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

Partial (In)Completeness in Abstract Interpretation

In the abstract interpretation framework, completeness represents an optimal simulation by the abstract operators over the behavior of the concrete operators. This corresponds to an ideal (often rare) feature where there is no loss of information accumulated in abstract computations with respect to the properties encoded by the underlying abstract domains. In this thesis, we deal with the opposite notion of completeness in abstract interpretation, that is, incompleteness, applied to two different contexts: static program analysis and formal languages over the Chomsky's hierarchy. In static program analysis, completeness is a very rare condition to be satisfied in practice and only the straightforward abstractions are complete for all programs, thus, we usually deal with incompleteness. For this reason, we introduce the notion of partial completeness. Partial completeness is a weaker notion of completeness which requires the imprecision of the analysis to be limited. A partially complete abstract interpretation allows some false alarms to be reported, but their number is bounded by a constant. We collect in partial completeness classes all the programs whose abstract interpretations share the same upper bound of imprecision. We then focus on the investigation of the computational limits of the class of partially complete programs with respect to a given abstract domain. Moreover, we show that the class of all partially complete programs is non-recursively enumerable, and its complement is productive whenever we allow an unlimited imprecision in the abstract domain. Finally, we formalize the local partial completeness class within which we require partial completeness only on some specific inputs. We prove that this last class of programs is a recursively enumerable set under a structural hypothesis on the underlying abstract domain, by showing an algorithm capable of proving the local partial completeness of a program with respect to a given abstract domain and an upper bound of imprecision. In formal language theory, we want to study a possible reformulation, by abstract interpretation, of classes of languages in the Chomsky's hierarchy, and, by exploiting the incompleteness of languages abstractions, we want to define separation results between classes of languages. To this end, we do a first step into this direction by studying the relation between indexed languages (recognized by indexed grammars) and context-free languages. Indexed grammars are a generalization of context-free grammars which recognize a proper subset of context-sensitive languages, the so called indexed languages. %The class of languages recognized by indexed grammars is called indexed languages and they correspond to the languages recognized by nested stack automata. For example, indexed grammars can recognize the language ${a^nb^nc^n mid ngeq 1 }$ which is not context-free, but they cannot recognize ${ (ab^n)^n mid ngeq 1}$ which is context-sensitive. Indexed grammars identify a set of languages that are more expressive than context-free languages, while having decidability results that lie in between the ones of context-free and context-sensitive languages. We provide a fixpoint characterization of the languages recognized by an indexed grammar and we study possible ways to abstract, in the abstract interpretation sense, these languages and their grammars into context-free and regular languages. We formalize the separation class between indexed and context-free languages, i.e., all the languages that cannot be generated by a context-free grammar, as an instance of incompleteness of stack elimination abstraction over indexed grammars

Complexity and modeling power of insertion-deletion systems

SISTEMAS DE INSERCIÓN Y BORRADO: COMPLEJIDAD Y CAPACIDAD DE MODELADO El objetivo central de la tesis es el estudio de los sistemas de inserción y borrado y su capacidad computacional. Más concretamente, estudiamos algunos modelos de generación de lenguaje que usan operaciones de reescritura de dos cadenas. También consideramos una variante distribuida de los sistemas de inserción y borrado en el sentido de que las reglas se separan entre un número finito de nodos de un grafo. Estos sistemas se denominan sistemas controlados mediante grafo, y aparecen en muchas áreas de la Informática, jugando un papel muy importante en los lenguajes formales, la lingüística y la bio-informática. Estudiamos la decidibilidad/ universalidad de nuestros modelos mediante la variación de los parámetros de tamaño del vector. Concretamente, damos respuesta a la cuestión más importante concerniente a la expresividad de la capacidad computacional: si nuestro modelo es equivalente a una máquina de Turing o no. Abordamos sistemáticamente las cuestiones sobre los tamaños mínimos de los sistemas con y sin control de grafo.COMPLEXITY AND MODELING POWER OF INSERTION-DELETION SYSTEMS The central object of the thesis are insertion-deletion systems and their computational power. More specifically, we study language generating models that use two string rewriting operations: contextual insertion and contextual deletion, and their extensions. We also consider a distributed variant of insertion-deletion systems in the sense that rules are separated among a finite number of nodes of a graph. Such systems are refereed as graph-controlled systems. These systems appear in many areas of Computer Science and they play an important role in formal languages, linguistics, and bio-informatics. We vary the parameters of the vector of size of insertion-deletion systems and we study decidability/universality of obtained models. More precisely, we answer the most important questions regarding the expressiveness of the computational model: whether our model is Turing equivalent or not. We systematically approach the questions about the minimal sizes of the insertiondeletion systems with and without the graph-control

