Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Undecidability of Interval Temporal Logics withthe Overlap Modality

D. Bresolin1, D. Della Monica2, V. Goranko3, A. Montanari2,G. Sciavicco4

1University of Verona, Italy

2University of Udine, Italy

3Technical University of Denmark

4Universidad de Murcia, Spain

Brixen, 23th July - TIME 2009

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Outline

1 Introduction to Interval Temporal Logics

2 Classifying HS fragments

3 Undecidability of logics with Overlap modality

4 Conclusions and future works

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Outline

1 Introduction to Interval Temporal Logics

2 Classifying HS fragments

3 Undecidability of logics with Overlap modality

4 Conclusions and future works

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Interval Temporal Logics

The time period, instead of the time intstant, is the primitivetemporal entity

Propositional letters are evaluated over pairs of points(instead of individual points)

Relations between worlds are more complicate than thepoint-based case

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Allen’s relations

J. F. Allen

Maintaining knowledge about temporal intervals.

Communications of the ACM, 1983.

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Allen’s relations

J. F. Allen

Maintaining knowledge about temporal intervals.

Communications of the ACM, 1983.

later

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Allen’s relations

J. F. Allen

Maintaining knowledge about temporal intervals.

Communications of the ACM, 1983.

later

after/meets

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Allen’s relations

J. F. Allen

Maintaining knowledge about temporal intervals.

Communications of the ACM, 1983.

later

after/meets

overlaps

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Allen’s relations

J. F. Allen

Maintaining knowledge about temporal intervals.

Communications of the ACM, 1983.

later

after/meets

overlaps

ends/finishes

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Allen’s relations

J. F. Allen

Maintaining knowledge about temporal intervals.

Communications of the ACM, 1983.

later

after/meets

overlaps

ends/finishes

during

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Allen’s relations

J. F. Allen

Maintaining knowledge about temporal intervals.

Communications of the ACM, 1983.

later

after/meets

overlaps

ends/finishes

during

begins/starts

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Allen’s relations

J. F. Allen

Maintaining knowledge about temporal intervals.

Communications of the ACM, 1983.

later 〈L〉

after/meets 〈A〉

overlaps 〈O〉

ends/finishes 〈E〉

during 〈D〉

begins/starts 〈B〉

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Allen’s relations

J. F. Allen

Maintaining knowledge about temporal intervals.

Communications of the ACM, 1983.

later 〈L〉 〈L〉

after/meets 〈A〉 〈A〉

overlaps 〈O〉 〈O〉

ends/finishes 〈E〉 〈E〉

during 〈D〉 〈D〉

begins/starts 〈B〉 〈B〉

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Some ontological choices

Time structure:linear or branching !discrete or dense !with or without beginning/end !

Nature of intervals:can or cannot intervals be unbounded !Are intervals with coinciding endpoints admissible or notadmissible !

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Some ontological choices

Time structure:linear or branching ?discrete or dense ?with or without beginning/end ?

Nature of intervals:can or cannot intervals be unbounded !Are intervals with coinciding endpoints admissible or notadmissible !

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Some ontological choices

Time structure:linear or branching !discrete or dense !with or without beginning/end !

Nature of intervals:can or cannot intervals be unbounded !Are intervals with coinciding endpoints admissible or notadmissible !

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Some ontological choices

Time structure:linear or branching !discrete or dense !with or without beginning/end !

Nature of intervals:can or cannot intervals be unbounded !Are intervals with coinciding endpoints admissible or notadmissible !

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Some ontological choices

Time structure:linear or branching !discrete or dense !with or without beginning/end !

Nature of intervals:can or cannot intervals be unbounded ?Are intervals with coinciding endpoints admissible or notadmissible ?

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Some ontological choices

Time structure:linear or branching !discrete or dense !with or without beginning/end !

Nature of intervals:can or cannot intervals be unbounded !Are intervals with coinciding endpoints admissible or notadmissible !

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Outline

1 Introduction to Interval Temporal Logics

2 Classifying HS fragments

3 Undecidability of logics with Overlap modality

4 Conclusions and future works

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

First discouraging undecidability results

HS is undecidable

J. Halpern and Y. Shoham

A propositional modal interval logic.

Journal of the ACM, 1991.

