Neighborhood Semantics for Modal Logic - Lecture 1epacuit/classes/esslli/nbhd-lec1.pdf · Eric Pacuit: Neighborhood Semantics, Lecture 1 7. ... Pragmatics. 1968. Eric Pacuit: Neighborhood

Neighborhood Semantics forModal Logic

Lecture 1

Eric Pacuit

ILLC, Universiteit van Amsterdamstaff.science.uva.nl/∼epacuit

August 13, 2007

Eric Pacuit: Neighborhood Semantics, Lecture 1 1

staff.science.uva.nl/~epacuit

Plan for the Course

Lecture 1: Introduction, Motivation and BackgroundInformation

Lecture 2: Basic Concepts, Non-normal Modal Logics,Completeness, Decidability, Complexity,Incompleteness, Relation with Relational Semantics

Lecture 3: Advanced Topics — Topological Semantics forModal Logic, some Model Theory


Lecture 5: Neighborhood Semantics in Action: Game Logic,Coalgebra, Common Knowledge, First-Order ModalLogic


Plan for the Course







Plan for the Course







Plan for the Course







Plan for the Course







Plan for the Course







Plan for the Course

Course Websitestaff.science.uva.nl/∼epacuit/nbhd esslli.html

Reading Material

X Modal Logic: an Introduction, Chapters 7 - 9, by Brian Chellas

X Monotonic Modal Logics by Helle Hvid Hansen, available atwww.few.vu.nl/∼hhansen/papers/scriptie pic.pdf

X The course reader (updated version available on the website)

Concerning Modal LogicModal Logic by P. Blackburn, M. de Rijke and Y. Venema.


staff.science.uva.nl/~epacuit/nbhd_esslli.html

www.few.vu.nl/~hhansen/papers/scriptie_pic.pdf

Lecture 1

� Background

� Introduction

� Motivating Examples

� A Primer on Modal Logic


Background

The Basic Modal Language: L

p | ¬ϕ | ϕ ∧ ψ | �ϕ | ♦ϕ

where p is an atomic proposition (At)


Background

Kripke (Relational) Models

M = 〈W ,R,V 〉

I W 6= ∅I R ⊆ W ×W

I V : At → ℘(W )


Background

Kripke (Relational) Models

M = 〈W ,R,V 〉

I W 6= ∅I R ⊆ W ×W

I V : At → ℘(W )


Background

Truth in a Kripke Model

1. M,w |= p iff w ∈ V (p)

2. M,w |= ¬ϕ iff M,w 6|= ϕ

3. M,w |= ϕ ∧ ψ iff M,w |= ϕ and M,w |= ψ

4. M,w |= �ϕ iff for each v ∈ W , if wRv then M, v |= ϕ

5. M,w |= ♦ϕ iff there is a v ∈ W such that wRv and M, v |= ϕ


Background

Some Validities

(M) �(ϕ ∧ ψ) → �ϕ ∧�ψ

(C) �ϕ ∧�ψ → �(ϕ ∧ ψ)

(N) �>

(K) �(ϕ→ ψ) → (�ϕ→ �ψ)

(Dual) �ϕ↔ ¬♦¬ϕ

(Nec) from ` ϕ infer ` �ϕ

(Re) from ` ϕ↔ ψ infer ` �ϕ↔ �ψ

(Mon)` ϕ→ ψ

` �ϕ→ �ψ


Background

Some Validities

(M) �(ϕ ∧ ψ) → �ϕ ∧�ψ

(C) �ϕ ∧�ψ → �(ϕ ∧ ψ)

(N) �>

(K) �(ϕ→ ψ) → (�ϕ→ �ψ)

(Dual) �ϕ↔ ¬♦¬ϕ

(Nec) from ` ϕ infer ` �ϕ

(Re) from ` ϕ↔ ψ infer ` �ϕ↔ �ψ

(Mon)` ϕ→ ψ

` �ϕ→ �ψ


Introduction

� Background

� Introduction




Introduction

Neighborhoods in Topology

In a topology, a neighborhood of a point x is any set A containingx such that you can “wiggle” x without leaving A.

