The Many Faces of Modal Logic Day 1: Examples

The Many Faces of Modal Logic Day 1: Examples

Dirk Pattinson

Australian National University, Canberra

(Slides based on a NASSLLI 2014 Tutorial and are joint work with Lutz Schr ¨oder)

LAC 2018

The Plan for Today

I Brief recap of propositional logic

I Good old modal logic

I A zoo of modal logics

I Syntax

I Semantics

I Reasoning principles

Propositional Logic

That is, classical propositional logic:

φ::=⊥ |p| ¬φ|φ₁∧φ₂ (p∈V) (∨,→,↔,>defined).

Semantics:valuationsκ:V →2={⊥,>}

κ6|=⊥

κ|=p iff κ(p) =>

κ|=¬φ iff κ6|=φ

κ|=φ₁∧φ₂ iff κ|=φ₁andκ|=φ₂

Satisfiability and All That

I φsatisfiableifκ|=φ for someκ

I φvalid ortautology (|=φ) ifκ|=φ for allκ

I φvalid iff¬φ unsatisfiable

I Φset of formulas:

Φ|=ψ iff ∀κ.(κ|= Φ→κ|=ψ)

iff ∃φ₁, . . . ,φ_n∈Φ.|=φ₁∧ · · · ∧φ_n→ψ (compactness)

Boolean Algebra

Alternative semantics: valuationsκ:V →A,ABoolean algebra;

κ|=φ iff κ(φ) =>

Stone duality:

Ais a Boolean subalgebra of some 2^X. This implies:

Validity in Boolean algebras = validity over2.

Normal Forms

Negation normal form (NNF):¬only in front of atoms – φ::=⊥ | > |p| ¬p|φ₁∧φ₂|φ₁∨φ₂. Conjunctive normal form (CNF):

I Literal = atom or negated atom

I Clause = finite disjunction (set) of literals

I CNF = finite conjunction (set) of clauses.

Dual: conjunctive clause, DNF.

Modal Logic

φ::=⊥ |p| ¬φ|φ1∧φ2|2φ (p∈P) with2φ read

I ‘necessarilyφ’

I ‘it is believed thatφ’ (doxastic)

I ‘it is known thatφ’ (epistemic)

I ‘it is obligatory thatφ⁰ (deontic)

I . . .

Dual: 3=¬2¬‘possibly’

Kripke Models

Kripke models(X,R,π), where

I (X,R)Kripke frame

I X set ofstates/worlds

I R⊆X×X accessibility/transitionrelation

I π:P→ P(X)valuation

Satisfaction in Kripke Models

x|=_Mφ ‘statexin modelM satisfiesφ’;[[φ]]_M ={x∈M|x|=_Mφ}. x6|=_M⊥

x|=_M p ⇐⇒ x∈V(p)

x|=_M φ∧ψ ⇐⇒ x|=_M φandx|=_M ψ x|=_M ¬φ ⇐⇒ x6|=_M φ

x|=_M2φ ⇐⇒ y|=_M φ for ally∈X such that(x,y)∈R

hence

x|=_M 3φ ⇐⇒ y|=_M φ for somey∈X such that(x,y)∈R.

The Modal Logic K

Classes of frames induce modal logics; for now: all frames.

Axiomatization:

I Propositional Reasoning:

I Modus ponensφ;φ→ψ/ψ

I all instances of propositional tautologies

I Necessitation

φ 2φ

I All instances of axiom

(K) 2(φ→ψ)→2φ→2ψ.

Soundness and Completeness

Φ`ψ ifφ<sub>1</sub>∧ · · · ∧φ<sub>n</sub>→ψ is derivable for someφ<sub>1</sub>, . . . ,φ<sub>n</sub>∈Φ. SoundnessIfΦ`ψ thenΦ|=ψ(Proof: Induction over`) (Strong) CompletenessIfΦ|=ψ thenΦ`ψ

Reword: Φconsistent ifΦ6` ⊥

I Soundness:Φsatisfiable→Φconsistent

I (Strong) Completeness:Φconsistent→Φsatisfiable

I Proof: Satisfy consistentΦincanonical model, with worlds = maximally consistent sets.

