/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).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, 32 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: filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M)) filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M)) sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y))) sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(n__filter(activate(Y), N, N))) nats(N) -> cons(N, n__nats(n__s(N))) zprimes -> sieve(nats(s(s(0)))) filter(X1, X2, X3) -> n__filter(X1, X2, X3) sieve(X) -> n__sieve(X) nats(X) -> n__nats(X) s(X) -> n__s(X) activate(n__filter(X1, X2, X3)) -> filter(activate(X1), activate(X2), activate(X3)) activate(n__sieve(X)) -> sieve(activate(X)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X 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: filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M)) filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M)) sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y))) sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(n__filter(activate(Y), N, N))) nats(N) -> cons(N, n__nats(n__s(N))) zprimes -> sieve(nats(s(s(0)))) filter(X1, X2, X3) -> n__filter(X1, X2, X3) sieve(X) -> n__sieve(X) nats(X) -> n__nats(X) s(X) -> n__s(X) activate(n__filter(X1, X2, X3)) -> filter(activate(X1), activate(X2), activate(X3)) activate(n__sieve(X)) -> sieve(activate(X)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X 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 activate(n__s(X)) ->^+ s(activate(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / n__s(X)]. 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: filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M)) filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M)) sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y))) sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(n__filter(activate(Y), N, N))) nats(N) -> cons(N, n__nats(n__s(N))) zprimes -> sieve(nats(s(s(0)))) filter(X1, X2, X3) -> n__filter(X1, X2, X3) sieve(X) -> n__sieve(X) nats(X) -> n__nats(X) s(X) -> n__s(X) activate(n__filter(X1, X2, X3)) -> filter(activate(X1), activate(X2), activate(X3)) activate(n__sieve(X)) -> sieve(activate(X)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X 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: filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M)) filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M)) sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y))) sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(n__filter(activate(Y), N, N))) nats(N) -> cons(N, n__nats(n__s(N))) zprimes -> sieve(nats(s(s(0)))) filter(X1, X2, X3) -> n__filter(X1, X2, X3) sieve(X) -> n__sieve(X) nats(X) -> n__nats(X) s(X) -> n__s(X) activate(n__filter(X1, X2, X3)) -> filter(activate(X1), activate(X2), activate(X3)) activate(n__sieve(X)) -> sieve(activate(X)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X S is empty. Rewrite Strategy: FULL ---------------------------------------- (9) DecreasingLoopProof (FINISHED) The following loop(s) give(s) rise to the lower bound EXP: The rewrite sequence activate(n__nats(X)) ->^+ cons(activate(X), n__nats(n__s(activate(X)))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / n__nats(X)]. The result substitution is [ ]. The rewrite sequence activate(n__nats(X)) ->^+ cons(activate(X), n__nats(n__s(activate(X)))) gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0]. The pumping substitution is [X / n__nats(X)]. The result substitution is [ ]. ---------------------------------------- (10) BOUNDS(EXP, INF)