Future event logic - axioms and complexity Hans van Ditmarsch - - PowerPoint PPT Presentation

Future event logic - axioms and complexity

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universit de Rennes 1, France AiML’10 24-27 August, Moscow, Russia

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 1 / 19

Informative Events

Consider a system that consists of a set of agents and a set of facts. The facts are known to be static so they do not change, although whether an agent knows a proposition is true may change. An agent may experience an informative event where their uncertainty in the system is reduced. An informative event is any change that updates a model in such a way that it is consistent with at least one of the “possibilities” inherent in the original model. Examples of informative events include announcements (public or private), message passing systems and action models.

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 2 / 19

Example

Alice and Bob have both applied for a tenured lecturing position, and are waiting outside the Dean’s office to hear which one has won the position. The Dean asks Alice to come into the office. He tells her she has won the position and she leaves. This is an informative event. However, from Bob’s point of view it has increased uncertainty. Previously he knew Alice did not know who had the job, but he considers it possible that she knows she has the job, that she knows she does not have the job, or that the Dean told her he has not yet made his decision.

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 3 / 19

Example

a

Alice Alice Alice Bob Bob Bob

b b a b b b

This graphic represents the effect of the informative event for Alice. Each circle represents a world where either Alice or Bob got the job, and an agent’s uncertainty between which world is the actual world is represented by the relations. The underlined world is the actual world.

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 4 / 19

Technical preliminaries

A finite set of agents A and a countably infinite set of atoms P

Structures

M = (S, R, V ) S ∋ s, t, . . . a domain of states R : A → P(S × S) accessibility relation; write Ra(s, t) V : P → P(S) a valuation For s ∈ S, Ms is a state or a pointed Kripke model.

Bisimulation

M = (S, R, V ) and M′ = (S′, R′, V ′). R ⊆ S × S′ is a bisimulation whenever (s, s′) ∈ R if for all a ∈ A: atoms s ∈ V (p) iff s′ ∈ V ′(p) for all p ∈ P forth-a ∀t ∈ sRa, ∃t′ ∈ s′R′

a with (t, t′) ∈ R

back-a vice versa

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 5 / 19

Technical preliminaries

A finite set of agents A and a countably infinite set of atoms P

Bisimulation

M = (S, R, V ) and M′ = (S′, R′, V ′). R ⊆ S × S′ is a bisimulation whenever (s, s′) ∈ R if for all a ∈ A: atoms s ∈ V (p) iff s′ ∈ V ′(p) for all p ∈ P forth-a ∀t ∈ sRa, ∃t′ ∈ s′R′

a with (t, t′) ∈ R

back-a vice versa

Simulation

A relation that satisfies atoms s ∈ V (p) iff s′ ∈ V ′(p) for all p ∈ P forth-a for every b ∈ A is a simulation. In that case M′

s′ is a simulation of Ms, and Ms is a

refinement of M′

s′, and we write Ms M′ s′

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 5 / 19

Technical preliminaries

A finite set of agents A and a countably infinite set of atoms P

Simulation

A relation that satisfies atoms s ∈ V (p) iff s′ ∈ V ′(p) for all p ∈ P forth-a for every b ∈ A is a simulation. In that case M′

s′ is a simulation of Ms, and Ms is a

refinement of M′

s′, and we write Ms M′ s′

a-simulation (a ∈ A)

A relation that satisfies atoms forth-a for every b ∈ A back-a for every b ∈ A − {a} is an a-simulation. In that case Ms is an a-refinement of M′

s′, and we write Ms a M′ s′

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 5 / 19

Technical preliminaries

A finite set of agents A and a countably infinite set of atoms P

a-simulation (a ∈ A)

A relation that satisfies atoms forth-a for every b ∈ A back-a for every b ∈ A − {a} is an a-simulation. In that case Ms is an a-refinement of M′

s′, and we write Ms a M′ s′

Here refinement corresponds to the diminishing uncertainty of agents as

• pposed to program refinement where detail is added to a specification.

Still programme refinement is a more deterministic system which agrees with the notion of diminishing uncertainty.

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 5 / 19

Back to the example refines_a Alice Alice Alice Bob Bob Bob

b b a b b b a

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 6 / 19

Future event logic: the language L⊲

Syntax

Given a finite set of agents A and a set of propositional atoms P, the language of L⊲ is inductively defined as ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | aϕ | ◮aϕ where a ∈ A and p ∈ P.

