[PPT] - Horn Fragments of Halpern-Shoham Interval Temporal Logic Agi Kurucz PowerPoint Presentation

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Horn Fragments of Halpern-Shoham Interval Temporal Logic

Agi Kurucz

Department of Informatics, King’s College London

Joint work with

D. Bresolin, E. Mu˜

noz-Velasco, V. Ryzhikov, G. Sciavicco and M. Zakharyaschev

ACM Transactions on Computational Logic, vol. 18(3) (2017), 22:1-22:39

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Allen’s interval relations

Time: linear order (T, ≤) Intervals: i = x, y with x ≤ y

♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣

i i A j j i B j j i E j j i D j j i L j j i O j j i ¯ A j j i ¯ B j j i ¯ E j j i ¯ D j j i ¯ L j j i ¯ O j j After Begins Ends During Later Overlaps

Agi Kurucz — Logic Colloquium 2018, Udine 1

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Allen’s interval relations – 2D representation

♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣

i i A j j i B j j i E j j i D j j i L j j i O j j i ¯ A j j i ¯ B j j i ¯ E j j i ¯ D j j i ¯ L j j i ¯ O j j

✲ ✻

(T, ≤) (T, ≤)

x

y

s

L ¯ B A ¯ D O ¯ E E ¯ O D B ¯ A ¯ L ¯ BE-fragment = ‘expanding’ modal product (T, ≤)×exp(T, ≤)

Agi Kurucz — Logic Colloquium 2018, Udine 2

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Halpern-Shoham interval temporal logic

HS: propositional multi-modal logic over the 12 Allen relations

formulas: ϕ ::= ⊤ | ⊥ | p | ¬ϕ | ϕ1 ∧ ϕ2 | Rϕ | [R]ϕ

p ∈ Variables, R ∈ {L, B, A, D, O, E, ¯ L, ¯ B, ¯ A, ¯ D, ¯ O, ¯ E}

models: M = (T, ≤, ν)

where ν : Intervals → 2Variables

truth-relation: M, i | = Rϕ ⇐ ⇒ M, j | = ϕ

for some interval j with iRj

M, i | = [R]ϕ ⇐ ⇒ M, j | = ϕ

for all intervals j with iRj

Variants:

discrete, dense, finite, . . . linear orders
open, closed, semi-closed intervals
‘punctual’

x, x intervals allowed/disallowed

‘reflexive/irreflexive’ interpretation of the Allen relations

E.g.: x1, y1Bx2, y2 ⇐ ⇒ x1 = x2 and y2 ≤ y1 x1, y1Bx2, y2 ⇐ ⇒ x1 = x2 and y2 < y1 p ∧ [R]¬p is not satisfiable with reflexive semantics

Agi Kurucz — Logic Colloquium 2018, Udine 3

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The satisfiability problem of HS

Task: Given a formula ϕ and a class C

f linear orders,

are there some model M based on a timeline from C and interval i such that M, i | = ϕ ? Undecidable

ver any ‘unbounded’ class of linear orders

Halpern–Shoham 1991 ‘Taming’ attempts so far:

constraining the underlying linear orders
relativisations
softening the semantics
restricting the set of available modal operators
coarsening the interval relations
restricting the nesting of modal operators

Agi Kurucz — Logic Colloquium 2018, Udine 4

SLIDE 6

Horn fragments of HS

clausal normal form

f HS-formulas:

ϕ ::= λ | ¬λ | ∀(λ0 ∧ · · · ∧ λn → λn+1 ∨ · · · ∨ λn+m) | ϕ1 ∧ ϕ2

where ∀ψ =

R(ψ ∧ [R]ψ ∧ [¯

R]ψ) λ ::= ⊤ | ⊥ | p | Rλ | [R]λ

HShorn :

ϕ ::= λ | ∀(λ0 ∧ · · · ∧ λn → λ) | ϕ1 ∧ ϕ2

HScore :

ϕ ::= λ | ∀(λ0 → λ1) | ∀(λ0 ∧ λ1 → ⊥) | ϕ1 ∧ ϕ2

HS✷

horn and HS✷ core :

NO diamond operators R in λ

Agi Kurucz — Logic Colloquium 2018, Udine 5

SLIDE 7

What can we express?

