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


--------------------------------------------------------------------------------


YES
We consider the system theBenchmark.

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

  a : [o] --> o 
  b : [o] --> o 
  f : [o] --> o 
  g : [o] --> o 

  f(f(X)) => f(a(b(f(X)))) 
  f(a(g(X))) => b(X) 
  b(X) => a(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(f(X)) >? f(a(b(f(X)))) 
  f(a(g(X))) >? b(X) 
  b(X) >? a(X) 

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

The following interpretation satisfies the requirements:

  a = \y0.y0 
  b = \y0.y0 
  f = \y0.y0 
  g = \y0.3 + 3y0 

Using this interpretation, the requirements translate to:

  [[f(f(_x0))]] = x0 >= x0 = [[f(a(b(f(_x0))))]] 
  [[f(a(g(_x0)))]] = 3 + 3x0 > x0 = [[b(_x0)]] 
  [[b(_x0)]] = x0 >= x0 = [[a(_x0)]] 

We can thus remove the following rules:

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

  Rules R_0:

    f(f(X)) => f(a(b(f(X)))) 
    b(X) => a(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 :  
    * 2 : 0, 1, 2 

This graph has the following strongly connected components:

  P_1:

    f#(f(X)) =#> 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 apply the subterm criterion with the following projection function:

  nu(f#) = 1 

Thus, we can orient the dependency pairs as follows:

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

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_1, R_0, minimal, f) by ({}, R_0, minimal, f).  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.