Semantics

Ms | = ◮aϕ iff for all M′

s′ a Ms, M′ s′ |

= ϕ Write ⊲aϕ for ¬◮a¬ϕ. It is true now, iff there is an unspecified informative event for agent a, or a-refinement, after which ϕ is true.

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 7 / 19

Future event logic: the language Lµ⊲

Syntax

Given a finite set of agents A and a set of propositional atoms P, the language of L⊲ is inductively defined as ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | aϕ | ◮aϕ | µx.ϕ where a ∈ A and p ∈ P.

Semantics

Ms | = ◮aϕ iff for all M′

s′ a Ms, M′ s′ |

= ϕ Write ⊲aϕ for ¬◮a¬ϕ. It is true now, iff there is an unspecified informative event for agent a, or a-refinement, after which ϕ is true. Write νx.ϕ for ¬µx.¬ϕ(¬x)

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 7 / 19

Example: Knowledge and belief

a b a b b p p p

• An informative event is possible after which agent a knows that p but

agent b does not know that. ⊲a(ap ∧ ¬bap)

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 8 / 19

Example:

Let S be a discrete-event system with two possible actions c and action u. Fix a formula ϕ (say in the modal µ-calculus).

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 9 / 19

Example: Open system – Module Checking

Let S be a discrete-event system with two possible actions c and action u. Fix a formula ϕ (say in the modal µ-calculus). Interprete action c as the moves of the system and action u as the moves

• f an environment.

S | = ◮u(ϕ) iff S satisfies ϕ in any environment

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 9 / 19

Example: Open system – Module Checking

Let S be a discrete-event system with two possible actions c and action u. Fix a formula ϕ (say in the modal µ-calculus). Interprete action c as the moves of the system and action u as the moves

• f an environment.

S | = ◮u(LiveEnv ⇒ ϕ) iff S satisfies ϕ in any “live” environment where LiveEnv = νx.♦u⊤ ∧ x

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 9 / 19

Example: Basic Control Problems

Let S be a discrete-event system with two possible actions c and action u. Fix a formula ϕ (say in the modal µ-calculus). Interprete action c as controllable and action u as uncontrollable. S | = ⊲c(ϕ) iff there is a way to control S to guarantee ϕ

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 9 / 19

Example: Controller Problems for Open Systems

Let S be a discrete-event system with two possible actions c and action u. Fix a formula ϕ (say in the modal µ-calculus). Interprete action c as controllable and action u as uncontrollable. S | = ⊲c◮u(LiveEnv ⇒ ϕ) iff there is a way to control the open system S in any live environment so that the resulting satisfies ϕ

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 9 / 19

And now?

FEL: Axiomatization of L⊲

◮ Soudness ◮ Completeness

FELµ: Axiomatization of Lµ⊲

◮ Soudness ◮ Completeness

Complexity upper-bound for L⊲ Succinctness of L⊲ The logics LC

⊲ where C is a fixed class of models (S5, K4, ...)

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 10 / 19

FEL: Axiomatization of L⊲ (one agent)

P All tautologies of propositional logic K (ϕ → ψ) → ϕ → ψ MP From ⊢ ϕ → ψ and ⊢ ϕ infer ⊢ ψ Nec1 From ⊢ ϕ infer ⊢ ϕ

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 11 / 19

FEL: Axiomatization of L⊲ (one agent)

P All tautologies of propositional logic K (ϕ → ψ) → ϕ → ψ MP From ⊢ ϕ → ψ and ⊢ ϕ infer ⊢ ψ Nec1 From ⊢ ϕ infer ⊢ ϕ Nec2 From ⊢ ϕ infer ⊢ ◮ϕ

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 11 / 19

FEL: Axiomatization of L⊲ (one agent)

P All tautologies of propositional logic K (ϕ → ψ) → ϕ → ψ MP From ⊢ ϕ → ψ and ⊢ ϕ infer ⊢ ψ Nec1 From ⊢ ϕ infer ⊢ ϕ Nec2 From ⊢ ϕ infer ⊢ ◮ϕ G0 ◮(ϕ → ψ) → ◮ϕ → ◮ψ

