Expressiveness, decidability, and undecidability of Interval Temporal Logic

Transcription

1 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. Montanari co-supervisors: G. Sciavicco and V. Goranko Udine - April 1, 2011

2 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

3 Outline Introduction The Halpern and Shoham s logic HS Expressiveness of HS The satisfiability problem for HS Undecidability Classical extensions Metric extensions Hybrid extensions First-order extensions Summary and perspectives

4 Outline Introduction The Halpern and Shoham s logic HS Expressiveness of HS The satisfiability problem for HS Undecidability Classical extensions Metric extensions Hybrid extensions First-order extensions Summary and perspectives

5 Interval-based temporal reasoning: origins and applications Interval-based temporal reasoning: reasoning about time, where the primary concept is time interval, rather than time instant. Origins: Philosophy, in particular philosophy and ontology of time. Linguistics: analysis of progressive tenses, semantics of natural languages. Artificial intelligence: temporal knowledge representation, temporal planning, theory of events, etc. Computer science: specification and design of hardware components, concurrent real-time processes, temporal databases, etc.

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

7 Motivations Some properties are intrinsically related to a time interval instead of a (set of) time instant, each of which has not duration. Think about the event: traveling from A to B :

8 Motivations Some properties are intrinsically related to a time interval instead of a (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.)

9 Motivations Some properties are intrinsically related to a time interval instead of a (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:

10 Motivations Some properties are intrinsically related to a time interval instead of a (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 flying arrow stands still, how is movement possible? ) The dividing instant dilemma ( if the light is on and it is turned off, what is its state at the instant between the two events? )

11 Motivations Some properties are intrinsically related to a time interval instead of a (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 flying arrow stands still, how is movement possible? ) The dividing instant dilemma ( if the light is on and it is turned off, what is its state at the instant between the two events? ) The truth of a formula over an interval does not necessarily depend on its truth over subintervals.

12 Interval temporal reasoning and temporal ontologies Interval-based temporal reasoning is subject to the same ontological dilemmas regarding the nature of Time as the instant-based temporal reasoning:

13 Interval temporal reasoning and temporal ontologies Interval-based temporal reasoning is subject to the same ontological dilemmas regarding the nature of Time as the instant-based temporal reasoning: linear or branching?

14 Interval temporal reasoning and temporal ontologies Interval-based temporal reasoning is subject to the same ontological dilemmas regarding the nature of Time as the instant-based temporal reasoning: linear or branching? discrete or dense?

15 Interval temporal reasoning and temporal ontologies Interval-based temporal reasoning is subject to the same ontological dilemmas regarding the nature of Time as the instant-based temporal reasoning: linear or branching? discrete or dense? with or without beginning/end?, etc.

16 Interval temporal reasoning and temporal ontologies Interval-based temporal reasoning is subject to the same ontological dilemmas regarding the nature of Time as the instant-based temporal reasoning: linear or branching? discrete or dense? with or without beginning/end?, etc. New issues arise regarding the nature of the intervals:

17 Interval temporal reasoning and temporal ontologies Interval-based temporal reasoning is subject to the same ontological dilemmas regarding the nature of Time as the instant-based temporal 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?

18 Interval temporal reasoning and temporal ontologies Interval-based temporal reasoning is subject to the same ontological dilemmas regarding the nature of Time as the instant-based temporal 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?

19 Intervals and interval structures D = D, < : partially ordered set. An interval in D: ordered pair [a, b], where a, b D and a b.

20 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.

21 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 all intervals over D.

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

23 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 all intervals 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 with their usual orders.

24 Binary interval relations on linear orders J. F. Allen Maintaining knowledge about temporal intervals. Communications of the ACM, volume 26(11), pages , 1983.

25 Binary interval relations on linear orders Later J. F. Allen Maintaining knowledge about temporal intervals. Communications of the ACM, volume 26(11), pages , 1983.

26 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 , 1983.

27 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 , 1983.

28 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 , 1983.

29 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 , 1983.

30 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 , 1983.

31 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 , 1983.

32 Outline Introduction The Halpern and Shoham s logic HS Expressiveness of HS The satisfiability problem for HS Undecidability Classical extensions Metric extensions Hybrid extensions First-order extensions Summary and perspectives

33 Halpern-Shoham s modal logic of interval relations Every interval relation gives rise to a modal operator over relational interval structures.

34 Halpern-Shoham s modal logic of interval relations Every interval relation gives rise to a modal operator over relational interval 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 , 1991.

