Liyi li and Elsa Gunter

a method to translate order-sorted algebras to many-sorted algebras

Order-Sorted Algebra (B)

Many-Sorted Algebra (A)

(S,\Phi,\Sigma,E,R)

(S,O,\Phi,\Sigma,E,R)

Example

The O relation (  ) can be viewed as a direct graph where each relation is an edge.

The graph may have different connected components.

For any tow sorts in a connected component in a order-sorted algebra, there is a unique top supersort of them.

\small s \equiv_{\leq} s' := \text{s and s' have a common supersort }

\leq

The operator f and f' are Argument compatible iif:

(\forall_{s_{i}, s'_{i}} | f:\Pi_{1}^{n} s_{i} \rightarrow s \wedge f':\Pi_{1}^{n} s'_{i} \rightarrow s' : s_i \equiv_{\leq} s'_{i} )

AC(f,f') := f and f' are arugment comptaible

An Order-Sorted algebra is sensible iif:

(\forall_{f,f'}| ac(f,f'): s\equiv_{\leq}s')

An Order-Sorted algebra is strictly sensible iif:

(\forall_{f,f'} \ \ | \ \ ac(f,f'): s = s')

(\forall_{f} \ \ | \ \ (\exists_{f'} \ \ | \ \ f':\Pi_{1}^{n} s'_{i} \rightarrow s' \ : \\ (\forall_{f'':\Pi_{1}^{n} s''_{i}} \ | \ ac(f,f'') \ : ac(f',f'') \wedge s'' \leq s' ) ) )

Translation TR

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Translating Operators
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Translating Equations and Terms
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Translating Semantic Rules
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(tr_{\Sigma})

The Order-Sorted algebra must be strictly sensible

Translating Operators

tr_{\Sigma} = tr^{\#}_{\Sigma} \circ tr^{'}_{\Sigma}

1.

tr_{\Sigma}^{'}

find a maximal argument-bounding operator f' for every operator f. Eliminate f if f' is different from f.

2.

tr_{\Sigma}^{\#}

for each pair (s,s') in O, create a unary relation:

\text{Cast\_s\_to\_s'}: s \rightarrow s'

non-core constructors

Translating Equations

E^{\#} = E \cup C_{E}

C_{E}

Set of core equality equations. For every two nodes in    , if there are more than one paths from the first node to the second, a equation is added to connect them.

\leq

Translating Equations (Example)

nat < real, real < AExp, nat < int, int < AExp

There is a new equation

Cast\_nat\_to\_int(Cast\_int\_to\_AExp(A:nat)) = Cast\_nat\_to\_real(Cast\_real\_to\_AExp(A:nat))

Translating terms (Example)

s(A:nat) + B:nat = A:nat + s(B:nat)

There is a new equation

Cast\_int\_to\_AExp(Cast\_nat\_to\_int(s(A:nat))) + Cast\_int\_to\_AExp(Cast\_nat\_to\_int(B:nat)) \\ = \\ Cast\_int\_to\_AExp(Cast\_nat\_to\_int(A:nat)) + Cast\_int\_to\_AExp(Cast\_nat\_to\_int(s(B:nat)))