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D. Della Monica
University of Udine
Department of Mathematics and Computer Science
Expressiveness, decidability, and
undecidability of Interval Temporal Logic
ITL - Beyond the end of the light
Ph.D. Defence
Dario Della Monica
supervisor: A. Montanarico-supervisors: G. Sciavicco and V. Goranko
Udine - April 1, 2011
D. Della Monica
At the beginning...
At the beginning, it was the darkness...
Then, logicians made the light,
they became curious,
and moved toward the darkness...
... as close as they could
D. Della Monica
Outline
IntroductionThe Halpern and Shoham’s logic HS
Expressiveness of HS
The satisfiability problem for HSUndecidability
Classical extensionsMetric extensionsHybrid extensionsFirst-order extensions
Summary and perspectives
D. Della Monica
Outline
IntroductionThe Halpern and Shoham’s logic HS
Expressiveness of HS
The satisfiability problem for HSUndecidability
Classical extensionsMetric extensionsHybrid extensionsFirst-order extensions
Summary and perspectives
D. Della Monica
Interval-based temporal reasoning: origins and applications
Interval-based temporal reasoning: reasoning about time, where theprimary concept is ‘time interval’, rather than ‘time instant’.
Origins:
◮ Philosophy, in particular philosophy and ontology of time.
◮ Linguistics: analysis of progressive tenses, semantics of naturallanguages.
◮ Artificial intelligence: temporal knowledge representation,temporal planning, theory of events, etc.
◮ Computer science: specification and design of hardwarecomponents, concurrent real-time processes, temporaldatabases, etc.
D. Della Monica
Motivations
Some properties are intrinsically related to a time interval instead ofa (set of) time instant, each of which has not duration.
D. Della Monica
Motivations
Some properties are intrinsically related to a time interval instead ofa (set of) time instant, each of which has not duration.
Think about the event: “traveling from A to B”:
D. Della Monica
Motivations
Some properties are intrinsically related to a time interval instead ofa (set of) time instant, each of which has not duration.
Think about the event: “traveling from A to B”:
◮ it is true over a precise interval of time
◮ it is not true over any other interval (starting/ending interval,inner interval, ecc.)
D. Della Monica
Motivations
Some properties are intrinsically related to a time interval instead ofa (set of) time instant, each of which has not duration.
Think about the event: “traveling from A to B”:
◮ it is true over a precise interval of time
◮ it is not true over any other interval (starting/ending interval,inner interval, ecc.)
Several philosophical and logical paradoxes disappear:
D. Della Monica
Motivations
Some properties are intrinsically related to a time interval instead ofa (set of) time instant, each of which has not duration.
Think about the event: “traveling from A to B”:
◮ it is true over a precise interval of time
◮ it is not true over any other interval (starting/ending interval,inner interval, ecc.)
Several philosophical and logical paradoxes disappear:
◮ Zeno’s flying arrow paradox (“if at each instant the flyingarrow stands still, how is movement possible?”)
◮ The dividing instant dilemma (“if the light is on and it is turnedoff, what is its state at the instant between the two events?”)
D. Della Monica
Motivations
Some properties are intrinsically related to a time interval instead ofa (set of) time instant, each of which has not duration.
Think about the event: “traveling from A to B”:
◮ it is true over a precise interval of time
◮ it is not true over any other interval (starting/ending interval,inner interval, ecc.)
Several philosophical and logical paradoxes disappear:
◮ Zeno’s flying arrow paradox (“if at each instant the flyingarrow stands still, how is movement possible?”)
◮ The dividing instant dilemma (“if the light is on and it is turnedoff, what is its state at the instant between the two events?”)
The truth of a formula over an interval does not necessarily dependon its truth over subintervals.
D. Della Monica
Interval temporal reasoning and temporal ontologies
Interval-based temporal reasoning is subject to the same ontologicaldilemmas regarding the nature of Time as the instant-basedtemporal reasoning:
D. Della Monica
Interval temporal reasoning and temporal ontologies
Interval-based temporal reasoning is subject to the same ontologicaldilemmas regarding the nature of Time as the instant-basedtemporal reasoning:
◮ linear or branching?
D. Della Monica
Interval temporal reasoning and temporal ontologies
Interval-based temporal reasoning is subject to the same ontologicaldilemmas regarding the nature of Time as the instant-basedtemporal reasoning:
◮ linear or branching?
◮ discrete or dense?
D. Della Monica
Interval temporal reasoning and temporal ontologies
Interval-based temporal reasoning is subject to the same ontologicaldilemmas regarding the nature of Time as the instant-basedtemporal reasoning:
◮ linear or branching?
◮ discrete or dense?
◮ with or without beginning/end?, etc.
D. Della Monica
Interval temporal reasoning and temporal ontologies
Interval-based temporal reasoning is subject to the same ontologicaldilemmas regarding the nature of Time as the instant-basedtemporal reasoning:
◮ linear or branching?
◮ discrete or dense?
◮ with or without beginning/end?, etc.
New issues arise regarding the nature of the intervals:
D. Della Monica
Interval temporal reasoning and temporal ontologies
Interval-based temporal reasoning is subject to the same ontologicaldilemmas regarding the nature of Time as the instant-basedtemporal reasoning:
◮ linear or branching?
◮ discrete or dense?
◮ with or without beginning/end?, etc.
New issues arise regarding the nature of the intervals:
◮ Can intervals be unbounded?
D. Della Monica
Interval temporal reasoning and temporal ontologies
Interval-based temporal reasoning is subject to the same ontologicaldilemmas regarding the nature of Time as the instant-basedtemporal reasoning:
◮ linear or branching?
◮ discrete or dense?
◮ with or without beginning/end?, etc.
New issues arise regarding the nature of the intervals:
◮ Can intervals be unbounded?
◮ Are intervals with coinciding endpoints admissible or not?
D. Della Monica
Intervals and interval structures
D = 〈D, <〉: partially ordered set.
An interval in D: ordered pair [a, b], where a, b ∈ D and a ≤ b.
D. Della Monica
Intervals and interval structures
D = 〈D, <〉: partially ordered set.
An interval in D: ordered pair [a, b], where a, b ∈ D and a ≤ b.
If a < b then [a, b] is a strict interval; [a, a] is a point interval.
D. Della Monica
Intervals and interval structures
D = 〈D, <〉: partially ordered set.
An interval in D: ordered pair [a, b], where a, b ∈ D and a ≤ b.
If a < b then [a, b] is a strict interval; [a, a] is a point interval.
I(D): the interval structure over D, consisting of the set of allintervals over D.
D. Della Monica
Intervals and interval structures
D = 〈D, <〉: partially ordered set.
An interval in D: ordered pair [a, b], where a, b ∈ D and a ≤ b.
If a < b then [a, b] is a strict interval; [a, a] is a point interval.
I(D): the interval structure over D, consisting of the set of allintervals over D.
In this talk I will restrict attention to linear interval structures,i.e. interval structures over linear orders.
D. Della Monica
Intervals and interval structures
D = 〈D, <〉: partially ordered set.
An interval in D: ordered pair [a, b], where a, b ∈ D and a ≤ b.
If a < b then [a, b] is a strict interval; [a, a] is a point interval.
I(D): the interval structure over D, consisting of the set of allintervals over D.
In this talk I will restrict attention to linear interval structures,i.e. interval structures over linear orders.
In particular, standard interval structures on N,Z,Q, and R withtheir usual orders.
D. Della Monica
Binary interval relations on linear orders
J. F. Allen
Maintaining knowledge about temporal intervals.
Communications of the ACM, volume 26(11), pages 832-843, 1983.
D. Della Monica
Binary interval relations on linear orders
Later
J. F. Allen
Maintaining knowledge about temporal intervals.
Communications of the ACM, volume 26(11), pages 832-843, 1983.
D. Della Monica
Binary interval relations on linear orders
Later
After (right neighbour)
J. F. Allen
Maintaining knowledge about temporal intervals.
Communications of the ACM, volume 26(11), pages 832-843, 1983.
D. Della Monica
Binary interval relations on linear orders
Later
After (right neighbour)
Overlaps (to right)
J. F. Allen
Maintaining knowledge about temporal intervals.
Communications of the ACM, volume 26(11), pages 832-843, 1983.
D. Della Monica
Binary interval relations on linear orders
Later
After (right neighbour)
Overlaps (to right)
Ends
J. F. Allen
Maintaining knowledge about temporal intervals.
Communications of the ACM, volume 26(11), pages 832-843, 1983.
D. Della Monica
Binary interval relations on linear orders
Later
After (right neighbour)
Overlaps (to right)
Ends
During (subinterval)
J. F. Allen
Maintaining knowledge about temporal intervals.
Communications of the ACM, volume 26(11), pages 832-843, 1983.
D. Della Monica
Binary interval relations on linear orders
Later
After (right neighbour)
Overlaps (to right)
Ends
During (subinterval)
Begins
J. F. Allen
Maintaining knowledge about temporal intervals.
Communications of the ACM, volume 26(11), pages 832-843, 1983.
D. Della Monica
Binary interval relations on linear orders
Later
After (right neighbour)
Overlaps (to right)
Ends
During (subinterval)
Begins
6 relations + their inverses + equality = 13 Allen’s relations.