35 Halpern-Shoham s modal logic of interval relations Every interval relation gives rise to a modal operator over relational interval 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 , non-strict semantics: All modalities are definable in terms of B, E, B, E HS BEBE

36 Halpern-Shoham s modal logic of interval relations Every interval relation gives rise to a modal operator over relational interval 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 , non-strict semantics: All modalities are definable in terms of B, E, B, E HS BEBE B B E E

37 Halpern-Shoham s modal logic of interval relations Every interval relation gives rise to a modal operator over relational interval 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 , non-strict semantics: strict semantics: All modalities are definable in terms Also needed additional modalities of B, E, B, E HS BEBE A, A HS BEBE+AA A A

38 Halpern-Shoham s modal logic of interval relations Every interval relation gives rise to a modal operator over relational interval 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 , non-strict semantics: strict semantics: All modalities are definable in terms Also needed additional modalities of B, E, B, E HS BEBE 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 ).

39 Models for propositional interval logics AP: a set of atomic propositions (over intervals).

40 Models for propositional interval logics AP: a set of atomic propositions (over intervals). Non-strict interval model: M + = I(D) +,V, where V : AP 2 I(D)+.

41 Models for propositional interval logics AP: a set of atomic propositions (over intervals). Non-strict interval model: M + = I(D) +,V, where V : AP 2 I(D)+. Strict interval model: M = I(D),V, where V : AP 2 I(D).

42 Formal semantics of HS B : M,[d 0,d 1] B φ iff there exists d 2 such that d 0 d 2 < d 1 and M,[d 0,d 2] φ. B : M,[d 0,d 1] B φ iff there exists d 2 such that d 1 < d 2 and M,[d 0,d 2] φ. current interval: B φ: B φ: φ φ

43 Formal semantics of HS B : M,[d 0,d 1] B φ iff there exists d 2 such that d 0 d 2 < d 1 and M,[d 0,d 2] φ. B : M,[d 0,d 1] B φ iff there exists d 2 such that d 1 < d 2 and M,[d 0,d 2] φ. E : M,[d 0,d 1] E φ iff there exists d 2 such that d 0 < d 2 d 1 and M,[d 2,d 1] φ. E : M,[d 0,d 1] E φ iff there exists d 2 such that d 2 < d 0 and M,[d 2,d 1] φ. current interval: E φ: E φ: φ φ

44 Formal semantics of HS B : M,[d 0,d 1] B φ iff there exists d 2 such that d 0 d 2 < d 1 and M,[d 0,d 2] φ. B : M,[d 0,d 1] B φ iff there exists d 2 such that d 1 < d 2 and M,[d 0,d 2] φ. E : M,[d 0,d 1] E φ iff there exists d 2 such that d 0 < d 2 d 1 and M,[d 2,d 1] φ. E : M,[d 0,d 1] E φ iff there exists d 2 such that d 2 < d 0 and M,[d 2,d 1] φ. A : M,[d 0,d 1] A φ iff there exists d 2 such that d 1 < d 2 and M,[d 1,d 2] φ. A : M,[d 0,d 1] A φ iff there exists d 2 such that d 2 < d 0 and M,[d 2,d 0] φ. current interval: A φ: A φ: φ φ

45 Formal semantics of HS - contd L : M,[d 0,d 1] L φ iff there exists d 2,d 3 such that d 1 < d 2 < d 3 and M,[d 2,d 3] φ. L : M,[d 0,d 1] L φ iff there exists d 2,d 3 such that d 2 < d 3 < d 0 and M,[d 2,d 3] φ. current interval: L φ: L φ: φ φ

46 Formal semantics of HS - contd L : M,[d 0,d 1] L φ iff there exists d 2,d 3 such that d 1 < d 2 < d 3 and M,[d 2,d 3] φ. L : M,[d 0,d 1] L φ iff there exists d 2,d 3 such that d 2 < d 3 < d 0 and M,[d 2,d 3] φ. D : M,[d 0,d 1] D φ iff there exists d 2,d 3 such that d 0 < d 2 < d 3 < d 1 and M,[d 2,d 3] φ. D : M,[d 0,d 1] D φ iff there exists d 2,d 3 such that d 2 < d 0 < d 1 < d 3 and M,[d 2,d 3] φ. current interval: D φ: D φ: φ φ

47 Formal semantics of HS - contd L : M,[d 0,d 1] L φ iff there exists d 2,d 3 such that d 1 < d 2 < d 3 and M,[d 2,d 3] φ. L : M,[d 0,d 1] L φ iff there exists d 2,d 3 such that d 2 < d 3 < d 0 and M,[d 2,d 3] φ. D : M,[d 0,d 1] D φ iff there exists d 2,d 3 such that d 0 < d 2 < d 3 < d 1 and M,[d 2,d 3] φ. D : M,[d 0,d 1] D φ iff there exists d 2,d 3 such that d 2 < d 0 < d 1 < d 3 and M,[d 2,d 3] φ. O : M,[d 0,d 1] O φ iff there exists d 2,d 3 such that d 0 < d 2 < d 1 < d 3 and M,[d 2,d 3] φ. O : M,[d 0,d 1] O φ iff there exists d 2,d 3 such that d 2 < d 0 < d 3 < d 1 and M,[d 2,d 3] φ. current interval: O φ: O φ: φ φ

48 Defining the other interval modalities in HS A useful new symbol is the modal constant π for point-intervals: M,[d 0,d 1 ] π iff d 0 = d 1.

