On some modifications and applications of the post correspondence problem

The Post Correspondence Problem was introduced by Emil Post in 1946. The problem considers pairs of lists of sequences of symbols, or words, where each word has its place on the list determined by its index. The Post Correspondence Problem asks does there exist a sequence of indices so that, when we write the words in the order of the sequence as single words from both lists, the two resulting words are equal. Post proved the problem to be undecidable, that is, no algorithm deciding it can exist. A variety of restrictions and modifications have been introduced to the original formulation of the problem, that have then been shown to be either decidable or undecidable. Both the original Post Correspondence Problem and its modifications have been widely used in proving other decision problems undecidable. In this thesis we consider some modifications of the Post Correspondence Problem as well as some applications of it in undecidability proofs. We consider a modification for sequences of indices that are infinite to two directions. We also consider a modification to the original Post Correspondence Problem where instead of the words being equal for a sequence of indices, we take two sequences that are conjugates of each other. Two words are conjugates if we can write one word by taking the other and moving some part of that word from the end to the beginning. Both modifications are shown to be undecidable. We also use the Post Correspondence Problem and its modification for injective morphisms in proving two problems from formal language theory to be undecidable; the first problem is on special shuffling of words and the second problem on fixed points of rational functions

Representing Graph Families with Edge Grammars

An edge grammar is a formal mechanism for representing families of related graphs (binary trees, hypercubes, meshes, etc.). Given an edge grammar, larger graphs in the family are derived from simple basis graphs using edge rewriting rules. A drawback to many graph grammars is that they cannot represent some important, highly regular graph families such as the family of shuffie-exchange graphs. Edge grammars, however, exist for all "computable " graph families, and simple edge gramma.rs exist for most regular graph families. In this paper, we define and illuskate edge grammars and analyze them in the context of formal language theory. Our results include hierarchy and decidability properties. Since this work originally was motivated by a need to represent graph families found in parallel computation, the application of edge grammars in this context is also discussed

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

Freezing 1-Tag Systems with States

We study 1-tag systems with states obeying the freezing property that only allows constant bounded number of rewrites of symbols. We look at examples of languages accepted by such systems, the accepting power of the model, as well as certain closure properties and decision problems. Finally we discuss a restriction of the system where the working alphabet must match the input alphabet.Comment: In Proceedings AFL 2023, arXiv:2309.0112

On the decidability of homomorphism equivalence for languages

AbstractWe consider decision problems of the following type. Given a language L and two homomorphisms h1 and h2, one has to determine to what extent h1 and h2 agree on L. For instance, we say that h1 and h2 are equivalent on L if h1(ω) = h2(ω) holds for each ω ε L. In our main theorem we present an algorithm for deciding whether two given homomorphisms are equivalent on a given context-free language. This result also gives an algorithm for deciding whether the translations defined by two deterministic gsm mappings agree on a given context-free language

Asynchronous Multiparty Session Type Implementability is Decidable - Lessons Learned from Message Sequence Charts

Multiparty session types (MSTs) provide efficient means to specify and verify asynchronous message-passing systems. For a global type, which specifies all interactions between roles in a system, the implementability problem asks whether there are local specifications for all roles such that their composition is deadlock-free and generates precisely the specified executions. Decidability of the implementability problem is an open question. We answer it positively for global types with sender-driven choice, which allow a sender to send to different receivers upon branching and a receiver to receive from different senders. To achieve this, we generalise results from the domain of high-level message sequence charts (HMSCs). This connection also allows us to comprehensively investigate how HMSC techniques can be adapted to the MST setting. This comprises techniques to make the problem algorithmically more tractable as well as a variant of implementability that may open new design space for MSTs. Inspired by potential performance benefits, we introduce a generalisation of the implementability problem that we, unfortunately, prove to be 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