J. F. Allen
Maintaining knowledge about temporal intervals.
Communications of the ACM, volume 26(11), pages 832-843, 1983.
D. Della Monica
Outline
IntroductionThe Halpern and Shoham’s logic HS
Expressiveness of HS
The satisfiability problem for HSUndecidability
Classical extensionsMetric extensionsHybrid extensionsFirst-order extensions
Summary and perspectives
D. Della Monica
Halpern-Shoham’s modal logic of interval relationsEvery interval relation gives rise to a modal operator over relationalinterval structures.
D. Della Monica
Halpern-Shoham’s modal logic of interval relationsEvery interval relation gives rise to a modal operator over relationalinterval structures. Thus, a multimodal logic arises:
J. Halpern and Y. Shoham
A propositional modal logic of time intervals.
Journal of the ACM, volume 38(4), pages 935-962, 1991.
D. Della Monica
Halpern-Shoham’s modal logic of interval relationsEvery interval relation gives rise to a modal operator over relationalinterval structures. Thus, a multimodal logic arises:
J. Halpern and Y. Shoham
A propositional modal logic of time intervals.
Journal of the ACM, volume 38(4), pages 935-962, 1991.
non-strict semantics:All modalities are definable in terms
of 〈B〉,〈E〉,〈B〉,〈E〉
HS ≡ BEBE
D. Della Monica
Halpern-Shoham’s modal logic of interval relationsEvery interval relation gives rise to a modal operator over relationalinterval structures. Thus, a multimodal logic arises:
J. Halpern and Y. Shoham
A propositional modal logic of time intervals.
Journal of the ACM, volume 38(4), pages 935-962, 1991.
non-strict semantics:All modalities are definable in terms
of 〈B〉,〈E〉,〈B〉,〈E〉
HS ≡ BEBE
〈B〉
〈B〉〈E〉
〈E〉
D. Della Monica
Halpern-Shoham’s modal logic of interval relationsEvery interval relation gives rise to a modal operator over relationalinterval structures. Thus, a multimodal logic arises:
J. Halpern and Y. Shoham
A propositional modal logic of time intervals.
Journal of the ACM, volume 38(4), pages 935-962, 1991.
non-strict semantics:All modalities are definable in terms
of 〈B〉,〈E〉,〈B〉,〈E〉
HS ≡ BEBE
strict semantics:Also needed additional modalities
〈A〉,〈A〉
HS ≡ BEBE + AA
〈A〉
〈A〉
D. Della Monica
Halpern-Shoham’s modal logic of interval relationsEvery interval relation gives rise to a modal operator over relationalinterval structures. Thus, a multimodal logic arises:
J. Halpern and Y. Shoham
A propositional modal logic of time intervals.
Journal of the ACM, volume 38(4), pages 935-962, 1991.
non-strict semantics:All modalities are definable in terms
of 〈B〉,〈E〉,〈B〉,〈E〉
HS ≡ BEBE
strict semantics:Also needed additional modalities
〈A〉,〈A〉
HS ≡ BEBE + AA
〈A〉
〈A〉
Syntax of Halpern-Shoham’s logic, hereafter called HS :
φ ::= p | ¬φ | φ ∧ ψ | 〈B〉φ | 〈E〉φ | 〈B〉φ | 〈E〉φ(
| 〈A〉φ | 〈A〉φ)
.
D. Della Monica
Models for propositional interval logics
AP : a set of atomic propositions (over intervals).
D. Della Monica
Models for propositional interval logics
AP : a set of atomic propositions (over intervals).
Non-strict interval model:
M+ = 〈I(D)+,V 〉,
where V : AP 7→ 2I(D)+.
D. Della Monica
Models for propositional interval logics
AP : a set of atomic propositions (over intervals).
Non-strict interval model:
M+ = 〈I(D)+,V 〉,
where V : AP 7→ 2I(D)+.
Strict interval model:
M− = 〈I(D)−,V 〉,
where V : AP 7→ 2I(D)−.
D. Della Monica
Formal semantics of HS
〈B〉: M, [d0, d1] 〈B〉φ iff there exists d2 such that d0 ≤ d2 < d1 andM, [d0, d2] φ.
〈B〉: M, [d0, d1] 〈B〉φ iff there exists d2 such that d1 < d2 andM, [d0, d2] φ.
current interval:
〈B〉φ:φ
〈B〉φ:φ
D. Della Monica
Formal semantics of HS
〈B〉: M, [d0, d1] 〈B〉φ iff there exists d2 such that d0 ≤ d2 < d1 andM, [d0, d2] φ.
〈B〉: M, [d0, d1] 〈B〉φ iff there exists d2 such that d1 < d2 andM, [d0, d2] φ.
〈E〉: M, [d0, d1] 〈E〉φ iff there exists d2 such that d0 < d2 ≤ d1 andM, [d2, d1] φ.
〈E〉: M, [d0, d1] 〈E〉φ iff there exists d2 such that d2 < d0 andM, [d2, d1] φ.
current interval:
〈E〉φ:φ
〈E〉φ:φ
D. Della Monica
Formal semantics of HS
〈B〉: M, [d0, d1] 〈B〉φ iff there exists d2 such that d0 ≤ d2 < d1 andM, [d0, d2] φ.
〈B〉: M, [d0, d1] 〈B〉φ iff there exists d2 such that d1 < d2 andM, [d0, d2] φ.
〈E〉: M, [d0, d1] 〈E〉φ iff there exists d2 such that d0 < d2 ≤ d1 andM, [d2, d1] φ.
〈E〉: M, [d0, d1] 〈E〉φ iff there exists d2 such that d2 < d0 andM, [d2, d1] φ.
〈A〉: M, [d0, d1] 〈A〉φ iff there exists d2 such that d1 < d2 andM, [d1, d2] φ.
〈A〉: M, [d0, d1] 〈A〉φ iff there exists d2 such that d2 < d0 andM, [d2, d0] φ.
current interval:
〈A〉φ:φ
〈A〉φ:φ
D. Della Monica
Formal semantics of HS - contd’
〈L〉: M, [d0, d1] 〈L〉φ iff there exists d2, d3 such that d1 < d2 < d3 andM, [d2, d3] φ.
〈L〉: M, [d0, d1] 〈L〉φ iff there exists d2, d3 such that d2 < d3 < d0 andM, [d2, d3] φ.
current interval:
〈L〉φ:φ
〈L〉φ:φ
D. Della Monica
Formal semantics of HS - contd’
〈L〉: M, [d0, d1] 〈L〉φ iff there exists d2, d3 such that d1 < d2 < d3 andM, [d2, d3] φ.
〈L〉: M, [d0, d1] 〈L〉φ iff there exists d2, d3 such that d2 < d3 < d0 andM, [d2, d3] φ.
〈D〉: M, [d0, d1] 〈D〉φ iff there exists d2, d3 such that d0 < d2 < d3 < d1 andM, [d2, d3] φ.
〈D〉: M, [d0, d1] 〈D〉φ iff there exists d2, d3 such that d2 < d0 < d1 < d3 andM, [d2, d3] φ.
current interval:
〈D〉φ:φ
〈D〉φ:φ
D. Della Monica
Formal semantics of HS - contd’
〈L〉: M, [d0, d1] 〈L〉φ iff there exists d2, d3 such that d1 < d2 < d3 andM, [d2, d3] φ.
〈L〉: M, [d0, d1] 〈L〉φ iff there exists d2, d3 such that d2 < d3 < d0 andM, [d2, d3] φ.
〈D〉: M, [d0, d1] 〈D〉φ iff there exists d2, d3 such that d0 < d2 < d3 < d1 andM, [d2, d3] φ.
〈D〉: M, [d0, d1] 〈D〉φ iff there exists d2, d3 such that d2 < d0 < d1 < d3 andM, [d2, d3] φ.
〈O〉: M, [d0, d1] 〈O〉φ iff there exists d2, d3 such that d0 < d2 < d1 < d3 andM, [d2, d3] φ.
〈O〉: M, [d0, d1] 〈O〉φ iff there exists d2, d3 such that d2 < d0 < d3 < d1 andM, [d2, d3] φ.
current interval:
〈O〉φ:φ
〈O〉φ:φ
D. Della Monica
Defining the other interval modalities in HS
A useful new symbol is the modal constant π for point-intervals:
M, [d0, d1] π iff d0 = d1.
D. Della Monica
Defining the other interval modalities in HS
A useful new symbol is the modal constant π for point-intervals:
M, [d0, d1] π iff d0 = d1.
It is definable as either [B ]⊥ or [E ]⊥, so it is only needed in weakerfragments of HS.
D. Della Monica
Defining the other interval modalities in HS
A useful new symbol is the modal constant π for point-intervals:
M, [d0, d1] π iff d0 = d1.
It is definable as either [B ]⊥ or [E ]⊥, so it is only needed in weakerfragments of HS.
In general, it is possible defining HS modalities in terms of others
D. Della Monica
The zoo of fragments of HS
Technically, there are 212 = 4096 fragments of HSOf them, several hundreds are of essentially different expressiveness
D. Della Monica
The zoo of fragments of HS
Technically, there are 212 = 4096 fragments of HSOf them, several hundreds are of essentially different expressiveness
Each of these, considered with respect to some parameters:
1. over special classes of interval structures (all, dense, discrete, finite, etc.)