49 Defining the other interval modalities in HS A useful new symbol is the modal constant π for point-intervals: M,[d 0,d 1 ] π iff d 0 = d 1. It is definable as either [B] or [E], so it is only needed in weaker fragments of HS.

50 Defining the other interval modalities in HS A useful new symbol is the modal constant π for point-intervals: M,[d 0,d 1 ] π iff d 0 = d 1. It is definable as either [B] or [E], so it is only needed in weaker fragments of HS. In general, it is possible defining HS modalities in terms of others

51 The zoo of fragments of HS Technically, there are 2 12 = 4096 fragments of HS Of them, several hundreds are of essentially different expressiveness

52 The zoo of fragments of HS Technically, there are 2 12 = 4096 fragments of HS Of 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)

53 Outline Introduction The Halpern and Shoham s logic HS Expressiveness of HS The satisfiability problem for HS Undecidability Classical extensions Metric extensions Hybrid extensions First-order extensions Summary and perspectives

54 Comparing the expressiveness of fragments of HS Expressiveness classification problem: classify the fragments of HS with respect to their expressiveness, relative to important classes of interval models.

55 The problem of comparing expressive power of HS fragments L 1,L 2 HS-fragments L 1 L 2

56 The problem of comparing expressive power of HS fragments L 1,L 2 HS-fragments L 1 {,,, } L 2

57 The problem of comparing expressive power of HS fragments L 1,L 2 HS-fragments L 1 {,,, } L 2 does L 1 translate into L 2? yes does L 2 translate into L 1? no does L 2 translate into L 1? yes no yes no

58 Truth-preserving translation There exists a truth-preserving translation of L 1 into L 2 iff L 2 is at least as expressive as L 1 (L 1 L 2 )

59 Truth-preserving translation There exists a truth-preserving translation of L 1 into L 2 iff L 2 is at least as expressive as L 1 (L 1 L 2 ) For each modal operator X of L 1 there exists a L 2 -formula ϕ s.t. X p ϕ

60 Truth-preserving translation There exists a truth-preserving translation of L 1 into L 2 iff L 2 is at least as expressive as L 1 (L 1 L 2 ) For each modal operator X of L 1 there exists a L 2 -formula ϕ s.t. X p ϕ 2 12 fragments (2 12 1) 2 comparisons

61 Inter-definability equations Notation: X 1 X 2...X n will denote the fragment of HS containing the modalities X 1, X 2,..., X n L p A A p O p E B p D p E B p

62 Inter-definability equations Notation: X 1 X 2...X n will denote the fragment of HS containing the modalities X 1, X 2,..., X n L p A A p O p E B p D p E B p L A O EB D EB

63 Inter-definability equations Notation: X 1 X 2...X n will denote the fragment of HS containing the modalities X 1, X 2,..., X n 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

64 Inter-definability equations Notation: X 1 X 2...X n will denote the fragment of HS containing the modalities X 1, X 2,..., X n 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

65 Inter-definability equations Notation: X 1 X 2...X n will denote the fragment of HS containing the modalities X 1, X 2,..., X n 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???

66 Inter-definability equations Notation: X 1 X 2...X n will denote the fragment of HS containing the modalities X 1, X 2,..., X n 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

67 Inter-definability equations Notation: X 1 X 2...X n will denote the fragment of HS containing the modalities X 1, X 2,..., X n 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

68 Inter-definability equations Notation: X 1 X 2...X n will denote the fragment of HS containing the modalities X 1, X 2,..., X n 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

69 Inter-definability equations Notation: X 1 X 2...X n will denote the fragment of HS containing the modalities X 1, X 2,..., X n 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 BISIMULATIONS

70 Bisimulation between interval structures Z M 1 M 2 is a bisimulations wrt the fragment X 1 X 2...X n iff

71 Bisimulation between interval structures Z M 1 M 2 is a bisimulations wrt the fragment X 1 X 2...X n iff 1. Z-related intervals satisfy the same propositional letters, i.e.: (i 1,i 2 ) Z (p is true over i 1 p is true over i 2 )