Graded Modal Logic

(Fine 1972)Count successors:

x|=_M 2kφ ⇐⇒ |{y∈X |(x,y)∈Randy|=_M ¬φ}| ≤k x|=_M3kφ ⇐⇒ |{y∈X |(x,y)∈Randy|=_M φ}|>k. (Note2=2^0,3⁰=3^).

Alternative semantics: Multigraphs(D’Agostino/Visser 2002) (X,µ:X×X →N∪ {∞},π)

x|=2kφ ⇐⇒ µ(x,[[¬φ]])≤k

x|=3kφ ⇐⇒ µ(x,[[φ]])>k.

Equivalent, i.e. makes the same (sets of) formulas satisfiable.

Axiomatizing Graded Modal Logic

(Fine 1972)

Propositional reasoning plus

φ/20φ

20(φ→ψ)→20φ→20ψ 3^kφ→3^lφ (l<k) 3kφ↔

k

_

i=−1

(3i(φ∧ψ)∧3^k−1−i(φ∧ ¬ψ)) 20(φ→ψ)→3kφ→3kψ

(with3⁻¹φ≡ >)

Description Logics

. . . feature ingredients from graded modal logic:

∃R≡3^R

∀R≡2^R

≥n R≡3^Rn−1

≤n R≡ ¬3^Rn

E.g.

Elephant= (=2 hasPart.tusk)u(=1 hasPart.trunk)u(=4 hasPart.leg)

Probabilistic Modal Logic

Exchangeµ :X×X →N∪ {∞}forMarkov Chain(X,µ,π)

I µ(x,·)probability measure on X for eachx∈X OperatorsL_p ‘with probability≥p’ (p∈Q∩[0,1])

x|=L_pφ ⇐⇒ µ(x,[[φ]])≥p Define ‘with probability≤p’:

M_p=L_1−p¬ E.g.

I Probabilistic concurrent systems: finished∨M_0.1error

Draghi Yellen

(No) Compactness

I LogicLcompactif every finitely satisfiable set ofL-formulas is satisfiable.

I Strong completeness implies compactness.

Probabilistic modal logic fails to be compact:

{L_p−1/na|n∈N} ∪ {¬L_pa} is finitely satisfiable but not satisfiable.

Axiomatizing Probabilistic Modal Logic: The Problem

Have

L_p(φ∧ψ)∧L_q(φ∧ ¬ψ)→L_p+qφ but no reasonable converse

Axiomatizing Probabilistic Modal Logic: The Easy Part

L_oφ L_p>

L_pφ→ ¬L_q¬φ(p+q>1)

¬L_pφ→M_pφ φ→ψ/L_pφ→L_pψ

Axiomatizing Probabilistic Modal Logic: The Hard Part

(Heifetz/Mongin 2001)

Forφ= (φ1, . . . ,φm),ψ = (ψ1, . . . ,ψn)put φ^(k)≡ ^_

1≤l₁<···<l_k≤m

(φ_l1∨ · · · ∨φ_lk)

φ↔ψ≡

max(m,n)

^

k=1

φ^(k)↔ψ^(k). Rule (B):

(φ₁, . . . ,φ_m)↔(ψ₁, . . . ,ψ_n) V_m

i=1L_p1φ_i∧^Vn_j=2M_qjψ_j→L_{p1+...+pm−q2−...−qn}ψ₁

Linear Inequalities

(Halpern/Fagin/Megiddo 1990)n-ary modal operators

n

∑

i=1

a_il(φ_i)≥b (a_i,b∈Q) interpreted by

[[l(φ)]]_x =µ(x,[[φ]]) E.g.

l(MunichChampion)≥10·l(NurembergChampion)

Same for graded logic→Presburger modal logic(Demri/Lugiez 2006):

3·#_hasGamewon+1·#_hasGametied≥37→ ¬relegated. E.g.majority logic(Pacuit/Salame 2004)

Wφ= #(φ)≥#(¬φ)

Neighbourhood Semantics

Compositional semantics of2^{overX requires} [[2]] :2^X→2^X or, transposing,

N:X →2^(2X) – that’s aneighbourhood frame:

Aneighbourhood ofx ⇐⇒ A∈N(x).