2. with strict or non-strict semantics
3. including or excluding π operator (whenever it cannot be defined)
D. Della Monica
Outline
IntroductionThe Halpern and Shoham’s logic HS
Expressiveness of HS
The satisfiability problem for HSUndecidability
Classical extensionsMetric extensionsHybrid extensionsFirst-order extensions
Summary and perspectives
D. Della Monica
Comparing the expressiveness of fragments of HS
Expressiveness classification problem: classify the fragments of HSwith respect to their expressiveness, relative to important classes ofinterval models.
D. Della Monica
The problem of comparing expressive power of HS fragmentsL1, L2 HS-fragments
L1 L2
D. Della Monica
The problem of comparing expressive power of HS fragmentsL1, L2 HS-fragments
L1 {≺,≡,≻, 6≈} L2
D. Della Monica
The problem of comparing expressive power of HS fragmentsL1, L2 HS-fragments
L1 {≺,≡,≻, 6≈} L2
does L1 translateinto L2?
does L2 translateinto L1?
does L2 translateinto L1?
≡ ≺ ≻ 6≈
yes no
yes no yes no
D. Della Monica
Truth-preserving translation
There exists a truth-preserving translation of L1 into L2
iffL2 is at least as expressive as L1
(L1 � L2)
D. Della Monica
Truth-preserving translation
There exists a truth-preserving translation of L1 into L2
iffL2 is at least as expressive as L1
(L1 � L2)
For each modal operator 〈X 〉 of L1 there exists a L2-formula ϕ s.t.
〈X 〉p ≡ ϕ
D. Della Monica
Truth-preserving translation
There exists a truth-preserving translation of L1 into L2
iffL2 is at least as expressive as L1
(L1 � L2)
For each modal operator 〈X 〉 of L1 there exists a L2-formula ϕ s.t.
〈X 〉p ≡ ϕ
212 fragments... 212·(212−1)2 comparisons
D. Della Monica
Inter-definability equations
Notation: X1X2 . . .Xn will denote the fragment of
HS containing the modalities 〈X1〉, 〈X2〉, . . . , 〈Xn〉
〈L〉p ≡ 〈A〉〈A〉p〈O〉p ≡ 〈E 〉〈B〉p〈D〉p ≡ 〈E 〉〈B〉p
D. Della Monica
Inter-definability equations
Notation: X1X2 . . .Xn will denote the fragment of
HS containing the modalities 〈X1〉, 〈X2〉, . . . , 〈Xn〉
〈L〉p ≡ 〈A〉〈A〉p 〈L〉 ⊑ A
〈O〉p ≡ 〈E 〉〈B〉p 〈O〉 ⊑ EB
〈D〉p ≡ 〈E 〉〈B〉p 〈D〉 ⊑ EB
D. Della Monica
Inter-definability equations
Notation: X1X2 . . .Xn will denote the fragment of
HS containing the modalities 〈X1〉, 〈X2〉, . . . , 〈Xn〉
〈L〉p ≡ 〈A〉〈A〉p 〈L〉 ⊑ A 〈L〉 ⊑ A
〈O〉p ≡ 〈E 〉〈B〉p 〈O〉 ⊑ EB 〈O〉 ⊑ EB
〈D〉p ≡ 〈E 〉〈B〉p 〈D〉 ⊑ EB 〈D〉 ⊑ EB
D. Della Monica
Inter-definability equations
Notation: X1X2 . . .Xn will denote the fragment of
HS containing the modalities 〈X1〉, 〈X2〉, . . . , 〈Xn〉
〈L〉p ≡ 〈A〉〈A〉p 〈L〉 ⊑ A 〈L〉 ⊑ A
〈O〉p ≡ 〈E 〉〈B〉p 〈O〉 ⊑ EB 〈O〉 ⊑ EB
〈D〉p ≡ 〈E 〉〈B〉p 〈D〉 ⊑ EB 〈D〉 ⊑ EB
〈L〉p ≡ 〈B〉[E ]〈B〉〈E 〉p 〈L〉 ⊑ BE 〈L〉 ⊑ BE
D. Della Monica
Inter-definability equations
Notation: X1X2 . . .Xn will denote the fragment of
HS containing the modalities 〈X1〉, 〈X2〉, . . . , 〈Xn〉
〈L〉p ≡ 〈A〉〈A〉p 〈L〉 ⊑ A 〈L〉 ⊑ A
〈O〉p ≡ 〈E 〉〈B〉p 〈O〉 ⊑ EB 〈O〉 ⊑ EB
〈D〉p ≡ 〈E 〉〈B〉p 〈D〉 ⊑ EB 〈D〉 ⊑ EB
〈L〉p ≡ 〈B〉[E ]〈B〉〈E 〉p 〈L〉 ⊑ BE 〈L〉 ⊑ BE
Soundness and completeness???
D. Della Monica
Inter-definability equations
Notation: X1X2 . . .Xn will denote the fragment of
HS containing the modalities 〈X1〉, 〈X2〉, . . . , 〈Xn〉
〈L〉p ≡ 〈A〉〈A〉p 〈L〉 ⊑ A 〈L〉 ⊑ A
〈O〉p ≡ 〈E 〉〈B〉p 〈O〉 ⊑ EB 〈O〉 ⊑ EB
〈D〉p ≡ 〈E 〉〈B〉p 〈D〉 ⊑ EB 〈D〉 ⊑ EB
〈L〉p ≡ 〈B〉[E ]〈B〉〈E 〉p 〈L〉 ⊑ BE 〈L〉 ⊑ BE
Soundness and completeness???
Soundness: all equations are valid
D. Della Monica
Inter-definability equations
Notation: X1X2 . . .Xn will denote the fragment of
HS containing the modalities 〈X1〉, 〈X2〉, . . . , 〈Xn〉
〈L〉p ≡ 〈A〉〈A〉p 〈L〉 ⊑ A 〈L〉 ⊑ A
〈O〉p ≡ 〈E 〉〈B〉p 〈O〉 ⊑ EB 〈O〉 ⊑ EB
〈D〉p ≡ 〈E 〉〈B〉p 〈D〉 ⊑ EB 〈D〉 ⊑ EB
〈L〉p ≡ 〈B〉[E ]〈B〉〈E 〉p 〈L〉 ⊑ BE 〈L〉 ⊑ BE
Soundness and completeness???
Soundness: all equations are valid SIMPLE
D. Della Monica
Inter-definability equations
Notation: X1X2 . . .Xn will denote the fragment of
HS containing the modalities 〈X1〉, 〈X2〉, . . . , 〈Xn〉
〈L〉p ≡ 〈A〉〈A〉p 〈L〉 ⊑ A 〈L〉 ⊑ A
〈O〉p ≡ 〈E 〉〈B〉p 〈O〉 ⊑ EB 〈O〉 ⊑ EB
〈D〉p ≡ 〈E 〉〈B〉p 〈D〉 ⊑ EB 〈D〉 ⊑ EB
〈L〉p ≡ 〈B〉[E ]〈B〉〈E 〉p 〈L〉 ⊑ BE 〈L〉 ⊑ BE
Soundness and completeness???
Soundness: all equations are valid SIMPLE
Completeness:there are no more
inter-definability equations
D. Della Monica
Inter-definability equations
Notation: X1X2 . . .Xn will denote the fragment of
HS containing the modalities 〈X1〉, 〈X2〉, . . . , 〈Xn〉
〈L〉p ≡ 〈A〉〈A〉p 〈L〉 ⊑ A 〈L〉 ⊑ A
〈O〉p ≡ 〈E 〉〈B〉p 〈O〉 ⊑ EB 〈O〉 ⊑ EB
〈D〉p ≡ 〈E 〉〈B〉p 〈D〉 ⊑ EB 〈D〉 ⊑ EB
〈L〉p ≡ 〈B〉[E ]〈B〉〈E 〉p 〈L〉 ⊑ BE 〈L〉 ⊑ BE
Soundness and completeness???