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 11 / 19

FEL: Axiomatization of L⊲ (one agent)

P All tautologies of propositional logic K (ϕ → ψ) → ϕ → ψ MP From ⊢ ϕ → ψ and ⊢ ϕ infer ⊢ ψ Nec1 From ⊢ ϕ infer ⊢ ϕ Nec2 From ⊢ ϕ infer ⊢ ◮ϕ G0 ◮(ϕ → ψ) → ◮ϕ → ◮ψ G1 α ↔ ◮α where α is a propositional formula

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 11 / 19

FEL: Axiomatization of L⊲ (one agent)

P All tautologies of propositional logic K (ϕ → ψ) → ϕ → ψ MP From ⊢ ϕ → ψ and ⊢ ϕ infer ⊢ ψ Nec1 From ⊢ ϕ infer ⊢ ϕ Nec2 From ⊢ ϕ infer ⊢ ◮ϕ G0 ◮(ϕ → ψ) → ◮ϕ → ◮ψ G1 α ↔ ◮α where α is a propositional formula GK

• γ∈Γ ♦⊲γ ↔ ⊲ (
• γ∈Γ

γ ∧

• γ∈Γ

♦γ)

• ∇Γ Cover operator

(where Γ is a finite set of formulas)

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 11 / 19

Soundness of FEL

P All tautologies of propositional logic K (ϕ → ψ) → ϕ → ψ MP From ⊢ ϕ → ψ and ⊢ ϕ infer ⊢ ψ Nec1 From ⊢ ϕ infer ⊢ ϕ Nec2 From ⊢ ϕ infer ⊢ ◮ϕ G0 ◮(ϕ → ψ) → ◮ϕ → ◮ψ G1 α ↔ ◮α where α is a propositional formula GK

• γ∈Γ ♦⊲γ ↔ ⊲∇Γ

Theorem

The axiomatization FEL is sound for L⊲.

Proof sketch

P, K, MP and Nec1 are sound (all models of L⊲ are models of L)

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 12 / 19

Soundness of FEL

Nec2 From ⊢ ϕ infer ⊢ ◮ϕ G0 ◮(ϕ → ψ) → ◮ϕ → ◮ψ G1 α ↔ ◮α where α is a propositional formula GK

• γ∈Γ ♦⊲γ ↔ ⊲∇Γ

Theorem

The axiomatization FEL is sound for L⊲.

Proof sketch

Nec2, G0 and G1 are easy

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 12 / 19

Soundness of FEL

Nec2 From ⊢ ϕ infer ⊢ ◮ϕ G0 ◮(ϕ → ψ) → ◮ϕ → ◮ψ G1 α ↔ ◮α where α is a propositional formula GK

• γ∈Γ ♦⊲γ ↔ ⊲∇Γ

Theorem

The axiomatization FEL is sound for L⊲.

Proof sketch

GK not that hard

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 12 / 19

Completeness of FEL

Lemma

Every formula of L⊲ is logically equivalent to a formula of L.

Proof

First, use cover logic instead of L ([Bilkova, M., A. Palmigiano and Y. Venema]) ϕ ::= ⊥ | ⊤ | ϕ ∨ ϕ | p ∧ ϕ | ¬p ∧ ϕ | ∇Γ Indeed ϕ iff ∇∅ ∨ ∇{ϕ}, and ♦ϕ iff ∇{ϕ, ⊤}. Axiom GK now takes shape GK

• γ∈Γ

∇{⊲γ, ⊤} ↔ ⊲∇Γ

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 13 / 19

Completeness of FEL

Lemma

Every formula of L⊲ is logically equivalent to a formula of L.

Proof

Second, show that given ψ in cover logic with refinement, ψ is equivalent to an ⊲-free formula, and therefore to a formula in L. We use equivalences: ⊲⊥ iff ⊥, ⊲⊤ iff ⊤, ⊲(p ∧ ϕ) iff p ∧ ⊲ϕ (refinements do not affect atoms), ⊲(¬p ∧ ϕ) iff ¬p ∧ ⊲ϕ. ⊲(ϕ ∨ ψ) iff ⊲ϕ ∨ ⊲ψ (directly from the semantics of ⊲). ⊲∇Γ iff

γ∈Γ ∇{⊲γ, ⊤} (GK)

Corollary

Let ϕ ∈ L⊲ be given and ψ ∈ L be equivalent to ϕ. If ψ is a theorem in K, then ϕ is a theorem in FEL.

