Question

I am reading a paper “Termination of term rewriting using dependency pairs” (Thomas Arts, Jurgen Giesl). In the example:

minus (x,0) -> x
minus (s(x), s(y)) -> minus (x,y)
quot (0, s(y)) -> 0
quote (s(x), s(y)) -> s (quot (minus (x, y), s(y)))

It said: “However, the TRS above is not compatible with a simplification ordering, because the left-hand side of the last quot-rule is embedded in its right-hand side if y is instantiated with s (x). Therefore these techniques cannot prove termination of this TRS"

I am not understand about the “if y is instantiated with s (x)“. Could you please if possible help me to understand it?
PS: if this side is not the place to ask this kind of question, could you please help me to know where I can ask? Thank you very much for your help

Answer 1

I am a PhD student in Jürgen Giesl’s group, so let’s see 🙂

Consider the case that you “arrive” at this rule with a term quot(s(x), s(s(x)). This is exactly what the sentence “if y is istantiated with s(x)” means.

Then the rule can be applied, giving s(quot(minus(x, s(x)), s(s(x)))). As you can see, the original term is embedded in the resulting term (read: by only considering subterms one can yield the original term). Whenever this is the case, no simplification order can be used to prove termination (but Dependency Pairs help).

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