Soundness: all equations are valid SIMPLE
Completeness:there are no more
BISIMULATIONSinter-definability equations
D. Della Monica
Bisimulation between interval structuresZ ⊆ M1 × M2 is a bisimulations wrt the fragment X1X2 . . .Xn iff
D. Della Monica
Bisimulation between interval structuresZ ⊆ M1 × M2 is a bisimulations wrt the fragment X1X2 . . .Xn iff
1. Z -related intervals satisfy the same propositional letters, i.e.:
(i1, i2) ∈ Z ⇒ (p is true over i1 ⇔ p is true over i2)
D. Della Monica
Bisimulation between interval structuresZ ⊆ M1 × M2 is a bisimulations wrt the fragment X1X2 . . .Xn iff
1. Z -related intervals satisfy the same propositional letters, i.e.:
(i1, i2) ∈ Z ⇒ (p is true over i1 ⇔ p is true over i2)
2. the bisimulation relation is “preserved” by modal operators,i.e., for every modal operator 〈X 〉:
M1
M2
i1
i2
i ′1
D. Della Monica
Bisimulation between interval structuresZ ⊆ M1 × M2 is a bisimulations wrt the fragment X1X2 . . .Xn iff
1. Z -related intervals satisfy the same propositional letters, i.e.:
(i1, i2) ∈ Z ⇒ (p is true over i1 ⇔ p is true over i2)
2. the bisimulation relation is “preserved” by modal operators,i.e., for every modal operator 〈X 〉:
(i1, i2) ∈ Z
M1
M2
i1
i2
i ′1
Z
D. Della Monica
Bisimulation between interval structuresZ ⊆ M1 × M2 is a bisimulations wrt the fragment X1X2 . . .Xn iff
1. Z -related intervals satisfy the same propositional letters, i.e.:
(i1, i2) ∈ Z ⇒ (p is true over i1 ⇔ p is true over i2)
2. the bisimulation relation is “preserved” by modal operators,i.e., for every modal operator 〈X 〉:
(i1, i2) ∈ Z
(i1, i′1) ∈ X
M1
M2
i1
i2
i ′1
Z
X
D. Della Monica
Bisimulation between interval structuresZ ⊆ M1 × M2 is a bisimulations wrt the fragment X1X2 . . .Xn iff
1. Z -related intervals satisfy the same propositional letters, i.e.:
(i1, i2) ∈ Z ⇒ (p is true over i1 ⇔ p is true over i2)
2. the bisimulation relation is “preserved” by modal operators,i.e., for every modal operator 〈X 〉:
(i1, i2) ∈ Z
(i1, i′1) ∈ X
}
⇒ ∃i ′2 s.t.
M1
M2
i1
i2
i ′1
i ′2
Z
X
D. Della Monica
Bisimulation between interval structuresZ ⊆ M1 × M2 is a bisimulations wrt the fragment X1X2 . . .Xn iff
1. Z -related intervals satisfy the same propositional letters, i.e.:
(i1, i2) ∈ Z ⇒ (p is true over i1 ⇔ p is true over i2)
2. the bisimulation relation is “preserved” by modal operators,i.e., for every modal operator 〈X 〉:
(i1, i2) ∈ Z
(i1, i′1) ∈ X
}
⇒ ∃i ′2 s.t.
{
(i ′1, i′2) ∈ Z
(i2, i′2) ∈ X
M1
M2
i1
i2
i ′1
i ′2
Z
X
Z
X
D. Della Monica
Bisimulation between interval structures - cont’d
Theorem Let Z be a bisimulation between M1 and M2 for thelanguage L and let i1 and i2 be intervals in M1 and M2,respectively. Then, truth of L-formulae is preserved by Z , i.e.,
If (i1, i2) ∈ Z , then for every formula ϕ of L:
M1, i1 ϕ iff M2, i2 ϕ
D. Della Monica
How to use bisimulations to disprove definability
Suppose that we want to prove:
〈X 〉 is not definable in terms of L
D. Della Monica
How to use bisimulations to disprove definability
Suppose that we want to prove:
〈X 〉 is not definable in terms of L
We must provide:
1. two models M1 and M2
D. Della Monica
How to use bisimulations to disprove definability
Suppose that we want to prove:
〈X 〉 is not definable in terms of L
We must provide:
1. two models M1 and M2
2. a bisimulation Z ⊆ M1 × M2 wrt fragment L
D. Della Monica
How to use bisimulations to disprove definability
Suppose that we want to prove:
〈X 〉 is not definable in terms of L
We must provide:
1. two models M1 and M2
2. a bisimulation Z ⊆ M1 × M2 wrt fragment L
3. two interval i1 ∈ M1 and i2 ∈ M2 such that
a. i1 and i2 are Z -relatedb. M1, i1 〈X 〉p and M2, i2 ¬〈X 〉p
D. Della Monica
How to use bisimulations to disprove definability
Suppose that we want to prove:
〈X 〉 is not definable in terms of L
We must provide:
1. two models M1 and M2
2. a bisimulation Z ⊆ M1 × M2 wrt fragment L
3. two interval i1 ∈ M1 and i2 ∈ M2 such that
a. i1 and i2 are Z -relatedb. M1, i1 〈X 〉p and M2, i2 ¬〈X 〉p
By contradiction
If 〈X 〉 is definable in terms of L, then 〈X 〉p is.
D. Della Monica
How to use bisimulations to disprove definability
Suppose that we want to prove:
〈X 〉 is not definable in terms of L
We must provide:
1. two models M1 and M2
2. a bisimulation Z ⊆ M1 × M2 wrt fragment L
3. two interval i1 ∈ M1 and i2 ∈ M2 such that
a. i1 and i2 are Z -relatedb. M1, i1 〈X 〉p and M2, i2 ¬〈X 〉p
By contradiction
If 〈X 〉 is definable in terms of L, then 〈X 〉p is.Truth of 〈X 〉p should have been preserved by Z , but 〈X 〉p is true ini1 (in M1) and false in i2 (in M2)
D. Della Monica
How to use bisimulations to disprove definability
Suppose that we want to prove:
〈X 〉 is not definable in terms of L
We must provide:
1. two models M1 and M2
2. a bisimulation Z ⊆ M1 × M2 wrt fragment L
3. two interval i1 ∈ M1 and i2 ∈ M2 such that
a. i1 and i2 are Z -relatedb. M1, i1 〈X 〉p and M2, i2 ¬〈X 〉p
By contradiction
If 〈X 〉 is definable in terms of L, then 〈X 〉p is.Truth of 〈X 〉p should have been preserved by Z , but 〈X 〉p is true ini1 (in M1) and false in i2 (in M2) ⇒ contradiction
D. Della Monica
An example: the operator 〈D〉
Semantics:
M, [a, b] 〈D〉ϕdef⇔ ∃c , d such that a < c < d < b and M, [c , d ] ϕ
〈D〉ϕ
ϕ
D. Della Monica
An example: the operator 〈D〉
Semantics:
M, [a, b] 〈D〉ϕdef⇔ ∃c , d such that a < c < d < b and M, [c , d ] ϕ
〈D〉ϕ
ϕ
Operator 〈D〉 is definable in terms of BE 〈D〉ϕ ≡ 〈B〉〈E 〉ϕ
D. Della Monica
An example: the operator 〈D〉
Semantics:
M, [a, b] 〈D〉ϕdef⇔ ∃c , d such that a < c < d < b and M, [c , d ] ϕ
〈D〉ϕ
ϕ
Operator 〈D〉 is definable in terms of BE 〈D〉ϕ ≡ 〈B〉〈E 〉ϕ
To prove that 〈D〉 is not definable in terms of any other fragment,we must prove that:
1) 〈D〉 is not definable in terms of ALBOALBEDO
2) 〈D〉 is not definable in terms of ALEOALBEDO
D. Della Monica
〈D〉 is not definable in terms of AA bisimulation wrt fragment A but not D
Bisimulation wrt A (AP = {p}):◮ models: M1 = 〈I(N),V1〉,M2 = 〈I(N),V2〉
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
D. Della Monica
〈D〉 is not definable in terms of AA bisimulation wrt fragment A but not D
Bisimulation wrt A (AP = {p}):◮ models: M1 = 〈I(N),V1〉,M2 = 〈I(N),V2〉
◮ V1(p) = {[1, 2]}
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
p
D. Della Monica
〈D〉 is not definable in terms of AA bisimulation wrt fragment A but not D
Bisimulation wrt A (AP = {p}):◮ models: M1 = 〈I(N),V1〉,M2 = 〈I(N),V2〉
◮ V1(p) = {[1, 2]}◮ V2(p) = ∅
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
p
¬p
D. Della Monica
〈D〉 is not definable in terms of AA bisimulation wrt fragment A but not D
Bisimulation wrt A (AP = {p}):◮ models: M1 = 〈I(N),V1〉,M2 = 〈I(N),V2〉
◮ V1(p) = {[1, 2]}◮ V2(p) = ∅
◮ bisimulation relation Z : ([x , y ], [w , z]) ∈ Z iff
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
p
¬p
D. Della Monica
〈D〉 is not definable in terms of AA bisimulation wrt fragment A but not D
Bisimulation wrt A (AP = {p}):◮ models: M1 = 〈I(N),V1〉,M2 = 〈I(N),V2〉
◮ V1(p) = {[1, 2]}◮ V2(p) = ∅
◮ bisimulation relation Z : ([x , y ], [w , z]) ∈ Z iff
1. [x , y ] = [w , z] = [0, 3]
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
p
¬p
D. Della Monica
〈D〉 is not definable in terms of AA bisimulation wrt fragment A but not D
Bisimulation wrt A (AP = {p}):◮ models: M1 = 〈I(N),V1〉,M2 = 〈I(N),V2〉
◮ V1(p) = {[1, 2]}◮ V2(p) = ∅
◮ bisimulation relation Z : ([x , y ], [w , z]) ∈ Z iff
1. [x , y ] = [w , z] = [0, 3]2. [x , y ] = [w , z] and x ≥ 3
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
p
¬p
D. Della Monica
〈D〉 is not definable in terms of AA bisimulation wrt fragment A but not D
Bisimulation wrt A (AP = {p}):◮ models: M1 = 〈I(N),V1〉,M2 = 〈I(N),V2〉
◮ V1(p) = {[1, 2]}◮ V2(p) = ∅
◮ bisimulation relation Z : ([x , y ], [w , z]) ∈ Z iff
1. [x , y ] = [w , z] = [0, 3]2. [x , y ] = [w , z] and x ≥ 3
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
p
¬p
M1, [0, 3] 〈D〉p and M2, [0, 3] ¬〈D〉p
D. Della Monica
〈D〉 is not definable in terms of AA bisimulation wrt fragment A but not D
Bisimulation wrt A (AP = {p}):◮ models: M1 = 〈I(N),V1〉,M2 = 〈I(N),V2〉
◮ V1(p) = {[1, 2]}◮ V2(p) = ∅
◮ bisimulation relation Z : ([x , y ], [w , z]) ∈ Z iff
1. [x , y ] = [w , z] = [0, 3]2. [x , y ] = [w , z] and x ≥ 3
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
p
¬p
M1, [0, 3] 〈D〉p and M2, [0, 3] ¬〈D〉p ⇒ the thesis
D. Della Monica
Outline
IntroductionThe Halpern and Shoham’s logic HS
Expressiveness of HS
The satisfiability problem for HSUndecidability
Classical extensionsMetric extensionsHybrid extensionsFirst-order extensions
Summary and perspectives
D. Della Monica
The satisfiability problem for HS
Satisfiability problem for a logic L Given an L-formula ϕ, is ϕsatisfiable, i.e., there exists a model and an interval in which ϕ istrue?
