Free Monads, Intrinsic Scoping, and Higher-Order Preunification

Type checking algorithms and theorem provers rely on unification algorithms. In presence of type families or higher-order logic, higher-order (pre)unification (HOU) is required. Many HOU algorithms are expressed in terms of $\lambda$-calculus and require encodings, such as higher-order abstract syntax, which are sometimes not comfortable to work with for language implementors. To facilitate implementations of languages, proof assistants, and theorem provers, we propose a novel approach based on the second-order abstract syntax of Fiore, data types \`a la carte of Swierstra, and intrinsic scoping of Bird and Patterson. With our approach, an object language is generated freely from a given bifunctor. Then, given an evaluation function and making a few reasonable assumptions on it, we derive a higher-order preunification procedure on terms in the object language. More precisely, we apply a variant of $E$-unification for second-order syntax. Finally, we briefly demonstrate an application of this technique to implement type checking (with type inference) for Martin-L\"of Type Theory, a dependent type theory

Proceedings of Sixth International Workshop on Unification

Swiss National Science Foundation; Austrian Federal Ministry of Science and Research; Deutsche Forschungsgemeinschaft (SFB 314); Christ Church, Oxford; Oxford University Computing Laborator

Automated Deduction – CADE 28

This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions

Assertion level proof planning with compiled strategies

This book presents new techniques that allow the automatic verification and generation of abstract human-style proofs. The core of this approach builds an efficient calculus that works directly by applying definitions, theorems, and axioms, which reduces the size of the underlying proof object by a factor of ten. The calculus is extended by the deep inference paradigm which allows the application of inference rules at arbitrary depth inside logical expressions and provides new proofs that are exponentially shorter and not available in the sequent calculus without cut. In addition, a strategy language for abstract underspecified declarative proof patterns is developed. Together, the complementary methods provide a framework to automate declarative proofs. The benefits of the techniques are illustrated by practical applications.Die vorliegende Arbeit beschäftigt sich damit, das Formalisieren von Beweisen zu vereinfachen, indem Methoden entwickelt werden, um informale Beweise formal zu verifizieren und erzeugen zu können. Dazu wird ein abstrakter Kalkül entwickelt, der direkt auf der Faktenebene arbeitet, welche von Menschen geführten Beweisen relativ nahe kommt. Anhand einer Fallstudie wird gezeigt, dass die abstrakte Beweisführung auf der Fakteneben vorteilhaft für automatische Suchverfahren ist. Zusätzlich wird eine Strategiesprache entwickelt, die es erlaubt, unterspezifizierte Beweismuster innerhalb des Beweisdokumentes zu spezifizieren und Beweisskizzen automatisch zu verfeinern. Fallstudien zeigen, dass komplexe Beweismuster kompakt in der entwickelten Strategiesprache spezifiziert werden können. Zusammen bilden die einander ergänzenden Methoden den Rahmen zur Automatisierung von deklarativen Beweisen auf der Faktenebene, die bisher überwiegend manuell entwickelt werden mussten

Assertion level proof planning with compiled strategies

This book presents new techniques that allow the automatic verification and generation of abstract human-style proofs. The core of this approach builds an efficient calculus that works directly by applying definitions, theorems, and axioms, which reduces the size of the underlying proof object by a factor of ten. The calculus is extended by the deep inference paradigm which allows the application of inference rules at arbitrary depth inside logical expressions and provides new proofs that are exponentially shorter and not available in the sequent calculus without cut. In addition, a strategy language for abstract underspecified declarative proof patterns is developed. Together, the complementary methods provide a framework to automate declarative proofs. The benefits of the techniques are illustrated by practical applications.Die vorliegende Arbeit beschäftigt sich damit, das Formalisieren von Beweisen zu vereinfachen, indem Methoden entwickelt werden, um informale Beweise formal zu verifizieren und erzeugen zu können. Dazu wird ein abstrakter Kalkül entwickelt, der direkt auf der Faktenebene arbeitet, welche von Menschen geführten Beweisen relativ nahe kommt. Anhand einer Fallstudie wird gezeigt, dass die abstrakte Beweisführung auf der Fakteneben vorteilhaft für automatische Suchverfahren ist. Zusätzlich wird eine Strategiesprache entwickelt, die es erlaubt, unterspezifizierte Beweismuster innerhalb des Beweisdokumentes zu spezifizieren und Beweisskizzen automatisch zu verfeinern. Fallstudien zeigen, dass komplexe Beweismuster kompakt in der entwickelten Strategiesprache spezifiziert werden können. Zusammen bilden die einander ergänzenden Methoden den Rahmen zur Automatisierung von deklarativen Beweisen auf der Faktenebene, die bisher überwiegend manuell entwickelt werden mussten

Studies related to the process of program development

The submitted work consists of a collection of publications arising from research carried out at Rhodes University (1970-1980) and at Heriot-Watt University (1980-1992). The theme of this research is the process of program development, i.e. the process of creating a computer program to solve some particular problem. The papers presented cover a number of different topics which relate to this process, viz. (a) Programming methodology programming. (b) Properties of programming languages. aspects of structured. (c) Formal specification of programming languages. (d) Compiler techniques. (e) Declarative programming languages. (f) Program development aids. (g) Automatic program generation. (h) Databases. (i) Algorithms and applications