A neighborhood system of a point x is the collection ofneighborhoods of x .

J. Dugundji. Topology. 1966.


Introduction

Neighborhoods in Modal Logic

Neighborhood Structure: 〈W ,N,V 〉

I W 6= ∅I N : W → ℘(℘(W ))

I V : At → ℘(W )


Introduction

Some Notation

Given ϕ ∈ L and a model M, the

I proposition expressed by ϕ

I extension of ϕ

I truth set of ϕ

is

(ϕ)M = {w ∈ W | M,w |= ϕ}


Introduction

Some Notation

Given ϕ ∈ L and a model M, the

I proposition expressed by ϕ

I extension of ϕ

I truth set of ϕ

is

(ϕ)M = {w ∈ W | M,w |= ϕ}


Brief History

w |= �ϕ if the truth set of ϕ is a neighborhood of w

neighborhood in some topology.J. McKinsey and A. Tarski. The Algebra of Topology. 1944.

contains all the immediate neighbors in some graphS. Kripke. A Semantic Analysis of Modal Logic. 1963.

an element of some distinguished collection of setsD. Scott. Advice on Modal Logic. 1970.

R. Montague. Pragmatics. 1968.


Brief History


What does it mean to be a neighborhood?






Brief History







Brief History







Brief History







Brief History

To see the necessity of the more general approach, wecould consider probability operators, conditionalnecessity, or, to invoke an especially perspicuous exampleof Dana Scott, the present progressive tense....Thus Nmight receive the awkward reading ‘it is being the casethat’, in the sense in which ‘it is being the case thatJones leaves’ is synonymous with ‘Jones is leaving’.

(Montague, pg. 73)

R. Montague. Pragmatics and Intentional Logic. 1970.


Brief History

Segerberg’s Essay

K. Segerberg. An Essay on Classical Modal Logic. Uppsula Technical Report,1970.

This essay purports to deal with classical modal logic.The qualification “classical” has not yet been given anestablished meaning in connection with modallogic....Clearly one would like to reserve the label“classical” for a category of modal logics which—ifpossible—is large enough to contain all or most of thesystems which for historical or theoretical reasons havecome to be regarded as important, and which also possesa high degree of naturalness and homogeneity.

(pg. 1)


Brief History

Segerberg’s Essay

K. Segerberg. An Essay on Classical Modal Logic. Uppsula Technical Report,1970.

This essay purports to deal with classical modal logic.The qualification “classical” has not yet been given anestablished meaning in connection with modallogic....Clearly one would like to reserve the label“classical” for a category of modal logics which—ifpossible—is large enough to contain all or most of thesystems which for historical or theoretical reasons havecome to be regarded as important, and which also possesa high degree of naturalness and homogeneity.

(pg. 1)


Brief History

R. Goldblatt. Mathematical Modal Logic: A View of its Evolution. Handbookof the History of Logic, 2005.


Motivating Examples

� Background

� Introduction




Motivating Examples

Logics of High Probability

�ϕ means “ϕ is assigned ‘high’ probability”, where high meansabove some threshold r ∈ [0, 1].

Claim: Mon is a valid rule of inference.

Claim: C is not valid.

H. Kyburg and C.M. Teng. The Logic of Risky Knowledge. Proceedings ofWoLLIC (2002).

A. Herzig. Modal Probability, Belief, and Actions. Fundamenta Informaticae(2003).


Motivating Examples








Motivating Examples








Motivating Examples








Motivating Examples

Games

�iϕ means “player i has a strategy that guarantees ϕ is true.”


