Complexity of Hybrid Logics over Transitive Frames Martin Mundhenk, - - PowerPoint PPT Presentation

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Complexity of Hybrid Logics over Transitive Frames Martin Mundhenk, - - PowerPoint PPT Presentation

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Complexity of Hybrid Logics over Transitive Frames Martin Mundhenk, Thomas Schneider { mundhenk,schneider } @cs.uni-jena.de Institut f ur Informatik, Friedrich-Schiller-Universit at Jena, Germany Thomas Schwentick, Volker Weber {

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Complexity of Hybrid Logics

  • ver Transitive Frames

Martin Mundhenk, Thomas Schneider {mundhenk,schneider}@cs.uni-jena.de Institut f¨ ur Informatik, Friedrich-Schiller-Universit¨ at Jena, Germany Thomas Schwentick, Volker Weber {thomas.schwentick,volker.weber}@udo.edu Fachbereich Informatik, Universit¨ at Dortmund, Germany 30 January 2006

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Complexity of Hybrid Logics

  • ver Transitive Frames

Modal Propositional Logic Temporal Logic Why Transitive Frames? Hybrid Logic Overview and Open Questions

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SLIDE 3

Modal Propositional Logic

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SLIDE 4

Modal Logic

Syntax

  • Formulas:

ϕ ::= p | ¬ϕ | ϕ1 ∧ ϕ2 | ✸ϕ, where p is an atomic proposition

  • Abbreviations ∨, →, ↔ as usual; ✷ϕ = ¬✸¬ϕ
  • Language: ML
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SLIDE 5

Modal Logic

Semantics

  • Models M = (W, R, V)
  • Frames F = (W, R)

arbitrary frame

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SLIDE 6

Modal Logic

Semantics

  • Models M = (W, R, V)
  • Frames F = (W, R)

transitive frame

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SLIDE 7

Modal Logic

Semantics

  • Models M = (W, R, V)
  • Frames F = (W, R)

transitive frame

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SLIDE 8

Modal Logic

Semantics

  • Models M = (W, R, V)
  • Frames F = (W, R)

transitive tree

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SLIDE 9

Modal Logic

Semantics

  • Models M = (W, R, V)
  • Frames F = (W, R)

linear order

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SLIDE 10

Modal Logic

Truth and Satisfiability

  • Truth is defined as usual.
  • We consider the satisfiability problem ML-SAT:

Given a formula ϕ, is there a model M = (W, R, V) and a point w ∈ W, such that M, w | = ϕ ?

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SLIDE 11

Modal Logic

Truth and Satisfiability

  • Truth is defined as usual.
  • We consider the satisfiability problem ML-SAT:

Given a formula ϕ, is there a model M = (W, R, V) and a point w ∈ W, such that M, w | = ϕ ?

  • ML-SAT is PSPACE-complete. [LADNER 1977]
  • Under restricted frame classes:
  • PSPACE-complete over transitive or reflexive frames
  • NP-complete over equivalence relations

[LADNER 1977]

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SLIDE 12

Temporal Logic

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SLIDE 13

Temporal Logic

Basic Temporal Operators

  • F, G (“Future”, “Going to”) — other names for ✸, ✷
  • P, H (“Past”, “Has been”)

— correspond to ✸−, ✷−

  • Example:

ϕ ϕ ϕ

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SLIDE 14

Temporal Logic

Basic Temporal Operators

  • F, G (“Future”, “Going to”) — other names for ✸, ✷
  • P, H (“Past”, “Has been”)

— correspond to ✸−, ✷−

  • Example:

ϕ ϕ ϕ Fϕ

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SLIDE 15

Temporal Logic

Basic Temporal Operators

  • F, G (“Future”, “Going to”) — other names for ✸, ✷
  • P, H (“Past”, “Has been”)

— correspond to ✸−, ✷−

  • Example:

ϕ ϕ ϕ Fϕ

¬Gϕ

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SLIDE 16

Temporal Logic

Basic Temporal Operators

  • F, G (“Future”, “Going to”) — other names for ✸, ✷
  • P, H (“Past”, “Has been”)

— correspond to ✸−, ✷−

  • Example:

ϕ ϕ ϕ Fϕ

¬Gϕ

Pϕ

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SLIDE 17

Temporal Logic

Basic Temporal Operators

  • F, G (“Future”, “Going to”) — other names for ✸, ✷
  • P, H (“Past”, “Has been”)

— correspond to ✸−, ✷−

  • Example:

