H, T) : H ⊆ T ⊆ P}: the set of all HT models

interpretation of formulas and programs in D-HT:

for a formula ϕ, ϕD-HT ⊆ HT. examples:

1

¬pD-HT = {(H, T) : p T} (and so p H by the heredity constraint in HT logic, i.e., H ⊆ T)

2

p ∨ ¬pD-HT = {(H, T) : p ∈ H or p T}

3

(trivial, but important) ¬¬pD-HT = {(H, T) : p ∈ T} (and therefore, upshot: pD-HT ⊂ ¬¬pD-HT )

for a program π, πD-HT is a relation on HT.

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Dynamic here-and-there logic (1)

notation:

HT = {(H, T) : H ⊆ T ⊆ P}: the set of all HT models

interpretation of formulas and programs in D-HT:

for a formula ϕ, ϕD-HT ⊆ HT. examples:

1

¬pD-HT = {(H, T) : p T} (and so p H by the heredity constraint in HT logic, i.e., H ⊆ T)

2

p ∨ ¬pD-HT = {(H, T) : p ∈ H or p T}

3

(trivial, but important) ¬¬pD-HT = {(H, T) : p ∈ T} (and therefore, upshot: pD-HT ⊂ ¬¬pD-HT )

for a program π, πD-HT is a relation on HT.

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Dynamic here-and-there logic (2)

interpretation of atomic update operations: upgrade p:+p executable, viz. when p H (‘here’) (ex: all, but black below) downgrade p: same for −p, viz. when p ∈ T(‘there’) (ex: all, but blue below)

• ∅, {p, q}
• {p}, {p, q}
• −pD-HT
• ∅, {p, q, r}
• +rD-HT
• ∅, {q}
• −pD-HT
• {r}, {p, q, r}
• +rD-HT

remark: not allowed to apply −p to blue again, neither +r to green... Similarly, neither +p to black, nor −r to orange...

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Dynamic here-and-there logic (3)

Mixed and more sophisticated examples:

1

+p⊤D-HT = {(H, T) : p H}

2

+p⊤ ∧ −p⊤D-HT = {(H, T) : p H and p ∈ T}

3

¬p?D-HT =

• (H, T), (H, T)
• : p T
• 4

(¬p ∨ q)? ∪ (−p; +q)D-HT =

• (H1, T1), (H2, T2)
• : (H2, T2) ∈ p → qD-HT
• ϕ is D-HT valid if and only if ϕD-HT = HT.

examples:

neither +p⊤ nor −p⊤ is valid, but +p ∪ −p⊤ is. [+p][+p]p and [−p][−p]¬p are valid. [p? ∪ ¬p?](p ∨ ¬p) is valid too. finally, [π]ϕ → ¬π¬ϕ is valid, but the converse is not.

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Dynamic here-and-there logic (3)

Mixed and more sophisticated examples:

1

+p⊤D-HT = {(H, T) : p H}

2

+p⊤ ∧ −p⊤D-HT = {(H, T) : p H and p ∈ T}

3

¬p?D-HT =

• (H, T), (H, T)
• : p T
• 4

(¬p ∨ q)? ∪ (−p; +q)D-HT =

• (H1, T1), (H2, T2)
• : (H2, T2) ∈ p → qD-HT
• ϕ is D-HT valid if and only if ϕD-HT = HT.

examples:

neither +p⊤ nor −p⊤ is valid, but +p ∪ −p⊤ is. [+p][+p]p and [−p][−p]¬p are valid. [p? ∪ ¬p?](p ∨ ¬p) is valid too. finally, [π]ϕ → ¬π¬ϕ is valid, but the converse is not.

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

D-HT is more expressive than HT itself.

and even more interesting examples, but why?:

[−p]⊥ ↔ ¬p, −p⊤ ↔ ¬¬p, and [+p]⊥ ↔ p

are all valid.

1

+p⊤ cannot be expressed in LHT, but why?

no formula LHT that conveys p ∈ T \ H.

2

‘heredity property of intuitionistic logic’ does not hold in D-HT (in general), but always in HT.

Definition

if (H, T) ∈ ϕD-HT then (T, T) ∈ ϕD-HT. counterex: consider (H, T) = (∅, {p}) and ϕ = +p⊤

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

D-HT is more expressive than HT itself.

and even more interesting examples, but why?:

[−p]⊥ ↔ ¬p, −p⊤ ↔ ¬¬p, and [+p]⊥ ↔ p

are all valid.

1

+p⊤ cannot be expressed in LHT, but why?

no formula LHT that conveys p ∈ T \ H.

2

‘heredity property of intuitionistic logic’ does not hold in D-HT (in general), but always in HT.

Definition

if (H, T) ∈ ϕD-HT then (T, T) ∈ ϕD-HT. counterex: consider (H, T) = (∅, {p}) and ϕ = +p⊤

13 / 22

Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

D-HT is more expressive than HT itself.

and even more interesting examples, but why?:

[−p]⊥ ↔ ¬p, −p⊤ ↔ ¬¬p, and [+p]⊥ ↔ p

are all valid.