Motivating Examples

Games


As1

Bs2 B s3

p p, q p, q q

p p, q

s1 |= �Ap

p, q q

s1 |= �Ap ∧�Aq

p, q p, q

s1 |= �Ap ∧�Aq ∧ ¬�A(p ∧ q)


Motivating Examples

Games


As1

Bs2 B s3

p p, q p, q qp p, q

s1 |= �Ap

p, q q

s1 |= �Ap ∧�Aq

p, q p, q

s1 |= �Ap ∧�Aq ∧ ¬�A(p ∧ q)


Motivating Examples

Games


As1

Bs2 B s3

p p, q p, q q

p p, q

s1 |= �Ap

p, q q

s1 |= �Ap ∧�Aq

p, q p, q

s1 |= �Ap ∧�Aq ∧ ¬�A(p ∧ q)


Motivating Examples

Games


As1

Bs2 B s3

p p, q p, q q

p p, q

s1 |= �Ap

p, q q

s1 |= �Ap ∧�Aq

p, q p, q

s1 |= �Ap ∧�Aq ∧ ¬�A(p ∧ q)


Motivating Examples

Games


R. Parikh. The Logic of Games and its Applications. Annals of DiscreteMathematics (1985).

M. Pauly and R. Parikh. Game Logic — An Overview. Studia Logica (2003).

J. van Benthem. Logic and Games. Course notes (2007).


Motivating Examples

Logic of Deduction

Let L0 ⊆ L be the set of propositional formulas.

Let Σ ⊆ L0 be the universe

Interpretation: (·)∗ : At → ℘(Σ)

I (ϕ ∨ ψ)∗ = (ϕ)∗ ∪ (ψ)∗

I (¬ϕ)∗ = Σ− (ϕ)∗

I (�ϕ)∗ = {α ∈ Σ | (ϕ)∗ ` α} (the deductive closure of ϕ)

Fact: �(ϕ→ ψ) → �ϕ→ �ψ is not valid.

P. Naumov. On modal logic of deductive closure. APAL (2005).


Motivating Examples

Logic of Deduction




I (ϕ ∨ ψ)∗ = (ϕ)∗ ∪ (ψ)∗

I (¬ϕ)∗ = Σ− (ϕ)∗





Motivating Examples

Logic of Deduction




I (ϕ ∨ ψ)∗ = (ϕ)∗ ∪ (ψ)∗

I (¬ϕ)∗ = Σ− (ϕ)∗

I (�ϕ)∗ = {α ∈ Σ | (ϕ)∗ ` α}

(the deductive closure of ϕ)




Motivating Examples

Logic of Deduction




I (ϕ ∨ ψ)∗ = (ϕ)∗ ∪ (ψ)∗

I (¬ϕ)∗ = Σ− (ϕ)∗





Motivating Examples

Logic of Deduction




I (ϕ ∨ ψ)∗ = (ϕ)∗ ∪ (ψ)∗

I (¬ϕ)∗ = Σ− (ϕ)∗





Motivating Examples

Logic of Deduction




I (ϕ ∨ ψ)∗ = (ϕ)∗ ∪ (ψ)∗

I (¬ϕ)∗ = Σ− (ϕ)∗


Fact: �ϕ ∧�ψ → �(ϕ ∧ ψ) is not valid.



Motivating Examples

Logic of Deduction




I (ϕ ∨ ψ)∗ = (ϕ)∗ ∪ (ψ)∗

I (¬ϕ)∗ = Σ− (ϕ)∗


Validities: ϕ→ �ϕ,



Motivating Examples

Logic of Deduction




I (ϕ ∨ ψ)∗ = (ϕ)∗ ∪ (ψ)∗

I (¬ϕ)∗ = Σ− (ϕ)∗


Validities: ϕ→ �ϕ, (Mon),



Motivating Examples

Logic of Deduction




I (ϕ ∨ ψ)∗ = (ϕ)∗ ∪ (ψ)∗

I (¬ϕ)∗ = Σ− (ϕ)∗


Validities: ϕ→ �ϕ, (Mon), �(ϕ ∨�ϕ) → �ϕ



Motivating Examples

Deontic Logic

�ϕ mean “it is obliged that ϕ.”

1. Jones murders Smith

2. Jones ought not to murder Smith

3. If Jones murders Smith, then Jones ought to murder Smithgently

4. Jones ought to murder Smith gently

5. If Jones murders Smith gently, then Jones murders Smith.

6. If Jones ought to muder Smith gently, then Jones ought tomurder Smith

7. Jones ought to murder Smith

J. Forrester. Paradox of Gentle Murder. 1984.

L. Goble. Murder Most Gentle: The Paradox Deepens. 1991.


Motivating Examples

Deontic Logic












Motivating Examples

Deontic Logic


X Jones murders Smith


X If Jones murders Smith, then Jones ought to murder Smithgently








Motivating Examples

Deontic Logic






⇒ If Jones murders Smith gently, then Jones murders Smith.