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

First discouraging undecidability results

HS is undecidable

J. Halpern and Y. Shoham

A propositional modal interval logic.

Journal of the ACM, 1991.

Undecidability of a small fragment of HS: BE

K. Lodaya

Sharpening the Undecidability of Interval Temporal Logic.

ASIAN 2000, volume 1961 of LNCS, pages 290-298. Springer, 2000.

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

First decidable fragments

Restrictions of the interval-based semanticslocality: truth of atomic propositions over an interval is definedas truth at its initial pointhomogeneity: truth of a formula over an interval implies truth ofthat formula over every sub-interval

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

First decidable fragments

Restrictions of the interval-based semanticslocality: truth of atomic propositions over an interval is definedas truth at its initial pointhomogeneity: truth of a formula over an interval implies truth ofthat formula over every sub-interval

Restrictions of the underlying structuressplit logic: each interval can be divided in subintervals in onlyone way

A. Montanari, G. Sciavicco, and N. Vitacolonna

Decidability of interval temporal logics over split-frames via granularity.

JELIA 2002, volume 2424 of LNCS, pages 259-270. Springer, 2002.

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

First decidable fragments

Restrictions of the interval-based semanticslocality: truth of atomic propositions over an interval is definedas truth at its initial pointhomogeneity: truth of a formula over an interval implies truth ofthat formula over every sub-interval

Restrictions of the underlying structuressplit logic: each interval can be divided in subintervals in onlyone way

Simple fragments of HSBB, EE

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

More meaningful decidable fragments

RPNL (A)

D. Bresolin, A. Montanari, and G. Sciavicco

An optimal decision procedure for Right Propositional NeighborhoodLogic.

Journal of Automated Reasoning, 2007.

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

More meaningful decidable fragments

RPNL (A)

PNL (AA)

D. Bresolin, A. Montanari, and P. Sala

An optimal tableau-based decision algorithm for PropositionalNeighborhood Logic.

STACS 2007, volume 4393 of LNCS, pages 549-560. Springer, 2007.

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

More meaningful decidable fragments

RPNL (A)

PNL (AA)

Subinterval logic (D)

D. Bresolin, V. Goranko, A. Montanari, P. Sala

Tableau-based decision procedures for the logics of subintervalstructures over dense orderings.

Journal of Logic and Computation, December 2008.

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

State of the art

DD is decidable over dense linear orders

Most extensions of A (resp., A) are undecidable (except forthe ones with BB and EE)

The class of fragments B∗E∗(= BE, BE, BE, BE) isundecidable

A. Montanari, G. Puppis and P. Sala

A Decidable Spatial Logic with Cone-shaped Cardinal Directions.

CSL 2009 (in press).

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

State of the art

DD is decidable over dense linear orders

Most extensions of A (resp., A) are undecidable (except forthe ones with BB and EE)

The class of fragments B∗E∗(= BE, BE, BE, BE) isundecidable

D. Bresolin, D. Della Monica, V. Goranko, A. Montanari, and G. Sciavicco

Decidable and Undecidable Fragments of Halpern and ShohamsInterval Temporal Logic: Towards a Complete Classification.

LPAR 2008, volume 5330 of LNCS, pages 590-604. Springer, 2008.

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

State of the art

DD is decidable over dense linear orders

Most extensions of A (resp., A) are undecidable (except forthe ones with BB and EE)

The class of fragments B∗E∗(= BE, BE, BE, BE) isundecidable

D. Bresolin, D. Della Monica, V. Goranko, A. Montanari, and G. Sciavicco

Decidable and Undecidable Fragments of Halpern and ShohamsInterval Temporal Logic: Towards a Complete Classification.

LPAR 2008, volume 5330 of LNCS, pages 590-604. Springer, 2008.