Some syntactic sugar: for ψ = λ1 ∧ · · · ∧ λk

∀(ψ → ¬λ)

≡ ∀(ψ ∧ λ → ⊥)

∀
ψ → λ′

1 ∧ · · · ∧ λ′ n)

≡ n

i=1 ∀

ψ → λ′

i)

∀
ψ → [R](λ′

1∧· · ·∧λ′ n → λ)

≡

∀(ψ → [R]p) ∧ ∀(p∧λ′

1∧· · ·∧λ′ n → λ)

∀
ψ → R(λ′

1∧· · ·∧λ′ n → λ)

≡

∀(ψ → Rp) ∧ ∀(p∧λ′

1∧· · ·∧λ′ n → λ)

∀(Rλ ∧ ψ → λ′)

≡ ∀(λ → [¯ R](ψ → λ′)) Advanced courses cannot be given during the morning sessions.

∀(¯ DMorningSession ∧ AdvancedCourse → ⊥) ∀(¯ BLectureDay ∧ ALunch ↔ MorningSession)

Teaching is both downward and upward hereditary.

∀(teaching → [D]teaching) ∀

([D]Oteaching ∨¯

Dteaching)∧Bteaching∧Eteaching → teaching

Agi Kurucz — Logic Colloquium 2018, Udine

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Our results on the satisfiability problem irreflexive semantics reflexive semantics HShorn

undecidable∗

HScore

undecidable∗ PSPACE-hard∗

decidable?

discrete: undecidable

HS✷

horn

PTIME-complete∗∗ dense: PTIME-complete discrete: PSPACE-hard

HS✷

core

decidable?

in PTIME∗∗ dense: in PTIME

∗over any ‘unbounded’ class of linear orders ∗∗over any nonempty class of linear orders

Note: propositional Datalog is PTIME-complete its core fragment is NLOGSPACE-complete

Agi Kurucz — Logic Colloquium 2018, Udine 7

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Proving PSPACE lower bounds

In both

HS✷

core

ver discrete linear orders with irreflexive semantics
HScore
ver arbitrary linear orders with arbitrary semantics

it is possible

to identify/generate an infinite (or unbounded finite) sequence of ‘units’
to pass polynomial-size information from one unit to the next

❀

PSPACE-hardness

HS✷

core

ver dense linear orders OR with reflexive semantics is in PTIME
HScore
ver arbitrary linear orders with irreflexive semantics

is even undecidable

Agi Kurucz — Logic Colloquium 2018, Udine

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SLIDE 10

PSPACE lower bounds – the tricks

in HS✷

core

ver

discrete linear orders with irreflexive semantics: identifying units: ∀(unit ↔ [E] ⊥) passing poly-size info to next unit: [λ1&λ2 ⇒ λ] ∀(λ1 → [A] µ) ∀(λ2 → [A] [¯ E] µ) ∀([A] [¯ A] µ → λ)

λ1, λ2 ← λ

µ µ

in HScore
ver arbitrary linear orders with arbitrary semantics:

generating units: unit ∧ ∀(unit → Aunit) ∀(unit → ¬[D]unit) ‘compressing’ info to be passed: [λ1&λ2 ⇒ λ] ∀(λ1 → Aµ1) ∧ ∀(λ2 → Aµ2) ∀(µ2 → ¬¯ Bµ1) ∀(µ1 → µ ∧ [¯ B]µ) ∧ ∀(µ2 → [B]µ) ∀([A] µ → λ)

λ1, λ2 ← λ µ1 µ2

µ

Agi Kurucz — Logic Colloquium 2018, Udine 9

SLIDE 11

Proving undecidability: encoding ‘grid-based’ problems

generating the ω×ω-grid (or arbitrarily large finite ‘initial segment’ of it)
passing poly-size info to right- and up-neighbours

❀ encoding

tilings
Turing machine computations
counter machine computations
Post correspondence problem
. . .

Agi Kurucz — Logic Colloquium 2018, Udine 10

SLIDE 12

Often ‘half-grid’ is enough nwω×ω = {(i, j) : i ≤ j < ω}

the nwω×ω tiling problem: given a finite set T

f tile types

t = (left(t), right(t), up(t), down(t))

. . decide whether there exists τ : nwω×ω → T such that, for all i ≤ j < ω, . .

up(τ(i, j)) = down(τ(i, j + 1)) and left(τ(i, j)) = right(τ(i + 1, j)) whenever i < j

(Berger 1966): the ω×ω tiling problem is undecidable . . ❀ the nwω×ω tiling problem is undecidable

Agi Kurucz — Logic Colloquium 2018, Udine 11

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Often ‘half-grid’ is enough

Turing machine computations starting with empty tape

n nwω×ω :

→ tape ↑

steps

$ $ $ $ A A C B B

Agi Kurucz — Logic Colloquium 2018, Udine

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Diagonal encoding of nwω×ω

HS -models are kind of grid-like BUT HS has NO ‘right-neighbour’ and ‘up-neighbor’ operators not even over discrete orders with irreflexive semantics Idea:

