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Runtime_Complexity_Innermost_Rewriting 2019-04-01 06.40 pair #433313911
details
property
value
status
complete
benchmark
qsort.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n057.star.cs.uiowa.edu
space
AProVE_09_Inductive
run statistics
property
value
solver
AProVE
configuration
complexity
runtime (wallclock)
291.616 seconds
cpu usage
1088.7
user time
1073.92
system time
14.7788
max virtual memory
1.91584E7
max residence set size
1.4975276E7
stage attributes
key
value
starexec-result
WORST_CASE(Omega(n^1), ?)
output
1088.49/291.55 WORST_CASE(Omega(n^1), ?) 1088.49/291.55 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1088.49/291.55 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1088.49/291.55 1088.49/291.55 1088.49/291.55 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1088.49/291.55 1088.49/291.55 (0) CpxTRS 1088.49/291.55 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1088.49/291.55 (2) TRS for Loop Detection 1088.49/291.55 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1088.49/291.55 (4) BEST 1088.49/291.55 (5) proven lower bound 1088.49/291.55 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1088.49/291.55 (7) BOUNDS(n^1, INF) 1088.49/291.55 (8) TRS for Loop Detection 1088.49/291.55 1088.49/291.55 1088.49/291.55 ---------------------------------------- 1088.49/291.55 1088.49/291.55 (0) 1088.49/291.55 Obligation: 1088.49/291.55 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1088.49/291.55 1088.49/291.55 1088.49/291.55 The TRS R consists of the following rules: 1088.49/291.55 1088.49/291.55 qsort(nil) -> nil 1088.49/291.55 qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, cons(x, xs))))) 1088.49/291.55 filterlow(n, nil) -> nil 1088.49/291.55 filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) 1088.49/291.55 if1(true, n, x, xs) -> filterlow(n, xs) 1088.49/291.55 if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) 1088.49/291.55 filterhigh(n, nil) -> nil 1088.49/291.55 filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) 1088.49/291.55 if2(true, n, x, xs) -> filterhigh(n, xs) 1088.49/291.55 if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) 1088.49/291.55 ge(x, 0) -> true 1088.49/291.55 ge(0, s(x)) -> false 1088.49/291.55 ge(s(x), s(y)) -> ge(x, y) 1088.49/291.55 append(nil, ys) -> ys 1088.49/291.55 append(cons(x, xs), ys) -> cons(x, append(xs, ys)) 1088.49/291.55 1088.49/291.55 S is empty. 1088.49/291.55 Rewrite Strategy: INNERMOST 1088.49/291.55 ---------------------------------------- 1088.49/291.55 1088.49/291.55 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1088.49/291.55 Transformed a relative TRS into a decreasing-loop problem. 1088.49/291.55 ---------------------------------------- 1088.49/291.55 1088.49/291.55 (2) 1088.49/291.55 Obligation: 1088.49/291.55 Analyzing the following TRS for decreasing loops: 1088.49/291.55 1088.49/291.55 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1088.49/291.55 1088.49/291.55 1088.49/291.55 The TRS R consists of the following rules: 1088.49/291.55 1088.49/291.55 qsort(nil) -> nil 1088.49/291.55 qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, cons(x, xs))))) 1088.49/291.55 filterlow(n, nil) -> nil 1088.49/291.55 filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) 1088.49/291.55 if1(true, n, x, xs) -> filterlow(n, xs) 1088.49/291.55 if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) 1088.49/291.55 filterhigh(n, nil) -> nil 1088.49/291.55 filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) 1088.49/291.55 if2(true, n, x, xs) -> filterhigh(n, xs) 1088.49/291.55 if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) 1088.49/291.55 ge(x, 0) -> true 1088.49/291.55 ge(0, s(x)) -> false 1088.49/291.55 ge(s(x), s(y)) -> ge(x, y) 1088.49/291.55 append(nil, ys) -> ys 1088.49/291.55 append(cons(x, xs), ys) -> cons(x, append(xs, ys)) 1088.49/291.55 1088.49/291.55 S is empty. 1088.49/291.55 Rewrite Strategy: INNERMOST 1088.49/291.55 ---------------------------------------- 1088.49/291.55 1088.49/291.55 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1088.49/291.55 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1088.49/291.55 1088.49/291.55 The rewrite sequence 1088.49/291.55 1088.49/291.55 append(cons(x, xs), ys) ->^+ cons(x, append(xs, ys)) 1088.49/291.55 1088.49/291.55 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 1088.49/291.55 1088.49/291.55 The pumping substitution is [xs / cons(x, xs)]. 1088.49/291.55 1088.49/291.55 The result substitution is [ ]. 1088.49/291.55 1088.49/291.55 1088.49/291.55 1088.49/291.55 1088.49/291.55 ---------------------------------------- 1088.49/291.55 1088.49/291.55 (4) 1088.49/291.55 Complex Obligation (BEST)
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