Expressiveness, decidability, and undecidability of...

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

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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

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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

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Outline

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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.

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Motivations

Some properties are intrinsically related to a time interval instead ofa (set of) time instant, each of which has not duration.

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Motivations

Think about the event: “traveling from A to B”:

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Motivations

◮ it is true over a precise interval of time

◮ it is not true over any other interval (starting/ending interval,inner interval, ecc.)

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Motivations

Several philosophical and logical paradoxes disappear:

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Motivations

◮ 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?”)

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Motivations

◮ 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.

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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:

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◮ linear or branching?

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◮ discrete or dense?

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◮ with or without beginning/end?, etc.

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New issues arise regarding the nature of the intervals:

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◮ Can intervals be unbounded?

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◮ Can intervals be unbounded?

◮ Are intervals with coinciding endpoints admissible or not?

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Intervals and interval structures

D = 〈D, <〉: partially ordered set.

An interval in D: ordered pair [a, b], where a, b ∈ D and a ≤ b.

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If a < b then [a, b] is a strict interval; [a, a] is a point interval.

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I(D): the interval structure over D, consisting of the set of allintervals over D.

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In this talk I will restrict attention to linear interval structures,i.e. interval structures over linear orders.

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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.

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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.

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J. F. Allen

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After (right neighbour)

J. F. Allen

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Overlaps (to right)

J. F. Allen

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Overlaps (to right)

J. F. Allen

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Overlaps (to right)

During (subinterval)

J. F. Allen

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Overlaps (to right)

Begins

J. F. Allen

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Overlaps (to right)

Begins

6 relations + their inverses + equality = 13 Allen’s relations.

J. F. Allen

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Outline

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Halpern-Shoham’s modal logic of interval relationsEvery interval relation gives rise to a modal operator over relationalinterval structures.

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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.

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non-strict semantics:All modalities are definable in terms

of 〈B〉,〈E〉,〈B〉,〈E〉

HS ≡ BEBE

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of 〈B〉,〈E〉,〈B〉,〈E〉

HS ≡ BEBE

〈B〉

〈B〉〈E〉

〈E〉

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of 〈B〉,〈E〉,〈B〉,〈E〉

HS ≡ BEBE

strict semantics:Also needed additional modalities

〈A〉,〈A〉

HS ≡ BEBE + AA

〈A〉

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of 〈B〉,〈E〉,〈B〉,〈E〉

HS ≡ BEBE

strict semantics:Also needed additional modalities

〈A〉,〈A〉

HS ≡ BEBE + AA

〈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)

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Models for propositional interval logics

AP : a set of atomic propositions (over intervals).

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Non-strict interval model:

M+ = 〈I(D)+,V 〉,

where V : AP 7→ 2I(D)+.

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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)−.

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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

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〈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

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〈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

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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.

current interval:

〈L〉φ:φ

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〈D〉: M, [d0, d1] 〈D〉φ iff there exists d2, d3 such that d0 < d2 < d3 < d1 andM, [d2, d3] φ.

current interval:

〈D〉φ:φ

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〈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

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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.

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M, [d0, d1] π iff d0 = d1.

It is definable as either [B ]⊥ or [E ]⊥, so it is only needed in weakerfragments of HS.

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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

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The zoo of fragments of HS

Technically, there are 212 = 4096 fragments of HSOf them, several hundreds are of essentially different expressiveness

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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)

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Outline

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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.

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The problem of comparing expressive power of HS fragmentsL1, L2 HS-fragments

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L1 {≺,≡,≻, 6≈} L2

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L1 {≺,≡,≻, 6≈} L2

does L1 translateinto L2?

does L2 translateinto L1?

≡ ≺ ≻ 6≈

yes no

yes no yes no

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Truth-preserving translation

There exists a truth-preserving translation of L1 into L2

iffL2 is at least as expressive as L1

(L1 � L2)

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(L1 � L2)

For each modal operator 〈X 〉 of L1 there exists a L2-formula ϕ s.t.