ϕ ϕ ϕ Fϕ

¬Gϕ

Pϕ Hϕ

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SLIDE 18

Temporal Logic

Basic Temporal Operators

  • F, G (“Future”, “Going to”) — other names for ✸, ✷
  • P, H (“Past”, “Has been”)

— correspond to ✸−, ✷−

  • Example:

ϕ ϕ ϕ Fϕ

¬Gϕ

Pϕ Hϕ

  • MLF,P-SAT remains PSPACE-complete. [SPAAN 1993]
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SLIDE 19

Temporal Logic

Until and Since

  • “There will be a point in the future, at which it will be spring,

and from now until then it will always be cold.”

cold cold cold spring

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Temporal Logic

Until and Since

  • “There will be a point in the future, at which it will be spring,

and from now until then it will always be cold.”

cold cold cold spring

U(spring, cold)

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SLIDE 21

Temporal Logic

Until and Since

  • “There will be a point in the future, at which it will be spring,

and from now until then it will always be cold.”

cold cold cold spring

U(spring, cold)

  • Analogously:

S(ϕ, ψ)

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SLIDE 22

Temporal Logic

Until and Since

  • “There will be a point in the future, at which it will be spring,

and from now until then it will always be cold.”

cold cold cold spring

U(spring, cold)

  • Analogously:

S(ϕ, ψ)

  • MLU,S-SAT over linear orders: PSPACE-complete.

(ML-SAT over linear orders: NP-complete.) [SISTLA, CLARKE 1985 / ONO, NAKAMURA 1980]

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SLIDE 23

Why Transitive Frames?

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SLIDE 24

Why Transitive Frames?

  • Transitivity is a property most temporal applications have in

common.

  • Can we exactly locate the decrease in complexity taking place

when proceeding from arbitrary frames to linear orders? arbitrary . . . linear Logic frames

  • rders

ML PSPACE . . . NP P PSPACE . . . NP i, @, P EXP . . . NP i, ↓ coRE . . . NP

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Hybrid Logic I

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Hybrid Logic I

Nominals

  • Allow for explicit naming of points.
  • Atomic propositions i, j, . . . that hold at exactly one point.
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Hybrid Logic I

Nominals

  • Allow for explicit naming of points.
  • Atomic propositions i, j, . . . that hold at exactly one point.
  • Example:
  • p → Fp defines reflexivity:
  • valid on all reflexive frames
  • not valid on any other frame
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SLIDE 28

Hybrid Logic I

Nominals

  • Allow for explicit naming of points.
  • Atomic propositions i, j, . . . that hold at exactly one point.
  • Example:
  • p → Fp defines reflexivity:
  • valid on all reflexive frames
  • not valid on any other frame
  • p → ¬Fp does not define irreflexivity.

p

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SLIDE 29

Hybrid Logic I

Nominals

  • Allow for explicit naming of points.
  • Atomic propositions i, j, . . . that hold at exactly one point.
  • Example:
  • p → Fp defines reflexivity:
  • valid on all reflexive frames
  • not valid on any other frame
  • p → ¬Fp does not define irreflexivity.

p p

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SLIDE 30

Hybrid Logic I

Nominals

  • Allow for explicit naming of points.
  • Atomic propositions i, j, . . . that hold at exactly one point.
  • Example:
  • p → Fp defines reflexivity:
  • valid on all reflexive frames
  • not valid on any other frame
  • p → ¬Fp does not define irreflexivity.

i

  • i → ¬Fi does!
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SLIDE 31

Hybrid Logic I

Nominals

  • Allow for explicit naming of points.
  • Atomic propositions i, j, . . . that hold at exactly one point.
  • Example:
  • p → Fp defines reflexivity:
  • valid on all reflexive frames
  • not valid on any other frame
  • p → ¬Fp does not define irreflexivity.

i

  • i → ¬Fi does!
  • HL = ML “plus” nominals.
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SLIDE 32

Hybrid Logic I

The @ Operator

  • “Jumps” to named points.
  • M, w |

= @

iϕ

iff M, V(i) | = ϕ

  • Example:

i ϕ @i ϕ

Complexity of satisfiability?