72 Bisimulation between interval structures Z M 1 M 2 is a bisimulations wrt the fragment X 1 X 2...X n iff 1. Z-related intervals satisfy the same propositional letters, i.e.: (i 1,i 2 ) Z (p is true over i 1 p is true over i 2 ) 2. the bisimulation relation is preserved by modal operators, i.e., for every modal operator X : M 1 i 1 i 1 M 2 i 2

73 Bisimulation between interval structures Z M 1 M 2 is a bisimulations wrt the fragment X 1 X 2...X n iff 1. Z-related intervals satisfy the same propositional letters, i.e.: (i 1,i 2 ) Z (p is true over i 1 p is true over i 2 ) 2. the bisimulation relation is preserved by modal operators, i.e., for every modal operator X : (i 1,i 2 ) Z M 1 i 1 i 1 Z M 2 i 2

74 Bisimulation between interval structures Z M 1 M 2 is a bisimulations wrt the fragment X 1 X 2...X n iff 1. Z-related intervals satisfy the same propositional letters, i.e.: (i 1,i 2 ) Z (p is true over i 1 p is true over i 2 ) 2. the bisimulation relation is preserved by modal operators, i.e., for every modal operator X : (i 1,i 2 ) Z (i 1,i 1 ) X X M 1 i 1 i 1 Z M 2 i 2

75 Bisimulation between interval structures Z M 1 M 2 is a bisimulations wrt the fragment X 1 X 2...X n iff 1. Z-related intervals satisfy the same propositional letters, i.e.: (i 1,i 2 ) Z (p is true over i 1 p is true over i 2 ) 2. the bisimulation relation is preserved by modal operators, i.e., for every modal operator X : } (i 1,i 2 ) Z (i 1,i 1 ) X i 2 s.t. X M 1 i 1 i 1 Z M 2 i 2 i 2

76 Bisimulation between interval structures Z M 1 M 2 is a bisimulations wrt the fragment X 1 X 2...X n iff 1. Z-related intervals satisfy the same propositional letters, i.e.: (i 1,i 2 ) Z (p is true over i 1 p is true over i 2 ) 2. the bisimulation relation is preserved by modal operators, i.e., for every modal operator X : } { (i 1,i 2 ) Z (i 1,i 1 ) X i 2 (i s.t. 1,i 2 ) Z (i 2,i 2 ) X X M 1 i 1 i 1 Z Z M 2 i 2 X i 2

77 Bisimulation between interval structures - cont d Theorem Let Z be a bisimulation between M 1 and M 2 for the language L and let i 1 and i 2 be intervals in M 1 and M 2, respectively. Then, truth of L-formulae is preserved by Z, i.e., If (i 1,i 2 ) Z, then for every formula ϕ of L: M 1,i 1 ϕ iff M 2,i 2 ϕ

78 How to use bisimulations to disprove definability Suppose that we want to prove: X is not definable in terms of L

79 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 M 1 and M 2

80 How to use bisimulations to disprove definability Suppose that we want to prove: We must provide: X is not definable in terms of L 1. two models M 1 and M 2 2. a bisimulation Z M 1 M 2 wrt fragment L

81 How to use bisimulations to disprove definability Suppose that we want to prove: We must provide: X is not definable in terms of L 1. two models M 1 and M 2 2. a bisimulation Z M 1 M 2 wrt fragment L 3. two interval i 1 M 1 and i 2 M 2 such that a. i 1 and i 2 are Z-related b. M 1, i 1 X p and M 2, i 2 X p

82 How to use bisimulations to disprove definability Suppose that we want to prove: We must provide: X is not definable in terms of L 1. two models M 1 and M 2 2. a bisimulation Z M 1 M 2 wrt fragment L 3. two interval i 1 M 1 and i 2 M 2 such that a. i 1 and i 2 are Z-related b. M 1, i 1 X p and M 2, i 2 X p By contradiction If X is definable in terms of L, then X p is.

83 How to use bisimulations to disprove definability Suppose that we want to prove: We must provide: X is not definable in terms of L 1. two models M 1 and M 2 2. a bisimulation Z M 1 M 2 wrt fragment L 3. two interval i 1 M 1 and i 2 M 2 such that a. i 1 and i 2 are Z-related b. M 1, i 1 X p and M 2, i 2 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 in i 1 (in M 1 ) and false in i 2 (in M 2 )

84 How to use bisimulations to disprove definability Suppose that we want to prove: We must provide: X is not definable in terms of L 1. two models M 1 and M 2 2. a bisimulation Z M 1 M 2 wrt fragment L 3. two interval i 1 M 1 and i 2 M 2 such that a. i 1 and i 2 are Z-related b. M 1, i 1 X p and M 2, i 2 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 in i 1 (in M 1 ) and false in i 2 (in M 2 ) contradiction