1

+p⊤ cannot be expressed in LHT, but why?

no formula LHT that conveys p ∈ T \ H.

2

‘heredity property of intuitionistic logic’ does not hold in D-HT (in general), but always in HT.

Definition

if (H, T) ∈ ϕD-HT then (T, T) ∈ ϕD-HT. counterex: consider (H, T) = (∅, {p}) and ϕ = +p⊤

13 / 22

Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

D-HT is more expressive than HT itself.

and even more interesting examples, but why?:

[−p]⊥ ↔ ¬p, −p⊤ ↔ ¬¬p, and [+p]⊥ ↔ p

are all valid.

1

+p⊤ cannot be expressed in LHT, but why?

no formula LHT that conveys p ∈ T \ H.

2

‘heredity property of intuitionistic logic’ does not hold in D-HT (in general), but always in HT.

Definition

if (H, T) ∈ ϕD-HT then (T, T) ∈ ϕD-HT. counterex: consider (H, T) = (∅, {p}) and ϕ = +p⊤

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Dynamic equilibrium logic

same definition as in HT for equilibrium models of D-HT... examples:

1

T = ∅ is the only equi. model for all valid formulas of D-HT.

2

¬¬p has no equi. model.

3

T = {q} is the only equi. model for ¬p → q.

4

T = {p} is the only equi. model for −p(¬p → q) and +q; +q(p ∧ q).

5

however, −q(p ∧ q) does not have any, because neither even does a D-HT model.

again same definition for logical consequence... examples:

1

¬¬p | ≈ ϕ, for every ϕ

2

p ∨ q | ≈ [¬p?]q

3

p ∨ q | ≈ [¬p?]+p; +p(p ∧ q)

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Outline

1

Introduction

2

HT logic and equilibrium logic

3

A dynamic extension of HT logic and of equilibrium logic

4

DL-PA: dynamic logic of propositional assignments

5

Relating D-HT and DL-PA

6

Conclusion

15 / 22

Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

A bit DlPa...

recently studied in [BalbianiHerzigTroquard-Lics13] (here) expansion of assignments to arbitrary formulas as ‘atomic programs’, and without converse operator, still same expresivity and complexity results... same abbreviations as before... moreover, [π]ϕ ∼π∼ϕ skip ⊤? (“nothing happens”) language:

π

• p:=ϕ | π; π | π ∪ π | π∗ | ϕ?

ϕ

• p | ⊥ | ϕ ∧ ϕ | ϕ ∨ ϕ | ϕ ⊃ ϕ | πϕ

where p ranges over a fixed set of propositional variables P.

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

DlPa continues...

semantics: a ‘valuation’ V is a subset of P.

ϕDL-PA ∈ 2P πDL-PA: relation between valuations Definition ϕ is DL-PA valid if and only if ϕDL-PA = 2P. ϕ is DL-PA satisfiable if ϕDL-PA ∅.

examples:

1

p:=⊤⊤, p:=⊤p and p:=⊥∼p are all valid.

2

as well as, ψ ∧ [ψ?]ϕ ⊃ ϕ and [p:=⊤ ∪ q:=⊤](p ∨ q)...

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Complexity of the full language (including the Kleene star)

proved in [BalbianiHerzigTroquard-Lics13] that:

model and satisfiability checking are both EXPTIME complete.

⇒ we expect: both problems for our DL-PA are also EXPTIME

complete because lower bounds for both problems clearly transfer. upper bounds for both problems can be established as in [BalbianiHerzigTroquard-Lics13].

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Outline

1

Introduction

2

HT logic and equilibrium logic

3

A dynamic extension of HT logic and of equilibrium logic

4

DL-PA: dynamic logic of propositional assignments

5

Relating D-HT and DL-PA

6

Conclusion

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

What we do hereafter: a quick summary

1

define a polynomial translation, tr1 from D-HT to DL-PA

2

embed some notions of D-HT into DL-PA such as:

D-HT satisfiability, consequence in equilibrium models,etc.

3

check the problems of D-HT validity and consequence in equilibrium models

4

establish an EXPTIME upper bound for the complexity of latter problem

5

then define another translation, tr2, the other way around and establish the EXPTIME hardness of these problems

20 / 22

Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Outline

1

Introduction

2

HT logic and equilibrium logic

3

A dynamic extension of HT logic and of equilibrium logic

4

DL-PA: dynamic logic of propositional assignments

5

Relating D-HT and DL-PA

6

Conclusion

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Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional

Key points of the talk

propose a neat (sound and complete), but also simple logic: D-HT notice: strong relation between DL-PA and D-HT construct the correspondence via polynomial translations even prove mathematical properties of extensions Further goals: first, epistemic extension then go on with reexamining the logical foundations of equilibrium logic, and ASP ....

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