Complexity Results for Confluence Problems

Abstract. We study the complexity of the confluence problem for re-stricted kinds of semi–Thue systems, vector replacement systems and general trace rewriting systems. We prove that confluence for length– reducing semi–Thue systems is P–complete and that this complexity reduces to NC2 in the monadic case. For length–reducing vector re-placement systems we prove that the confluence problem is PSPACE– complete and that the complexity reduces to NP and P for monadic sys-tems and special systems, respectively. Finally we prove that for special trace rewriting systems, confluence can be decided in polynomial time and that the extended word problem for special trace rewriting systems is undecidable.

Formal models of the extension activity of DNA polymerase enzymes

The study of formal language operations inspired by enzymatic actions on DNA is part of ongoing efforts to provide a formal framework and rigorous treatment of DNA-based information and DNA-based computation. Other studies along these lines include theoretical explorations of splicing systems, insertion-deletion systems, substitution, hairpin extension, hairpin reduction, superposition, overlapping concatenation, conditional concatenation, contextual intra- and intermolecular recombinations, as well as template-guided recombination. First, a formal language operation is proposed and investigated, inspired by the naturally occurring phenomenon of DNA primer extension by a DNA-template-directed DNA polymerase enzyme. Given two DNA strings u and v, where the shorter string v (called the primer) is Watson-Crick complementary and can thus bind to a substring of the longer string u (called the template) the result of the primer extension is a DNA string that is complementary to a suffix of the template which starts at the binding position of the primer. The operation of DNA primer extension can be abstracted as a binary operation on two formal languages: a template language L1 and a primer language L2. This language operation is called L1-directed extension of L2 and the closure properties of various language classes, including the classes in the Chomsky hierarchy, are studied under directed extension. Furthermore, the question of finding necessary and sufficient conditions for a given language of target strings to be generated from a given template language when the primer language is unknown is answered. The canonic inverse of directed extension is used in order to obtain the optimal solution (the minimal primer language) to this question. The second research project investigates properties of the binary string and language operation overlap assembly as defined by Csuhaj-Varju, Petre and Vaszil as a formal model of the linear self-assembly of DNA strands: The overlap assembly of two strings, xy and yz, which share an overlap y, results in the string xyz. In this context, we investigate overlap assembly and its properties: closure properties of various language families under this operation, and related decision problems. A theoretical analysis of the possible use of iterated overlap assembly to generate combinatorial DNA libraries is also given. The third research project continues the exploration of the properties of the overlap assembly operation by investigating closure properties of various language classes under iterated overlap assembly, and the decidability of the completeness of a language. The problem of deciding whether a given string is terminal with respect to a language, and the problem of deciding if a given language can be generated by an overlap assembly operation of two other given languages are also investigated

The productivity of polymorphic stream equations and the composition of circular traversals