Harel 1985, Halpern–Shoham 1991, Marx–Reynolds 1999, Reynolds–Zakharyaschev 2001

(0, 0)

1

(0, 1)

2

(0, 2)

3 4 5

(0, 3)

6 7 8 9

(0, 4)

10

(1, 4) (2, 4) (3, 4)

generate an infinite sequence of ‘units’
right-neighbour of each non-diagonal unit is the next unit in the sequence
use ‘local’ pointers to access the units representing

the up-neighbour of each grid-location

Agi Kurucz — Logic Colloquium 2018, Udine 13

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Diagonal encoding of nwω×ω – version 1

Halpern–Shoham 1991

1 line1 → 2 line2 → 3 4 5 line3 → 6 7 8 9 line4 → 10 wall ↓ diagonal ւ

start of line1 is 1, and up of(0) = 1
start of linei+1 is the end of linei + 1, for all i > 0
every line starts with some n on the wall and

ends with some m on the diagonal

if n is in linei then up of(n) is in linei+1
if m < n then up of(m) < up of(n)
if n > 0 is on the wall then there is m with up of(m) = n
if n is neither on the wall nor on the diagonal

then there is m with up of(m) = n

Agi Kurucz — Logic Colloquium 2018, Udine 14

SLIDE 16

Version 1 is doable in HScore with irreflexive semantics

✉= unit

✒

t ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉

diag up

t t

up up

t

line

t t t

up up up

t

line wall diag wall diag wall diag wall

t

line

t t t

up up up

♣♣♣

FOR EXAMPLE:

if n is neither on the wall nor on the diagonal then there is m with up of(m) = n
¯

Dline & unit ⇒ A¯ Aup

Agi Kurucz — Logic Colloquium 2018, Udine

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Diagonal encoding of nwω×ω – version 2

Marx–Reynolds 1999 Reynolds–Zakharyaschev 2001

1 2 3 4 5 6 7 8 9 10 diagonal ւ

0 is on the diagonal, and up of(0) = 1
if n is on the diagonal, then up of(n) + 1 is on the diagonal
if n is the up-neighbour of some location, then n is not on the diagonal
if n is not on the diagonal, then up of(n + 1) = up of(n) + 1
if n is on the diagonal, then up of(n + 1) = up of(n) + 2

Agi Kurucz — Logic Colloquium 2018, Udine 16

SLIDE 18

Version 2 is doable in HShorn

✒
✒

t ✉ ✉ ✉ ✉

up

t t

up up

t t t

up up up

t t t

up up up

♣♣♣ ✉ ❡ ✉ ✉ ❡ ❡ ✉ ✉ ✉ ❡ ❡ ❡ ✉ ✉ ✉ ❡ ❡ ❡ ✉= unit ∧ diag ❡= unit ∧ diag

FOR EXAMPLE:

if n is the up-neighbour of some location, then n is not on the diagonal

∀

up → [E] (unit → diag)
∧

∀(diag ∧ diag → ⊥)

Agi Kurucz — Logic Colloquium 2018, Udine 17

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Chessboard – a trick for reflexive semantics/arbitrary orders

With these choices it is not possible to generate a unique unit-sequence, BUT ‘discretising’ HS-models can be done in HShorn : Spaan 1993 Artale–Kontchakov–Lutz– Wolter–Zakharyaschev 2007 p

tickh ↓ tickh ↓ tickh ↓ tickh ↓ tickv ← tickv ← tickv ← tickv ←

using horizontal and vertical ‘tick’-variables
forcing the ‘stability’ of relevant variables in each chessboard square
combining chessboards with the previous techniques

❀ HShorn is undecidable over arbitrary linear orders with arbitrary semantics

Agi Kurucz — Logic Colloquium 2018, Udine 18

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Some more open questions on the satisfiability problem

mixing Horn fragments with other ‘taming’ techniques
Halpern–Shoham 1991:

more general temporal structures

(T, ≤) is a partial order
for every t ≤ t′, the interval [t, t′] is linearly ordered

❀ (T, ≤) is forest-like What about Horn fragments of HS over these? Gabelaia–K–Wolter–Zakharyaschev 2005: bimodal products of transitive structures are undecidable

Can we make use of the grid-like nature of HS-models directly,

without diagonally encoding grids? Hampson-K 2015: bimodal product logics ‘difference operator’×‘linear order’ are undecidable proof is by encoding counter machine computations directly on the grid-like models

Agi Kurucz — Logic Colloquium 2018, Udine 19