85 An example: the operator D Semantics: M,[a,b] D ϕ def c,d such that a < c < d < b and M,[c,d] ϕ D ϕ ϕ

86 An example: the operator D Semantics: M,[a,b] D ϕ def 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

87 An example: the operator D Semantics: M,[a,b] D ϕ def 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

88 D is not definable in terms of A A bisimulation wrt fragment A but not D Bisimulation wrt A (AP = {p}): models: M 1 = I(N),V 1,M 2 = I(N),V

89 D is not definable in terms of A A bisimulation wrt fragment A but not D Bisimulation wrt A (AP = {p}): models: M 1 = I(N),V 1,M 2 = I(N),V 2 V1 (p) = {[1, 2]} 0 1 p

90 D is not definable in terms of A A bisimulation wrt fragment A but not D Bisimulation wrt A (AP = {p}): models: M 1 = I(N),V 1,M 2 = I(N),V 2 V1 (p) = {[1, 2]} V2 (p) = 0 1 p p

91 D is not definable in terms of A A bisimulation wrt fragment A but not D Bisimulation wrt A (AP = {p}): models: M 1 = I(N),V 1,M 2 = I(N),V 2 V1 (p) = {[1, 2]} V2 (p) = bisimulation relation Z: ([x,y],[w,z]) Z iff 0 1 p p

92 D is not definable in terms of A A bisimulation wrt fragment A but not D Bisimulation wrt A (AP = {p}): models: M 1 = I(N),V 1,M 2 = I(N),V 2 V1 (p) = {[1, 2]} V2 (p) = bisimulation relation Z: ([x,y],[w,z]) Z iff 1. [x, y] = [w, z] = [0, 3] 0 1 p p

93 D is not definable in terms of A A bisimulation wrt fragment A but not D Bisimulation wrt A (AP = {p}): models: M 1 = I(N),V 1,M 2 = I(N),V 2 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 p p

94 D is not definable in terms of A A bisimulation wrt fragment A but not D Bisimulation wrt A (AP = {p}): models: M 1 = I(N),V 1,M 2 = I(N),V 2 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 p p M 1,[0,3] D p and M 2,[0,3] D p

95 D is not definable in terms of A A bisimulation wrt fragment A but not D Bisimulation wrt A (AP = {p}): models: M 1 = I(N),V 1,M 2 = I(N),V 2 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 p p M 1,[0,3] D p and M 2,[0,3] D p the thesis

96 Outline Introduction The Halpern and Shoham s logic HS Expressiveness of HS The satisfiability problem for HS Undecidability Classical extensions Metric extensions Hybrid extensions First-order extensions Summary and perspectives

97 The satisfiability problem for HS Satisfiability problem for a logic L Given an L-formula ϕ, is ϕ satisfiable, i.e., there exists a model and an interval in which ϕ is true?

98 The satisfiability problem for HS Satisfiability problem for a logic L Given an L-formula ϕ, is ϕ satisfiable, i.e., there exists a model and an interval in which ϕ is true? L is decidable (wrt the satisfiability problem) iff for each formula it is possible to answer the question

99 The satisfiability problem for HS Satisfiability problem for a logic L Given an L-formula ϕ, is ϕ satisfiable, i.e., there exists a model and an interval in which ϕ is true? L is decidable (wrt the satisfiability problem) iff for each formula it is possible to answer the question there exists a terminating algorithm that answer yes / not for any ϕ

100 The satisfiability problem for HS Satisfiability problem for a logic L Given an L-formula ϕ, is ϕ satisfiable, i.e., there exists a model and an interval in which ϕ is true? L is decidable (wrt the satisfiability problem) iff for each formula it is possible to answer the question there exists a terminating algorithm that answer yes / not for any ϕ Expressive enough, yet decidable, HS fragments

101 The satisfiability problem for HS Satisfiability problem for a logic L Given an L-formula ϕ, is ϕ satisfiable, i.e., there exists a model and an interval in which ϕ is true? L is decidable (wrt the satisfiability problem) iff for each formula it is possible to answer the question there 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

102 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,

103 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.

