details

property	value
status	complete
benchmark	4.31.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n025.star.cs.uiowa.edu
space	SK90

run statistics

property	value
solver	Wanda
configuration	FirstOrder
runtime (wallclock)	0.0631659030914 seconds
cpu usage	0.058707202
max memory	3489792.0

stage attributes

key	value
output-size	3920
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. !870 : [o * o] --> o !dot : [o * o] --> o if : [o * o * o] --> o nil : [] --> o purge : [o] --> o remove : [o * o] --> o purge(nil) => nil purge(!dot(X, Y)) => !dot(X, purge(remove(X, Y))) remove(X, nil) => nil remove(X, !dot(Y, Z)) => if(!870(X, Y), remove(X, Z), !dot(Y, remove(X, Z))) As the system is orthogonal, it is terminating if it is innermost terminating by [Gra95]. Then, by [FuhGieParSchSwi11], it suffices to prove (innermost) termination of the typed system, with sort annotations chosen to respect the rules, as follows: !870 : [xa * xa] --> oa !dot : [xa * xa] --> xa if : [oa * xa * xa] --> xa nil : [] --> xa purge : [xa] --> xa remove : [xa * xa] --> xa 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] purge#(!dot(X, Y)) =#> purge#(remove(X, Y)) 1] purge#(!dot(X, Y)) =#> remove#(X, Y) 2] remove#(X, !dot(Y, Z)) =#> remove#(X, Z) 3] remove#(X, !dot(Y, Z)) =#> remove#(X, Z) Rules R_0: purge(nil) => nil purge(!dot(X, Y)) => !dot(X, purge(remove(X, Y))) remove(X, nil) => nil remove(X, !dot(Y, Z)) => if(!870(X, Y), remove(X, Z), !dot(Y, remove(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 : 2, 3 * 2 : 2, 3 * 3 : 2, 3 This graph has the following strongly connected components: P_1: remove#(X, !dot(Y, Z)) =#> remove#(X, Z) remove#(X, !dot(Y, Z)) =#> remove#(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). We apply the subterm criterion with the following projection function: nu(remove#) = 2 Thus, we can orient the dependency pairs as follows: nu(remove#(X, !dot(Y, Z))) = !dot(Y, Z) |> Z = nu(remove#(X, Z)) nu(remove#(X, !dot(Y, Z))) = !dot(Y, Z) |> Z = nu(remove#(X, Z)) By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_1, R_0, minimal, f) by ({}, R_0, minimal, f). By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed. As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination. +++ Citations +++ [FuhGieParSchSwi11] C. Fuhs, J. Giesl, M. Parting, P. Schneider-Kamp, and S. Swiderski. Proving Termination by Dependency Pairs and Inductive Theorem Proving. In volume 47(2) of Journal of Automated Reasoning. 133--160, 2011. [Gra95] B. Gramlich. Abstract Relations Between Restricted Termination and Confluence Properties of Rewrite Systems. In volume 24(1-2) of Fundamentae Informaticae. 3--23, 1995. [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012. [KusIsoSakBla09] K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui. Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems. In volume 92(10) of IEICE Transactions on Information and Systems. 2007--2015, 2009. popout

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