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The Complexity of Counting Models of Linear-time Temporal Logic Joint work with Hazem Torfah Martin Zimmermann Saarland University September 4th, 2014 Highlights 2014, Paris, France Martin Zimmermann Saarland University The Complexity of

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The Complexity of Counting Models of Linear-time Temporal Logic

Joint work with Hazem Torfah

Martin Zimmermann

Saarland University

September 4th, 2014

Highlights 2014, Paris, France

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 1/7

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SLIDE 2

Counting Complexity

f : Σ∗ → N is in #P if there is an NP machine M such that f (w) is equal to the number of accepting runs of M on w.

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 2/7

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Counting Complexity

f : Σ∗ → N is in #P if there is an NP machine M such that f (w) is equal to the number of accepting runs of M on w. For complexity class C: f : Σ∗ → N is in #C if there is an NP machine M with oracle in C such that f (w) is equal to the number of accepting runs

  • f M on w.

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 2/7

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SLIDE 4

Counting Complexity

f : Σ∗ → N is in #P if there is an NP machine M such that f (w) is equal to the number of accepting runs of M on w. For complexity class C: f : Σ∗ → N is in #C if there is an NP machine M with oracle in C such that f (w) is equal to the number of accepting runs

  • f M on w.

Remark: f ∈ #C implies f (w) ∈ O(2p(|w|)) for some polynomial p.

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 2/7

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SLIDE 5

Counting Complexity

f : Σ∗ → N is in #P if there is an NP machine M such that f (w) is equal to the number of accepting runs of M on w. For complexity class C: f : Σ∗ → N is in #C if there is an NP machine M with oracle in C such that f (w) is equal to the number of accepting runs

  • f M on w.

Remark: f ∈ #C implies f (w) ∈ O(2p(|w|)) for some polynomial p. We need larger counting classes. f : Σ∗ → N is in #

dPspace, if there is a nondeterministic

polynomial-space Turing machine M such that f (w) is equal to the number of accepting runs of M on w.

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 2/7

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Counting Complexity

f : Σ∗ → N is in #P if there is an NP machine M such that f (w) is equal to the number of accepting runs of M on w. For complexity class C: f : Σ∗ → N is in #C if there is an NP machine M with oracle in C such that f (w) is equal to the number of accepting runs

  • f M on w.

Remark: f ∈ #C implies f (w) ∈ O(2p(|w|)) for some polynomial p. We need larger counting classes. f : Σ∗ → N is in #

dPspace, if there is a nondeterministic

polynomial-space Turing machine M such that f (w) is equal to the number of accepting runs of M on w. Analogously: #

dExptime, # dExpspace, and # d2Exptime.

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 2/7

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Counting Complexity

Lemma

#P

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7

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Counting Complexity

Lemma

#P #Pspace ⊆

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7

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Counting Complexity

Lemma

#P #Pspace ⊆ #Exptime ⊆

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7

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SLIDE 10

Counting Complexity

Lemma

#P #Pspace ⊆ #Exptime ⊆ #NExptime ⊆

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7

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SLIDE 11

Counting Complexity

Lemma

#P #Pspace ⊆ #Exptime ⊆ #NExptime ⊆ #Expspace ⊆

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7

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Counting Complexity

Lemma

#P #Pspace ⊆ #Exptime ⊆ #NExptime ⊆ #Expspace ⊆ #2Exptime ⊆

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7

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Counting Complexity

Lemma

#P #Pspace ⊆ #Exptime ⊆ #NExptime ⊆ #Expspace ⊆ #2Exptime ⊆ #

dPspace Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7

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Counting Complexity

Lemma

#P #Pspace ⊆ #Exptime ⊆ #NExptime ⊆ #Expspace ⊆ #2Exptime ⊆ #

dPspace

#

dExptime

⊆

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7

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Counting Complexity

Lemma

#P #Pspace ⊆ #Exptime ⊆ #NExptime ⊆ #Expspace ⊆ #2Exptime ⊆ #

dPspace

#

dExptime

⊆ #

dExpspace

  • Martin Zimmermann

Saarland University The Complexity of Counting LTL Models 3/7

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SLIDE 16

Counting Complexity

Lemma

#P #Pspace ⊆ #Exptime ⊆ #NExptime ⊆ #Expspace ⊆ #2Exptime ⊆ #

dPspace

#

dExptime

⊆ #

dExpspace

  • #

d2Exptime

⊆

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7

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SLIDE 17

Counting Complexity

Lemma

#P #Pspace ⊆ #Exptime ⊆ #NExptime ⊆ #Expspace ⊆ #2Exptime ⊆ #

dPspace

#

dExptime

⊆ #

dExpspace

  • #

d2Exptime

⊆

  • Martin Zimmermann

Saarland University The Complexity of Counting LTL Models 3/7

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SLIDE 18

Counting Complexity

Lemma

#P #Pspace ⊆ #Exptime ⊆ #NExptime ⊆ #Expspace ⊆ #2Exptime ⊆ #

dPspace

#

dExptime

⊆ #

dExpspace

  • #

d2Exptime

⊆

  • Reductions:

f is #P-hard, if there is a polynomial time computable function r s. t. f (r(M, w)) is equal to the number of accepting runs of M on w.