〈X 〉p ≡ ϕ

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(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

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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

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〈L〉p ≡ 〈A〉〈A〉p 〈L〉 ⊑ A

〈O〉p ≡ 〈E 〉〈B〉p 〈O〉 ⊑ EB

〈D〉p ≡ 〈E 〉〈B〉p 〈D〉 ⊑ EB

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〈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

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〈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

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〈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???

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〈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: all equations are valid

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〈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: all equations are valid SIMPLE

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〈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

Completeness:there are no more

inter-definability equations

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〈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

Completeness:there are no more

BISIMULATIONSinter-definability equations

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Bisimulation between interval structuresZ ⊆ M1 × M2 is a bisimulations wrt the fragment X1X2 . . .Xn iff

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1. Z -related intervals satisfy the same propositional letters, i.e.:

(i1, i2) ∈ Z ⇒ (p is true over i1 ⇔ p is true over i2)

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2. the bisimulation relation is “preserved” by modal operators,i.e., for every modal operator 〈X 〉:

i ′1

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(i1, i2) ∈ Z

i ′1

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(i1, i2) ∈ Z

(i1, i′1) ∈ X

i ′1

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(i1, i2) ∈ Z

(i1, i′1) ∈ X

⇒ ∃i ′2 s.t.

i ′1

i ′2

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(i1, i2) ∈ Z

(i1, i′1) ∈ X

⇒ ∃i ′2 s.t.

(i ′1, i′2) ∈ Z

(i2, i′2) ∈ X

i ′1

i ′2

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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

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How to use bisimulations to disprove definability

Suppose that we want to prove:

〈X 〉 is not definable in terms of L

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We must provide:

1. two models M1 and M2

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We must provide:

2. a bisimulation Z ⊆ M1 × M2 wrt fragment L

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We must provide:

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

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We must provide:

By contradiction

If 〈X 〉 is definable in terms of L, then 〈X 〉p is.

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We must provide:

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)

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We must provide:

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

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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

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Semantics:

〈D〉ϕ

Operator 〈D〉 is definable in terms of BE 〈D〉ϕ ≡ 〈B〉〈E 〉ϕ

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Semantics:

〈D〉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

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〈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

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◮ V1(p) = {[1, 2]}

0 1 2 3 4 5 6 7 8 9

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◮ V1(p) = {[1, 2]}◮ V2(p) = ∅

0 1 2 3 4 5 6 7 8 9

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◮ 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

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◮ V1(p) = {[1, 2]}◮ V2(p) = ∅

1. [x , y ] = [w , z] = [0, 3]

0 1 2 3 4 5 6 7 8 9

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◮ V1(p) = {[1, 2]}◮ V2(p) = ∅

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

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◮ V1(p) = {[1, 2]}◮ V2(p) = ∅

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

M1, [0, 3] 〈D〉p and M2, [0, 3] ¬〈D〉p

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◮ V1(p) = {[1, 2]}◮ V2(p) = ∅

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

M1, [0, 3] 〈D〉p and M2, [0, 3] ¬〈D〉p ⇒ the thesis

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Outline

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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?

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L is decidable (wrt the satisfiability problem)iff

for each formula it is possible to answer the question

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for each formula it is possible to answer the questionthere exists a terminating algorithm that answer yes / not for any ϕ

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Expressive enough, yet decidable, HS fragments

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Expressive enough, yet decidable, HS fragments

Classification of all HS fragments wrt (un)decidability

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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.

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◮ 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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◮ 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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◮ ABBA (and AEEA) over finite structures

◮ DDBBLL over Q

P. Sala

PhD thesis

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Weakest undecidable HS fragments

AD,AD,AD,AD

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AD,AD,AD,AD

BE,BE,BE,BE

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AD,AD,AD,AD

BE,BE,BE,BE

O (and O)

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AD,AD,AD,AD

BE,BE,BE,BE

O (and O)

D (and D) over discrete

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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

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Outline

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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.