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Outline

1 Introduction to Interval Temporal Logics

2 Classifying HS fragments

3 Undecidability of logics with Overlap modality

4 Conclusions and future works

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

In this paper

We study the satisfiability problem for logics containing theoverlap modality

We provide a number of undecidability results

All the extensions of the fragments O and O (except for theextensions with L and L) are undecidable

The logic OO is undecidable over discrete linear orders

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Proof overview

Reduction from the Octant Tiling Problem

This is the problem of establishing whether a given finite set of tiletypes T = {t1, . . . , tk} can tile O = {(i, j) : i, j ∈ N ∧ 0 ≤ i ≤ j}respecting the color constraints.

by König’s Lemma

N × N → O

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Proof overview

Reduction from the Octant Tiling Problem

This is the problem of establishing whether a given finite set of tiletypes T = {t1, . . . , tk} can tile O = {(i, j) : i, j ∈ N ∧ 0 ≤ i ≤ j}respecting the color constraints.

by König’s Lemma

N × N → O

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Proof overview

Reduction from the Octant Tiling Problem

This is the problem of establishing whether a given finite set of tiletypes T = {t1, . . . , tk} can tile O = {(i, j) : i, j ∈ N ∧ 0 ≤ i ≤ j}respecting the color constraints.

by König’s Lemma

N × N → O

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Proof overview (cont’d)

We focus on the proof for the fragment AO

We build a formula φT ∈ AO s.t.φT is satisfiable ⇔ T can tile the octant.

Op. Semantics〈A〉 M, [a, b] 〈A〉φ ⇔ ∃c(b < c.M, [b, c] φ)〈O〉 M, [a, b] 〈O〉φ ⇔ ∃c, d(a < c < b < d .M, [c, d] φ)

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Proof overview (cont’d)

1 Encoding the octant2 Encoding the neighbourhood relations

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Proof overview (cont’d)

1 Encoding the octant2 Encoding the neighbourhood relations

u → 〈A〉u

u u u u u u u u u u u u u u u

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Proof overview (cont’d)

1 Encoding the octant2 Encoding the neighbourhood relations

u u u u u u u u u u u u u u u∗ t ∗ t t ∗ t t t ∗ t t t t ∗

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Proof overview (cont’d)

1 Encoding the octant2 Encoding the neighbourhood relations

u u u u u u u u u u u u u u u∗ t ∗ t t ∗ t t t ∗ t t t t ∗

t11

t21 t22

t31 t32 t

33

t41 t42 t

43 t

44

t51 t52 t

53 t

54 t

55

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Proof overview (cont’d)

1 Encoding the octant2 Encoding the neighbourhood relations

u u u u u u u u u u u u u u u∗ t ∗ t t ∗ t t t ∗ t t t t ∗

t11

t21 t22

t31 t32 t

33

t41 t42 t

43 t

44

t51 t52 t

53 t

54 t

55

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Proof overview (cont’d)

1 Encoding the octant2 Encoding the neighbourhood relations

u u u u u u u u u u u u u u u∗ t ∗ t t ∗ t t t ∗ t t t t ∗

t11

t21 t22

t31 t32 t

33

t41 t42 t

43 t

44

t51 t52 t

53 t

54 t

55

b

f

b

f

b

z }| {z }| {z }| {z }| {b f b f

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Proof overview (cont’d)

1 Encoding the octant2 Encoding the neighbourhood relations

u u u u u u u u u u u u u u u∗ t ∗ t t ∗ t t t ∗ t t t t ∗

t11

t21 t22

t31 t32 t

33

t41 t42 t

43 t

44

t51 t52 t

53 t

54 t

55

b

f

b

f

b

z }| {z }| {z }| {z }| {b f b f

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Proof overview (cont’d)

1 Encoding the octant2 Encoding the neighbourhood relations

t11

t21 t22

t31 t32 t

33

t41 t42 t

43 t

44

t51 t52 t

53 t

54 t

55

b

f

b

f

b

∗ t11

z }| {z }| {z }| {z }| {b f b f

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Proof overview (cont’d)

1 Encoding the octant2 Encoding the neighbourhood relations

t11

t21 t22

t31 t32 t

33

t41 t42 t

43 t

44

t51 t52 t

53 t

54 t

55

b

f

b

f

b

∗ t11 ∗ t

21 t

22

z }| {z }| {z }| {z }| {b f b f

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Proof overview (cont’d)

1 Encoding the octant2 Encoding the neighbourhood relations

t11

t21 t22

t31 t32 t

33

t41 t42 t

43 t

44

t51 t52 t

53 t

54 t

55

b

f

b

f

b

∗ t11 ∗ t

21 t

22 ∗ t

33 t

32 t

31

z }| {z }| {z }| {z }| {b f b f

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Proof overview (cont’d)

