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TRS Stand 20472 pair #381710054
details
property
value
status
complete
benchmark
022.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n023.star.cs.uiowa.edu
space
AotoYamada_05
run statistics
property
value
solver
Wanda
configuration
FirstOrder
runtime (wallclock)
2.43996286392 seconds
cpu usage
2.40969149
max memory
7.5272192E7
stage attributes
key
value
output-size
10615
starexec-result
YES
output
/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES We consider the system theBenchmark. We are asked to determine termination of the following first-order TRS. app : [o * o] --> o cons : [] --> o leaf : [] --> o mapt : [] --> o maptlist : [] --> o nil : [] --> o node : [] --> o app(app(mapt, X), app(leaf, Y)) => app(leaf, app(X, Y)) app(app(mapt, X), app(node, Y)) => app(node, app(app(maptlist, X), Y)) app(app(maptlist, X), nil) => nil app(app(maptlist, X), app(app(cons, Y), Z)) => app(app(cons, app(app(mapt, X), Y)), app(app(maptlist, X), Z)) We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs in [Kop12, Ch. 7.8]). We thus obtain the following dependency pair problem (P_0, R_0, minimal, formative): Dependency Pairs P_0: 0] app#(app(mapt, X), app(leaf, Y)) =#> app#(leaf, app(X, Y)) 1] app#(app(mapt, X), app(leaf, Y)) =#> app#(X, Y) 2] app#(app(mapt, X), app(node, Y)) =#> app#(node, app(app(maptlist, X), Y)) 3] app#(app(mapt, X), app(node, Y)) =#> app#(app(maptlist, X), Y) 4] app#(app(mapt, X), app(node, Y)) =#> app#(maptlist, X) 5] app#(app(maptlist, X), app(app(cons, Y), Z)) =#> app#(app(cons, app(app(mapt, X), Y)), app(app(maptlist, X), Z)) 6] app#(app(maptlist, X), app(app(cons, Y), Z)) =#> app#(cons, app(app(mapt, X), Y)) 7] app#(app(maptlist, X), app(app(cons, Y), Z)) =#> app#(app(mapt, X), Y) 8] app#(app(maptlist, X), app(app(cons, Y), Z)) =#> app#(mapt, X) 9] app#(app(maptlist, X), app(app(cons, Y), Z)) =#> app#(app(maptlist, X), Z) 10] app#(app(maptlist, X), app(app(cons, Y), Z)) =#> app#(maptlist, X) Rules R_0: app(app(mapt, X), app(leaf, Y)) => app(leaf, app(X, Y)) app(app(mapt, X), app(node, Y)) => app(node, app(app(maptlist, X), Y)) app(app(maptlist, X), nil) => nil app(app(maptlist, X), app(app(cons, Y), Z)) => app(app(cons, app(app(mapt, X), Y)), app(app(maptlist, X), Z)) Thus, the original system is terminating if (P_0, R_0, minimal, formative) is finite. We consider the dependency pair problem (P_0, R_0, minimal, formative). We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows: * 0 : * 1 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 * 2 : * 3 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 * 4 : * 5 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 * 6 : * 7 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 * 8 : * 9 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 * 10 : This graph has the following strongly connected components: P_1: app#(app(mapt, X), app(leaf, Y)) =#> app#(X, Y) app#(app(mapt, X), app(node, Y)) =#> app#(app(maptlist, X), Y) app#(app(maptlist, X), app(app(cons, Y), Z)) =#> app#(app(cons, app(app(mapt, X), Y)), app(app(maptlist, X), Z)) app#(app(maptlist, X), app(app(cons, Y), Z)) =#> app#(app(mapt, X), Y) app#(app(maptlist, X), app(app(cons, Y), Z)) =#> app#(app(maptlist, X), Z) By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f). Thus, the original system is terminating if (P_1, R_0, minimal, formative) is finite. We consider the dependency pair problem (P_1, R_0, minimal, formative). The formative rules of (P_1, R_0) are R_1 ::= app(app(mapt, X), app(leaf, Y)) => app(leaf, app(X, Y)) app(app(mapt, X), app(node, Y)) => app(node, app(app(maptlist, X), Y)) app(app(maptlist, X), app(app(cons, Y), Z)) => app(app(cons, app(app(mapt, X), Y)), app(app(maptlist, X), Z)) By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_1, R_0, minimal, formative) by (P_1, R_1, minimal, formative). Thus, the original system is terminating if (P_1, R_1, minimal, formative) is finite. We consider the dependency pair problem (P_1, R_1, minimal, formative). We will use the reduction pair processor [Kop12, Thm. 7.16]. It suffices to find a standard reduction pair [Kop12, Def. 6.69]. Thus, we must orient: app#(app(mapt, X), app(leaf, Y)) >? app#(X, Y) app#(app(mapt, X), app(node, Y)) >? app#(app(maptlist, X), Y) app#(app(maptlist, X), app(app(cons, Y), Z)) >? app#(app(cons, app(app(mapt, X), Y)), app(app(maptlist, X), Z))
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