D. Della Monica
The satisfiability problem for HS
Satisfiability problem for a logic L Given an L-formula ϕ, is ϕsatisfiable, i.e., there exists a model and an interval in which ϕ istrue?
L is decidable (wrt the satisfiability problem)iff
for each formula it is possible to answer the question
D. Della Monica
The satisfiability problem for HS
Satisfiability problem for a logic L Given an L-formula ϕ, is ϕsatisfiable, i.e., there exists a model and an interval in which ϕ istrue?
L is decidable (wrt the satisfiability problem)iff
for each formula it is possible to answer the questionthere exists a terminating algorithm that answer yes / not for any ϕ
D. Della Monica
The satisfiability problem for HS
Satisfiability problem for a logic L Given an L-formula ϕ, is ϕsatisfiable, i.e., there exists a model and an interval in which ϕ istrue?
L is decidable (wrt the satisfiability problem)iff
for each formula it is possible to answer the questionthere exists a terminating algorithm that answer yes / not for any ϕ
Expressive enough, yet decidable, HS fragments
D. Della Monica
The satisfiability problem for HS
Satisfiability problem for a logic L Given an L-formula ϕ, is ϕsatisfiable, i.e., there exists a model and an interval in which ϕ istrue?
L is decidable (wrt the satisfiability problem)iff
for each formula it is possible to answer the questionthere exists a terminating algorithm that answer yes / not for any ϕ
Expressive enough, yet decidable, HS fragments
Classification of all HS fragments wrt (un)decidability
D. Della Monica
Maximal decidable HS fragments
◮ PNL (≡ AA) in general case
D. Bresolin, V. Goranko, A. Montanari, G. Sciavicco
Propositional Interval Neighborhood Logic: Decidability,Expressiveness, and Undecidable Extensions.
Annals of Pure and Applied Logics, 2009, 161, 289-304.
D. Della Monica
Maximal decidable HS fragments
◮ PNL (≡ AA) in general case
◮ ABBL (and AEEL) in general case
D. Bresolin, A. Montanari, P. Sala, G. Sciavicco
What’s decidable about Halpern and Shoham’s interval logic?The maximal fragment ABBL.
LICS 2011, 2011.
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Maximal decidable HS fragments
◮ PNL (≡ AA) in general case
◮ ABBL (and AEEL) in general case
◮ ABBA (and AEEA) over finite structures
A. Montanari, G. Puppis, P. Sala
Maximal Decidable Fragments of Halpern and Shoham’sModal Logic of Intervals.
ICALP 2010, 2010, LNCS 6199, 2010, 345-356.
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Maximal decidable HS fragments
◮ PNL (≡ AA) in general case
◮ ABBL (and AEEL) in general case
◮ ABBA (and AEEA) over finite structures
◮ DDBBLL over Q
P. Sala
PhD thesis
2010
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Weakest undecidable HS fragments
AD,AD,AD,AD
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Weakest undecidable HS fragments
AD,AD,AD,AD
BE,BE,BE,BE
D. Della Monica
Weakest undecidable HS fragments
AD,AD,AD,AD
BE,BE,BE,BE
O (and O)
D. Della Monica
Weakest undecidable HS fragments
AD,AD,AD,AD
BE,BE,BE,BE
O (and O)
D (and D) over discrete
D. Della Monica
Weakest undecidable HS fragments
AD,AD,AD,AD [in this thesis]
BE,BE,BE,BE [in this thesis]
O (and O) [in this thesis]
D (and D) over discrete [Michaliszyn, Marcinkowski]
The Ultimate Undecidability Result
for the Halpern-Shoham Logic
LICS 2011
D. Della Monica
Outline
IntroductionThe Halpern and Shoham’s logic HS
Expressiveness of HS
The satisfiability problem for HSUndecidability
Classical extensionsMetric extensionsHybrid extensionsFirst-order extensions
Summary and perspectives
D. Della Monica
The Octant Tiling ProblemThis is the problem of establishing whether a given finite set of tiletypes T = {t1, . . . , tk} can tile the 2nd octant of the integer plane:
O = {(i , j) : i , j ∈ N ∧ 0 ≤ i ≤ j},
while respecting the color constraints.
D. Della Monica
The Octant Tiling ProblemThis is the problem of establishing whether a given finite set of tiletypes T = {t1, . . . , tk} can tile the 2nd octant of the integer plane:
O = {(i , j) : i , j ∈ N ∧ 0 ≤ i ≤ j},
while respecting the color constraints.
D. Della Monica
The Octant Tiling ProblemThis is the problem of establishing whether a given finite set of tiletypes T = {t1, . . . , tk} can tile the 2nd octant of the integer plane:
O = {(i , j) : i , j ∈ N ∧ 0 ≤ i ≤ j},
while respecting the color constraints.
Proposition The Octant Tiling Problem is undecidable.
D. Della Monica
The Octant Tiling ProblemThis is the problem of establishing whether a given finite set of tiletypes T = {t1, . . . , tk} can tile the 2nd octant of the integer plane:
O = {(i , j) : i , j ∈ N ∧ 0 ≤ i ≤ j},
while respecting the color constraints.
Proposition The Octant Tiling Problem is undecidable.
Proof: by reduction from the tiling problem for N× N, usingKönig’s Lemma.
D. Della Monica
Undecidability of the interval logics via tiling:generic construction
1. Encoding of the octant
2. Encoding of the neighborhood relations◮ Right-neighborhood relation SIMPLE◮ Above-neighborhood relation HARD
D. Della Monica
Undecidability of the interval logics via tiling:generic construction
1. Encoding of the octant
2. Encoding of the neighborhood relations◮ Right-neighborhood relation SIMPLE◮ Above-neighborhood relation HARD
Encoding of the octant
◮ Force the existence of a unique infinite chain of unit-intervalson the linear order, which covers an initial segment of theinterval model. (propositional letter u)
Unit intervals are used to place tiles and delimiting symbols.
D. Della Monica
Undecidability of the interval logics via tiling:generic construction
1. Encoding of the octant
2. Encoding of the neighborhood relations◮ Right-neighborhood relation SIMPLE◮ Above-neighborhood relation HARD
Encoding of the octant
◮ Force the existence of a unique infinite chain of unit-intervalson the linear order, which covers an initial segment of theinterval model. (propositional letter u)
Unit intervals are used to place tiles and delimiting symbols.
◮ ID-intervals are then introduced to represent the layers of tiles.(propositional letter Id)
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Undecidability of the interval logics via tiling:generic construction cont’d
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Undecidability of the interval logics via tiling:generic construction cont’d
Each ID-interval must have the right number of tiles
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Undecidability of the interval logics via tiling:generic construction cont’d
Each ID-interval must have the right number of tiles
The most challenging part usually is to ensure that the consecutiveID-intervals match vertically: the Above-Neighbour relation.
D. Della Monica
Undecidability of the interval logics via tiling:generic construction cont’d
Each ID-interval must have the right number of tiles
The most challenging part usually is to ensure that the consecutiveID-intervals match vertically: the Above-Neighbour relation.
For that, auxiliary propositional letter up_rel can be used toconnecting (endpoints of) two intervals representing tiles that areabove connected in the octant
D. Della Monica
Undecidability of the interval logics via tiling:generic construction completed
Eventually, we encode the given Octant tiling problem by specifyingthe matching conditions between intervals that are right-connectedor above-connected.
D. Della Monica
Undecidability of the interval logics via tiling:generic construction completed
Eventually, we encode the given Octant tiling problem by specifyingthe matching conditions between intervals that are right-connectedor above-connected.
The specific part of the construction is to use the given fragment ofHS to set the chain of unit intervals and to express all necessaryproperties of IDs, the propositional letters for correspondenceintervals, and the tile matching conditions.
D. Della Monica
Summary of (un)decidability results and outlook
◮ In summary: interval logics are generally undecidable, evenunder very weak assumptions.
D. Della Monica
Summary of (un)decidability results and outlook
◮ In summary: interval logics are generally undecidable, evenunder very weak assumptions.
