Higher-Order Horn Clauses

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 λ-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

A mechanization of sorted higher-order logic based on the resolution principle

The usage of sorts in first-order automated deduction has brought greater conciseness of representation and a considerable gain in efficiency by reducing the search spaces involved. This suggests that sort information can be employed in higher-order theorem proving with similar results. This thesis develops a sorted higher-order logic SUM HOL suitable for automatic theorem proving applications. SUM HOL is based on a sorted Lambda-calculus SUM A->, which is obtained by extending Church';s simply typed Lambda-calculus by a higher-order sort concept including term declarations and functional base sorts. The term declaration mechanism studied here is powerful enough to allow convenient formalization of a large body of mathematics, since it offers natural primitives for domains and codomains of functions, and allows to treat function restriction. Furthermore, it subsumes most other mechanisms for the declaration of sort information known from the literature, and can thus serve as a general framework for the study of sorted higher-order logics. For instance, the term declaration mechanism of SUM HOL subsumes the subsorting mechanism as a derived notion, and hence justifies our special form of subsort inference. We present sets of transformations for sorted higher-order unification and pre-unification, and prove the nondeterministic completeness of the algorithm induced by these transformations. The main technical difficulty of unification in ! is that the analysis of general bindings is much more involved than in the unsorted case, since in the presence of term declarations well-sortedness is not a structural property. This difficulty is overcome by a structure theorem that links the structure of a formula to the structure of its sorting derivation. We develop two notions of set-theoretic semantics for SUM HOL. General SUM-models are a direct generalization of Henkin';s general models to the sorted setting. Since no known machine-oriented calculus can adequately mechanize full extensionality, we generalize general SUM-models further to SUM-model structures, which allow full extensionality to fail. The notions of SUM-model structures and general SUM-models allow us to prove model existence theorems for them. These model-theoretic variants of Andrews unifying principle for type theory'; can be used as a powerful tool in completeness proofs of higher-order calculi. Finally, we use our pre-unification algorithms as a central inference procedure for a sorted higherorder resolution calculus in the spirit of Huet';s Constrained Resolution. This calculus is proven sound and complete with respect to our semantics. It differs from Huet';s calculus by allowing early unification strategies and using variable dependencies. For the completeness proof we make use of our model existence theorem, and prove a strong lifting lemma

A mechanization of sorted higher-order logic based on the resolution principle

The usage of sorts in first-order automated deduction has brought greater conciseness of representation and a considerable gain in efficiency by reducing the search spaces involved. This suggests that sort information can be employed in higher-order theorem proving with similar results. This thesis develops a sorted higher-order logic SUM HOL suitable for automatic theorem proving applications. SUM HOL is based on a sorted Lambda-calculus SUM A->, which is obtained by extending Church\u27;s simply typed Lambda-calculus by a higher-order sort concept including term declarations and functional base sorts. The term declaration mechanism studied here is powerful enough to allow convenient formalization of a large body of mathematics, since it offers natural primitives for domains and codomains of functions, and allows to treat function restriction. Furthermore, it subsumes most other mechanisms for the declaration of sort information known from the literature, and can thus serve as a general framework for the study of sorted higher-order logics. For instance, the term declaration mechanism of SUM HOL subsumes the subsorting mechanism as a derived notion, and hence justifies our special form of subsort inference. We present sets of transformations for sorted higher-order unification and pre-unification, and prove the nondeterministic completeness of the algorithm induced by these transformations. The main technical difficulty of unification in ! is that the analysis of general bindings is much more involved than in the unsorted case, since in the presence of term declarations well-sortedness is not a structural property. This difficulty is overcome by a structure theorem that links the structure of a formula to the structure of its sorting derivation. We develop two notions of set-theoretic semantics for SUM HOL. General SUM-models are a direct generalization of Henkin\u27;s general models to the sorted setting. Since no known machine-oriented calculus can adequately mechanize full extensionality, we generalize general SUM-models further to SUM-model structures, which allow full extensionality to fail. The notions of SUM-model structures and general SUM-models allow us to prove model existence theorems for them. These model-theoretic variants of Andrews unifying principle for type theory\u27; can be used as a powerful tool in completeness proofs of higher-order calculi. Finally, we use our pre-unification algorithms as a central inference procedure for a sorted higherorder resolution calculus in the spirit of Huet\u27;s Constrained Resolution. This calculus is proven sound and complete with respect to our semantics. It differs from Huet\u27;s calculus by allowing early unification strategies and using variable dependencies. For the completeness proof we make use of our model existence theorem, and prove a strong lifting lemma

A Labelled Analytic Theorem Proving Environment for Categorial Grammar

We present a system for the investigation of computational properties of categorial grammar parsing based on a labelled analytic tableaux theorem prover. This proof method allows us to take a modular approach, in which the basic grammar can be kept constant, while a range of categorial calculi can be captured by assigning different properties to the labelling algebra. The theorem proving strategy is particularly well suited to the treatment of categorial grammar, because it allows us to distribute the computational cost between the algorithm which deals with the grammatical types and the algebraic checker which constrains the derivation.Comment: 11 pages, LaTeX2e, uses examples.sty and a4wide.st

Constraint Handling Rules with Binders, Patterns and Generic Quantification

Constraint Handling Rules provide descriptions for constraint solvers. However, they fall short when those constraints specify some binding structure, like higher-rank types in a constraint-based type inference algorithm. In this paper, the term syntax of constraints is replaced by $\lambda$-tree syntax, in which binding is explicit; and a new $\nabla$ generic quantifier is introduced, which is used to create new fresh constants.Comment: Paper presented at the 33nd International Conference on Logic Programming (ICLP 2017), Melbourne, Australia, August 28 to September 1, 2017 16 pages, LaTeX, no PDF figure

Elaboration in Dependent Type Theory

To be usable in practice, interactive theorem provers need to provide convenient and efficient means of writing expressions, definitions, and proofs. This involves inferring information that is often left implicit in an ordinary mathematical text, and resolving ambiguities in mathematical expressions. We refer to the process of passing from a quasi-formal and partially-specified expression to a completely precise formal one as elaboration. We describe an elaboration algorithm for dependent type theory that has been implemented in the Lean theorem prover. Lean's elaborator supports higher-order unification, type class inference, ad hoc overloading, insertion of coercions, the use of tactics, and the computational reduction of terms. The interactions between these components are subtle and complex, and the elaboration algorithm has been carefully designed to balance efficiency and usability. We describe the central design goals, and the means by which they are achieved

Scavenger 0.1: A Theorem Prover Based on Conflict Resolution

This paper introduces Scavenger, the first theorem prover for pure first-order logic without equality based on the new conflict resolution calculus. Conflict resolution has a restricted resolution inference rule that resembles (a first-order generalization of) unit propagation as well as a rule for assuming decision literals and a rule for deriving new clauses by (a first-order generalization of) conflict-driven clause learning.Comment: Published at CADE 201