Thue Systems Research Articles

On the spectrum between reaction systems and string rewriting

On the spectrum between reaction systems and string rewriting

Regular Abstractions for Array Systems

Verifying safety and liveness over array systems is a highly challenging problem. Array systems naturally capture parameterized systems such as distributed protocols with an unbounded number of processes. Such distributed protocols often exploit process IDs during their computation, resulting in array systems whose element values range over an infinite domain. In this paper, we develop a novel framework for proving safety and liveness over array systems. The crux of the framework is to overapproximate an array system as a string rewriting system (i.e. over a finite alphabet) by means of a new predicate abstraction that exploits the so-called indexed predicates. This allows us to tap into powerful verification methods for string rewriting systems that have been heavily developed in the last two decades or so (e.g. regular model checking). We demonstrate how our method yields simple, automatically verifiable proofs of safety and liveness properties for challenging examples, including Dijkstra's self-stabilizing protocol and the Chang-Roberts leader election protocol.

A morphological study of fivefold Islamic geometric patterns using formal grammar for computer-aided design applications

This paper presents a morphological study of Islamic geometric patterns (IGP) and their role in application of formal grammar in computational modeling of IGPs. Through a comprehensive literature review and data collection, we analyze the morphological properties of these patterns using techniques such as geometric transformations, pattern classification, and symmetry analysis. Based on our findings, we explore how these properties can be used in constructing a formal grammar through string rewriting system for a CAD application. Building on a study on the potential of a string rewriting system for modeling IGPs, the current research suggests an update to the previous system and introduces a new morphological structure for IGPs. The new method has an expanded sample set and is tested on a class of 5-fold star patterns with 12 members and demonstrates successful development. The results are implemented in a Grasshopper add-on, providing a flexible platform to generate the patterns through strings and to control their parameters. This tool opens up new possibilities to bridge traditional patterns with contemporary technologies and make them more accessible. Furthermore, it contributes to the preservation of IGPs as a significant cultural and architectural heritage, while also advancing the evolution of these patterns to a new and contemporary generation.

An Automated Approach to the Collatz Conjecture

We explore the Collatz conjecture and its variants through the lens of termination of string rewriting. We construct a rewriting system that simulates the iterated application of the Collatz function on strings corresponding to mixed binary–ternary representations of positive integers. We prove that the termination of this rewriting system is equivalent to the Collatz conjecture. We also prove that a previously studied rewriting system that simulates the Collatz function using unary representations does not admit termination proofs via natural matrix interpretations, even when used in conjunction with dependency pairs. To show the feasibility of our approach in proving mathematically interesting statements, we implement a minimal termination prover that uses natural/arctic matrix interpretations and we find automated proofs of nontrivial weakenings of the Collatz conjecture. Although we do not succeed in proving the Collatz conjecture, we believe that the ideas here represent an interesting new approach.

On Causal Equivalence by Tracing in String Rewriting

We introduce proof terms for string rewrite systems and, using these, show that various notions of equivalence on reductions known from the literature can be viewed as different perspectives on the notion of causal equivalence. In particular, we show that permutation equivalence classes (as known from the lambda-calculus and term rewriting) are uniquely represented both by trace graphs (known from physics as causal graphs) and by so-called greedy multistep reductions (as known from algebra). We present effective maps from the former to the latter, topological multi-sorting TM, and vice versa, the proof term algebra [[ ]].

On Parsing Programming Languages with Turing-Complete Parser

A new parsing method based on the semi-Thue system is described. Similar to, but with more efficient implementation than Markov normal algorithms, it can be used for parsing any recursively enumerable language. Despite its computational power, it is meant to be used primarily for parsing programming and domain-specific languages. It enables a straightforward simulation of a number of existing parsing algorithms based on context-free grammars. The list includes both top-down shift-produce methods (such as SLL and LL) and bottom-up shift-reduce methods (such as LALR and LR), as well as mixed top-down-and-bottom-up methods such as LLLR. To justify the use of the new parsing method, the paper provides numerous examples of how a parser can actually be made in practice. It is advised that the main part of the parser is based on some simple well-established approach, e.g., SLL(1), while syntactically more complicated phrases can be parsed by exploiting the full power of the new parser. These phrases may either be extensions to the original language or some embedded domain-specific language. In all such and similar cases, no part of the language is restricted to be context-free. In fact, context-sensitive languages can be handled quite efficiently.