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7

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Counting Complexity

Lemma

#P #Pspace ⊆ #Exptime ⊆ #NExptime ⊆ #Expspace ⊆ #2Exptime ⊆ #

dPspace

#

dExptime

⊆ #

dExpspace

  • #

d2Exptime

⊆

  • Reductions:

f is #P-hard, if there is a polynomial time computable function r s. t. f (r(M, w)) is equal to the number of accepting runs of M on w. Hardness for other classes analogously. Completeness as usual.

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7

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Counting Word-Models

Theorem

The following problem is #P-complete: Given an LTL formula ϕ and a bound k (in unary), how many k-word-models does ϕ have?

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 4/7

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Counting Word-Models

Theorem

The following problem is #P-complete: Given an LTL formula ϕ and a bound k (in unary), how many k-word-models does ϕ have? The following problem is #

dPspace-complete: Given an LTL

formula ϕ and a bound k (in binary), how many k-word-models does ϕ have?

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 4/7

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Counting Word-Models

Theorem

The following problem is #P-complete: Given an LTL formula ϕ and a bound k (in unary), how many k-word-models does ϕ have? The following problem is #

dPspace-complete: Given an LTL

formula ϕ and a bound k (in binary), how many k-word-models does ϕ have? Lower bound: Pspace-hardness of LTL satisfiability [SC85] made

  • ne-to-one

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 4/7

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Counting Word-Models

Theorem

The following problem is #P-complete: Given an LTL formula ϕ and a bound k (in unary), how many k-word-models does ϕ have? The following problem is #

dPspace-complete: Given an LTL

formula ϕ and a bound k (in binary), how many k-word-models does ϕ have? Lower bound: Pspace-hardness of LTL satisfiability [SC85] made

  • ne-to-one

Upper bound: Guess word of length k and model-check it

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 4/7

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Counting Tree-Models with Unary Bounds

Theorem

The following problem is #

dExptime-complete: Given an LTL

formula ϕ and a bound k (in unary), how many k-tree-models does ϕ have?

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 5/7

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Counting Tree-Models with Unary Bounds

Theorem

The following problem is #

dExptime-complete: Given an LTL

formula ϕ and a bound k (in unary), how many k-tree-models does ϕ have? Lower bound:

2p(n) 2p(n) c1 c2 c2p(n)−1 c2p(n) p(n) p(n) left right

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 5/7

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Counting Tree-Models with Unary Bounds

Theorem

The following problem is #

dExptime-complete: Given an LTL

formula ϕ and a bound k (in unary), how many k-tree-models does ϕ have? Lower bound:

2p(n) 2p(n) c1 c2 c2p(n)−1 c2p(n) p(n) p(n) left right

Upper bound: Guess tree of height k and model-check it.

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 5/7

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Counting Tree-Models with Binary Bounds

Theorem

The following problem is #

dExpspace-hard and in # d2Exptime:

Given an LTL formula ϕ and a bound k (in binary), how many k-tree-models does ϕ have?

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 6/7

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Counting Tree-Models with Binary Bounds

Theorem

The following problem is #

dExpspace-hard and in # d2Exptime:

Given an LTL formula ϕ and a bound k (in binary), how many k-tree-models does ϕ have? Lower bound:

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15

left right each inner tree has exponentially many leaves tree has exponential height (thus, doubly-exponentially many inner trees)

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 6/7

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Counting Tree-Models with Binary Bounds

Theorem

The following problem is #

dExpspace-hard and in # d2Exptime:

Given an LTL formula ϕ and a bound k (in binary), how many k-tree-models does ϕ have? Lower bound:

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15

left right each inner tree has exponentially many leaves tree has exponential height (thus, doubly-exponentially many inner trees) Upper bound: Guess tree of height k and model-check it

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 6/7

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Conclusion

Overview of results:

unary binary words #P-compl. #

dPspace-compl.

trees #

dExptime-compl.

#

dExpspace-hard/# d2Exptime

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 7/7

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Conclusion

Overview of results:

unary binary words #P-compl. #

dPspace-compl.

trees #

dExptime-compl.

#

dExpspace-hard/# d2Exptime

Lower bounds: safety LTL, upper bounds: full LTL

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 7/7

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Conclusion

Overview of results:

unary binary words #P-compl. #

dPspace-compl.

trees #

dExptime-compl.

#

dExpspace-hard/# d2Exptime

Lower bounds: safety LTL, upper bounds: full LTL Open problems: Close the gap!

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 7/7

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Conclusion

Overview of results:

unary binary words #P-compl. #

dPspace-compl.

trees #

dExptime-compl.

#

dExpspace-hard/# d2Exptime

Lower bounds: safety LTL, upper bounds: full LTL Open problems: Close the gap! Lowering the upper bound: how to guess and model-check doubly-exponentially sized trees in exponential space?

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 7/7

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Conclusion

Overview of results:

unary binary words #P-compl. #

dPspace-compl.

trees #

dExptime-compl.

#

dExpspace-hard/# d2Exptime

Lower bounds: safety LTL, upper bounds: full LTL Open problems: Close the gap! Lowering the upper bound: how to guess and model-check doubly-exponentially sized trees in exponential space? Raising the lower bound: how to encode doubly-exponentially sized configurations using polynomially sized formulas? Do games help?

Martin Zimmermann Saarland University The Complexity of Counting LTL Models 7/7