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TRS Stand 20472 pair #381709739
details
property
value
status
complete
benchmark
lindau.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n190.star.cs.uiowa.edu
space
Rubio_04
run statistics
property
value
solver
Wanda
configuration
FirstOrder
runtime (wallclock)
0.0677947998047 seconds
cpu usage
0.064240206
max memory
2920448.0
stage attributes
key
value
output-size
2721
starexec-result
YES
output
/export/starexec/sandbox/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES We consider the system theBenchmark. We are asked to determine termination of the following first-order TRS. a : [o] --> o b : [o] --> o c : [o] --> o e : [] --> o c(b(a(X))) => a(a(b(b(c(c(X)))))) a(X) => e b(X) => e c(X) => e 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] c#(b(a(X))) =#> a#(a(b(b(c(c(X)))))) 1] c#(b(a(X))) =#> a#(b(b(c(c(X))))) 2] c#(b(a(X))) =#> b#(b(c(c(X)))) 3] c#(b(a(X))) =#> b#(c(c(X))) 4] c#(b(a(X))) =#> c#(c(X)) 5] c#(b(a(X))) =#> c#(X) Rules R_0: c(b(a(X))) => a(a(b(b(c(c(X)))))) a(X) => e b(X) => e c(X) => e 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 : * 4 : * 5 : 0, 1, 2, 3, 4, 5 This graph has the following strongly connected components: P_1: c#(b(a(X))) =#> c#(X) 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(c#) = 1 Thus, we can orient the dependency pairs as follows: nu(c#(b(a(X)))) = b(a(X)) |> X = nu(c#(X)) 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 +++ [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.
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