/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.

  and : [o * o] --> o 
  not : [o] --> o 
  or : [o * o] --> o 

  not(not(X)) => X 
  not(or(X, Y)) => and(not(not(not(X))), not(not(not(Y)))) 
  not(and(X, Y)) => or(not(not(not(X))), not(not(not(Y)))) 

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] not#(or(X, Y)) =#> not#(not(not(X)))   
    1] not#(or(X, Y)) =#> not#(not(X))   
    2] not#(or(X, Y)) =#> not#(X)   
    3] not#(or(X, Y)) =#> not#(not(not(Y)))   
    4] not#(or(X, Y)) =#> not#(not(Y))   
    5] not#(or(X, Y)) =#> not#(Y)   
    6] not#(and(X, Y)) =#> not#(not(not(X)))   
    7] not#(and(X, Y)) =#> not#(not(X))   
    8] not#(and(X, Y)) =#> not#(X)   
    9] not#(and(X, Y)) =#> not#(not(not(Y)))   
    10] not#(and(X, Y)) =#> not#(not(Y))   
    11] not#(and(X, Y)) =#> not#(Y)   

  Rules R_0:

    not(not(X)) => X 
    not(or(X, Y)) => and(not(not(not(X))), not(not(not(Y)))) 
    not(and(X, Y)) => or(not(not(not(X))), not(not(not(Y)))) 

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 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:

  not#(or(X, Y)) >? not#(not(not(X))) 
  not#(or(X, Y)) >? not#(not(X)) 
  not#(or(X, Y)) >? not#(X) 
  not#(or(X, Y)) >? not#(not(not(Y))) 
  not#(or(X, Y)) >? not#(not(Y)) 
  not#(or(X, Y)) >? not#(Y) 
  not#(and(X, Y)) >? not#(not(not(X))) 
  not#(and(X, Y)) >? not#(not(X)) 
  not#(and(X, Y)) >? not#(X) 
  not#(and(X, Y)) >? not#(not(not(Y))) 
  not#(and(X, Y)) >? not#(not(Y)) 
  not#(and(X, Y)) >? not#(Y) 
  not(not(X)) >= X 
  not(or(X, Y)) >= and(not(not(not(X))), not(not(not(Y)))) 
  not(and(X, Y)) >= or(not(not(not(X))), not(not(not(Y)))) 

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

The following interpretation satisfies the requirements:

  and = \y0y1.1 + y1 + 2y0 
  not = \y0.y0 
  not# = \y0.2y0 
  or = \y0y1.1 + y1 + 2y0 

Using this interpretation, the requirements translate to:

  [[not#(or(_x0, _x1))]] = 2 + 2x1 + 4x0 > 2x0 = [[not#(not(not(_x0)))]] 
  [[not#(or(_x0, _x1))]] = 2 + 2x1 + 4x0 > 2x0 = [[not#(not(_x0))]] 
  [[not#(or(_x0, _x1))]] = 2 + 2x1 + 4x0 > 2x0 = [[not#(_x0)]] 
  [[not#(or(_x0, _x1))]] = 2 + 2x1 + 4x0 > 2x1 = [[not#(not(not(_x1)))]] 
  [[not#(or(_x0, _x1))]] = 2 + 2x1 + 4x0 > 2x1 = [[not#(not(_x1))]] 
  [[not#(or(_x0, _x1))]] = 2 + 2x1 + 4x0 > 2x1 = [[not#(_x1)]] 
  [[not#(and(_x0, _x1))]] = 2 + 2x1 + 4x0 > 2x0 = [[not#(not(not(_x0)))]] 
  [[not#(and(_x0, _x1))]] = 2 + 2x1 + 4x0 > 2x0 = [[not#(not(_x0))]] 
  [[not#(and(_x0, _x1))]] = 2 + 2x1 + 4x0 > 2x0 = [[not#(_x0)]] 
  [[not#(and(_x0, _x1))]] = 2 + 2x1 + 4x0 > 2x1 = [[not#(not(not(_x1)))]] 
  [[not#(and(_x0, _x1))]] = 2 + 2x1 + 4x0 > 2x1 = [[not#(not(_x1))]] 
  [[not#(and(_x0, _x1))]] = 2 + 2x1 + 4x0 > 2x1 = [[not#(_x1)]] 
  [[not(not(_x0))]] = x0 >= x0 = [[_x0]] 
  [[not(or(_x0, _x1))]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[and(not(not(not(_x0))), not(not(not(_x1))))]] 
  [[not(and(_x0, _x1))]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[or(not(not(not(_x0))), not(not(not(_x1))))]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_0, 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.