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 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 2 #C implies f (w) 2 O(2p(jwj)) 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. 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 Remark: f 2 #C implies f (w) 2 O(2p(jwj)) 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. 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 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. 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 2 #C implies f (w) 2 O(2p(jwj)) for some polynomial p. Martin Zimmermann Saarland University The Complexity of Counting LTL Models 2/7 Analogously: #d Exptime,#d Expspace, and #d 2Exptime. 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 2 #C implies f (w) 2 O(2p(jwj)) 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 2 #C implies f (w) 2 O(2p(jwj)) 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 #d Pspace ⊆#d Exptime ( #d Expspace ⊆ #d 2Exptime ( ( ( ( ⊆ #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. Counting Complexity Lemma #P Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7 #d Pspace ⊆#d Exptime ( #d Expspace ⊆ #d 2Exptime ( ( ( ( ⊆ #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. Counting Complexity Lemma #P ⊆ #Pspace Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7 #d Pspace ⊆#d Exptime ( #d Expspace ⊆ #d 2Exptime ( ( ( ( ⊆ #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. Counting Complexity Lemma #P ⊆ #Pspace ⊆ #Exptime Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7 #d Pspace ⊆#d Exptime ( #d Expspace ⊆ #d 2Exptime ( ( ( ( ⊆ #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. Counting Complexity Lemma #P ⊆ #Pspace ⊆ #Exptime ⊆ #NExptime Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7 #d Pspace ⊆#d Exptime ( #d Expspace ⊆ #d 2Exptime ( ( ( ( ⊆ #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. Counting Complexity Lemma #P ⊆ #Pspace ⊆ #Exptime ⊆ #NExptime ⊆ #Expspace Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7 #d Pspace ⊆#d Exptime ( #d Expspace ⊆ #d 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. Counting Complexity Lemma #P ⊆ #Pspace ⊆ #Exptime ⊆ #NExptime ⊆ #Expspace ⊆ #2Exptime Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7 ⊆#d Exptime ( #d Expspace ⊆ #d 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. Counting Complexity Lemma #d Pspace #P ⊆ #Pspace ⊆ #Exptime ⊆ #NExptime ⊆ #Expspace ⊆ #2Exptime Martin Zimmermann Saarland University The Complexity of Counting LTL Models 3/7 ( #d Expspace ⊆ #d 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. 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 ⊆ #d 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. 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 ( ( ( ( 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. 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 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. 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 Hardness for other classes analogously. Completeness as usual. 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.