We consider the system 02Ackermann.

  Alphabet:

    ack : [N * N] --> N 
    s : [N] --> N 
    z : [] --> N 

  Rules:

    ack(z, x) => s(x) 
    ack(s(x), z) => ack(x, s(z)) 
    ack(s(x), s(y)) => ack(x, ack(s(x), y)) 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs.

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

  Dependency Pairs P_0:

    0] ack#(s(X), z) =#> ack#(X, s(z))   
    1] ack#(s(X), s(Y)) =#> ack#(X, ack(s(X), Y))   
    2] ack#(s(X), s(Y)) =#> ack#(s(X), Y)   

  Rules R_0:

    ack(z, X) => s(X) 
    ack(s(X), z) => ack(X, s(z)) 
    ack(s(X), s(Y)) => ack(X, ack(s(X), 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 apply the subterm criterion with the following projection function:

  nu(ack#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(ack#(s(X), z)) = s(X) |> X = nu(ack#(X, s(z))) 
  nu(ack#(s(X), s(Y))) = s(X) |> X = nu(ack#(X, ack(s(X), Y))) 
  nu(ack#(s(X), s(Y))) = s(X) = s(X) = nu(ack#(s(X), Y)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_0, R_0, minimal, f) by (P_1, R_0, minimal, f), where P_1 contains:

  ack#(s(X), s(Y)) =#> ack#(s(X), Y)   

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(ack#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(ack#(s(X), s(Y))) = s(Y) |> Y = nu(ack#(s(X), Y)) 

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.