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# setfunctions: Encapsulate non-deterministic computations

This package contains an implementation of set functions. The general idea of set functions is described in:

S. Antoy, M. Hanus: Set Functions for Functional Logic Programming Proc. 11th International Conference on Principles and Practice of Declarative Programming (PPDP’09), pp. 73-82, ACM Press, 2009

The general concept of set functions is as follows. If `f` is an n-ary function, then `(setn f)` is a set-valued function that collects all non-determinism caused by f (but not the non-determinism caused by evaluating arguments!) in a set. Thus, `(setn f a1 ... an)` returns the set of all values of `(f b1 ... bn)` where `b1`,…,`bn` are values of the arguments `a1`,…,`an` (i.e., the arguments are evaluated “outside” this capsule so that the non-determinism caused by evaluating these arguments is not captured in this capsule but yields several results for `(setn...)`. Similarly, logical variables occuring in `a1`,…,`an` are not bound inside this capsule (in PAKCS they cause a suspension until they are bound).

Remark: Since there is no special syntax for set functions, one has to write `(setn f)` for the set function of the n-ary top-level function `f`. The correct usage of set functions is currently not checked by the compiler, i.e., one can also write unintended uses like `set0 ((+1) (1 ? 2))`. In order to check the correct use of set functions, it is recommended to apply the tool CurryCheck on Curry programs which reports illegal uses of set functions (among other properties).

The set of values returned by a set function is represented by an abstract type `Values` on which several operations are defined in this module. Actually, it is a multiset of values, i.e., duplicates are not removed.

The handling of failures and nested occurrences of set functions is not specified in the previous paper. Thus, a detailed description of the semantics of set functions as implemented in this library can be found in the paper

J. Christiansen, M. Hanus, F. Reck, D. Seidel: A Semantics for Weakly Encapsulated Search in Functional Logic Programs Proc. 15th International Conference on Principles and Practice of Declarative Programming (PPDP’13), pp. 49-60, ACM Press, 2013

Note that the implementation of this library uses multisets instead of sets. Thus, the result of a set function might contain multiple values. From a declarative point of view, this is not relevant. It has the advantage that equality is not required on values, i.e., encapsulated values can also be functional.

The PAKCS implementation of set functions has several restrictions, in particular:

1. The multiset of values is completely evaluated when demanded. Thus, if it is infinite, its evaluation will not terminate even if only some elements (e.g., for a containment test) are demanded. However, for the emptiness test, at most one value will be computed
2. The arguments of a set function are strictly evaluated before the set functions itself will be evaluated.
3. If the multiset of values contains unbound variables, the evaluation suspends.