/export/starexec/sandbox/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES
We consider the system theBenchmark.

We are asked to determine termination of the following first-order TRS.

  a : [] --> o 
  f : [o] --> o 
  g : [o] --> o 
  h : [o] --> o 
  k : [o * o * o] --> o 

  f(a) => g(h(a)) 
  h(g(X)) => g(h(f(X))) 
  k(X, h(X), a) => h(X) 
  k(f(X), Y, 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]):

  f(a) >? g(h(a)) 
  h(g(X)) >? g(h(f(X))) 
  k(X, h(X), a) >? h(X) 
  k(f(X), Y, X) >? f(X) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  a = 0 
  f = \y0.y0 
  g = \y0.y0 
  h = \y0.y0 
  k = \y0y1y2.3 + 3y0 + 3y1 + 3y2 

Using this interpretation, the requirements translate to:

  [[f(a)]] = 0 >= 0 = [[g(h(a))]] 
  [[h(g(_x0))]] = x0 >= x0 = [[g(h(f(_x0)))]] 
  [[k(_x0, h(_x0), a)]] = 3 + 6x0 > x0 = [[h(_x0)]] 
  [[k(f(_x0), _x1, _x0)]] = 3 + 3x1 + 6x0 > x0 = [[f(_x0)]] 

We can thus remove the following rules:

  k(X, h(X), a) => h(X) 
  k(f(X), Y, X) => f(X) 

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs in [Kop12, Ch. 7.8]).

We thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):

  Dependency Pairs P_0:

    0] f#(a) =#> h#(a)   
    1] h#(g(X)) =#> h#(f(X))   
    2] h#(g(X)) =#> f#(X)   

  Rules R_0:

    f(a) => g(h(a)) 
    h(g(X)) => g(h(f(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).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 :  
    * 1 : 1, 2 
    * 2 : 0 

This graph has the following strongly connected components:

  P_1:

    h#(g(X)) =#> h#(f(X))   

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, formative) is finite.

We consider the dependency pair problem (P_1, R_0, minimal, formative).

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:

  h#(g(X)) >? h#(f(X)) 
  f(a) >= g(h(a)) 
  h(g(X)) >= g(h(f(X))) 

We orient these requirements with a polynomial interpretation in the natural numbers.

We consider usable_rules with respect to the following argument filtering:


This leaves the following ordering requirements:

  h#(g(X)) > h#(f(X)) 
  f(a) >= g(h(a)) 

The following interpretation satisfies the requirements:

  a = 2 
  f = \y0.1 + y0 
  g = \y0.3 + 2y0 
  h = \y0.0 
  h# = \y0.3y0 

Using this interpretation, the requirements translate to:

  [[h#(g(_x0))]] = 9 + 6x0 > 3 + 3x0 = [[h#(f(_x0))]] 
  [[f(a)]] = 3 >= 3 = [[g(h(a))]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_1, R_0) by ({}, R_0).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.


+++ Citations +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
[KusIsoSakBla09]  K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui.  Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems.  In volume 92(10) of IEICE Transactions on Information and Systems.  2007--2015, 2009.