Then

x|=2φ ⇐⇒ [[φ]]∈N(x)

Axiomatizing Neighbourhood Logic

Propositional reasoning plusreplacement of equivalents:

φ↔ψ 2φ↔2ψ

Monotone Neighbourhoods

Imposemonotonicity

φ→ψ 2φ→2ψ

→monotone neighbourhood frames:

A∈N(x)∧A⊆B→B∈N(x)

Game Logic

Monotonicity plusseriality:

2> 3>

Semantically:

X ∈N(x),/0∈/N(x).

Temporalized multi-agent version: game logic(Parikh 1983) [a]φ ‘Angel can enforceφin gamea’

haiφ ‘Demon can enforceφ in gamea’

Conditional Logics

Binary operator· ⇒ ·for non-material implication, e.g. ‘if – then normally’ (default implication):

|= (Monday ⇒ work)6→((Monday ∧ sick)⇒ work) (non-monotonic conditional)

Selection Function Semantics

Conditional frame(X,(R_A),π):

R_A⊆X×X (A⊆X) xR_Ay‘atx,yis most typical for conditionA’.

φ⇒ ·is box overR_[[φ]]:

x|=φ⇒ψ iff ∀y.(xR_φy→y|=ψ).

The Conditional Logic CK

= The logic of all conditional frames

I replacement of equivalents on the left:

φ↔φ⁰ (φ⇒ψ)↔(φ⁰⇒ψ)

I normality on the right:

(φ⇒(ψ→χ))→(φ⇒ψ)→(φ⇒χ).

Reasoning Principles = Frame Conditions

ID φ⇒φ xR_φy→y|=φ

CEM (φ⇒ψ)∨(φ⇒ ¬ψ) xR_φy∧xR_φy⁰→y=y⁰ MP (φ⇒ψ)→φ→ψ xR_φx

KLM

(Burgess 1981; Kraus/Lehmann/Magidor 1990) IDplus

DIS (φ⇒χ)→(ψ⇒χ)→((φ∨ψ)⇒χ) CM (φ⇒χ)→(φ⇒ψ)→((φ∧χ)⇒ψ) Cautious Monotony:

(Monday ⇒ work)→(Monday ⇒ sick)→((Monday ∧ sick)⇒ work)

Preferential Models

KLM complete forpreferential models(X,R,π):

I R⊆X³

I Rxyz: ‘y more typical thanzas an alternative tox’

Rxyz→Rxyy

(Rxyz∧Rxzw)→Rxyw with

x |=φ ⇒ψ iff ∀y.(Rxyy→ ∃z.(Rxzy∧ ∀t.(Rxtz →M,t |=ψ))).

Coalition Logic / Alternating-Time Temporal Logic

I N={1, . . . ,n}set ofagents

I Concurrent game structure: at each statex,

I setsS_i^x ofmoves(i∈N)

I outcome function

f_x : (

∏

i∈N

S_i^x)→X

I Operators[C]‘coalitionCcan force’

x|= [C]φ ⇐⇒ ∃σ_C ∈

∏

i∈C

S^x_i.∀σ_N−C∈

∏

i∈N−C

S_i^x.f_xhσ_C,σN−Ci |=φ.

Axiomatizing Coalition Logic

¬[C]⊥

[C]>

¬[/0]¬φ→[N]φ [C](φ∧ψ)→[C]φ

[C]φ∧[D]ψ→[C∪D](φ∧ψ) if C∩D6=/0.

Summing up

I Modalities are just logical operators

I Broad range of syntactic and semantic phenomena

I Common features:

I Models consist ofworlds/states

I Worlds are connected bytransitions, broadly construed:

I relational

I weighted

I probabilistic

I game-based

I proposition-indexed

I proximity-based

Tomorrow:

I Everything is coalgebraic :)