 # Homework Assignments Week 1.8: Constraint Resolution

## Unification

1. Given the following equalities:

`````` f(a,g()) == d,
h(d) == e,
k(c,x) == k(g(),c),
e == h(f(b,c))
``````

and the following unifier:

``````a -> h(y),
b -> h(y),
c -> g(),
d -> f(h(y), g()),
e -> h(f(h(y), g())),
x -> g()
``````

explain the concept of a most general unifier, and state if the given unifier is most general or not. Illustrate your explanation by giving another unifier for these equalities (less general if the given unifier is most general, or more general otherwise).

2. Given the following equalities:

``````f(g(a,b),h(b,c)) == f(c,e),
g(h(e,a),b) == g(d,b),
h(a,i()) == h(a,b)
``````

give a unifying substitution.

3. Given the following type equalities:

``````f(a) == b,
g(b,d) == g(f(c),f(e)),
h() == e
``````

give a substitution that satisfies the equalities.

4. Explain the concept of principality in terms of your previous answer. Illustrate your answer with a different unifier that would also satisfy the constraints of (3).

## Constraint Semantics

Explain what constraint semantics are, and discuss the relation between semantics and solver in terms of soundness, completeness, and principality.