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	n138.star.cs.uiowa.edu
space	Hamana_17

run statistics

property	value
solver	Wanda 2.2a
configuration	default
runtime (wallclock)	0.082929 seconds
cpu usage	0.082956
user time	0.07366
system time	0.009296
max virtual memory	113188.0
max residence set size	3588.0

stage attributes

key	value
starexec-result	YES

output

YES We consider the system theBenchmark. Alphabet: bind : [] --> Ta -> (a -> Ta) -> Ta return : [] --> a -> Ta Rules: bind (return x) (/\y.f y) => f x bind x (/\y.return y) => x bind (bind x (/\y.f y)) (/\z.g z) => bind x (/\u.bind (f u) (/\v.g v)) 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: bind : [Ta * a -> Ta] --> Ta return : [a] --> Ta ~AP1 : [a -> Ta * a] --> Ta Rules: bind(return(X), /\x.~AP1(F, x)) => ~AP1(F, X) bind(X, /\x.return(x)) => X bind(bind(X, /\x.~AP1(F, x)), /\y.~AP1(G, y)) => bind(X, /\z.bind(~AP1(F, z), /\u.~AP1(G, u))) bind(return(X), /\x.return(x)) => return(X) bind(bind(X, /\x.return(x)), /\y.~AP1(F, y)) => bind(X, /\z.bind(return(z), /\u.~AP1(F, u))) bind(bind(X, /\x.~AP1(F, x)), /\y.return(y)) => bind(X, /\z.bind(~AP1(F, z), /\u.return(u))) bind(bind(X, /\x.return(x)), /\y.return(y)) => bind(X, /\z.bind(return(z), /\u.return(u))) ~AP1(F, X) => F X 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: Alphabet: bind : [Ta * a -> Ta] --> Ta return : [a] --> Ta Rules: bind(return(X), /\x.Y(x)) => Y(X) bind(X, /\x.return(x)) => X bind(bind(X, /\x.Y(x)), /\y.Z(y)) => bind(X, /\z.bind(Y(z), /\u.Z(u))) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): bind(return(X), /\x.Y(x)) >? Y(X) bind(X, /\x.return(x)) >? X bind(bind(X, /\x.Y(x)), /\y.Z(y)) >? bind(X, /\z.bind(Y(z), /\u.Z(u))) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: bind = \y0G1.3 + 3y0 + G1(y0) + 2y0G1(y0) return = \y0.3 + y0 Using this interpretation, the requirements translate to: [[bind(return(_x0), /\x._x1(x))]] = 12 + 3x0 + 2x0F1(3 + x0) + 7F1(3 + x0) > F1(x0) = [[_x1(_x0)]] [[bind(_x0, /\x.return(x))]] = 6 + 2x0x0 + 10x0 > x0 = [[_x0]] [[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)))]] We can thus remove the following rules: bind(return(X), /\x.Y(x)) => Y(X) bind(X, /\x.return(x)) => X bind(bind(X, /\x.Y(x)), /\y.Z(y)) => bind(X, /\z.bind(Y(z), /\u.Z(u))) 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 +++ [Kop11] C. Kop. Simplifying Algebraic Functional Systems. In Proceedings of CAI 2011, volume 6742 of LNCS. 201--215, Springer, 2011. [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012. popout

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