/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). (0) CpxTRS (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (2) TRS for Loop Detection (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (4) BEST (5) proven lower bound (6) LowerBoundPropagationProof [FINISHED, 0 ms] (7) BOUNDS(n^1, INF) (8) TRS for Loop Detection (9) DecreasingLoopProof [FINISHED, 28 ms] (10) BOUNDS(EXP, INF) ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__a -> a__c a__b -> a__c a__c -> e a__k -> l a__d -> m a__a -> a__d a__b -> a__d a__c -> l a__k -> m a__A -> a__h(a__f(a__a), a__f(a__b)) a__h(X, X) -> a__g(mark(X), mark(X), a__f(a__k)) a__g(d, X, X) -> a__A a__f(X) -> a__z(mark(X), X) a__z(e, X) -> mark(X) mark(A) -> a__A mark(a) -> a__a mark(b) -> a__b mark(c) -> a__c mark(d) -> a__d mark(k) -> a__k mark(z(X1, X2)) -> a__z(mark(X1), X2) mark(f(X)) -> a__f(mark(X)) mark(h(X1, X2)) -> a__h(mark(X1), mark(X2)) mark(g(X1, X2, X3)) -> a__g(mark(X1), mark(X2), mark(X3)) mark(e) -> e mark(l) -> l mark(m) -> m a__A -> A a__a -> a a__b -> b a__c -> c a__d -> d a__k -> k a__z(X1, X2) -> z(X1, X2) a__f(X) -> f(X) a__h(X1, X2) -> h(X1, X2) a__g(X1, X2, X3) -> g(X1, X2, X3) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (2) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__a -> a__c a__b -> a__c a__c -> e a__k -> l a__d -> m a__a -> a__d a__b -> a__d a__c -> l a__k -> m a__A -> a__h(a__f(a__a), a__f(a__b)) a__h(X, X) -> a__g(mark(X), mark(X), a__f(a__k)) a__g(d, X, X) -> a__A a__f(X) -> a__z(mark(X), X) a__z(e, X) -> mark(X) mark(A) -> a__A mark(a) -> a__a mark(b) -> a__b mark(c) -> a__c mark(d) -> a__d mark(k) -> a__k mark(z(X1, X2)) -> a__z(mark(X1), X2) mark(f(X)) -> a__f(mark(X)) mark(h(X1, X2)) -> a__h(mark(X1), mark(X2)) mark(g(X1, X2, X3)) -> a__g(mark(X1), mark(X2), mark(X3)) mark(e) -> e mark(l) -> l mark(m) -> m a__A -> A a__a -> a a__b -> b a__c -> c a__d -> d a__k -> k a__z(X1, X2) -> z(X1, X2) a__f(X) -> f(X) a__h(X1, X2) -> h(X1, X2) a__g(X1, X2, X3) -> g(X1, X2, X3) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence mark(z(X1, X2)) ->^+ a__z(mark(X1), X2) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / z(X1, X2)]. The result substitution is [ ]. ---------------------------------------- (4) Complex Obligation (BEST) ---------------------------------------- (5) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__a -> a__c a__b -> a__c a__c -> e a__k -> l a__d -> m a__a -> a__d a__b -> a__d a__c -> l a__k -> m a__A -> a__h(a__f(a__a), a__f(a__b)) a__h(X, X) -> a__g(mark(X), mark(X), a__f(a__k)) a__g(d, X, X) -> a__A a__f(X) -> a__z(mark(X), X) a__z(e, X) -> mark(X) mark(A) -> a__A mark(a) -> a__a mark(b) -> a__b mark(c) -> a__c mark(d) -> a__d mark(k) -> a__k mark(z(X1, X2)) -> a__z(mark(X1), X2) mark(f(X)) -> a__f(mark(X)) mark(h(X1, X2)) -> a__h(mark(X1), mark(X2)) mark(g(X1, X2, X3)) -> a__g(mark(X1), mark(X2), mark(X3)) mark(e) -> e mark(l) -> l mark(m) -> m a__A -> A a__a -> a a__b -> b a__c -> c a__d -> d a__k -> k a__z(X1, X2) -> z(X1, X2) a__f(X) -> f(X) a__h(X1, X2) -> h(X1, X2) a__g(X1, X2, X3) -> g(X1, X2, X3) S is empty. Rewrite Strategy: FULL ---------------------------------------- (6) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (7) BOUNDS(n^1, INF) ---------------------------------------- (8) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__a -> a__c a__b -> a__c a__c -> e a__k -> l a__d -> m a__a -> a__d a__b -> a__d a__c -> l a__k -> m a__A -> a__h(a__f(a__a), a__f(a__b)) a__h(X, X) -> a__g(mark(X), mark(X), a__f(a__k)) a__g(d, X, X) -> a__A a__f(X) -> a__z(mark(X), X) a__z(e, X) -> mark(X) mark(A) -> a__A mark(a) -> a__a mark(b) -> a__b mark(c) -> a__c mark(d) -> a__d mark(k) -> a__k mark(z(X1, X2)) -> a__z(mark(X1), X2) mark(f(X)) -> a__f(mark(X)) mark(h(X1, X2)) -> a__h(mark(X1), mark(X2)) mark(g(X1, X2, X3)) -> a__g(mark(X1), mark(X2), mark(X3)) mark(e) -> e mark(l) -> l mark(m) -> m a__A -> A a__a -> a a__b -> b a__c -> c a__d -> d a__k -> k a__z(X1, X2) -> z(X1, X2) a__f(X) -> f(X) a__h(X1, X2) -> h(X1, X2) a__g(X1, X2, X3) -> g(X1, X2, X3) S is empty. Rewrite Strategy: FULL ---------------------------------------- (9) DecreasingLoopProof (FINISHED) The following loop(s) give(s) rise to the lower bound EXP: The rewrite sequence mark(f(X)) ->^+ a__z(mark(mark(X)), mark(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. The pumping substitution is [X / f(X)]. The result substitution is [ ]. The rewrite sequence mark(f(X)) ->^+ a__z(mark(mark(X)), mark(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [X / f(X)]. The result substitution is [ ]. ---------------------------------------- (10) BOUNDS(EXP, INF)