104 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 s Modal Logic of Intervals. ICALP 2010, 2010, LNCS 6199, 2010,

105 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

106 Weakest undecidable HS fragments AD,AD,AD,AD

107 Weakest undecidable HS fragments AD,AD,AD,AD BE,BE,BE,BE

108 Weakest undecidable HS fragments AD,AD,AD,AD BE,BE,BE,BE O (and O)

109 Weakest undecidable HS fragments AD,AD,AD,AD BE,BE,BE,BE O (and O) D (and D) over discrete

110 Weakest undecidable HS fragments AD, AD, AD, AD BE, BE, BE, BE O (and O) D (and D) over discrete [in this thesis] [in this thesis] [in this thesis] [Michaliszyn, Marcinkowski] The Ultimate Undecidability Result for the Halpern-Shoham Logic LICS 2011

111 Outline Introduction The Halpern and Shoham s logic HS Expressiveness of HS The satisfiability problem for HS Undecidability Classical extensions Metric extensions Hybrid extensions First-order extensions Summary and perspectives

112 The Octant Tiling Problem This is the problem of establishing whether a given finite set of tile types T = {t 1,...,t k } can tile the 2nd octant of the integer plane: O = {(i,j) : i,j N 0 i j}, while respecting the color constraints.

113 The Octant Tiling Problem This is the problem of establishing whether a given finite set of tile types T = {t 1,...,t k } can tile the 2nd octant of the integer plane: O = {(i,j) : i,j N 0 i j}, while respecting the color constraints.

114 The Octant Tiling Problem This is the problem of establishing whether a given finite set of tile types T = {t 1,...,t k } 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.

115 The Octant Tiling Problem This is the problem of establishing whether a given finite set of tile types T = {t 1,...,t k } 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, using König s Lemma.

116 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

117 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-intervals on the linear order, which covers an initial segment of the interval model. (propositional letter u) Unit intervals are used to place tiles and delimiting symbols.

118 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-intervals on the linear order, which covers an initial segment of the interval 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)

119 Undecidability of the interval logics via tiling: generic construction cont d

120 Undecidability of the interval logics via tiling: generic construction cont d Each ID-interval must have the right number of tiles

121 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 consecutive ID-intervals match vertically: the Above-Neighbour relation.

122 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 consecutive ID-intervals match vertically: the Above-Neighbour relation. For that, auxiliary propositional letter up_rel can be used to connecting (endpoints of) two intervals representing tiles that are above connected in the octant

123 Undecidability of the interval logics via tiling: generic construction completed Eventually, we encode the given Octant tiling problem by specifying the matching conditions between intervals that are right-connected or above-connected.

124 Undecidability of the interval logics via tiling: generic construction completed Eventually, we encode the given Octant tiling problem by specifying the matching conditions between intervals that are right-connected or above-connected. The specific part of the construction is to use the given fragment of HS to set the chain of unit intervals and to express all necessary properties of IDs, the propositional letters for correspondence intervals, and the tile matching conditions.

125 Summary of (un)decidability results and outlook In summary: interval logics are generally undecidable, even under very weak assumptions.

126 Summary of (un)decidability results and outlook In summary: interval logics are generally undecidable, even under very weak assumptions. In particular, most fragments of HS ( 90%) have been proved undecidable over most of the natural classes of interval structures.

127 Summary of (un)decidability results and outlook In summary: interval logics are generally undecidable, even under very weak assumptions. In particular, most fragments of HS ( 90%) have been proved undecidable over most of the natural classes of interval structures. There are still some currently unknown cases (< 8%), and some are conjectured undecidable (over general and special classes), e.g., L D, where X {X,X}; etc.

128 Summary of (un)decidability results and outlook In summary: interval logics are generally undecidable, even under very weak assumptions. In particular, most fragments of HS ( 90%) have been proved undecidable over most of the natural classes of interval structures. There are still some currently unknown cases (< 8%), and some are conjectured undecidable (over general and special classes), e.g., L D, where X {X,X}; etc. Not all results transfer readily between the strict and the non-strict semantics, and between the classes of all, dense, discrete, etc. interval structures.

129 Summary of (un)decidability results and outlook In summary: interval logics are generally undecidable, even under very weak assumptions. In particular, most fragments of HS ( 90%) have been proved undecidable over most of the natural classes of interval structures. There are still some currently unknown cases (< 8%), and some are conjectured undecidable (over general and special classes), e.g., L D, where X {X,X}; etc. Not all results transfer readily between the strict and the non-strict semantics, and between the classes of all, dense, discrete, etc. interval structures. More statistics are available on the web page:

130 Outline Introduction The Halpern and Shoham s logic HS Expressiveness of HS The satisfiability problem for HS Undecidability Classical extensions Metric extensions Hybrid extensions First-order extensions Summary and perspectives

131 Outline Introduction The Halpern and Shoham s logic HS Expressiveness of HS The satisfiability problem for HS Undecidability Classical extensions Metric extensions Hybrid extensions First-order extensions Summary and perspectives

