Bounded Combinatory Logic

In combinatory logic one usually assumes a fixed set of basic combinators (axiom schemes), usually K and S. In this setting the set of provable formulas (inhabited types) is PSPACE-complete in simple types and undecidable in intersection types. When arbitrary sets of axiom schemes are considered, the inhabitation problem is undecidable even in simple types (this is known as Linial-Post theorem). k-bounded combinatory logic with intersection types arises from combinatory logic by imposing the bound k on the depth of types (formulae) which may be substituted for type variables in axiom schemes. We consider the inhabitation (provability) problem for k-bounded combinatory logic: Given an arbitrary set of typed combinators and a type tau, is there a combinatory term of type tau in k-bounded combinatory logic? Our main result is that the problem is (k+2)-EXPTIME complete for k-bounded combinatory logic with intersection types, for every fixed k (and hence non-elementary when k is a parameter). We also show that the problem is EXPTIME-complete for simple types, for all k. Theoretically, our results give new insight into the expressive power of intersection types. From an application perspective, our results are useful as a foundation for composition synthesis based on combinatory logic

Computability in constructive type theory

We give a formalised and machine-checked account of computability theory in the Calculus of Inductive Constructions (CIC), the constructive type theory underlying the Coq proof assistant. We first develop synthetic computability theory, pioneered by Richman, Bridges, and Bauer, where one treats all functions as computable, eliminating the need for a model of computation. We assume a novel parametric axiom for synthetic computability and give proofs of results like Rice’s theorem, the Myhill isomorphism theorem, and the existence of Post’s simple and hypersimple predicates relying on no other axioms such as Markov’s principle or choice axioms. As a second step, we introduce models of computation. We give a concise overview of definitions of various standard models and contribute machine-checked simulation proofs, posing a non-trivial engineering effort. We identify a notion of synthetic undecidability relative to a fixed halting problem, allowing axiom-free machine-checked proofs of undecidability. We contribute such undecidability proofs for the historical foundational problems of computability theory which require the identification of invariants left out in the literature and now form the basis of the Coq Library of Undecidability Proofs. We then identify the weak call-by-value λ-calculus L as sweet spot for programming in a model of computation. We introduce a certifying extraction framework and analyse an axiom stating that every function of type ℕ → ℕ is L-computable.Wir behandeln eine formalisierte und maschinengeprüfte Betrachtung von Berechenbarkeitstheorie im Calculus of Inductive Constructions (CIC), der konstruktiven Typtheorie die dem Beweisassistenten Coq zugrunde liegt. Wir entwickeln erst synthetische Berechenbarkeitstheorie, vorbereitet durch die Arbeit von Richman, Bridges und Bauer, wobei alle Funktionen als berechenbar behandelt werden, ohne Notwendigkeit eines Berechnungsmodells. Wir nehmen ein neues, parametrisches Axiom für synthetische Berechenbarkeit an und beweisen Resultate wie das Theorem von Rice, das Isomorphismus Theorem von Myhill und die Existenz von Post’s simplen und hypersimplen Prädikaten ohne Annahme von anderen Axiomen wie Markov’s Prinzip oder Auswahlaxiomen. Als zweiten Schritt führen wir Berechnungsmodelle ein. Wir geben einen kompakten Überblick über die Definition von verschiedenen Berechnungsmodellen und erklären maschinengeprüfte Simulationsbeweise zwischen diesen Modellen, welche einen hohen Konstruktionsaufwand beinhalten. Wir identifizieren einen Begriff von synthetischer Unentscheidbarkeit relativ zu einem fixierten Halteproblem welcher axiomenfreie maschinengeprüfte Unentscheidbarkeitsbeweise erlaubt. Wir erklären solche Beweise für die historisch grundlegenden Probleme der Berechenbarkeitstheorie, die das Identifizieren von Invarianten die normalerweise in der Literatur ausgelassen werden benötigen und nun die Basis der Coq Library of Undecidability Proofs bilden. Wir identifizieren dann den call-by-value λ-Kalkül L als sweet spot für die Programmierung in einem Berechnungsmodell. Wir führen ein zertifizierendes Extraktionsframework ein und analysieren ein Axiom welches postuliert dass jede Funktion vom Typ N→N L-berechenbar ist