◮ In particular, most fragments of HS (∼ 90%) have beenproved undecidable over most of the natural classes of intervalstructures.
D. Della Monica
Summary of (un)decidability results and outlook
◮ In summary: interval logics are generally undecidable, evenunder very weak assumptions.
◮ In particular, most fragments of HS (∼ 90%) have beenproved undecidable over most of the natural classes of intervalstructures.
◮ There are still some currently unknown cases (< 8%), andsome are conjectured undecidable (over general and specialclasses), e.g., L∗D∗, where X ∗ ∈ {X ,X}; etc.
D. Della Monica
Summary of (un)decidability results and outlook
◮ In summary: interval logics are generally undecidable, evenunder very weak assumptions.
◮ In particular, most fragments of HS (∼ 90%) have beenproved undecidable over most of the natural classes of intervalstructures.
◮ There are still some currently unknown cases (< 8%), andsome are conjectured undecidable (over general and specialclasses), e.g., L∗D∗, where X ∗ ∈ {X ,X}; etc.
◮ Not all results transfer readily between the strict and thenon-strict semantics, and between the classes of all, dense,discrete, etc. interval structures.
D. Della Monica
Summary of (un)decidability results and outlook
◮ In summary: interval logics are generally undecidable, evenunder very weak assumptions.
◮ In particular, most fragments of HS (∼ 90%) have beenproved undecidable over most of the natural classes of intervalstructures.
◮ There are still some currently unknown cases (< 8%), andsome are conjectured undecidable (over general and specialclasses), e.g., L∗D∗, where X ∗ ∈ {X ,X}; etc.
◮ Not all results transfer readily between the strict and thenon-strict semantics, and between the classes of all, dense,discrete, etc. interval structures.
◮ More statistics are available on the web page:https://itl.dimi.uniud.it/content/logic-hs
D. Della Monica
Outline
IntroductionThe Halpern and Shoham’s logic HS
Expressiveness of HS
The satisfiability problem for HSUndecidability
Classical extensionsMetric extensionsHybrid extensionsFirst-order extensions
Summary and perspectives
D. Della Monica
Outline
IntroductionThe Halpern and Shoham’s logic HS
Expressiveness of HS
The satisfiability problem for HSUndecidability
Classical extensionsMetric extensionsHybrid extensionsFirst-order extensions
Summary and perspectives
D. Della Monica
PNL: syntax and semantics
Syntax
◮ PNL: ϕ ::= p | ¬ϕ | ϕ ∨ ϕ | 〈A〉ϕ | 〈A〉ϕ
D. Della Monica
PNL: syntax and semantics
Syntax
◮ PNL: ϕ ::= p | ¬ϕ | ϕ ∨ ϕ | 〈A〉ϕ | 〈A〉ϕ
Semantics
◮ Operators meets (〈A〉) and met-by (〈A〉):
meets:〈A〉ϕ
D. Della Monica
PNL: syntax and semantics
Syntax
◮ PNL: ϕ ::= p | ¬ϕ | ϕ ∨ ϕ | 〈A〉ϕ | 〈A〉ϕ
Semantics
◮ Operators meets (〈A〉) and met-by (〈A〉):
meets:〈A〉ϕ ϕ
D. Della Monica
PNL: syntax and semantics
Syntax
◮ PNL: ϕ ::= p | ¬ϕ | ϕ ∨ ϕ | 〈A〉ϕ | 〈A〉ϕ
Semantics
◮ Operators meets (〈A〉) and met-by (〈A〉):
meets:〈A〉ϕ ϕ
met-by:〈A〉ϕ
D. Della Monica
PNL: syntax and semantics
Syntax
◮ PNL: ϕ ::= p | ¬ϕ | ϕ ∨ ϕ | 〈A〉ϕ | 〈A〉ϕ
Semantics
◮ Operators meets (〈A〉) and met-by (〈A〉):
meets:〈A〉ϕ ϕ
met-by:〈A〉ϕϕ
D. Della Monica
Metric interval logics
Two types of metric extensions of interval logics over the integers:
D. Della Monica
Metric interval logics
Two types of metric extensions of interval logics over the integers:
1. Extensions of the modal operators: 〈A〉=k , 〈A〉>k , 〈A〉[k,k′], . . .
D. Della Monica
Metric interval logics
Two types of metric extensions of interval logics over the integers:
1. Extensions of the modal operators: 〈A〉=k , 〈A〉>k , 〈A〉[k,k′], . . .
2. Atomic propositions for length constraints: len>k, len=k, . . .
D. Della Monica
Metric interval logics
Two types of metric extensions of interval logics over the integers:
1. Extensions of the modal operators: 〈A〉=k , 〈A〉>k , 〈A〉[k,k′], . . .
2. Atomic propositions for length constraints: len>k, len=k, . . .
The former are definable in terms of the latter in PNL, e.g.:
〈A〉>kp := 〈A〉(p ∧ len>k).
D. Della Monica
Metric interval logics
Two types of metric extensions of interval logics over the integers:
1. Extensions of the modal operators: 〈A〉=k , 〈A〉>k , 〈A〉[k,k′], . . .
2. Atomic propositions for length constraints: len>k, len=k, . . .
The former are definable in terms of the latter in PNL, e.g.:
〈A〉>kp := 〈A〉(p ∧ len>k).
MPNL: PNL extended with integer constraints for interval lengths.
D. Della Monica
Decidability of metric interval logic
Theorem Satisfiability in MPNL on N is decidable. It isNEXPTIME-complete if the metric constraints are represented inunary, and in between EXPSPACE and 2NEXPTIME if they arerepresented in binary.
D. Bresolin, D. Della Monica, V. Goranko, A. Montanari, G. Sciavicco
Metric Propositional Neighborhood Interval Logics: expressiveness,decidability, and undecidability, Proc. of the European Conference onArtificial Intelligence (ECAI), 2010.
D. Della Monica
Decidability of metric interval logic
Theorem Satisfiability in MPNL on N is decidable. It isNEXPTIME-complete if the metric constraints are represented inunary, and in between EXPSPACE and 2NEXPTIME if they arerepresented in binary.
D. Bresolin, D. Della Monica, V. Goranko, A. Montanari, G. Sciavicco
Metric Propositional Neighborhood Interval Logics: expressiveness,decidability, and undecidability, Proc. of the European Conference onArtificial Intelligence (ECAI), 2010.
Exact complexity is an open problem
D. Della Monica
Relative expressive power of logics in MPNL
MPNL<
MPNL=
MPNL>
MPNL()
MPNL=l MPNL
()
l
D. Della Monica
Relative expressive power of logics in MPNL
MPNLε ≡ PNL
MPNL<,ε MPNL>,ε
MPNL=,ε MPNL<,> MPNL(),ε
MPNL[]
MPNL
MPNL<
MPNL=
MPNL>
MPNL()
MPNL=l MPNL
()
l
D. Della Monica
Relative expressive power of logics in MPNL
MPNLε ≡ PNL
MPNL<,ε MPNL>,ε
MPNL=,ε MPNL<,> MPNL(),ε
MPNL[]
MPNL
MPNL<
MPNL=
MPNL>
MPNL()
MPNL=l MPNL
()
l
in 2NEXPTIME
EXPSPACE -hard
NEXPTIME -complete
D. Della Monica
Relative expressive power of logics in MPNL
MPNLε ≡ PNL
MPNL<,ε MPNL>,ε
MPNL=,ε MPNL<,> MPNL(),ε
MPNL[]
MPNL
MPNL<
MPNL=
MPNL>
MPNL()
MPNL=l MPNL
()
l
in 2NEXPTIME
EXPSPACE -hard
NEXPTIME -complete
Decidability of MPNL: by small model propertyComparing expressiveness of metric fragments: by bisimulations
D. Della Monica
Outline
IntroductionThe Halpern and Shoham’s logic HS
Expressiveness of HS
The satisfiability problem for HSUndecidability
Classical extensionsMetric extensionsHybrid extensionsFirst-order extensions
Summary and perspectives
D. Della Monica
Extending PNL
PNL
D. Della Monica
Extending PNL
PNL
NEXPTIME-co
D. Della Monica
Extending PNL
PNL
PNL +any HS operator
NEXPTIME-co
D. Della Monica
Extending PNL
PNL
PNL +any HS operator
NEXPTIME-co
Undecidable
D. Della Monica
Extending PNL
PNL
PNL +any HS operator
MPNL
NEXPTIME-co
Undecidable
D. Della Monica
Extending PNL
PNL
PNL +any HS operator
MPNL
NEXPTIME-co
Undecidable
between EXPSPACEand 2NEXPTIME
D. Della Monica
Extending PNL
PNL
PNL +any HS operator
MPNL MPNL+
NEXPTIME-co
Undecidable
between EXPSPACEand 2NEXPTIME
D. Della Monica
Extending PNL
PNL
PNL +any HS operator
MPNL MPNL+
NEXPTIME-co
Undecidable
between EXPSPACEand 2NEXPTIME
Undecidable
D. Della Monica
Extending PNL
PNL
PNL +any HS operator
MPNL MPNL+
Hybrid Extensions
NEXPTIME-co
Undecidable
between EXPSPACEand 2NEXPTIME
Undecidable
D. Della Monica
Extending PNL
PNL
PNL +any HS operator
MPNL MPNL+
Hybrid Extensions
NEXPTIME-co
Undecidable
between EXPSPACEand 2NEXPTIME
Undecidable
???