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 13 / 19

FELµ: Axiomatization of Lµ⊲ (one agent)

FEL                        P All tautologies of propositional logic K (ϕ → ψ) → ϕ → ψ MP From ⊢ ϕ → ψ and ⊢ ϕ infer ⊢ ψ Nec1 From ⊢ ϕ infer ⊢ ϕ Nec2 From ⊢ ϕ infer ⊢ ◮ϕ G0 ◮(ϕ → ψ) → ◮ϕ → ◮ψ G1 α ↔ ◮α where α is a propositional formula GK

• γ∈Γ ♦⊲γ ↔ ⊲∇Γ

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 14 / 19

FELµ: Axiomatization of Lµ⊲ (one agent)

FEL                        P All tautologies of propositional logic K (ϕ → ψ) → ϕ → ψ MP From ⊢ ϕ → ψ and ⊢ ϕ infer ⊢ ψ Nec1 From ⊢ ϕ infer ⊢ ϕ Nec2 From ⊢ ϕ infer ⊢ ◮ϕ G0 ◮(ϕ → ψ) → ◮ϕ → ◮ψ G1 α ↔ ◮α where α is a propositional formula GK

• γ∈Γ ♦⊲γ ↔ ⊲∇Γ

Axiom and rule for the modal µ-calculus: F1 ϕ[µx.ϕ\x] → µx.ϕ F2 From ϕ[ψ\x] → ψ infer µxϕ → ψ

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 14 / 19

FELµ: Axiomatization of Lµ⊲ (one agent)

FEL                        P All tautologies of propositional logic K (ϕ → ψ) → ϕ → ψ MP From ⊢ ϕ → ψ and ⊢ ϕ infer ⊢ ψ Nec1 From ⊢ ϕ infer ⊢ ϕ Nec2 From ⊢ ϕ infer ⊢ ◮ϕ G0 ◮(ϕ → ψ) → ◮ϕ → ◮ψ G1 α ↔ ◮α where α is a propositional formula GK

• γ∈Γ ♦⊲γ ↔ ⊲∇Γ

Axiom and rule for the modal µ-calculus: F1 ϕ[µx.ϕ\x] → µx.ϕ F2 From ϕ[ψ\x] → ψ infer µxϕ → ψ Two new interaction axioms: G3 ◮µx.ϕ ↔ µx.◮ϕ

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 14 / 19

FELµ: Axiomatization of Lµ⊲ (one agent)

FEL                        P All tautologies of propositional logic K (ϕ → ψ) → ϕ → ψ MP From ⊢ ϕ → ψ and ⊢ ϕ infer ⊢ ψ Nec1 From ⊢ ϕ infer ⊢ ϕ Nec2 From ⊢ ϕ infer ⊢ ◮ϕ G0 ◮(ϕ → ψ) → ◮ϕ → ◮ψ G1 α ↔ ◮α where α is a propositional formula GK

• γ∈Γ ♦⊲γ ↔ ⊲∇Γ

Axiom and rule for the modal µ-calculus: F1 ϕ[µx.ϕ\x] → µx.ϕ F2 From ϕ[ψ\x] → ψ infer µxϕ → ψ Two new interaction axioms: G3 ◮µx.ϕ ↔ µx.◮ϕ where µx.ϕ is a disjunctive formula

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 14 / 19

FELµ: Axiomatization of Lµ⊲ (one agent)

FEL                        P All tautologies of propositional logic K (ϕ → ψ) → ϕ → ψ MP From ⊢ ϕ → ψ and ⊢ ϕ infer ⊢ ψ Nec1 From ⊢ ϕ infer ⊢ ϕ Nec2 From ⊢ ϕ infer ⊢ ◮ϕ G0 ◮(ϕ → ψ) → ◮ϕ → ◮ψ G1 α ↔ ◮α where α is a propositional formula GK

