# HiLog

HiLog is a programming logic with higher-order syntax, which allows arbitrary terms to appear in predicate and function positions. However, the model theory of HiLog is first-order. Although syntactically HiLog strictly extends first order logic, HiLog can be embedded into this logic.

HiLog is described in detail in [1] .[2] It was later extended in the direction of many-sorted logic in.[3] Other contributions to the theory of HiLog include [4] .[5]

The XSB System parses HiLog syntax, but the integration of HiLog into XSB is only partial. In particular, HiLog is not integrated with the XSB module system. A full implementation of HiLog is available in the Flora-2 system.

In,[1] it has been shown that HiLog can be embedded into first-order logic through a fairly simple transformation. For instance, p(X)(Y,Z(V)(W)) gets embedded as the following first-order term:

``` apply(p(X),Y,apply(apply(Z,V),W))
```

Details can be found in.[1]

The Framework for Logic-Based Dialects (RIF-FLD) of the Rule Interchange Format (RIF) is largely based on the ideas underlying HiLog and F-logic.

## Examples

In all the examples, below, capitalized symbols denote variables and the comma denotes logical conjunction, as in most logic programming languages. The first and the second examples show that variables can appear in predicate positions. Predicates can even be complex terms, such as closure(P) or maplist(F) below. The third example shows that variables can also appear in place of atomic formulas, while the fourth example illustrates the use of variables in place of function symbols. The first example defines a generic transitive closure operator, which can be applied to an arbitrary binary predicate. The second example is similar. It defines a LISP-like mapping operator, which applies to an arbitrary binary predicate. The third example shows that the Prolog meta-predicate call/1 can be expressed in HiLog in a natural way and without the use of extra-logical features. The last example defines a predicate that traverses arbitrary binary trees represented as first-order terms.

```  closure(P)(X,Y) <- P(X,Y).
closure(P)(X,Y) <- P(X,Z), closure(P)(Z,Y).

maplist(F)([],[]).
maplist(F)([X|R],[Y|Z]) <- F(X,Y), maplist(F)(R,Z).

call(X) <- X.

traverse(X(L,R)) <- traverse(L), traverse(R).
```

## References

1. ^ a b c W. Chen, M. Kifer and D.S. Warren (1993), HiLog: A Foundation for Higher-Order Logic Programming. Journal of Logic Programming, 1993.
2. ^ W. Chen, M. Kifer and D.S. Warren (1989), HiLog: A first order semantics for higher-order logic programming constructs. Proc. North American Logic Programming Conference, 1989.
3. ^ W. Chen and M. Kifer (1994), Sorted HiLog: Sorts in Higher-Order Logic Data Languages. Int’l Conference on Database Theory, Springer Lecture Notes in Computer Science #893.
4. ^ K.A. Ross (1994), On Negation in HiLog. Journal of Logic Programming, 1994.
5. ^ J. de Bruijn and S. Heymans (2008), On the Relationship between Description Logic-based and F-Logic-based Ontologies. Fundamenta Informaticae 82:3, 2008, pp. 213-236.