D. Della Monica
Possible hybrid extension of PNL and MPNL
Nominals are definable in PNL(Basic Hybrid PNL)
D. Della Monica
Possible hybrid extension of PNL and MPNL
Binders over state variables (intervals)(Strongly Hybrid MPNL)
lead to undecidability
Nominals are definable in PNL(Basic Hybrid PNL)
D. Della Monica
Possible hybrid extension of PNL and MPNL
Binders over state variables (intervals)(Strongly Hybrid MPNL)
lead to undecidability
Nominals are definable in PNL(Basic Hybrid PNL)
D. Della Monica
Possible hybrid extension of PNL and MPNL
Binders over state variables (intervals)(Strongly Hybrid MPNL)
lead to undecidability
Binders over length of intervals(Weakly Hybrid MPNL)
Nominals are definable in PNL(Basic Hybrid PNL)
D. Della Monica
Weakly Hybrid MPNL (WHMPNL)
Metric constraints of MPNL use constants
len=5, len>2, . . .
WHMPNL allows one to store the length of the currentinterval and to refer to it in sub-formulae
↓x (. . . |=|x), ↓x (. . . |≤|x), . . .
D. Della Monica
WHMPNL fragments
Remark
◮ Constant metric constraints are inter-definable
◮ Hybrid metric constraints ARE NOT!!!(e.g.: you cannot define len≤x in terms of len=x)
D. Della Monica
WHMPNL fragments
Remark
◮ Constant metric constraints are inter-definable
◮ Hybrid metric constraints ARE NOT!!!(e.g.: you cannot define len≤x in terms of len=x)
Possible choices:
1. which subset of hybrid constraints among {<,≤,=,≥, >}
D. Della Monica
WHMPNL fragments
Remark
◮ Constant metric constraints are inter-definable
◮ Hybrid metric constraints ARE NOT!!!(e.g.: you cannot define len≤x in terms of len=x)
Possible choices:
1. which subset of hybrid constraints among {<,≤,=,≥, >}
2. constant metric constraints are allowed or not (WHPNL orWHMPNL)
D. Della Monica
WHMPNL fragments
Remark
◮ Constant metric constraints are inter-definable
◮ Hybrid metric constraints ARE NOT!!!(e.g.: you cannot define len≤x in terms of len=x)
Possible choices:
1. which subset of hybrid constraints among {<,≤,=,≥, >}
2. constant metric constraints are allowed or not (WHPNL orWHMPNL)
3. how many length variables
D. Della Monica
WHMPNL fragments
Remark
◮ Constant metric constraints are inter-definable
◮ Hybrid metric constraints ARE NOT!!!(e.g.: you cannot define len≤x in terms of len=x)
Possible choices:
1. which subset of hybrid constraints among {<,≤,=,≥, >}
2. constant metric constraints are allowed or not (WHPNL orWHMPNL)
3. how many length variables
set of hybrid constant # of lengthconstraints constraints variables
WHMPNL(<,≤,=,≥, >) {<,≤,=,≥, >} YES unboundedWHPNL(<,=) {<,=} NO unboundedWHPNL(<)1 {<} NO 1
D. Della Monica
WHMPNL fragments
Remark
◮ Constant metric constraints are inter-definable
◮ Hybrid metric constraints ARE NOT!!!(e.g.: you cannot define len≤x in terms of len=x)
Possible choices:
1. which subset of hybrid constraints among {<,≤,=,≥, >}
2. constant metric constraints are allowed or not (WHPNL orWHMPNL)
3. how many length variables
set of hybrid constant # of lengthconstraints constraints variables
WHMPNL(<,≤,=,≥, >) {<,≤,=,≥, >} YES unboundedWHPNL(<,=) {<,=} NO unboundedWHPNL(<)1 {<} NO 1
D. Della Monica
The fragment WHPNL(=)1
Reduction from the Finite Tiling ProblemThis is the problem of establishing whether, for a given finite set of tiletypes T = {t1, . . . , tk}, there exists a finite rectangle R having theborder colored with a fixed color such that T can tile R respecting thecolor constraints.
D. Della Monica
The fragment WHPNL(=)1
Reduction from the Finite Tiling ProblemThis is the problem of establishing whether, for a given finite set of tiletypes T = {t1, . . . , tk}, there exists a finite rectangle R having theborder colored with a fixed color such that T can tile R respecting thecolor constraints.
D. Della Monica
Outline
IntroductionThe Halpern and Shoham’s logic HS
Expressiveness of HS
The satisfiability problem for HSUndecidability
Classical extensionsMetric extensionsHybrid extensionsFirst-order extensions
Summary and perspectives
D. Della Monica
Extending PNL
PNL
PNL +any HS operator
MetricPNL
MPNL+
Hybrid Extensions
NEXPTIME-co
Undecidable
between EXPSPACEand 2NEXPTIME
Undecidable
D. Della Monica
Extending PNL
PNL
PNL +any HS operator
MetricPNL
MPNL+
Hybrid Extensions
NEXPTIME-co
Undecidable
between EXPSPACEand 2NEXPTIME
Undecidable
Undecidable
D. Della Monica
Extending PNL
PNL
PNL +any HS operator
MetricPNL
MPNL+
Hybrid Extensions
First-Order ExtensionsNEXPTIME-co
Undecidable
between EXPSPACEand 2NEXPTIME
Undecidable
Undecidable
D. Della Monica
Extending PNL
PNL
PNL +any HS operator
MetricPNL
MPNL+
Hybrid Extensions
First-Order ExtensionsNEXPTIME-co
Undecidable
between EXPSPACEand 2NEXPTIME
Undecidable
Undecidable
???
D. Della Monica
First-Order together with Propositional
F O R P N L
First-Order Right Propositional Neighborhood Logic
D. Della Monica
First-Order together with Propositional
F O R P N L
First-Order Right Propositional Neighborhood Logic
D. Della Monica
First-Order together with Propositional
F O R P N L
First-Order Right Propositional Neighborhood Logic
D. Della Monica
First-Order together with Propositional
F O R P N L
First-Order Right Propositional Neighborhood Logic
D. Della Monica
First-Order together with Propositional
F O R P N L
First-Order Right Propositional Neighborhood Logic
1. Propositional (modal) setting
w1
w2
w4
w3
D. Della Monica
First-Order together with Propositional
F O R P N L
First-Order Right Propositional Neighborhood Logic
1. Propositional (modal) setting
2. First-Order setting◮ predicates over elements◮ existential and universal
quantificationsw1
w2
w4
w3
D. Della Monica
First-Order together with Propositional
F O R P N L
First-Order Right Propositional Neighborhood Logic
1. Propositional (modal) setting
2. First-Order setting◮ predicates over elements◮ existential and universal
quantificationsw1
w4
w3w2
D. Della Monica
First-Order together with Propositional
F O R P N L
First-Order Right Propositional Neighborhood Logic
1. Propositional (modal) setting
2. First-Order setting◮ predicates over elements◮ existential and universal
quantificationsw1
w2
w4
w3
D. Della Monica
First-Order together with Propositional
F O R P N L
First-Order Right Propositional Neighborhood Logic
1. Propositional (modal) setting
2. First-Order setting◮ predicates over elements◮ existential and universal
quantifications
3. Propositional (modal) +First-Order setting
w1
w2
w3d1
d2
d3
w4
Propositional (modal)domain
First-Orderdomain
D. Della Monica
Parameters of the logic
◮ Temporal domain: discrete, dense, finite, bounded,unbounded, . . .
D. Della Monica
Parameters of the logic
◮ Temporal domain: discrete, dense, finite, bounded,unbounded, . . .
◮ First-order domain: finite, infinite, expanding, . . .
D. Della Monica
Parameters of the logic
◮ Temporal domain: discrete, dense, finite, bounded,unbounded, . . .
◮ First-order domain: finite, infinite, expanding, . . .
◮ First-order constructs:◮ predicates P(. . .),Q(. . .), . . .◮ individual variables x , y , . . .◮ individual constants a, b, . . .◮ function f (. . .), g(. . .), . . .◮ quantifiers◮ terms t1, t2, . . . (variables, constants, and functions)
D. Della Monica
Parameters of the logic
◮ Temporal domain: discrete, dense, finite, bounded,unbounded, . . .
◮ First-order domain: finite, infinite, expanding, . . .
◮ First-order constructs:◮ predicates P(. . .),Q(. . .), . . .◮ individual variables x , y , . . .◮ individual constants a, b, . . .◮ function f (. . .), g(. . .), . . .◮ quantifiers◮ terms t1, t2, . . . (variables, constants, and functions)
D. Della Monica
Parameters of the logic
◮ Temporal domain: discrete, dense, finite, bounded,unbounded, . . .
◮ First-order domain: finite, infinite, expanding, . . .
◮ First-order constructs:◮ predicates P(. . .),Q(. . .), . . .◮ individual variables x , y , . . .◮ individual constants a, b, . . .◮ function f (. . .), g(. . .), . . .◮ quantifiers◮ terms t1, t2, . . . (variables, constants, and functions)
D. Della Monica
Parameters of the logic
◮ Temporal domain: discrete, dense, finite, bounded,unbounded, . . .