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O = {(i , j) : i , j ∈ N ∧ 0 ≤ i ≤ j},

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O = {(i , j) : i , j ∈ N ∧ 0 ≤ i ≤ j},

Proposition The Octant Tiling Problem is undecidable.

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O = {(i , j) : i , j ∈ N ∧ 0 ≤ i ≤ j},

Proposition The Octant Tiling Problem is undecidable.

Proof: by reduction from the tiling problem for N× N, usingKönig’s Lemma.

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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

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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.

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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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Each ID-interval must have the right number of tiles

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The most challenging part usually is to ensure that the consecutiveID-intervals match vertically: the Above-Neighbour relation.

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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

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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.

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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.

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Summary of (un)decidability results and outlook

◮ In summary: interval logics are generally undecidable, evenunder very weak assumptions.

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◮ In particular, most fragments of HS (∼ 90%) have beenproved undecidable over most of the natural classes of intervalstructures.

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◮ 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.

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◮ Not all results transfer readily between the strict and thenon-strict semantics, and between the classes of all, dense,discrete, etc. interval structures.

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◮ 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

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Outline

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Outline

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PNL: syntax and semantics

Syntax

◮ PNL: ϕ ::= p | ¬ϕ | ϕ ∨ ϕ | 〈A〉ϕ | 〈A〉ϕ

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Syntax

◮ PNL: ϕ ::= p | ¬ϕ | ϕ ∨ ϕ | 〈A〉ϕ | 〈A〉ϕ

Semantics

◮ Operators meets (〈A〉) and met-by (〈A〉):

meets:〈A〉ϕ

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Syntax

◮ PNL: ϕ ::= p | ¬ϕ | ϕ ∨ ϕ | 〈A〉ϕ | 〈A〉ϕ

Semantics

meets:〈A〉ϕ ϕ

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Syntax

◮ PNL: ϕ ::= p | ¬ϕ | ϕ ∨ ϕ | 〈A〉ϕ | 〈A〉ϕ

Semantics

meets:〈A〉ϕ ϕ

met-by:〈A〉ϕ

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Syntax

◮ PNL: ϕ ::= p | ¬ϕ | ϕ ∨ ϕ | 〈A〉ϕ | 〈A〉ϕ

Semantics

meets:〈A〉ϕ ϕ

met-by:〈A〉ϕϕ

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Metric interval logics

Two types of metric extensions of interval logics over the integers:

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1. Extensions of the modal operators: 〈A〉=k , 〈A〉>k , 〈A〉[k,k′], . . .

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2. Atomic propositions for length constraints: len>k, len=k, . . .

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The former are definable in terms of the latter in PNL, e.g.:

〈A〉>kp := 〈A〉(p ∧ len>k).

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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.

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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.

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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

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Relative expressive power of logics in MPNL

MPNL()

MPNL=l MPNL

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MPNLε ≡ PNL

MPNL<,ε MPNL>,ε

MPNL=,ε MPNL<,> MPNL(),ε

MPNL[]

MPNL()

MPNL=l MPNL

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MPNLε ≡ PNL

MPNL<,ε MPNL>,ε

MPNL[]

MPNL()

MPNL=l MPNL

in 2NEXPTIME

EXPSPACE -hard

NEXPTIME -complete

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MPNLε ≡ PNL

MPNL<,ε MPNL>,ε

MPNL[]

MPNL()

MPNL=l MPNL

in 2NEXPTIME

EXPSPACE -hard

NEXPTIME -complete

Decidability of MPNL: by small model propertyComparing expressiveness of metric fragments: by bisimulations

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Outline

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Extending PNL

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Extending PNL

NEXPTIME-co

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Extending PNL

PNL +any HS operator

NEXPTIME-co

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Extending PNL

NEXPTIME-co

Undecidable

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Extending PNL

NEXPTIME-co

Undecidable

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Extending PNL

NEXPTIME-co

Undecidable

between EXPSPACEand 2NEXPTIME

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Extending PNL

MPNL MPNL+

NEXPTIME-co

Undecidable

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Extending PNL

MPNL MPNL+

NEXPTIME-co

Undecidable

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Extending PNL

MPNL MPNL+

Hybrid Extensions

NEXPTIME-co

Undecidable

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Extending PNL

MPNL MPNL+

Hybrid Extensions

NEXPTIME-co

Undecidable

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Possible hybrid extension of PNL and MPNL

