ack(0,

ack(s(

ack(s(

R

↳Dependency Pair Analysis

ACK(s(x), 0) -> ACK(x, s(0))

ACK(s(x), s(y)) -> ACK(x, ack(s(x),y))

ACK(s(x), s(y)) -> ACK(s(x),y)

Furthermore,

R

↳DPs

→DP Problem 1

↳Polynomial Ordering

**ACK(s( x), s(y)) -> ACK(s(x), y)**

ack(0,y) -> s(y)

ack(s(x), 0) -> ack(x, s(0))

ack(s(x), s(y)) -> ack(x, ack(s(x),y))

innermost

The following dependency pairs can be strictly oriented:

ACK(s(x), s(y)) -> ACK(x, ack(s(x),y))

ACK(s(x), 0) -> ACK(x, s(0))

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(ACK(x)_{1}, x_{2})= x _{1}_{ }^{ }_{ }^{ }POL(ack(x)_{1}, x_{2})= 0 _{ }^{ }_{ }^{ }POL(0)= 0 _{ }^{ }_{ }^{ }POL(s(x)_{1})= 1 + x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Polynomial Ordering

**ACK(s( x), s(y)) -> ACK(s(x), y)**

ack(0,y) -> s(y)

ack(s(x), 0) -> ack(x, s(0))

ack(s(x), s(y)) -> ack(x, ack(s(x),y))

innermost

The following dependency pair can be strictly oriented:

ACK(s(x), s(y)) -> ACK(s(x),y)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(ACK(x)_{1}, x_{2})= x _{2}_{ }^{ }_{ }^{ }POL(s(x)_{1})= 1 + x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Polo

...

→DP Problem 3

↳Dependency Graph

ack(0,y) -> s(y)

ack(s(x), 0) -> ack(x, s(0))

ack(s(x), s(y)) -> ack(x, ack(s(x),y))

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes