Isabelle [28, 30] is an interactive theorem prover that supports a variety of logics. It represents rules as propositions (not as functions) and builds proofs by combining rules. These operations constitute a meta-logic (or `logical framework') in which the object-logics are formalized. Isabelle is now based on higher-order logic --- a precise and well-understood foundation. Examples illustrate use of this meta-logic to formalize logics and proofs. Axioms for first-order logic are shown sound and complete. Backwards proof is formalized by meta-reasoning about object-level entailment. Higher-order logic has several practical advantages over other meta-logics. Many proof techniques are known, such as Huet's higher-order unification procedure. Key words: higher-order logic, higher-order unification, Isabelle, LCF, logical frameworks, meta-reasoning, natural deduction Contents 1 History and overview 2 2 The meta-logic M 4 2.1 Syntax of the meta-logic ......................... 4 2.2 ...

A proof-theoretic characterization of logical languages that form suitable bases for Prolog-like programming languages is provided. This characterization is based on the principle that the declarative meaning of a logic program, provided by provability in a logical system, should coincide with its operational meaning, provided by interpreting logical connectives as simple and fixed search instructions. The operational semantics is formalized by the identification of a class of cut-free sequent proofs called uniform proofs. A uniform proof is one that can be found by a goal-directed search that respects the interpretation of the logical connectives as search instructions. The concept of a uniform proof is used to define the notion of an abstract logic programming language, and it is shown that first-order and higher-order Horn clauses with classical provability are examples of such a language. Horn clauses are then generalized to hereditary Harrop formulas and it is shown that first-order and higher-order versions of this new class of formulas are also abstract logic programming languages if the inference rules are those of either intuitionistic or minimal logic. The programming language significance of the various generalizations to first-order Horn clauses is briefly discussed.

A meta programming language must be able to represent and manipulate such syntactic structures as programs, formulas, types, and proofs. A common characteristic of all these structures is that they involve notions of abstractions, scope, bound and free variables, substitution instances, and equality up to alphabetic changes of bound variables.

We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science --- LICS'96 (E. Clarke editor), pp. 264--275, New Brunswick, NJ, July 27--30 1996. mal basis for a conservative extension of the LF logical framework. LLF combines the expressive power of dependent types with linear logic to permit the natural and concise representation of a whole new class of deductive systems, namely those dealing with state. As an example we encode a version of Mini-ML with references including its type system, its operational semantics, and a proof of type preservation. Another example is the encoding of a sequent calculus for classical linear logic and its cut elimination theorem. LLF can also be given an operational interpretation as a logic programming language under which the representations above can be used for type inference, evaluation and cut-elimination. 1 Introduction A logical framework is a formal system desig...

Unification problems are identified with conjunctions of equations between simply typed λ-terms where free variables in the equations can be universally or existentially quantified. Two schemes for simplifying quantifier alternation, called Skolemization and raising (a dual of Skolemization), are presented. In this setting where variables of functional type can be quantified and not all types contain closed terms, the naive generalization of firstorder Skolemization has several technical problems that are addressed. The method of searching for pre-unifiers described by Huet is easily extended to the mixed prefix setting, although solving flexible-flexible unification problems is undecidable since types may be empty. Unification problems may have numerous incomparable unifiers. Occasionally, unifiers share common factors and several of these are presented. Various optimizations on the general unification search problem are as discussed. 1.

Abstract: λProlog is a logic programming language that extends Prolog by incorporating notions of higher-order functions, λ-terms, higher-order unification, polymorphic types, and mechanisms for building modules and secure abstract data types. These new features are provided in a principled fashion by extending the classical first-order theory of Horn clauses to the intuitionistic higher-order theory of hereditary Harrop formulas. The justification for considering this extension a satisfactory logic programming language is provided through the proof-theoretic notion of a uniform proof. The correspondence between each extension to Prolog and the new features in the stronger logical theory is discussed. Also discussed are various aspects of an experimental implementation of λProlog. Appears in the Fifth International Conference Symposium on Logic Programming, 15 – 19 August 1988, Seattle, Washington. This is a slightly corrected version of

We show that the problem of partial type inference in the nthb-order polymorphic X-calculus is equivalent to nth-order unification. On the one hand, this means that partial type inference in polymorphic X-calculi of order 2 or higher is undecidable. On the other hand, higher-order unification is often tractable in practice, and our translation entails a very useful algorithm for partial type inference in the w-order polymorphic X-calculus. We present an implementation in AProlog in full.

Formal reasoning about computer programs can be based directly on the semantics of the programming language, or done in a special purpose logic like Hoare logic. The advantage of the first approach is that it guarantees that the formal reasoning applies to the language being used (it is well known, for example, that Hoare’s assignment axiom fails to hold for most programming languages). The advantage of the second approach is that the proofs can be more direct and natural. In this paper, an attempt to get the advantages of both approaches is described. The rules of Hoare logic are mechanically derived from the semantics of a simple imperative programming language (using the HOL system). These rules form the basis for a simple program verifier in which verification conditions are generated by LCF-style tactics whose validations use the derived Hoare rules. Because Hoare logic is derived, rather than postulated, it is straightforward to mix semantic and axiomatic reasoning. It is also straightforward to combine the constructs of Hoare logic with other application-specific notations. This is briefly illustrated for various logical constructs, including termination statements, VDM-style ‘relational’ correctness specifications, weakest precondition statements and dynamic logic formulae. The theory underlying the work presented here is well known. Our contribution is to propose a way of mechanizing this theory in a way that makes certain practical details work out smoothly.

A generalization of Horn clauses to a higher-order logic is described and examined as a basis for logic programming. In qualitative terms, these higher-order Horn clauses are obtained from the first-order ones by replacing first-order terms with simply typed #-terms and by permitting quantification over all occurrences of function symbols and some occurrences of predicate symbols. Several proof-theoretic results concerning these extended clauses are presented. One result shows that although the substitutions for predicate variables can be quite complex in general, the substitutions necessary in the context of higher-order Horn clauses are tightly constrained. This observation is used to show that these higher-order formulas can specify computations in a fashion similar to first-order Horn clauses. A complete theorem proving procedure is also described for the extension. This procedure is obtained by interweaving higher-order unification with backchaining and goal reductions, and constitutes a higher-order generalization of SLD-resolution. These results have a practical realization in the higher-order logic programming language called λProlog.

: First-order Horn clause logic can be extended to a higher-order setting in which function and predicate symbols can be variables and terms are replaced with simply typed -terms. For such a logic programming language to be complete in principle, it must incorporate higher-order unification. Although higher-order unification is more complex than usual first-order unification, its availability makes writing certain kinds of programs far more straightforward. In this paper, we present such programs written in a higherorder version of Prolog called Prolog. These programs manipulate structures, such as formulas and programs, which contain abstractions or bound variables. We show how a simple natural deduction theorem prover can be implemented in this language. Similarly we demonstrate how several simple program transformers for a functional programming language can be written in Prolog. These Prolog programs exploit the availability of -conversion and higher-order unification to elegantly ...