Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-201)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by ttt2 @ termCOMP 2023)

1 String Reversal

1.1 Dependency Pair Transformation

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

• The 1^st component contains the pair

  c#(c(x1))	→	b#(x1)	(11)
  b#(a(x1))	→	b#(x1)	(7)
  b#(a(x1))	→	b#(b(x1))	(8)
  b#(a(x1))	→	c#(a(b(b(x1))))	(10)

  1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers

  [b#(x1)]	=	0 0 -∞ -∞ · x1 + -∞ -∞ -∞ -∞
  [b(x1)]	=	0 -∞ 1 0 · x1 + -∞ -∞ -∞ -∞
  [c#(x1)]	=	0 0 -∞ -∞ · x1 + -∞ -∞ -∞ -∞
  [a(x1)]	=	1 0 1 0 · x1 + -∞ -∞ -∞ -∞
  [c(x1)]	=	0 0 1 1 · x1 + -∞ -∞ -∞ -∞

  together with the usable rules

  a(x1)	→	x1	(1)
  b(a(x1))	→	c(a(b(b(x1))))	(5)
  b(x1)	→	x1	(3)
  c(c(x1))	→	a(b(x1))	(6)

  (w.r.t. the implicit argument filter of the reduction pair), the pair

  c#(c(x1))

  →

  b#(x1)

  (11)

  could be deleted.

  1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    b#(a(x1))	→	b#(b(x1))	(8)
    b#(a(x1))	→	b#(x1)	(7)

    1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers

    [b#(x1)]	=	0 0 -∞ -∞ · x1 + -∞ -∞ -∞ -∞
    [b(x1)]	=	0 0 0 0 · x1 + -∞ -∞ -∞ -∞
    [a(x1)]	=	0 0 2 2 · x1 + -∞ -∞ 3 -∞
    [c(x1)]	=	0 0 2 0 · x1 + 1 -∞ 0 -∞

    together with the usable rules

    a(x1)	→	x1	(1)
    b(a(x1))	→	c(a(b(b(x1))))	(5)
    b(x1)	→	x1	(3)
    c(c(x1))	→	a(b(x1))	(6)

    (w.r.t. the implicit argument filter of the reduction pair), the pairs

    b#(a(x1))	→	b#(b(x1))	(8)
    b#(a(x1))	→	b#(x1)	(7)

    could be deleted.

    1.1.1.1.1.1.1 P is empty

    There are no pairs anymore.