This thesis has two independent parts concerned with different aspects of laziness in functional programs. The first part is a theoretical study of productivity for very restricted stream programs. In the second part we define a programming abstraction over a recursive pattern for defining circular traversals modularly. Productivity is in general undecidable. By restricting ourselves to mutually recursive polymorphic stream equations having only three basic operations, namely "head", "tail", and "cons", we aim to prove interesting properties about productivity. Still undecidable for this restricted class of programs, productivity of polymorphic stream functions is equivalent to the totality of their indexing function, which characterise their behaviour in terms of operations on indices. We prove that our equations generate all possible polymorphic stream functions, and therefore their indexing functions are all the computable functions, whose totality problem is indeed undecidable. We then further restrict our language by reducing the numbers of equations and parameters, but despite those constraints the equations retain their expressiveness. In the end we establish that even two non-mutually recursive equations on unary stream functions are undecidable with complexity $Π_2^0$. However, the productivity of a single unary equation is decidable. Circular traversals have been used in the eighties as an optimisation to combine multiple traversals in a single traversal. In particular they provide more opportunities for applying deforestation techniques since it is the case that an intermediate datastructure can only be eliminated if it is consumed only once. Another use of circular programs is in the implementation of attribute grammars in lazy functional languages. There is a systematic transformation to define a circular traversal equivalent to multiple traversals. Programming with this technique is not modular since the individual traversals are merged together. Some tools exist to transform programs automatically and attribute grammars have been suggested as a way to describe the circular traversals modularly. Going to the root of the problem, we identify a recursive pattern that allows us to define circular programs modularly in a functional style. We give two successive implementations, the first one is based on algebras and has limited scope: not all circular traversals can be defined this way. We show that the recursive scheme underlying attribute grammars computation rules is essential to combine circular programs. We implement a generic recursive operation on a novel attribute grammar abstraction, using containers as a parametric generic representation of recursive datatypes. The abstraction makes attribute grammars first-class objects. Such a strongly typed implementation is novel and make it possible to implement a high level embedded language for defining attribute grammars, with many interesting new features promoting modularity

The productivity of polymorphic stream equations and the composition of circular traversals

This thesis has two independent parts concerned with different aspects of laziness in functional programs. The first part is a theoretical study of productivity for very restricted stream programs. In the second part we define a programming abstraction over a recursive pattern for defining circular traversals modularly. Productivity is in general undecidable. By restricting ourselves to mutually recursive polymorphic stream equations having only three basic operations, namely "head", "tail", and "cons", we aim to prove interesting properties about productivity. Still undecidable for this restricted class of programs, productivity of polymorphic stream functions is equivalent to the totality of their indexing function, which characterise their behaviour in terms of operations on indices. We prove that our equations generate all possible polymorphic stream functions, and therefore their indexing functions are all the computable functions, whose totality problem is indeed undecidable. We then further restrict our language by reducing the numbers of equations and parameters, but despite those constraints the equations retain their expressiveness. In the end we establish that even two non-mutually recursive equations on unary stream functions are undecidable with complexity $Π_2^0$. However, the productivity of a single unary equation is decidable. Circular traversals have been used in the eighties as an optimisation to combine multiple traversals in a single traversal. In particular they provide more opportunities for applying deforestation techniques since it is the case that an intermediate datastructure can only be eliminated if it is consumed only once. Another use of circular programs is in the implementation of attribute grammars in lazy functional languages. There is a systematic transformation to define a circular traversal equivalent to multiple traversals. Programming with this technique is not modular since the individual traversals are merged together. Some tools exist to transform programs automatically and attribute grammars have been suggested as a way to describe the circular traversals modularly. Going to the root of the problem, we identify a recursive pattern that allows us to define circular programs modularly in a functional style. We give two successive implementations, the first one is based on algebras and has limited scope: not all circular traversals can be defined this way. We show that the recursive scheme underlying attribute grammars computation rules is essential to combine circular programs. We implement a generic recursive operation on a novel attribute grammar abstraction, using containers as a parametric generic representation of recursive datatypes. The abstraction makes attribute grammars first-class objects. Such a strongly typed implementation is novel and make it possible to implement a high level embedded language for defining attribute grammars, with many interesting new features promoting modularity