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Higher Order Rewriting Union Beta pair #487094053
details
property
value
status
complete
benchmark
AotoYamada_05__005.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n143.star.cs.uiowa.edu
space
Uncurried_Applicative_11
run statistics
property
value
solver
Wanda 2.2a
configuration
default
runtime (wallclock)
0.0180879 seconds
cpu usage
0.018144
user time
0.009507
system time
0.008637
max virtual memory
113188.0
max residence set size
1804.0
stage attributes
key
value
starexec-result
YES
output
YES We consider the system theBenchmark. Alphabet: 0 : [] --> a add : [] --> a -> a -> a curry : [a -> a -> a] --> a -> a -> a plus : [] --> a -> a -> a s : [a] --> a Rules: plus 0 x => x plus s(x) y => s(plus x y) curry(f) x y => f x y add => curry(plus) This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). Symbol curry is an encoding for application that is only used in innocuous ways. We can simplify the program (without losing non-termination) by removing it. This gives: Alphabet: 0 : [] --> a add : [] --> a -> a -> a plus : [] --> a -> a -> a s : [a] --> a Rules: plus 0 X => X plus s(X) Y => s(plus X Y) add => plus We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): plus 0 X >? X plus s(X) Y >? s(plus X Y) add >? plus We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: 0 = 3 add = \y0y1.3 + 3y0 + 3y1 plus = \y0y1.3y0 + 3y1 s = \y0.3 + y0 Using this interpretation, the requirements translate to: [[plus 0 _x0]] = 12 + 4x0 > x0 = [[_x0]] [[plus s(_x0) _x1]] = 12 + 4x0 + 4x1 > 3 + 4x0 + 4x1 = [[s(plus _x0 _x1)]] [[add]] = \y0y1.3 + 3y0 + 3y1 > \y0y1.3y0 + 3y1 = [[plus]] We can thus remove the following rules: plus 0 X => X plus s(X) Y => s(plus X Y) add => plus All rules were succesfully removed. Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold. +++ Citations +++ [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012.
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