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Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270347
details
property
value
status
complete
benchmark
monad.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n106.star.cs.uiowa.edu
space
Hamana_17
run statistics
property
value
solver
Wanda 2.1c
configuration
default
runtime (wallclock)
0.082243 seconds
cpu usage
0.082276
user time
0.077075
system time
0.005201
max virtual memory
113176.0
max residence set size
3624.0
stage attributes
key
value
starexec-result
YES
output
0.00/0.07 YES 0.00/0.08 We consider the system theBenchmark. 0.00/0.08 0.00/0.08 Alphabet: 0.00/0.08 0.00/0.08 bind : [] --> Ta -> (a -> Ta) -> Ta 0.00/0.08 return : [] --> a -> Ta 0.00/0.08 0.00/0.08 Rules: 0.00/0.08 0.00/0.08 bind (return x) (/\y.f y) => f x 0.00/0.08 bind x (/\y.return y) => x 0.00/0.08 bind (bind x (/\y.f y)) (/\z.g z) => bind x (/\u.bind (f u) (/\v.g v)) 0.00/0.08 0.00/0.08 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. 0.00/0.08 0.00/0.08 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: 0.00/0.08 0.00/0.08 Alphabet: 0.00/0.08 0.00/0.08 bind : [Ta * a -> Ta] --> Ta 0.00/0.08 return : [a] --> Ta 0.00/0.08 ~AP1 : [a -> Ta * a] --> Ta 0.00/0.08 0.00/0.08 Rules: 0.00/0.08 0.00/0.08 bind(return(X), /\x.~AP1(F, x)) => ~AP1(F, X) 0.00/0.08 bind(X, /\x.return(x)) => X 0.00/0.08 bind(bind(X, /\x.~AP1(F, x)), /\y.~AP1(G, y)) => bind(X, /\z.bind(~AP1(F, z), /\u.~AP1(G, u))) 0.00/0.08 bind(return(X), /\x.return(x)) => return(X) 0.00/0.08 bind(bind(X, /\x.return(x)), /\y.~AP1(F, y)) => bind(X, /\z.bind(return(z), /\u.~AP1(F, u))) 0.00/0.08 bind(bind(X, /\x.~AP1(F, x)), /\y.return(y)) => bind(X, /\z.bind(~AP1(F, z), /\u.return(u))) 0.00/0.08 bind(bind(X, /\x.return(x)), /\y.return(y)) => bind(X, /\z.bind(return(z), /\u.return(u))) 0.00/0.08 ~AP1(F, X) => F X 0.00/0.08 0.00/0.08 Symbol ~AP1 is an encoding for application that is only used in innocuous ways. We can simplify the program (without losing non-termination) by removing it. Additionally, we can remove some (now-)redundant rules. This gives: 0.00/0.08 0.00/0.08 Alphabet: 0.00/0.08 0.00/0.08 bind : [Ta * a -> Ta] --> Ta 0.00/0.08 return : [a] --> Ta 0.00/0.08 0.00/0.08 Rules: 0.00/0.08 0.00/0.08 bind(return(X), /\x.Y(x)) => Y(X) 0.00/0.08 bind(X, /\x.return(x)) => X 0.00/0.08 bind(bind(X, /\x.Y(x)), /\y.Z(y)) => bind(X, /\z.bind(Y(z), /\u.Z(u))) 0.00/0.08 0.00/0.08 We use rule removal, following [Kop12, Theorem 2.23]. 0.00/0.08 0.00/0.08 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 0.00/0.08 0.00/0.08 bind(return(X), /\x.Y(x)) >? Y(X) 0.00/0.08 bind(X, /\x.return(x)) >? X 0.00/0.08 bind(bind(X, /\x.Y(x)), /\y.Z(y)) >? bind(X, /\z.bind(Y(z), /\u.Z(u))) 0.00/0.08 0.00/0.08 We orient these requirements with a polynomial interpretation in the natural numbers. 0.00/0.08 0.00/0.08 The following interpretation satisfies the requirements: 0.00/0.08 0.00/0.08 bind = \y0G1.3 + 3y0 + G1(y0) + 2y0G1(y0) 0.00/0.08 return = \y0.3 + y0 0.00/0.08 0.00/0.08 Using this interpretation, the requirements translate to: 0.00/0.08 0.00/0.08 [[bind(return(_x0), /\x._x1(x))]] = 12 + 3x0 + 2x0F1(3 + x0) + 7F1(3 + x0) > F1(x0) = [[_x1(_x0)]] 0.00/0.08 [[bind(_x0, /\x.return(x))]] = 6 + 2x0x0 + 10x0 > x0 = [[_x0]] 0.00/0.08 [[bind(bind(_x0, /\x._x1(x)), /\y._x2(y))]] = 12 + 9x0 + 2F1(x0)F2(3 + 3x0 + F1(x0) + 2x0F1(x0)) + 3F1(x0) + 4x0F1(x0)F2(3 + 3x0 + F1(x0) + 2x0F1(x0)) + 6x0F1(x0) + 6x0F2(3 + 3x0 + F1(x0) + 2x0F1(x0)) + 7F2(3 + 3x0 + F1(x0) + 2x0F1(x0)) > 6 + 9x0 + F2(F1(x0)) + 2x0F2(F1(x0)) + 2F1(x0)F2(F1(x0)) + 3F1(x0) + 4x0F1(x0)F2(F1(x0)) + 6x0F1(x0) = [[bind(_x0, /\x.bind(_x1(x), /\y._x2(y)))]] 0.00/0.08 0.00/0.08 We can thus remove the following rules: 0.00/0.08 0.00/0.08 bind(return(X), /\x.Y(x)) => Y(X) 0.00/0.08 bind(X, /\x.return(x)) => X 0.00/0.08 bind(bind(X, /\x.Y(x)), /\y.Z(y)) => bind(X, /\z.bind(Y(z), /\u.Z(u))) 0.00/0.08 0.00/0.08 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. 0.00/0.08 0.00/0.08 0.00/0.08 +++ Citations +++ 0.00/0.08 0.00/0.08 [Kop11] C. Kop. Simplifying Algebraic Functional Systems. In Proceedings of CAI 2011, volume 6742 of LNCS. 201--215, Springer, 2011. 0.00/0.08 [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012. 0.00/0.08 EOF
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