Proceedings of the 13 ESSLLI Student Session - Multiple Choices ...

Proceedings of the 13 th ESSLLI Student Session
Example 5 Reconsider the rewrite system Rmult from Example 1. We suppose that the
second argument of addition (add) is safe (safe(add) = {2}) and that all arguments of
multiplication (mult) are normal (safe(mult) = ∅). Furthermore let the precedence >
be defined as mult > add > s. Then Rmult is compatible with >pop∗. As a consequence
of Theorem 4, the number of rewrite steps starting from mult(�n�, �m�) is polynomially
bounded in n and m.
In order to verify compatibility for this particular instance >pop∗ we need to show that
all the rules in Rmult are strictly decreasing with respect to >pop∗, that is l >pop∗ r holds
for l → r ∈ Rmult. To exemplify this, consider the rule add(s(x), y) → s(add(x, y)).
We write 〈i〉 for the i-th case of Definition 3. From s(x) >pop∗ x by rule 〈2〉 we infer
[s(x)](>pop∗)mul[x]. Furthermore [y](> = pop∗)mul[y] holds and thus by rule 〈4〉 we obtain
add(s(x), y) >pop∗ add(x, y). Finally, from this and add > s we conclude by one application
of rule 〈3〉 that add(s(x), y) >pop∗ s(add(x, y)) holds.
2 A Propositional Encoding of POP ∗ with Finite Semantic Labeling
In (Zantema, 1995) the transformation technique semantic labeling is introduced. From
R a labeled TRS Rlab is obtained by labeling the function symbols in R with semantic
information. Semantics are given to R by defining a model. A model is a F-algebra
A, i.e. a carrier A equipped with operations fA : A n → A for every n-ary symbol
f ∈ F, such that for every rule l → r ∈ R and any assignment α : V → A, the equality
[α]A(l) = [α]A(r) holds. Here [α]A(t) denotes the interpretation of t with assignment α,
inductively defined by [α]A(t) = α(t) if t ∈ V and [α]A(t) = fA([α]A(t1), . . . , [α]A(tn))
if t = f(t1, . . . , tn). The system is then labeled according to a labeling ℓ for A, i.e. a set
of mappings ℓf : A n → A for every n-ary function symbol f ∈ F. 1
For every assignment α, the mapping labα(t) is defined by labα(t) = t if t ∈ V and
labα(f(t1, . . . , tn)) = fa(labα(t1), . . . , labα(tn)) where a = ℓf([α]A(t1), . . . , [α]A(tn)).
The labeled TRS Rlab is obtained by labeling all rules for all assignments α, that is
Rlab = {labα(l) → labα(r) | l → r ∈ R and assignment α}.
The main theorem from (Zantema, 1995) states that Rlab is terminating if and only if R
is terminating. In the following, we restrict to algebras B with carrier B = {true, false},
however the approach is extensible to arbitrary finite carriers.
To encode a Boolean function b : B n → B, we make use of unique propositional atoms
bw for every sequence of arguments w = w1, . . . , wn ∈ B n . The atom bw will denote
the result of applying w1, . . . , wn to b. Let a1, . . . , an be propositional formulas. To
impose restrictions on the encoded function b, we introduce the formula �b�(a1, . . . , an)
such that for a satisfying assignment ν the equality ν(�b�(a1, . . . , an)) = bν(a1),...,ν(an)
holds. For instance with �b�(a1, a2) ↔ r we assert that the encoded function b satisfies
b(ν(a1), ν(a2)) = ν(r).
For every assignment α : V → A and term t appearing in R we introduce the atoms
intα,t and labα,t for t �∈ V. The meaning of intα,t will be the result of [α]B(t), labα,t
will denote the label of the root symbol of t under α. In order to ensure this for t =
1 The definition from (Zantema, 1995) allows the labeling of a subset of F and leave other symbols
unchanged. In our context, this has no consequence and simplifies the translation.
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