132 PNL: syntax and semantics Syntax PNL: ϕ ::= p ϕ ϕ ϕ A ϕ A ϕ

133 PNL: syntax and semantics Syntax PNL: ϕ ::= p ϕ ϕ ϕ A ϕ A ϕ Semantics Operators meets ( A ) and met-by ( A ): A ϕ meets:

134 PNL: syntax and semantics Syntax PNL: ϕ ::= p ϕ ϕ ϕ A ϕ A ϕ Semantics Operators meets ( A ) and met-by ( A ): A ϕ ϕ meets:

135 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 A ϕ met-by:

136 PNL: syntax and semantics Syntax PNL: ϕ ::= p ϕ ϕ ϕ A ϕ A ϕ Semantics Operators meets ( A ) and met-by ( A ): A ϕ ϕ meets: ϕ A ϕ met-by:

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

138 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 ],...

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

140 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 ], Atomic propositions for length constraints: len >k, len =k,... The former are definable in terms of the latter in PNL, e.g.: A >k p := A (p len >k ).

141 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 ], Atomic propositions for length constraints: len >k, len =k,... The former are definable in terms of the latter in PNL, e.g.: A >k p := A (p len >k ). MPNL: PNL extended with integer constraints for interval lengths.

142 Decidability of metric interval logic Theorem Satisfiability in MPNL on N is decidable. It is NEXPTIME-complete if the metric constraints are represented in unary, and in between EXPSPACE and 2NEXPTIME if they are represented in binary. D. Bresolin,, V. Goranko, A. Montanari, G. Sciavicco Metric Propositional Neighborhood Interval Logics: expressiveness, decidability, and undecidability, Proc. of the European Conference on Artificial Intelligence (ECAI), 2010.

143 Decidability of metric interval logic Theorem Satisfiability in MPNL on N is decidable. It is NEXPTIME-complete if the metric constraints are represented in unary, and in between EXPSPACE and 2NEXPTIME if they are represented in binary. D. Bresolin,, V. Goranko, A. Montanari, G. Sciavicco Metric Propositional Neighborhood Interval Logics: expressiveness, decidability, and undecidability, Proc. of the European Conference on Artificial Intelligence (ECAI), Exact complexity is an open problem

144 Relative expressive power of logics in MPNL MPNL = l MPNL () l MPNL = MPNL () MPNL < MPNL >

145 Relative expressive power of logics in MPNL MPNL MPNL = l MPNL [] MPNL () l MPNL =,ε MPNL <,> MPNL (),ε MPNL = MPNL <,ε MPNL >,ε MPNL () MPNL < MPNL ε PNL MPNL >

146 Relative expressive power of logics in MPNL MPNL in 2NEXPTIME EXPSPACE-hard MPNL = l MPNL [] MPNL () l MPNL =,ε MPNL <,> MPNL (),ε MPNL = MPNL <,ε MPNL >,ε MPNL () MPNL < MPNL ε PNL MPNL > NEXPTIME-complete

147 Relative expressive power of logics in MPNL MPNL in 2NEXPTIME EXPSPACE-hard MPNL = l MPNL [] MPNL () l MPNL =,ε MPNL <,> MPNL (),ε MPNL = MPNL <,ε MPNL >,ε MPNL () MPNL < MPNL ε PNL MPNL > NEXPTIME-complete Decidability of MPNL: by small model property Comparing expressiveness of metric fragments: by bisimulations

148 Outline Introduction The Halpern and Shoham s logic HS Expressiveness of HS The satisfiability problem for HS Undecidability Classical extensions Metric extensions Hybrid extensions First-order extensions Summary and perspectives

149 Extending PNL PNL

150 Extending PNL NEXPTIME-co PNL

151 Extending PNL PNL + any HS operator NEXPTIME-co PNL

152 Extending PNL PNL + any HS operator Undecidable NEXPTIME-co PNL

153 Extending PNL PNL + any HS operator Undecidable NEXPTIME-co PNL MPNL

154 Extending PNL PNL + any HS operator Undecidable NEXPTIME-co PNL between EXPSPACE and 2NEXPTIME MPNL

155 Extending PNL PNL + any HS operator Undecidable NEXPTIME-co PNL between EXPSPACE and 2NEXPTIME MPNL MPNL +

156 Extending PNL PNL + any HS operator Undecidable NEXPTIME-co PNL MPNL MPNL + between EXPSPACE and 2NEXPTIME Undecidable

157 Extending PNL PNL + any HS operator Undecidable NEXPTIME-co PNL Hybrid Extensions MPNL MPNL + between EXPSPACE and 2NEXPTIME Undecidable

158 Extending PNL NEXPTIME-co PNL + any HS operator Undecidable??? PNL Hybrid Extensions MPNL MPNL + between EXPSPACE and 2NEXPTIME Undecidable

159 Possible hybrid extension of PNL and MPNL Nominals are definable in PNL (Basic Hybrid PNL)

160 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)

161 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)

162 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)

163 Weakly Hybrid MPNL (WHMPNL) Metric constraints of MPNL use constants len =5,len >2,... WHMPNL allows one to store the length of the current interval and to refer to it in sub-formulae x (... = x), x (... x),...

