Spaces
Explore
Communities
Statistics
Reports
Cluster
Status
Help
HRS union beta 16688 pair #381734367
details
property
value
status
complete
benchmark
restriction.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n095.star.cs.uiowa.edu
space
Hamana_17
run statistics
property
value
solver
Wanda
configuration
HigherOrder
runtime (wallclock)
7.5635201931 seconds
cpu usage
7.560705768
max memory
4.1803776E8
stage attributes
key
value
output-size
7048
starexec-result
YES
output
/export/starexec/sandbox/solver/bin/starexec_run_HigherOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES We consider the system theBenchmark. Alphabet: New : [] --> N -> A -> A Rules: New (/\x.y) => y New (/\x.New (/\y.f x y)) => New (/\z.New (/\u.f u z)) Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term. We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0). This leads to the following AFSM: Alphabet: New : [N -> A] --> A ~AP1 : [N -> N -> A * N] --> N -> A Rules: New(/\x.X) => X New(/\x.New(/\y.~AP1(F, x) y)) => New(/\z.New(/\u.~AP1(F, u) z)) ~AP1(F, X) => F X We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): New(/\x.X) >? X New(/\x.New(/\y.~AP1(F, x) y)) >? New(/\z.New(/\u.~AP1(F, u) z)) ~AP1(F, X) >? F X We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: New = \G0.3 + G0(0) ~AP1 = \G0y1y2.3 + y1 + G0(y1,y2) Using this interpretation, the requirements translate to: [[New(/\x._x0)]] = 3 + x0 > x0 = [[_x0]] [[New(/\x.New(/\y.~AP1(_F0, x) y))]] = 9 + F0(0,0) >= 9 + F0(0,0) = [[New(/\x.New(/\y.~AP1(_F0, y) x))]] [[~AP1(_F0, _x1)]] = \y0.3 + x1 + F0(x1,y0) > \y0.x1 + F0(x1,y0) = [[_F0 _x1]] We can thus remove the following rules: New(/\x.X) => X ~AP1(F, X) => F X We use the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs. We thus obtain the following dependency pair problem (P_0, R_0, minimal, all): Dependency Pairs P_0: 0] New#(/\x.New(/\y.~AP1(F, x, y))) =#> New#(/\z.New(/\u.~AP1(F, u, z))) 1] New#(/\x.New(/\y.~AP1(F, x, y))) =#> New#(/\z.~AP1(F, z, u)) Rules R_0: New(/\x.New(/\y.~AP1(F, x, y))) => New(/\z.New(/\u.~AP1(F, u, z))) 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 : 0, 1 * 1 : This graph has the following strongly connected components: P_1: New#(/\x.New(/\y.~AP1(F, x, y))) =#> New#(/\z.New(/\u.~AP1(F, u, 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, all) is finite. We consider the dependency pair problem (P_1, R_0, minimal, all). We will use the reduction pair processor [Kop12, Thm. 7.16]. It suffices to find a standard reduction pair [Kop12, Def. 6.69]. Thus, we must orient: New#(/\x.New(/\y.~AP1(F, x, y))) >? New#(/\z.New(/\u.~AP1(F, u, z))) New(/\x.New(/\y.~AP1(F, x, y))) >= New(/\z.New(/\u.~AP1(F, u, z))) Since this representation is not advantageous for the higher-order recursive path ordering, we present the strict requirements in their unextended form, which is not problematic since for any F, s and substituion gamma: (F s)gamma beta-reduces to F(s)gamma.)
popout
output may be truncated. 'popout' for the full output.
job log
popout
actions
all output
return to HRS union beta 16688