Logical Systems On the Concept, Expressive Power and Expressiveness Characterizations

[ES] Esta tesis es una investigación sobre los conceptos principales ocurriendo en los teoremas tipo-Lindström, esto es, el concepto de sistema lógico y el concepto de expresividad. Lindström, entre otros resultados similares, caracterizó la lógica de primer orden como siendo máximamente expresiva entre las lógicas que tienen compacidad y la propriedad de Löwenheim-Skolem. Para tal, él tenía que dar una definición precisa de qué es una lógica y qué es una relación de expresividad. Tales resultados suelen ser usados para extraer conclusiones fuertes sobre la naturaleza de la lógica y, específicamente, de la lógica de primer orden. No obstante, con la excepción de una discusión inicial en el libro sobre lógicas modelo-teóricas editado por Barwise y Feferman, hay pocas discusiones conceptuales de estos resultados de caracterización en la literatura. Específicamente, nos parece problemática la falta de justificación de por qué cierto concepto de sistema lógico y cierto concepto de expresividad fueran elegidos. Nuestro propósito es de contribuir a esta discusión y también proponer un criterio de expresividad más amplio, por medio de traducciones entre lógicas

A Study on the Epistemic Situation Calculus in the Framework of Granular Reasoning

公立千歳科学技術大学令和1年

Noncomparabilities & Non Standard Logics

Many normative theories set forth in the welfare economics, distributive justice and cognate literatures posit noncomparabilities or incommensurabilities between magnitudes of various kinds. In some cases these gaps are predicated on metaphysical claims, in others upon epistemic claims, and in still others upon political-moral claims. I show that in all such cases they are best given formal expression in nonstandard logics that reject bivalence, excluded middle, or both. I do so by reference to an illustrative case study: a contradiction known to beset John Rawls\u27s selection and characterization of primary goods as the proper distribuendum in any distributively just society. The contradiction is avoided only by reformulating Rawls\u27s claims in a nonstandard form, which form happens also to cohere quite attractively with Rawls\u27s intuitive argumentation on behalf of his claims

Decidability for Residuated Lattices and Substructural Logics

We present a number of results related to the decidability and undecidability of various varieties of residuated lattices and their corresponding substructural logics. The context of this analysis is the extension of residuated lattices by various simple equations, dually, the extension of substructural logics by simple structural rules, with the aim of classifying simple equations by the decidability properties shared by their extensions. We also prove a number of relationships among simple extensions by showing the equational theory of their idempotent semiring reducts coincides with simple extensions of idempotent semirings. On the decidability front, we develop both semantical and syntactical methods for establishing decidability as well as tractability of decision procedures. On the undecidability front, we develop a notion of algebraic machines for which the theory of residuated frames will allow us to encode decision problems within the theories of residuated lattices and their substructural analogues. We prove the undecidability of the word problem for a broad class of simple extensions for both commutative and non-commutative residuated lattices. Furthermore, through a deduction theorem we establish the undecidability of the equational theory for a broad class of simple extensions. Translated in terms of substructural logics, we prove that the undecidability of both provability and deducibility for a multitude of extensions of FLe by simple rules

Noncomparabilities & Non Standard Logics

Many normative theories set forth in the welfare economics, distributive justice and cognate literatures posit noncomparabilities or incommensurabilities between magnitudes of various kinds. In some cases these gaps are predicated on metaphysical claims, in others upon epistemic claims, and in still others upon political-moral claims. I show that in all such cases they are best given formal expression in nonstandard logics that reject bivalence, excluded middle, or both. I do so by reference to an illustrative case study: a contradiction known to beset John Rawls\u27s selection and characterization of primary goods as the proper distribuendum in any distributively just society. The contradiction is avoided only by reformulating Rawls\u27s claims in a nonstandard form, which form happens also to cohere quite attractively with Rawls\u27s intuitive argumentation on behalf of his claims