/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.

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

  g(h(g(X))) => g(X) 
  g(g(X)) => g(h(g(X))) 
  h(h(X)) => h(f(h(X), 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] g#(h(g(X))) =#> g#(X)   
    1] g#(g(X)) =#> g#(h(g(X)))   
    2] g#(g(X)) =#> h#(g(X))   
    3] g#(g(X)) =#> g#(X)   
    4] h#(h(X)) =#> h#(f(h(X), X))   
    5] h#(h(X)) =#> h#(X)   

  Rules R_0:

    g(h(g(X))) => g(X) 
    g(g(X)) => g(h(g(X))) 
    h(h(X)) => h(f(h(X), 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 : 0, 1, 2, 3 
    * 1 : 0 
    * 2 :  
    * 3 : 0, 1, 2, 3 
    * 4 :  
    * 5 : 4, 5 

This graph has the following strongly connected components:

  P_1:

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

  P_2:

    h#(h(X)) =#> h#(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) and (P_2, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative) and (P_2, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_2, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(h#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(h#(h(X))) = h(X) |> X = nu(h#(X)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_2, R_0, minimal, f) by ({}, R_0, minimal, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

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:

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

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

The following interpretation satisfies the requirements:

  f = \y0y1.0 
  g = \y0.1 + y0 
  g# = \y0.2y0 
  h = \y0.y0 

Using this interpretation, the requirements translate to:

  [[g#(h(g(_x0)))]] = 2 + 2x0 > 2x0 = [[g#(_x0)]] 
  [[g#(g(_x0))]] = 2 + 2x0 >= 2 + 2x0 = [[g#(h(g(_x0)))]] 
  [[g#(g(_x0))]] = 2 + 2x0 > 2x0 = [[g#(_x0)]] 
  [[g(h(g(_x0)))]] = 2 + x0 >= 1 + x0 = [[g(_x0)]] 
  [[g(g(_x0))]] = 2 + x0 >= 2 + x0 = [[g(h(g(_x0)))]] 
  [[h(h(_x0))]] = x0 >= 0 = [[h(f(h(_x0), _x0))]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_1, R_0, minimal, formative) by (P_3, R_0, minimal, formative), where P_3 consists of:

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

Thus, the original system is terminating if (P_3, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_3, 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 :  

This graph has no strongly connected components.  By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem.

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.