◮ First-order domain: finite, infinite, expanding, . . .
◮ First-order constructs:◮ predicates P(. . .),Q(. . .), . . .◮ individual variables x , y , . . .◮ individual constants a, b, . . .◮ function f (. . .), g(. . .), . . .◮ quantifiers◮ terms t1, t2, . . . (variables, constants, and functions)
terms = variables
D. Della Monica
Parameters of the logic
◮ Temporal domain: discrete, dense, finite, bounded,unbounded, . . .
◮ First-order domain: finite, infinite, expanding, . . .
◮ First-order constructs:◮ predicates P(. . .),Q(. . .), . . .◮ individual variables x , y , . . .◮ individual constants a, b, . . .◮ function f (. . .), g(. . .), . . .◮ quantifiers◮ terms t1, t2, . . . (variables, constants, and functions)
terms = variables
for tight undecidability only 1 variable (no free variables)
D. Della Monica
Undecidability of FORPNL
Reduction from the Finite Tiling ProblemThis is the problem of establishing whether, for a given finite set of tiletypes T = {t1, . . . , tk}, there exists a finite rectangle R having theborder colored with a fixed color such that T can tile R respecting thecolor constraints.
D. Della Monica
Undecidability of FORPNL
Reduction from the Finite Tiling ProblemThis is the problem of establishing whether, for a given finite set of tiletypes T = {t1, . . . , tk}, there exists a finite rectangle R having theborder colored with a fixed color such that T can tile R respecting thecolor constraints.
D. Della Monica
Undecidability of FORPNL
Reduction from the Finite Tiling ProblemThis is the problem of establishing whether, for a given finite set of tiletypes T = {t1, . . . , tk}, there exists a finite rectangle R having theborder colored with a fixed color such that T can tile R respecting thecolor constraints.
It is possible to simulate HS operators 〈B〉 〈E〉 〈D〉
D. Della Monica
Extending PNL: the final picture
PNL
PNL +any HS operator
MetricPNL
MPNL+
Hybrid Extensions
First-Order ExtensionsNEXPTIME-co
Undecidable
between EXPSPACEand 2NEXPTIME
Undecidable
Undecidable
Undecidable
D. Della Monica
Outline
IntroductionThe Halpern and Shoham’s logic HS
Expressiveness of HS
The satisfiability problem for HSUndecidability
Classical extensionsMetric extensionsHybrid extensionsFirst-order extensions
Summary and perspectives
D. Della Monica
Summary and perspectives
This talk outlined several major topics in the area of interval logics:
D. Della Monica
Summary and perspectives
This talk outlined several major topics in the area of interval logics:
◮ Expressiveness of HS fragments
D. Della Monica
Summary and perspectives
This talk outlined several major topics in the area of interval logics:
◮ Expressiveness of HS fragments
◮ Undecidability of HS fragments
D. Della Monica
Summary and perspectives
This talk outlined several major topics in the area of interval logics:
◮ Expressiveness of HS fragments
◮ Undecidability of HS fragments
◮ Classical extension of PNL
D. Della Monica
Summary and perspectives
This talk outlined several major topics in the area of interval logics:
◮ Expressiveness of HS fragments
◮ Undecidability of HS fragments
◮ Classical extension of PNL
The main research agenda so far: to complete the classifications ofexpressiveness and (un)decidability of fragments of HS.
D. Della Monica
Summary and perspectives
This talk outlined several major topics in the area of interval logics:
◮ Expressiveness of HS fragments
◮ Undecidability of HS fragments
◮ Classical extension of PNL
The main research agenda so far: to complete the classifications ofexpressiveness and (un)decidability of fragments of HS.
Not discussed, and not yet explored, but important:
D. Della Monica
Summary and perspectives
This talk outlined several major topics in the area of interval logics:
◮ Expressiveness of HS fragments
◮ Undecidability of HS fragments
◮ Classical extension of PNL
The main research agenda so far: to complete the classifications ofexpressiveness and (un)decidability of fragments of HS.
Not discussed, and not yet explored, but important:
◮ Model checking of Interval logics
D. Della Monica
Summary and perspectives
This talk outlined several major topics in the area of interval logics:
◮ Expressiveness of HS fragments
◮ Undecidability of HS fragments
◮ Classical extension of PNL
The main research agenda so far: to complete the classifications ofexpressiveness and (un)decidability of fragments of HS.
Not discussed, and not yet explored, but important:
◮ Model checking of Interval logics
◮ Automata-based techniques for interval logics
D. Della Monica
Exams and attended courses
◮ Exams◮ International Lipari Summer School 2008 on “Algorithms:
Science and Engineering” 14 - 25 July 2008, Lipari; 1.5 Credits◮ “Constraint Programming and NMR Constraints for
Determining Protein Structure”, A. Dovier◮ GAMES Spring School 2009◮ “Systems Biology”, A. Policriti/M. Miculan◮ “Computational Complexity (Complessità computazionale)”, R.
Rizzi◮ “Introduction to Software Configuration Management”, L.
Bendix
◮ Other courses◮ “(Meta-)Modeling with UML and OCL”, M. Gogolla◮ “Data Mining and Mathematical Programming”, P. Serafini◮ “Sistemi Reattivi: automi, logica, algoritmi” (Master Course),
A. Montanari◮ English course for academic purposes (CLAV)
D. Della Monica
Other activities
◮ Summer school◮ International Lipari Summer School 2008 on “Algorithms:
Science and Engineering”◮ GAMES Spring School 2009 (Bertinoro)
◮ Visiting◮ Oct - Dec 2009: University of Murcia - Murcia, Spain (G.
Sciavicco)◮ Sept - Nov 2010: Technical University of Denmark (DTU) -
Lyngby, Copenhagen, Denmark (V. Goranko)
◮ Events organization◮ Annual Workshop of the ESF Networking Programme on
Games for Design and Verification (GAMES 2009)◮ First International Symposium on Games, Automata, Logics
and Formal Verification (GandALF 2010)◮ Second International Symposium on Games, Automata, Logics
and Formal Verification (GandALF 2011)
D. Della Monica
Publications
1. D. Bresolin, D. Della Monica, V. Goranko, A. Montanari, and G.
Sciavicco. “Decidable and Undecidable Fragments of Halpernand Shohams Interval Temporal Logic: Towards a CompleteClassification”. In Proc. of 15th International Conference on Logic
for Programming, Artificial Intelligence, and Reasoning (LPAR
2008), 2008.
2. D. Bresolin, D. Della Monica, V. Goranko, A. Montanari, and G.
Sciavicco. “Undecidability of Interval Temporal Logics with theOverlap Modality”. In Proc. of 16th International Symposium on
Temporal Representation and Reasoning (TIME 2009), 2009.
3. D. Bresolin, D. Della Monica, V. Goranko, A. Montanari, and G.
Sciavicco. “Undecidability of the Logic of Overlap Relation overDiscrete Linear Orderings”. Electronic Notes in Theoretical
Computer Science (Proc. of the 6th Workshop on Methods for
Modalities (M4M-6 2009), 2010.
D. Della Monica
Publications - contd’
4. D. Della Monica, V. Goranko, and G. Sciavicco. “Hybrid MetricPropositional Neighborhood Logics with Interval LengthBinders”. In Proc. of International Workshop on Hybrid Logic and
Applications (HyLo 2010), 2010. To appear on ENTCS.
5. D. Della Monica and G. Sciavicco. “On First-Order PropositionalNeighborhood Logics: a First Attempt”. In Proc. of ECAI 2010
Workshop on Spatio-Temporal Dynamics (STeDY 2010), 2010.
6. D. Bresolin, D. Della Monica, V. Goranko, A. Montanari, and G.
Sciavicco. “Metric Propositional Neighborhood Logics:Expressiveness, Decidability, and Undecidability”. In Proc. of
19th European Conference on Artificial Intelligence (ECAI 2010),
2010.
7. D. Bresolin, D. Della Monica, A. Montanari, P. Sala, and G.
Sciavicco. “A decidable spatial generalization of Metric IntervalTemporal Logic”. In Proc. of 17th International Symposium on
Temporal Representation and Reasoning (TIME 2010), 2010.
D. Della Monica
Publications - contd’
8. D. Bresolin, D. Della Monica, V. Goranko, A. Montanari, G.
Sciavicco. “Metric propositional neighborhood logics on naturalnumbers”. Journal of Software & Systems Modeling (doi:
10.1007/s10270-011-0195-y, online since February 2011).
9. D. Della Monica, V. Goranko, A. Montanari, G. Sciavicco.
“Expressiveness of the Interval Logics of Allen’s Relations onthe Class of all Linear Orders: Complete Classification”.accepted to IJCAI 2011.
D. Della Monica
Publications - contd’
8. D. Bresolin, D. Della Monica, V. Goranko, A. Montanari, G.
Sciavicco. “Metric propositional neighborhood logics on naturalnumbers”. Journal of Software & Systems Modeling (doi:
10.1007/s10270-011-0195-y, online since February 2011).
9. D. Della Monica, V. Goranko, A. Montanari, G. Sciavicco.
“Expressiveness of the Interval Logics of Allen’s Relations onthe Class of all Linear Orders: Complete Classification”.accepted to IJCAI 2011.
The end.