Models and Equality for Logical Programming

1 I n t r o d u c t i o n This paper argues that some very significant benefits are available to logic programming from using certain concepts from first order model theory, namely: • order-sorted logic and models; • initial models; • interpretation into rLxed models for certain fLxed sorts, functions and relations; and • true semantic equality. These techniques, which are all standard in the theory of abstract data types [17, 22, 14], provide an attractive alternative to the more syntactical and operational approaches generally favored in logic programming. Moreover, they provide a powerful approach that supports: • user-defined abstract data types; • built-in data types; • combined logic and functional programming; and • constraint-based programming, in a way that can utilize standard algorithms for standard problems, such as linear programming. In addition, we suggest that the more recent theory of institutions [I0] may provide conceptual insight and clarification, as well as a broadening of the general scope of logic programming, so as to encompass any logical system satisfying certain simple restrictions. In a sense, this paper is an attempt to explicate our previous paper on Eqlog [II], by giving a fuller account of its mathematical semantics, as well as further details, polemics, and comparisons with the 1Supported in part by Office of Naval Research Contracts N00014-85-C-0417 and N00014-86-C-0450, and a gift from the System Development Foundation. existing literature° One reason that [11] may have been obscure to many readers, is the large number of new ideas that it tried to introduce all at once~ here, we attempt to highlight certain ideas by ignoring others. Among the features of Eqlog deliberately downplayed here are: modules~ both hierarchical and generic; theories and views; and "attributes" of operators (e.g., associativity and commutativity}. Although these features greatly increase the expressive power of Eqlog, they would also distract from the basic foundational and semantic issues that we wish to emphasize here. For similar reasons, this paper does not develop most issues concerning the operational semantics of the various systems that are discussed. Thus, unification, term rewriting, narrowing and resolution are only touched upon. They are discussed in somewhat more detail in [11], and will receive full treatment in [23] and [26]. 1.1 Order -Sor ted Logic Ordinary unsorted logic offers the dubious advantage that anything can be applied to anything; for example, 3 * f i r s t n a ~ e ( a g e ( f a l s e ) ) < 2 birth-placo(teuperature(329)) is a well-formed expression. Although beloved by hackers of Lisp and Prolog, unsorted logic is too permissive. The trouble is that the usual alternative, many-sorted logic, is too restrictive, since it does not support overloading of function symbols such a s _ * for integer, rational, and complex numbers. In addition, an expression like ( -4 / -2) does not, strictly speaking, parse (assuming that factorial only applies to natural numbers). Here, we suggest that order-sor ted logic, with subsorts and operator loading, plus the additional twist of r e t r a c t s (although they are not discussed here; see [14]), really does provide sufficient expressiveness, while still banishing the truly meaningless. Although the specialization of many-sorted logic to many-sorted algebra has been very successfully applied to the theory of abstract data types, many-sorted algebra can produce some very awkward specifications in practice, primarily due to difficulties in handling erroneous expressions, such as dividing by zero in the rationals, or taking the top of an empty stack. In fact there is no really satisfactory way to define either the rationals or stacks with MSA. However, order-sorted algebra overcomes these obstacles through its richer type system, which supports subsorts, overloaded operators, and total functions that would otherwise have to be partial. Moreover, order-sorted algebra is the basis of both OBJ [9] and Eqlog [11]. Finally, order-sorted algebra solves the constructor-selector problem, which, roughly speaking, is to define inverses, called selectors, for constructors; the solution is to restrict selectors to the largest subsorts where they make sense. For example, pop and top are only defined for non-empty stacks. [15] shows not only that order-sorted algebra solves this problem, but also that many-sorted algebra cannot solve it. The essence of order-sorted logic is to provide a subsort partial ordering among the sorts, and to interpret it semantically as subset inclusion, among the carriers of a model, and to support operator overloading that is interpreted as restricting functions to subsorts. Two happy facts are that ordersorted logic is only slightly more difficult than many-sorted logic, and that essentially all results generalize from the many-sorted to the order-sorted case without complication. See [14] for a comprehensive treatment of order-sorted algebra. This paper broadens the logical framework to allow not only algebras, but also models of arbitrary first-order signatures, with both function and predicate symbols, including equality, and gives rules of deduction for Horn clauses in such a logic~ proving their completeness and several other basic results that are directly relevant to our model-theoretic account of logic and functional programming, including initiality and Herbrand theorems.