• γ∈Γ ♦⊲γ ↔ ⊲∇Γ

Axiom and rule for the modal µ-calculus: F1 ϕ[µx.ϕ\x] → µx.ϕ F2 From ϕ[ψ\x] → ψ infer µxϕ → ψ Two new interaction axioms: G3 ◮µx.ϕ ↔ µx.◮ϕ where µx.ϕ is a disjunctive formula G4 ◮νx.ϕ ↔ νx.◮ϕ where νx.ϕ is a df

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 14 / 19

Disjunctive formulas

Definition

A disjunctive formula (df) is specified by the following abstract syntax: α ::= x | α ∨ α | µx.α | νx.α | π ∧ ∇Γ |◮α | ⊲α where π is a conjunction of free literals (atoms or negated atoms, but not fixed-point variables) µ-disjunctive formulas are disjunctive formulas of Lµ (the ones without ◮

• r ⊲ operators)

Proposition

Every formula ϕ of Lµ is equivalent to a µ-disjunctive formula

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 15 / 19

Soundness of FELµ

FEL +        F1 ϕ[µx.ϕ\x] → µx.ϕ F2 From ϕ[ψ\x] → ψ infer µxϕ → ψ G3 ◮µx.ϕ ↔ µx.◮ϕ where µx.ϕ is a disjunctive formula G4 ◮νx.ϕ ↔ νx.◮ϕ where νx.ϕ is a df The soundness of F1 and F2 are well known [Arnold, A. and D. Niwinski]

Theorem

The axioms G3 and G4 are sound.

Proof sketch

Very technical. Use bisimulation quantifiers [D’Agostino, G. and G. Lenzi, French, T.]

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 16 / 19

Soundness of FELµ

FEL +        F1 ϕ[µx.ϕ\x] → µx.ϕ F2 From ϕ[ψ\x] → ψ infer µxϕ → ψ G3 ◮µx.ϕ ↔ µx.◮ϕ where µx.ϕ is a disjunctive formula G4 ◮νx.ϕ ↔ νx.◮ϕ where νx.ϕ is a df The soundness of F1 and F2 are well known [Arnold, A. and D. Niwinski]

Theorem

The axioms G3 and G4 are sound.

Proof sketch

1

as µz.f (z) = {z | f (z) ⊆ z} µx.ϕ is equivalent to ∀x((ϕ → x) → x) “” quantifies over all states in the model

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 16 / 19

Soundness of FELµ

FEL +        F1 ϕ[µx.ϕ\x] → µx.ϕ F2 From ϕ[ψ\x] → ψ infer µxϕ → ψ G3 ◮µx.ϕ ↔ µx.◮ϕ where µx.ϕ is a disjunctive formula G4 ◮νx.ϕ ↔ νx.◮ϕ where νx.ϕ is a df The soundness of F1 and F2 are well known [Arnold, A. and D. Niwinski]

Theorem

The axioms G3 and G4 are sound.

Proof sketch

1

µx.ϕ is equivalent to ∀x((ϕ → x) → x)

2

νx.ϕ is equivalent to ∃x((x → ϕ) ∧ x) “” quantifies over all states in the model

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 16 / 19

Soundness of FELµ

FEL +        F1 ϕ[µx.ϕ\x] → µx.ϕ F2 From ϕ[ψ\x] → ψ infer µxϕ → ψ G3 ◮µx.ϕ ↔ µx.◮ϕ where µx.ϕ is a disjunctive formula G4 ◮νx.ϕ ↔ νx.◮ϕ where νx.ϕ is a df The soundness of F1 and F2 are well known [Arnold, A. and D. Niwinski]

Theorem

The axioms G3 and G4 are sound.

Proof sketch

1

µx.ϕ is equivalent to ∀x((ϕ → x) → x)

2

νx.ϕ is equivalent to ∃x((x → ϕ) ∧ x)

3

◮ϕ is equivalent to ∀rϕr and ⊲ϕ is equivalent to ∃rϕr “” quantifies over all states in the model “ϕr” is the relativization of ϕ to the atom r, which may be computed recursively by replacing every occurrence of ψ in ϕ with (r → ψr)

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 16 / 19

Soundness of FELµ

FEL +        F1 ϕ[µx.ϕ\x] → µx.ϕ F2 From ϕ[ψ\x] → ψ infer µxϕ → ψ G3 ◮µx.ϕ ↔ µx.◮ϕ where µx.ϕ is a disjunctive formula G4 ◮νx.ϕ ↔ νx.◮ϕ where νx.ϕ is a df The soundness of F1 and F2 are well known [Arnold, A. and D. Niwinski]

Theorem

The axioms G3 and G4 are sound.