Motivating Examples

Deontic Logic






X If Jones murders Smith gently, then Jones murders Smith.

(Mon) If Jones ought to muder Smith gently, then Jones ought tomurder Smith





Motivating Examples

Deontic Logic





X Jones ought to murder Smith gently


X If Jones ought to muder Smith gently, then Jones ought tomurder Smith





Motivating Examples

Deontic Logic



6 Jones ought not to murder Smith





6 Jones ought to murder Smith




Motivating Examples

Social Choice Theory

�α mean “the group accepts α.”


Motivating Examples



Note: the language is restricted so that ��α is not a wff.


Motivating Examples



Consensus: α is accepted provided everyone accepts α.

(E) �α↔ �β provided α↔ β is a tautology

(M) �(α ∧ β) → (�α ∧�β)

(C) (�α ∧�β) → (�α ∧�β)

(N) �>(D) ¬�⊥


Motivating Examples



Consensus: α is accepted provided everyone accepts α.


(M) �(α ∧ β) → (�α ∧�β)

(C) (�α ∧�β) → (�α ∧�β)

(N) �>(D) ¬�⊥

Theorem The above axioms axiomatize consensus (providedn ≥ 2|At|)).


Motivating Examples



Majority: α is accepted if a majority of the agents accept α.


Motivating Examples



Majority: α is accepted if a majority of the agents accept α.


(M) �(α ∧ β) → (�α ∧�β)

(S) �α→ ¬�¬α(T) ([≥]ϕ1 ∧ · · · ∧ [≥]ϕk ∧ [≤]ψ1 ∧ · · · ∧ [≤]ψk) →∧

1≤i≤k([=]ϕi ∧ [=]ψi ) where ∀v ∈ VI :|{i | v(ϕi ) = 1}| = |{i | v(ψi ) = 1}|

Theorem The above axioms axiomatize majority.


Motivating Examples



M. Pauly. Axiomatizing Collective Judgement Sets in a Minimal Logical Lan-guage. 2006.

T. Daniels. Social Choice and Logic via Simple Games. ILLC, Masters Thesis,2007.


Motivating Examples

Other Examples

I Epistemic Logic: the logical omniscience problem.M. Vardi. On Epistemic Logic and Logical Omniscience. TARK (1986).

I Reasoning about coalitionsM. Pauly. Logic for Social Software. Ph.D. Thesis, ILLC (2001).

I Knowledge RepresentationV. Padmanabhan, G. Governatori, K. Su . Knowledge Assesment: A ModalLogic Approach. KRAQ (2007).

I Program logics: modeling concurrent programsD. Peleg. Concurrent Dynamic Logic. J. ACM (1987).

I ??????


A Primer on Modal Logic

� Background

� Introduction





Slogan 1: Modal languages are simple yet expressive languagesfor talking about relational structures.

Slogan 2: Modal languages provide an internal, local perspectiveon relational structures.

P. Blackburn, M. de Rijke and Y. Venema. Modal Logic. 2001.



P. Blackburn and J. van Benthem. Modal Logic: A Semantics Perspective.Handbook of Modal Logic (2007).



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Notation

A Kripke frame is a pair 〈W ,R〉 where R ⊆ W ×W .

Let F = 〈W ,R〉 be a Kripke frame and M = 〈W ,R,V 〉 a modelbased on M.

ϕ is satisfiable in M if there exists w ∈ W such that M,w |= ϕ

ϕ is valid in M (M |= ϕ) if ∀w ∈ W , M,w |= ϕ

ϕ is valid on a frame F (F |= ϕ) if for all models M based on F,M |= ϕ.



Notation








Notation








Notation








Notation








Definable Properties

A modal formula ϕ defines a class of frames K provided

F ∈ K iff F |= ϕ

I �ϕ→ ��ϕ defines the class of transitive frames.

