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
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
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 of M on w . Martin Zimmermann Saarland University The Complexity of Counting LTL Models 2/7
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 of M on w . Remark: f ∈ # C implies f ( w ) ∈ O (2 p ( | w | ) ) for some polynomial p . Martin Zimmermann Saarland University The Complexity of Counting LTL Models 2/7
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 of M on w . Remark: f ∈ # C implies f ( w ) ∈ O (2 p ( | w | ) ) for some polynomial p . We need larger counting classes. f : Σ ∗ → N is in # d Pspace , 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
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 of M on w . Remark: f ∈ # C implies f ( w ) ∈ O (2 p ( | w | ) ) for some polynomial p . We need larger counting classes. f : Σ ∗ → N is in # d Pspace , 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: # d Exptime , # d Expspace , and # d 2Exptime . Martin Zimmermann Saarland University The Complexity of Counting LTL Models 2/7
Counting Complexity Lemma # P Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7
Counting Complexity Lemma # P ⊆ # Pspace Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7
Counting Complexity Lemma # P ⊆ # Pspace ⊆ # Exptime Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7
Counting Complexity Lemma # P ⊆ # Pspace ⊆ # Exptime ⊆ # NExptime Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7
Counting Complexity Lemma # P ⊆ # Pspace ⊆ # Exptime ⊆ # NExptime ⊆ # Expspace Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7
Counting Complexity Lemma # P ⊆ # Pspace ⊆ # Exptime ⊆ # NExptime ⊆ # Expspace ⊆ # 2Exptime Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7
Counting Complexity Lemma # d Pspace # P ⊆ # Pspace ⊆ # Exptime ⊆ # NExptime ⊆ # Expspace ⊆ # 2Exptime Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7
Counting Complexity Lemma # d Pspace # d Exptime ⊆ # P ⊆ # Pspace ⊆ # Exptime ⊆ # NExptime ⊆ # Expspace ⊆ # 2Exptime Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7
Counting Complexity Lemma # d Pspace # d Exptime � # d Expspace ⊆ # P ⊆ # Pspace ⊆ # Exptime ⊆ # NExptime ⊆ # Expspace ⊆ # 2Exptime Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7
Counting Complexity Lemma # d Pspace # d Exptime � # d Expspace # d 2Exptime ⊆ ⊆ # P ⊆ # Pspace ⊆ # Exptime ⊆ # NExptime ⊆ # Expspace ⊆ # 2Exptime Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7
Counting Complexity Lemma # d Pspace # d Exptime � # d Expspace # d 2Exptime ⊆ ⊆ � � � � # P ⊆ # Pspace ⊆ # Exptime ⊆ # NExptime ⊆ # Expspace ⊆ # 2Exptime Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7
Counting Complexity Lemma # d Pspace # d Exptime � # d Expspace # d 2Exptime ⊆ ⊆ � � � � # P ⊆ # Pspace ⊆ # Exptime ⊆ # NExptime ⊆ # Expspace ⊆ # 2Exptime 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
Counting Complexity Lemma # d Pspace # d Exptime � # d Expspace # d 2Exptime ⊆ ⊆ � � � � # P ⊆ # Pspace ⊆ # Exptime ⊆ # NExptime ⊆ # Expspace ⊆ # 2Exptime 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
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
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 # d Pspace -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
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 # d Pspace -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 one-to-one Martin Zimmermann Saarland University The Complexity of Counting LTL Models 4/7
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 # d Pspace -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 one-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
Counting Tree-Models with Unary Bounds Theorem The following problem is # d Exptime -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
Counting Tree-Models with Unary Bounds Theorem The following problem is # d Exptime -complete: Given an LTL formula ϕ and a bound k (in unary), how many k -tree-models does ϕ have? 2 p ( n ) Lower bound: p ( n ) left right p ( n ) c 1 c 2 c 2 p ( n ) − 1 c 2 p ( n ) 2 p ( n ) Martin Zimmermann Saarland University The Complexity of Counting LTL Models 5/7
Counting Tree-Models with Unary Bounds Theorem The following problem is # d Exptime -complete: Given an LTL formula ϕ and a bound k (in unary), how many k -tree-models does ϕ have? 2 p ( n ) Lower bound: p ( n ) left right p ( n ) c 1 c 2 c 2 p ( n ) − 1 c 2 p ( n ) 2 p ( n ) Upper bound: Guess tree of height k and model-check it. Martin Zimmermann Saarland University The Complexity of Counting LTL Models 5/7
Counting Tree-Models with Binary Bounds Theorem The following problem is # d Expspace -hard and in # d 2Exptime : 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
Counting Tree-Models with Binary Bounds Theorem The following problem is # d Expspace -hard and in # d 2Exptime : Given an LTL formula ϕ and a bound k (in binary), how many k-tree-models does ϕ have? Lower bound: right C 1 C 2 C 9 left C 3 C 6 C 10 C 13 C 4 C 5 C 7 C 8 C 11 C 12 C 14 C 15 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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