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Higher Order Rewriting Union Beta pair #487093815
details
property
value
status
complete
benchmark
noabs.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n150.star.cs.uiowa.edu
space
Mixed_HO_10
run statistics
property
value
solver
Wanda 2.2a
configuration
default
runtime (wallclock)
0.13725 seconds
cpu usage
0.136721
user time
0.101755
system time
0.034966
max virtual memory
113188.0
max residence set size
3412.0
stage attributes
key
value
starexec-result
YES
output
YES We consider the system theBenchmark. Alphabet: A : [term * term] --> term L : [term -> term] --> term V : [term] --> term noabs : [term] --> term Rules: noabs(A(x, y)) => A(noabs(x), noabs(y)) noabs(V(x)) => V(x) A(L(f), x) => f noabs(x) This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). We use the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs. After applying [Kop12, Thm. 7.22] to denote collapsing dependency pairs in an extended form, we thus obtain the following dependency pair problem (P_0, R_0, minimal, formative): Dependency Pairs P_0: 0] noabs#(A(X, Y)) =#> A#(noabs(X), noabs(Y)) 1] noabs#(A(X, Y)) =#> noabs#(X) 2] noabs#(A(X, Y)) =#> noabs#(Y) 3] A#(L(F), X) =#> F(noabs(X)) 4] A#(L(F), X) =#> noabs#(X) Rules R_0: noabs(A(X, Y)) => A(noabs(X), noabs(Y)) noabs(V(X)) => V(X) A(L(F), X) => F noabs(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). The formative rules of (P_0, R_0) are R_1 ::= noabs(A(X, Y)) => A(noabs(X), noabs(Y)) A(L(F), X) => F noabs(X) By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_0, R_0, minimal, formative) by (P_0, R_1, minimal, formative). Thus, the original system is terminating if (P_0, R_1, minimal, formative) is finite. We consider the dependency pair problem (P_0, R_1, minimal, formative). We will use the reduction pair processor [Kop12, Thm. 7.16]. As the system is abstraction-simple and the formative flag is set, it suffices to find a tagged reduction pair [Kop12, Def. 6.70]. Thus, we must orient: noabs#(A(X, Y)) >? A#(noabs(X), noabs(Y)) noabs#(A(X, Y)) >? noabs#(X) noabs#(A(X, Y)) >? noabs#(Y) A#(L(F), X) >? F(noabs(X)) A#(L(F), X) >? noabs#(X) noabs(A(X, Y)) >= A(noabs(X), noabs(Y)) A(L(F), X) >= F noabs(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: A = \y0y1.3y0 A# = \y0y1.2y0 + 2y1 L = \G0.3 + G0(0) noabs = \y0.0 noabs# = \y0.2 Using this interpretation, the requirements translate to: [[noabs#(A(_x0, _x1))]] = 2 > 0 = [[A#(noabs(_x0), noabs(_x1))]] [[noabs#(A(_x0, _x1))]] = 2 >= 2 = [[noabs#(_x0)]] [[noabs#(A(_x0, _x1))]] = 2 >= 2 = [[noabs#(_x1)]] [[A#(L(_F0), _x1)]] = 6 + 2x1 + 2F0(0) > F0(0) = [[_F0(noabs(_x1))]] [[A#(L(_F0), _x1)]] = 6 + 2x1 + 2F0(0) > 2 = [[noabs#(_x1)]] [[noabs(A(_x0, _x1))]] = 0 >= 0 = [[A(noabs(_x0), noabs(_x1))]] [[A(L(_F0), _x1)]] = 9 + 3F0(0) >= F0(0) = [[_F0 noabs(_x1)]] By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_0, R_1, minimal, formative) by (P_1, R_1, minimal, formative), where P_1 consists of: noabs#(A(X, Y)) =#> noabs#(X) noabs#(A(X, Y)) =#> noabs#(Y) 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 apply the subterm criterion with the following projection function: nu(noabs#) = 1 Thus, we can orient the dependency pairs as follows: nu(noabs#(A(X, Y))) = A(X, Y) |> X = nu(noabs#(X)) nu(noabs#(A(X, Y))) = A(X, Y) |> Y = nu(noabs#(Y)) By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_1, R_1, minimal, f) by ({}, R_1, minimal, f). By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.
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