Nominals are definable in PNL(Basic Hybrid PNL)

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Binders over state variables (intervals)(Strongly Hybrid MPNL)

lead to undecidability

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Binders over length of intervals(Weakly Hybrid MPNL)

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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), . . .

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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)

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WHMPNL fragments

Remark

Possible choices:

1. which subset of hybrid constraints among {<,≤,=,≥, >}

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WHMPNL fragments

Remark

Possible choices:

2. constant metric constraints are allowed or not (WHPNL orWHMPNL)

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WHMPNL fragments

Remark

Possible choices:

3. how many length variables

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WHMPNL fragments

Remark

Possible choices:

set of hybrid constant # of lengthconstraints constraints variables

WHMPNL(<,≤,=,≥, >) {<,≤,=,≥, >} YES unboundedWHPNL(<,=) {<,=} NO unboundedWHPNL(<)1 {<} NO 1

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WHMPNL fragments

Remark

Possible choices:

set of hybrid constant # of lengthconstraints constraints variables

WHMPNL(<,≤,=,≥, >) {<,≤,=,≥, >} YES unboundedWHPNL(<,=) {<,=} NO unboundedWHPNL(<)1 {<} NO 1

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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.

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The fragment WHPNL(=)1

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Outline

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Extending PNL

MetricPNL

Hybrid Extensions

NEXPTIME-co

Undecidable

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Extending PNL

MetricPNL

Hybrid Extensions

NEXPTIME-co

Undecidable

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Extending PNL

MetricPNL

Hybrid Extensions

First-Order ExtensionsNEXPTIME-co

Undecidable

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Extending PNL

MetricPNL

Hybrid Extensions

Undecidable

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First-Order together with Propositional

F O R P N L

First-Order Right Propositional Neighborhood Logic

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F O R P N L

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F O R P N L

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F O R P N L

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F O R P N L

1. Propositional (modal) setting

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F O R P N L

2. First-Order setting◮ predicates over elements◮ existential and universal

quantificationsw1

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F O R P N L

quantificationsw1

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F O R P N L

quantificationsw1

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F O R P N L

quantifications

3. Propositional (modal) +First-Order setting

Propositional (modal)domain

First-Orderdomain

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Parameters of the logic

◮ Temporal domain: discrete, dense, finite, bounded,unbounded, . . .

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◮ First-order domain: finite, infinite, expanding, . . .

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◮ 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)

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terms = variables

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terms = variables

for tight undecidability only 1 variable (no free variables)

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Undecidability of FORPNL

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It is possible to simulate HS operators 〈B〉〈E〉〈D〉

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Extending PNL: the final picture

MetricPNL

Hybrid Extensions

Undecidable

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Outline

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This talk outlined several major topics in the area of interval logics:

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◮ Expressiveness of HS fragments

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◮ Undecidability of HS fragments

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◮ Classical extension of PNL

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The main research agenda so far: to complete the classifications ofexpressiveness and (un)decidability of fragments of HS.

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Not discussed, and not yet explored, but important:

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◮ Model checking of Interval logics

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◮ Model checking of Interval logics

◮ Automata-based techniques for interval logics

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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)

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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.

Sciavicco. “Undecidability of Interval Temporal Logics with theOverlap Modality”. In Proc. of 16th International Symposium on

Temporal Representation and Reasoning (TIME 2009), 2009.

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.

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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.

Sciavicco. “Metric Propositional Neighborhood Logics:Expressiveness, Decidability, and Undecidability”. In Proc. of

19th European Conference on Artificial Intelligence (ECAI 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.

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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.

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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.

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