Curry: An Integrated Functional Logic Language

tion 3 with σ(t1 . . . tn) = σ(t ′ 1 . . . t′ n), then at least one of the following conditions holds:
(a) σ(e) = σ(e ′ ) (compatibility of right-hand sides).
(b) σ(c) and σ(c ′ ) are not simultaneously satisfiable (incompatibility of conditions). A decidable
approximation of this condition can be found in [30].
These conditions ensure the confluence of the rules if they are considered as a conditional term
rewriting system [47]. Implementations of Curry may check these conditions and warn the user if
they are not satisfied. There are also more general conditions to ensure confluence [47] which can
be checked instead of the above conditions.
Note that defining equations of higher-type, e.g., f = g if f and g are of type Bool -> Bool, are
formally excluded in order to define a simple operational semantics based on first-order rewriting. 4
For convenience, a defining equation f = g between functions is allowed but will be interpreted in
Curry as syntactic sugar for the corresponding defining equation f x = g x on base types.
2.3.1 Functions vs. Variables
In lazy functional languages, different occurrences of the same variable are shared to avoid multiple
evaluations of identical expressions. For instance, if we apply the rule
double x = x+x
to an expression double t, we obtain the new expression t+t but both occurrences of t denote the
identical expression, i.e., these subterms will be simultaneously evaluated. Thus, several occurrences
of the same variable are always shared, i.e., if one occurrence of an argument variable, which
might bound to an evaluable expression when the function is applied, is evaluated to some value,
all other occurrences of this variable are replaced by the same value (without evaluating these
other occurrences again). This sharing is necessary not only for efficiency reasons but it has also
an influence on the soundness of the operational semantics in the presence of non-deterministic
functions (see also [18]). For instance, consider the non-deterministic function coin defined by the
rules
coin = 0
coin = 1
Thus, the expression coin evaluates to 0 or 1. However, the result values of the expression
(double coin) depend on the sharing of the two occurrences of coin after applying the rule
for double: if both occurrences are shared (as in Curry), the results are 0 or 2, otherwise (without
sharing) the results are 0, 1, or 2. The sharing of argument variables corresponds to the so-called
“call time choice” in the declarative semantics [18]: if a rule is applied to a function call, a unique
value must be assigned to each argument (in the example above: either 0 or 1 must be assigned
to the occurrence of coin when the expression (double coin) is evaluated). This does not mean
3 A substitution σ is a mapping from variables into expressions which is extended to a homomorphism on expressions
by defining σ(f t1 . . . tn) = f σ(t1) . . . σ(tn). {x1 ↦→ e1, . . . , xk ↦→ ek} denotes the substitution σ with σ(xi) = ei
(i = 1, . . . , k) and σ(x) = x for all variables x �∈ {x1, . . . , xk}.
4 Note that this is also necessary in an extension of Curry which allows higher-order rewrite rules, since rules with
lambda-abstractions in left-hand sides which are not of base type may cause a gap between the standard notion of
higher-order rewriting and the corresponding equational theory [41].
6