164 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 )

165 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 {<,,=,,>}

166 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 or WHMPNL)

167 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 or WHMPNL) 3. how many length variables

168 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 or WHMPNL) 3. how many length variables set of hybrid constant # of length constraints constraints variables WHMPNL(<,,=,,>) {<,,=,,>} YES unbounded WHPNL(<, =) {<, =} NO unbounded WHPNL(<) 1 {<} NO 1

169 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 or WHMPNL) 3. how many length variables set of hybrid constant # of length constraints constraints variables WHMPNL(<,,=,,>) {<,,=,,>} YES unbounded WHPNL(<, =) {<, =} NO unbounded WHPNL(<) 1 {<} NO 1

170 The fragment WHPNL(=) 1 Reduction from the Finite Tiling Problem This is the problem of establishing whether, for a given finite set of tile types T = {t 1,...,t k }, there exists a finite rectangle R having the border colored with a fixed color such that T can tile R respecting the color constraints.

171 The fragment WHPNL(=) 1 Reduction from the Finite Tiling Problem This is the problem of establishing whether, for a given finite set of tile types T = {t 1,...,t k }, there exists a finite rectangle R having the border colored with a fixed color such that T can tile R respecting the color constraints.

172 Outline Introduction The Halpern and Shoham s logic HS Expressiveness of HS The satisfiability problem for HS Undecidability Classical extensions Metric extensions Hybrid extensions First-order extensions Summary and perspectives

173 Extending PNL PNL + any HS operator Undecidable NEXPTIME-co PNL Hybrid Extensions between EXPSPACE and 2NEXPTIME Metric PNL MPNL + Undecidable

174 Extending PNL PNL + any HS operator Undecidable NEXPTIME-co PNL Hybrid Extensions between EXPSPACE and 2NEXPTIME Metric PNL MPNL + Undecidable Undecidable

175 Extending PNL PNL + any HS operator Undecidable NEXPTIME-co First-Order Extensions PNL Hybrid Extensions between EXPSPACE and 2NEXPTIME Metric PNL MPNL + Undecidable Undecidable

176 Extending PNL PNL + any HS operator Undecidable??? NEXPTIME-co First-Order Extensions PNL Hybrid Extensions between EXPSPACE and 2NEXPTIME Metric PNL MPNL + Undecidable Undecidable

177 First-Order together with Propositional F O R P N L First-Order Right Propositional Neighborhood Logic

178 First-Order together with Propositional F O R P N L First-Order Right Propositional Neighborhood Logic

179 First-Order together with Propositional F O R P N L First-Order Right Propositional Neighborhood Logic

180 First-Order together with Propositional F O R P N L First-Order Right Propositional Neighborhood Logic

181 First-Order together with Propositional F O R P N L First-Order Right Propositional Neighborhood Logic 1. Propositional (modal) setting w 2 w 3 w 1 w 4

182 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 w 1 w 2 w 4 w 3

183 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 w 2 w 1 w 4 w 3

184 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 w 1 w 2 w 4 w 3

185 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 w 1 w 2 w 3 d 1 w 4 Propositional (modal) domain d 2 d 3 First-Order domain

186 Parameters of the logic Temporal domain: discrete, dense, finite, bounded, unbounded,...

187 Parameters of the logic Temporal domain: discrete, dense, finite, bounded, unbounded,... First-order domain: finite, infinite, expanding,...

188 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, t 2,... (variables, constants, and functions)

189 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, t 2,... (variables, constants, and functions)

190 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, t 2,... (variables, constants, and functions)

191 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, t 2,... (variables, constants, and functions) terms = variables

192 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, t 2,... (variables, constants, and functions) terms = variables for tight undecidability only 1 variable (no free variables)

193 Undecidability of FORPNL Reduction from the Finite Tiling Problem This is the problem of establishing whether, for a given finite set of tile types T = {t 1,...,t k }, there exists a finite rectangle R having the border colored with a fixed color such that T can tile R respecting the color constraints.

194 Undecidability of FORPNL Reduction from the Finite Tiling Problem This is the problem of establishing whether, for a given finite set of tile types T = {t 1,...,t k }, there exists a finite rectangle R having the border colored with a fixed color such that T can tile R respecting the color constraints.