I ϕ↔ �ϕ defines the class of frames consisting of isolatedreflexive points (∀x ∈ W , xRy → x = y).

I �(�ϕ→ ϕ) defines the class of secondary-reflexive frames(∀w , v ∈ W , if wRv then vRv).

Some modal formulas correspond to genuine second-orderproperties: Lob (�(�ϕ→ ϕ) → �ϕ), McKinsey (�♦ϕ→ ♦�ϕ)





F ∈ K iff F |= ϕ

X �ϕ→ ��ϕ defines the class of transitive frames.

I ϕ↔ �ϕ defines the class of frames consisting of isolatedreflexive points (∀x ∈ W , xRy → x = y).







F ∈ K iff F |= ϕ


X ϕ↔ �ϕ defines the class of frames consisting of isolatedreflexive points (∀x ∈ W , xRy → x = y).







F ∈ K iff F |= ϕ



X �(�ϕ→ ϕ) defines the class of secondary-reflexive frames(∀w , v ∈ W , if wRv then vRv).






F ∈ K iff F |= ϕ









F ∈ K iff F |= ϕ




The Sahlqvist Theorem gives an algorithm for finding a first-ordercorrespondant for certain modal formulas.



Slogan 3: Modal logics are not isolated formal systems.



The Standard Translation

stx : L → L1

stx : L → L1

First-order language with anappropriate signature

stx(p) = Px

stx(¬ϕ) = ¬stx(ϕ)stx(ϕ ∧ ψ) = stx(ϕ) ∧ stx(ψ)stx(�ϕ) = ∀y(xRy → sty (ϕ))stx(♦ϕ) = ∃y(xRy ∧ sty (ϕ))

Lemma For each w ∈ W , M,w |= ϕ iff M |= stx(ϕ)[x/w ].




stx : L → L1stx : L → L1


stx(p) = Pxstx(¬ϕ) = ¬stx(ϕ)

stx(ϕ ∧ ψ) = stx(ϕ) ∧ stx(ψ)stx(�ϕ) = ∀y(xRy → sty (ϕ))stx(♦ϕ) = ∃y(xRy ∧ sty (ϕ))





stx : L → L1

stx : L → L1


stx(p) = Pxstx(¬ϕ) = ¬stx(ϕ)stx(ϕ ∧ ψ) = stx(ϕ) ∧ stx(ψ)

stx(�ϕ) = ∀y(xRy → sty (ϕ))stx(♦ϕ) = ∃y(xRy ∧ sty (ϕ))





stx : L → L1

stx : L → L1


stx(p) = Pxstx(¬ϕ) = ¬stx(ϕ)stx(ϕ ∧ ψ) = stx(ϕ) ∧ stx(ψ)stx(�ϕ) = ∀y(xRy → sty (ϕ))

stx(♦ϕ) = ∃y(xRy ∧ sty (ϕ))





stx : L → L1

stx : L → L1


stx(p) = Pxstx(¬ϕ) = ¬stx(ϕ)stx(ϕ ∧ ψ) = stx(ϕ) ∧ stx(ψ)stx(�ϕ) = ∀y(xRy → sty (ϕ))stx(♦ϕ) = ∃y(xRy ∧ sty (ϕ))





stx : L → L1

stx : L → L1



Fact: Modal logic falls in the two-variable fragment of L1.




stx : L → L1

stx : L → L1



Lemma For each w ∈ W , M,w |= ϕ iff M stx(ϕ)[x/w ].



What can we say with modal logic? What about in comparisonwith first-order logic?



Disjoint Union

Definition Let M1 = 〈W1,R1,V1〉 and M2 = 〈W2,R2,V2〉. Thedisjoint union is the structure M1 ]M2 = 〈W ,R,V 〉 where

I W = W1 ∪W2

I R = R1 ∪ R2

I for all p ∈ At, V (p) = V1(p) ∪ V2(p)

Fact The universal modality is not definable in the basic modallanguage.



