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Higher Order Rewriting Union Beta pair #487093769
details
property
value
status
complete
benchmark
noneating.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n140.star.cs.uiowa.edu
space
Kop_11
run statistics
property
value
solver
Wanda 2.2a
configuration
default
runtime (wallclock)
0.0690531 seconds
cpu usage
0.065708
user time
0.038213
system time
0.027495
max virtual memory
113188.0
max residence set size
4676.0
stage attributes
key
value
starexec-result
YES
output
YES We consider the system theBenchmark. Alphabet: 0 : [] --> o a : [] --> o f : [o -> o] --> o g : [o] --> o h : [o * o] --> o s : [o] --> o Rules: a => f(/\x.g(x)) f(/\x.y) => a g(x) => h(x, x) h(0, x) => x h(s(x), 0) => g(x) This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): a >? f(/\x.g(x)) f(/\x.X) >? a g(X) >? h(X, X) h(0, X) >? X h(s(X), 0) >? g(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: 0 = 3 a = 0 f = \G0.G0(0) g = \y0.3y0 h = \y0y1.y0 + 2y1 s = \y0.3 + 3y0 Using this interpretation, the requirements translate to: [[a]] = 0 >= 0 = [[f(/\x.g(x))]] [[f(/\x._x0)]] = x0 >= 0 = [[a]] [[g(_x0)]] = 3x0 >= 3x0 = [[h(_x0, _x0)]] [[h(0, _x0)]] = 3 + 2x0 > x0 = [[_x0]] [[h(s(_x0), 0)]] = 9 + 3x0 > 3x0 = [[g(_x0)]] We can thus remove the following rules: h(0, X) => X h(s(X), 0) => g(X) We observe that the rules contain a first-order subset: g(X) => h(X, X) Moreover, the system is finitely branching. Thus, by [Kop12, Thm. 7.55], we may omit all first-order dependency pairs from the dependency pair problem (DP(R), R) if this first-order part is Ce-terminating when seen as a many-sorted first-order TRS. According to the external first-order termination prover, this system is indeed Ce-terminating: || Input TRS: || 1: g(PeRCenTX) -> h(PeRCenTX,PeRCenTX) || 2: TIlDePAIR(PeRCenTX,PeRCenTY) -> PeRCenTX || 3: TIlDePAIR(PeRCenTX,PeRCenTY) -> PeRCenTY || Number of strict rules: 3 || Direct POLO(bPol) ... removes: 1 3 2 || h w: x1 + x2 || TIlDePAIR w: 2 * x1 + 2 * x2 + 1 || g w: 2 * x1 + 1 || Number of strict rules: 0 || 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, all): Dependency Pairs P_0: 0] a# =#> f#(/\x.g(x)) 1] a# =#> g#(X) 2] f#(/\x.X) =#> a# Rules R_0: a => f(/\x.g(x)) f(/\x.X) => a g(X) => h(X, X) Thus, the original system is terminating if (P_0, R_0, minimal, all) is finite. We consider the dependency pair problem (P_0, R_0, minimal, all). 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 : 0, 1
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