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SLIDE 33

Hybrid Logic I

HL@-SAT Over arbitrary and transitive frames: PSPACE-complete. [ARECES, BLACKBURN, MARX 1999/2000]

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SLIDE 34

Hybrid Logic I

HL@-SAT Over arbitrary and transitive frames: PSPACE-complete. [ARECES, BLACKBURN, MARX 1999/2000] HL@

F,P-SAT

Over arbitrary and transitive frames: EXPTIME-complete. [ABM]

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SLIDE 35

Hybrid Logic I

HL@-SAT Over arbitrary and transitive frames: PSPACE-complete. [ARECES, BLACKBURN, MARX 1999/2000] HL@

F,P-SAT

Over arbitrary and transitive frames: EXPTIME-complete. [ABM] HL@

U,S-SAT

  • Over arbitrary frames: EXPTIME-complete. [ABM]
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SLIDE 36

Hybrid Logic I

HL@-SAT Over arbitrary and transitive frames: PSPACE-complete. [ARECES, BLACKBURN, MARX 1999/2000] HL@

F,P-SAT

Over arbitrary and transitive frames: EXPTIME-complete. [ABM] HL@

U,S-SAT

  • Over arbitrary frames: EXPTIME-complete. [ABM]
  • Over transitive frames:
  • EXPTIME-hard and in 2EXPTIME. [MSSW 2005]
  • Lower bound holds for MLU-SAT.
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Hybrid Logic I

HL@-SAT Over arbitrary and transitive frames: PSPACE-complete. [ARECES, BLACKBURN, MARX 1999/2000] HL@

F,P-SAT

Over arbitrary and transitive frames: EXPTIME-complete. [ABM] HL@

U,S-SAT

  • Over arbitrary frames: EXPTIME-complete. [ABM]
  • Over transitive frames:
  • EXPTIME-hard and in 2EXPTIME. [MSSW 2005]
  • Lower bound holds for MLU-SAT.
  • Over transitive trees:
  • EXPTIME-complete. [MSSW 2005]
  • Lower bound holds for MLU-SAT.
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SLIDE 38

Hybrid Logic II

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SLIDE 39

Hybrid Logic II

The ↓ Operator

  • ↓ x.ϕ: Name the current point x and evaluate ϕ, treating all
  • ccurrences of x in ϕ as nominals for this point.
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SLIDE 40

Hybrid Logic II

The ↓ Operator

  • ↓ x.ϕ: Name the current point x and evaluate ϕ, treating all
  • ccurrences of x in ϕ as nominals for this point.
  • Example: U can be expressed by means of ↓ and @:

U(ϕ, ψ) ≡ ↓ x.✸↓y.ϕ ∧ @x✷(✸y → ψ)

  • or, alternatively, by means of ↓ and past modalities:

U(ϕ, ψ) ≡ ↓ x.F

  • ϕ ∧ H(Px → ψ)
  • ψ

ψ ψ ϕ x

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SLIDE 41

Hybrid Logic II

Satisfiability for ↓ languages

  • Over arbitrary frames, HL↓ is undecidable.

[ARECES, BLACKBURN, MARX 1999]

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Hybrid Logic II

Satisfiability for ↓ languages

  • Over arbitrary frames, HL↓ is undecidable.

[ARECES, BLACKBURN, MARX 1999]

  • Over transitive frames:
  • HL↓ is NEXPTIME-complete. [MSSW 2005]
  • HL↓,@ and HL↓

F,P are undecidable. [MSSW 2005]

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SLIDE 43

Hybrid Logic II

Satisfiability for ↓ languages

  • Over arbitrary frames, HL↓ is undecidable.

[ARECES, BLACKBURN, MARX 1999]

  • Over transitive frames:
  • HL↓ is NEXPTIME-complete. [MSSW 2005]
  • HL↓,@ and HL↓

F,P are undecidable. [MSSW 2005]

  • Over transitive trees:
  • ↓ alone is useless.
  • HL↓,@ and HL↓

F,P are nonelementarily decidable.

[MSSW 2005]

  • ELEMENTARY = DTIME
  • 22. . .2n
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SLIDE 44

Overview and Open Questions

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SLIDE 45

Overview and Open Questions

arbitrary transitive transitive linear Logic frames frames trees

  • rders

i, @ PSPACE PSPACE PSPACE NP i, @, P EXP EXP PSPACE NP i, @, U, S EXP in 2EXP, EXP PSPACE- EXP-hard hard i, ↓ coRE NEXP PSPACE NP i, ↓, @ coRE coRE nonel. nonel. i, ↓, P coRE coRE nonel. nonel. i, ↓, @, P coRE coRE nonel. nonel.

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SLIDE 46

Overview and Open Questions

arbitrary transitive transitive linear Logic frames frames trees

  • rders

i, @ PSPACE PSPACE PSPACE NP i, @, P EXP EXP PSPACE NP i, @, U, S EXP in 2EXP, EXP PSPACE- EXP-hard hard i, ↓ coRE NEXP PSPACE NP i, ↓, @ coRE coRE nonel. nonel. i, ↓, P coRE coRE nonel. nonel. i, ↓, @, P coRE coRE nonel. nonel.

Thank you!