1 Encoding the octant2 Encoding the neighbourhood relations

t11

t21 t22

t31 t32 t

33

t41 t42 t

43 t

44

t51 t52 t

53 t

54 t

55

b

f

b

f

b

∗ t11 ∗ t

21 t

22 ∗ t

33 t

32 t

31 ∗ t

41 t

42 t

43 t

44 ∗

z }| {z }| {z }| {z }| {b f b f

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Proof overview (cont’d)

1 Encoding the octant2 Encoding the neighbourhood relations

t11

t21 t22

t31 t32 t

33

t41 t42 t

43 t

44

t51 t52 t

53 t

54 t

55

b

f

b

f

b

∗ t11 ∗ t

21 t

22 ∗ t

33 t

32 t

31 ∗ t

41 t

42 t

43 t

44 ∗

z }| {z }| {z }| {z }| {b f b f

up_relb → ¬〈O〉up_relb

up_relf → ¬〈O〉up_relf

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Theorems

Theorem [AO undecidability]

The satisfiability problem for the logic AO is undecidable overany class of linear orders that contains at least one linear orderwith an infinite ascending sequence.

Theorem [A∗O∗, B∗O∗, E∗O∗, D∗O∗ undecidability]

The satisfiability problem for the logics AO, BO, BO, EO, EO,DO, and DO (resp., AO, A O, BO, B O, EO, E O, DO, and D O)is undecidable over any class of linear orders that contains atleast one linear order with an infinite ascending (resp.,descending) sequence.

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Outline

1 Introduction to Interval Temporal Logics

2 Classifying HS fragments

3 Undecidability of logics with Overlap modality

4 Conclusions and future works

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Conclusions

Undecidable extensions of the fragments O and O:A∗O∗, B∗O∗, E∗O∗, D∗O∗

Fragment OO undecidable over discrete linear ordersExtended to the fragment O (resp., O) over discrete linearorders (unpublished)

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Conclusions

Undecidable extensions of the fragments O and O:A∗O∗, B∗O∗, E∗O∗, D∗O∗

Fragment OO undecidable over discrete linear ordersExtended to the fragment O (resp., O) over discrete linearorders (unpublished)

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Conclusions

Undecidable extensions of the fragments O and O:A∗O∗, B∗O∗, E∗O∗, D∗O∗

Fragment OO undecidable over discrete linear ordersExtended to the fragment O (resp., O) over discrete linearorders (unpublished)

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Conclusions

Undecidable extensions of the fragments O and O:A∗O∗, B∗O∗, E∗O∗, D∗O∗

Fragment OO undecidable over discrete linear ordersExtended to the fragment O (resp., O) over discrete linearorders (unpublished)

Current classification for the main classes of linear orders

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Future works

To complete the classification of HS fragments:

L∗O∗ and L∗D∗: conjecture is undecidability

O, O, and OO over dense linear orders: conjecture is ?

D, D, and DD over discrete linear orders: conjecture is ?

B∗D∗: conjecture is decidability at least over densestructures

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Future works

To complete the classification of HS fragments:

L∗O∗ and L∗D∗: conjecture is undecidability

O, O, and OO over dense linear orders: conjecture is ?

D, D, and DD over discrete linear orders: conjecture is ?

B∗D∗: conjecture is decidability at least over densestructures

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Future works

To complete the classification of HS fragments:

L∗O∗ and L∗D∗: conjecture is undecidability

O, O, and OO over dense linear orders: conjecture is ?

D, D, and DD over discrete linear orders: conjecture is ?

B∗D∗: conjecture is decidability at least over densestructures

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to ITLsClassifying HS fragments

Undecidability of logics with Overlap modalityConclusions and future works

Future works

To complete the classification of HS fragments:

L∗O∗ and L∗D∗: conjecture is undecidability

O, O, and OO over dense linear orders: conjecture is ?

D, D, and DD over discrete linear orders: conjecture is ?

B∗D∗: conjecture is decidability at least over densestructures

Bresolin, Della Monica, Goranko, Montanari, and Sciavicco Undecidability of ITLs with the Overlap Modality

Introduction to Interval Temporal LogicsClassifying HS fragmentsUndecidability of logics with Overlap modalityConclusions and future works