Proof sketch

Example of G3 (in contrapositive form ⊲νx.ϕ ↔ νx.⊲ϕ) ⊲νx.ϕ ↔ ∃r∃x((x → ϕ) ∧ x)r ↔ ∃x∃r((x → ϕr) ∧ x) ↔ ∃x(∃r(x → ϕr) ∧ x) → ∃x(∃r(x → ϕr) ∧ x) (you get ← only for disjunct. form.) ↔ ∃x((x → ∃rϕr) ∧ x) ↔ νx.⊲ϕ

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 16 / 19

Completeness of FELµ

Lemma

Every formula of Lµ

⊲ is equivalent in FELµ to a formula of the modal

µ-calculus Lµ.

Proof

Remove the ⊲ operators: use what we did for FEL and ⊲µx.ϕ iff µx.⊲ϕ (by G4 noting that all subformulas of a disjunctive formula are themselves disjunctive). ⊲νx.ϕ iff νx.⊲ϕ (by G3).

Corollary

FELµ is complete for the logic Lµ

⊲

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 17 / 19

Complexity

From previous

L⊲ expressively equivalent to L and Lµ

⊲ expressively equivalent to Lµ

Decidability from computable translations, BUT disjunctive normal forms yield a non-elementary decision procedure

We have shown

There is a 2EXPTIME upper-bound for L⊲: we use a non-trivial tableau construction using a finite iteration of two-player games on some initial tableau L⊲ and Lµ

⊲ are exponentially more succinct than L and Lµ respectively

Proposition

L⊲ is able to express the property that two binary trees are n-bisimilar, with a formula of size O(n2). (with Lµ you would need a formula of size exponential)

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 18 / 19

Perspectives

Complexity:

◮ For L⊲: what is the exact complexity? Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 19 / 19

Perspectives

Complexity:

◮ For L⊲: what is the exact complexity? ◮ For Lµ

⊲: a non-elementary upper-bound via bisimuation quantifiers. An

elementary upper-bound by adapting the tableau construction for L⊲? Is there a 2EXPTIME lower-bound from L⊲?

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 19 / 19

Perspectives

Complexity:

◮ For L⊲: what is the exact complexity? ◮ For Lµ

⊲: a non-elementary upper-bound via bisimuation quantifiers. An

elementary upper-bound by adapting the tableau construction for L⊲? Is there a 2EXPTIME lower-bound from L⊲?

Varying classes of models on the axiomatization given: We note that while the schema FEL is sound for L⊲, it is not the case that the axiom GK:

γ∈Γ ♦⊲γ ↔ ⊲∇Γ is sound for restricted classes

• f models:

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 19 / 19

Perspectives

Complexity:

◮ For L⊲: what is the exact complexity? ◮ For Lµ

⊲: a non-elementary upper-bound via bisimuation quantifiers. An

elementary upper-bound by adapting the tableau construction for L⊲? Is there a 2EXPTIME lower-bound from L⊲?

Varying classes of models on the axiomatization given: We note that while the schema FEL is sound for L⊲, it is not the case that the axiom GK:

γ∈Γ ♦⊲γ ↔ ⊲∇Γ is sound for restricted classes

• f models:

Consider the S5 frames ¬p p and Γ = {p ∧ ♦¬p}: ♦⊲{p ∧ ♦¬p} is true (on the left) but not ⊲(p ∧ ♦¬p) (since (p ∧ ♦¬p) is never true in reflexive frame)

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 19 / 19

Perspectives

Complexity:

◮ For L⊲: what is the exact complexity? ◮ For Lµ

⊲: a non-elementary upper-bound via bisimuation quantifiers. An

elementary upper-bound by adapting the tableau construction for L⊲? Is there a 2EXPTIME lower-bound from L⊲?

Examine axiomatizations and complexity for refinement quantifiers in logics such as S5, KD45 and K4.

Hans van Ditmarsch University of Sevilla, Spain Tim French University of Western Australia, Australia Sophie Pinchinat Universi Future event logic - axioms and complexity 19 / 19