220 research outputs found
On the Termination Problem for Probabilistic Higher-Order Recursive Programs
In the last two decades, there has been much progress on model checking of
both probabilistic systems and higher-order programs. In spite of the emergence
of higher-order probabilistic programming languages, not much has been done to
combine those two approaches. In this paper, we initiate a study on the
probabilistic higher-order model checking problem, by giving some first
theoretical and experimental results. As a first step towards our goal, we
introduce PHORS, a probabilistic extension of higher-order recursion schemes
(HORS), as a model of probabilistic higher-order programs. The model of PHORS
may alternatively be viewed as a higher-order extension of recursive Markov
chains. We then investigate the probabilistic termination problem -- or,
equivalently, the probabilistic reachability problem. We prove that almost sure
termination of order-2 PHORS is undecidable. We also provide a fixpoint
characterization of the termination probability of PHORS, and develop a sound
(but possibly incomplete) procedure for approximately computing the termination
probability. We have implemented the procedure for order-2 PHORSs, and
confirmed that the procedure works well through preliminary experiments that
are reported at the end of the article
Denotational and operational preciseness of subtyping: A roadmap
The notion of subtyping has gained an important role both in theoretical and applicative domains: in lambda and concurrent calculi as well as in object-oriented programming languages. The soundness and the completeness, together referred to as the preciseness of subtyping, can be considered from two different points of view: denotational and operational. The former preciseness is based on the denotation of a type, which is a mathematical object describing the meaning of the type in accordance with the denotations of other expressions from the language. The latter preciseness has been recently developed with respect to type safety, i.e. the safe replacement of a term of a smaller type when a term of a bigger type is expected. The present paper shows that standard proofs of operational preciseness imply denotational preciseness and gives an overview on this subject
Pregrammars and Intersection Types
A representation of intersection types in terms of pregrammars is presented. Pregrammar based rewriting relations, corresponding respectively to type checking and inhabitation are defined and the latter is used to implement a Wajsberg/Ben-Yelles style alternating semi-decision algorithm for inhabitation. The usefulness of the framework is illustrated by revisiting and partially extending standard inhabitation related results for intersection types, as well as establishing new ones. It is shown how the notion of bounded multiset dimension emerges naturally and the relation between the two settings is clarified. A meaningful rank independent superset of the set of rank 2 types is identified for which EXPSPACE-completeness for inhabitation as well as for counting is proved. Finally, a standard result on negatively non-duplicated simple types is extended to intersection types
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 Riceâs theorem, the Myhill isomorphism theorem, and the existence of Postâs simple and hypersimple predicates relying on no other axioms such as Markovâs 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 â â â is L-computable.Wir behandeln eine formalisierte und maschinengeprĂŒfte 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 fĂŒr synthetische Berechenbarkeit an und beweisen Resultate wie das Theorem von Rice, das Isomorphismus Theorem von Myhill und die Existenz von Postâs simplen und hypersimplen PrĂ€dikaten ohne Annahme von anderen Axiomen wie Markovâs Prinzip oder Auswahlaxiomen. Als zweiten Schritt fĂŒhren wir Berechnungsmodelle ein. Wir geben einen kompakten Ăberblick ĂŒber die Definition von verschiedenen Berechnungsmodellen und erklĂ€ren maschinengeprĂŒfte Simulationsbeweise zwischen diesen Modellen, welche einen hohen Konstruktionsaufwand beinhalten. Wir identifizieren einen Begriff von synthetischer Unentscheidbarkeit relativ zu einem fixierten Halteproblem welcher axiomenfreie maschinengeprĂŒfte Unentscheidbarkeitsbeweise erlaubt. Wir erklĂ€ren solche Beweise fĂŒr 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-KalkĂŒl L als sweet spot fĂŒr die Programmierung in einem Berechnungsmodell. Wir fĂŒhren ein zertifizierendes Extraktionsframework ein und analysieren ein Axiom welches postuliert dass jede Funktion vom Typ NâN L-berechenbar ist
Dedukti: a Logical Framework based on the -Calculus Modulo Theory
Dedukti is a Logical Framework based on the -Calculus Modulo
Theory. We show that many theories can be expressed in Dedukti: constructive
and classical predicate logic, Simple type theory, programming languages, Pure
type systems, the Calculus of inductive constructions with universes, etc. and
that permits to used it to check large libraries of proofs developed in other
proof systems: Zenon, iProver, FoCaLiZe, HOL Light, and Matita
Foundations of Software Science and Computation Structures
This open access book constitutes the proceedings of the 24th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2021, which was held during March 27 until April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The 28 regular papers presented in this volume were carefully reviewed and selected from 88 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems
Decidability and Algorithmic Analysis of Dependent Object Types (DOT)
Dependent Object Types, or DOT, is a family of calculi developed to study the Scala programming language. These calculi have path dependent types as a feature, and potentially intersection types, union types and recursive types. So far, the study of DOT calculi mostly focuses on the soundness proof, which does not directly contribute to development of
compilers. This thesis presents a detailed investigation of decidability and algorithmic properties of the family of DOT calculi.
In decidability analysis, the undecidability of subtyping of several calculi is formally established, including the D <: and D ⧠calculi. Prior to this investigation, the undecidability of subtyping of all DOT calculi including D <: was open. Decidability analysis puts emphasis on a particular form of subtyping rules, called normal form. It turns out that a normal form definition is not only as expressive, but also more suggestive than the original definition. A conceptual device, called small-step analysis, is introduced to assist converting a usual definition of subtyping to its normal form definition. Moreover, decidability analysis gives direct contributions to the algorithmic analysis, by revealing two decidable fragments of D <: in declarative form, called the kernels. Decidability analysis also suggests a novel subtyping algorithm framework, stare-at subtyping. Stare-at subtyping and an existing algorithm are shown to be sound and complete w.r.t. their corresponding kernels. In algorithmic analysis, stare-at subtyping is extended to other calculi, with more features than D <:, including D
â§, ÎŒDART and jDOT. In ÎŒDART and jDOT, bi-directional type assignment algorithms are developed. The algorithms developed in this thesis are all shown to be sound with respect to their target calculi and terminating.
During the development of the algorithms, analysis shows a number of ways in which the Wadlerfest DOT calculus does not directly correspond to the Scala language, while substantially increases the difficulties of algorithmic design. jDOT, therefore, is developed as an alternative formalization of Scala
- âŠ