Disjoint Union


I W = W1 ∪W2

I R = R1 ∪ R2


Lemma For each collection of Kripke structures {Mi | i ∈ I}, foreach w ∈ Wi , Mi ,w |= ϕ iff

⊎i∈I Mi ,w |= ϕ



Disjoint Union


I W = W1 ∪W2

I R = R1 ∪ R2





Generated Submodel

Definition M′ = 〈W ′,R ′,V ′〉 is a generated submodel ofM = 〈W ,R,V 〉 provided

I W ′ ⊆ W is R-closed:for each w ′ ∈ W and v ∈ W , if wRv then v ∈ W ′.

I R ′ = R ∩W ′ ×W ′

I for all p ∈ At, V ′(p) = V (p) ∩W ′




Generated Submodel



I R ′ = R ∩W ′ ×W ′


Lemma If M′ is a generated submodel of M then for eachw ∈ W ′, M′,w |= ϕ iff M,w |= ϕ



Generated Submodel



I R ′ = R ∩W ′ ×W ′


Fact The backwards looking modality is not definable in the basicmodal language.



Bounded Morphism

Definition A bounded morphism between models M = 〈W ,R,V 〉and M′ = 〈W ′,R ′,V ′〉 is a function f with domain W and rangeW ′ such that:

Atomic harmony: for each p ∈ At, w ∈ V (p) iff f (w) ∈ V ′(p)

Morphism: if wRv then f (w)Rf (v)

Zag: if f (w)R ′v ′ then ∃v ∈ W such that f (v) = v ′ and wRv




Bounded Morphism





Lemma If M′ is a bounded morphic image of M then for eachw ∈ W , M,w |= ϕ iff M′, f (w) |= ϕ



Bounded Morphism





Fact Counting modalities are not definable in the basic modallanguage (eg., ♦1ϕ iff ϕ is true in more than 1 accessible world).



Bisimulation

A bisimulation between M = 〈W ,R,V 〉and M′ = 〈W ′,R ′,V ′〉 is anon-empty binary relation Z ⊆ W ×W ′ such that whenever wZw ′:

Atomic harmony: for each p ∈ At, w ∈ V (p) iff w ′ ∈ V ′(p)

Zig: if wRv , then ∃v ′ ∈ W ′ such that vZv ′ and w ′R ′v ′

Zag: if w ′R ′v ′ then ∃v ∈ W such that vZv ′ and wRv

Fact The universal modality is not definable in the



Bisimulation





We write M,w ↔ M′,w ′ if there is a Z such that wZw ′.



Bisimulation





We write M,w ! M′,w ′ iff ∀ϕ ∈ L, M,w |= ϕ iff M′,w ′ |= ϕ.



Bisimulation





Lemma If M,w ↔ M′,w ′ then M,w ! M′,w ′.



Bisimulation





Lemma On finite frames, if M,w ! M′,w ′ then M,w ↔ M′,w ′.



The Van Benthem Characterization Theorem

Modal logic is the bisimulation invariant fragment of first-orderlogic.



The Van Benthem Characterization Theorem

For any first-order formula ϕ(x), TFAE:

1. ϕ(x) is invariant for bisimulation

2. ϕ(x) is equivalent to the standard translation of a basic modalformula.



The Goldblatt-Thomason Theorem

An elementary class of frames K is modally definiable iff

it is closedunder disjoint unions, bounded morphic images, generatedsubframes, and reflects ultrafilter extensions.




An elementary class of frames K is modally definiable iff it is closedunder disjoint unions,

bounded morphic images, generatedsubframes, and reflects ultrafilter extensions.




An elementary class of frames K is modally definiable iff it is closedunder disjoint unions, bounded morphic images,

generatedsubframes, and reflects ultrafilter extensions.




An elementary class of frames K is modally definiable iff it is closedunder disjoint unions, bounded morphic images, generatedsubframes,

and reflects ultrafilter extensions.




An elementary class of frames K is modally definiable iff it is closedunder disjoint unions, bounded morphic images, generatedsubframes, and reflects ultrafilter extensions.



End of lecture 1.


Documents

Neighborhood Semantics for Modal Logic - Lecture 1epacuit/classes/esslli/nbhd-lec1.pdf · Eric Pacuit: Neighborhood Semantics, Lecture 1 7. ... Pragmatics. 1968. Eric Pacuit: Neighborhood