Post on 15-Feb-2021
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