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TRS Stand 20472 pair #381710747
details
property
value
status
complete
benchmark
4.44.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n003.star.cs.uiowa.edu
space
SK90
run statistics
property
value
solver
Wanda
configuration
FirstOrder
runtime (wallclock)
0.0510549545288 seconds
cpu usage
0.03842904
max memory
1679360.0
stage attributes
key
value
output-size
2563
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 b : [] --> o f : [o] --> o g : [o] --> o h : [o] --> o i : [o] --> o f(h(X)) => f(i(X)) g(i(X)) => g(h(X)) h(a) => b i(a) => b We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): f(h(X)) >? f(i(X)) g(i(X)) >? g(h(X)) h(a) >? b i(a) >? b We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: a = 3 b = 0 f = \y0.y0 g = \y0.2y0 h = \y0.y0 i = \y0.y0 Using this interpretation, the requirements translate to: [[f(h(_x0))]] = x0 >= x0 = [[f(i(_x0))]] [[g(i(_x0))]] = 2x0 >= 2x0 = [[g(h(_x0))]] [[h(a)]] = 3 > 0 = [[b]] [[i(a)]] = 3 > 0 = [[b]] We can thus remove the following rules: h(a) => b i(a) => b 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] f#(h(X)) =#> f#(i(X)) 1] g#(i(X)) =#> g#(h(X)) Rules R_0: f(h(X)) => f(i(X)) g(i(X)) => g(h(X)) 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 : This graph has no strongly connected components. By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem. 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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