On bi-infinite and conjugate post correspondence problems

We study two modifications of the Post Correspondence Problem (PCP), namely (1) the bi-infinite version, where it is asked whether there exists a bi-infinite word such that two given morphisms agree on it, and (2) the conjugate version, where we require the images of a solution for two given morphisms are conjugates of each other. For the conjugate PCP we give an undecidability proof by reducing it to the word problem for a special type of semi-Thue systems and for the bi-infinite PCP we give a simple argument that it is in the class Σ20 of the arithmetical hierarchy.

From Hertzsprung’s problem to pattern-rewriting systems

Drawing on a problem posed by Hertzsprung in 1887, we say that a given permutation π∈𝒮 n contains the Hertzsprung pattern σ∈𝒮 k if there is a factor π(d+1)π(d+2)⋯π(d+k) of π such that π(d+1)-σ(1)=⋯=π(d+k)-σ(k). Using a combination of the Goulden–Jackson cluster method and the transfer-matrix method we determine the joint distribution of occurrences of any set of (incomparable) Hertzsprung patterns, thus substantially generalizing earlier results by Jackson et al. on the distribution of ascending and descending runs in permutations. We apply our results to the problem of counting permutations up to pattern-replacement equivalences, and using pattern-rewriting systems – a new formalism similar to the much studied string-rewriting systems – we solve a couple of open problems raised by Linton et al. in 2012.

Integer Weighted Automata on Infinite Words

In this paper we combine two classical generalisations of finite automata (weighted automata and automata on infinite words) into a model of integer weighted automata on infinite words and study the universality and the emptiness problems under zero weight acceptance. We show that the universality problem is undecidable for three-state automata by a direct reduction from the infinite Post correspondence problem. We also consider other more general acceptance conditions as well as their complements with respect to the universality and the emptiness problems. Additionally, we build a universal integer weighted automaton with fixed transitions. This automaton has an additional integer input that allows it to simulate any semi-Thue system.

Questions in algebra and mathematical logic. Scientific heritage of S. I. Adian

This is a survey of results on the Burnside problem and properties of Burnside groups, the finite basis problem for group identities, periodic products of groups and Malcev’s problem, construction of groups with special properties (Tarski monsters), constructive bounds in the Burnside- Magnus problem, and algorithmic problems: the problem of recognition of group properties, the word problem for semigroups with one relation, and semi-Thue systems. The focus is on the most important results obtained in papers of Adian and his students. Bibliography: 81 titles.

On prefixal one-rule string rewrite systems

Prefixal one-rule string rewrite systems are one-rule string rewrite systems for which the left-hand side of the rule is a prefix of the right-hand side of the rule. String rewrite systems induce a transformation over languages: from a starting word, one can associate all its descendants. We prove, in this work, that the transformation induced by a prefixal one-rule rewrite system always transforms a finite language into a context-free language, a property that is surprisingly not satisfied by arbitrary one-rule rewrite systems. We also give here a decidable characterization of the prefixal one-rule rewrite systems whose induced transformation is a rational transduction.

Lower Bounds for Synchronizing Word Lengths in Partial Automata

It was conjectured by Černý in 1964, that a synchronizing DFA on [Formula: see text] states always has a synchronizing word of length at most [Formula: see text], and he gave a sequence of DFAs for which this bound is reached. Until now a full analysis of all DFAs reaching this bound was only given for [Formula: see text], and with bounds on the number of symbols for [Formula: see text]. Here we give the full analysis for [Formula: see text], without bounds on the number of symbols. For PFAs (partial automata) on [Formula: see text] states we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding [Formula: see text] for [Formula: see text]. Where DFAs with long synchronization typically have very few symbols, for PFAs we observe that more symbols may increase the synchronizing word length. For PFAs on [Formula: see text] states and two symbols we investigate all occurring synchronizing word lengths. We give series of PFAs on two and three symbols, reaching the maximal possible length for some small values of [Formula: see text]. For [Formula: see text], the construction on two symbols is the unique one reaching the maximal length. For both series the growth is faster than [Formula: see text], although still quadratic. Based on string rewriting, for arbitrary size we construct a PFA on three symbols with exponential shortest synchronizing word length, giving significantly better bounds than earlier exponential constructions. We give a transformation of this PFA to a PFA on two symbols keeping exponential shortest synchronizing word length, yielding a better bound than applying a similar known transformation. Both PFAs are transitive. Finally, we show that exponential lengths are even possible with just one single undefined transition, again with transitive constructions.

The inclusion structure of partially lossy queue monoids and their trace submonoids

We model the behavior of a lossy fifo-queue as a monoid of transformations that are induced by sequences of writing and reading. To have a common model for reliable and lossy queues, we split the alphabet of the queue into two parts: the forgettable letters and the letters that are transmitted reliably. We describe this monoid by means of a confluent and terminating semi-Thue system and then study some of the monoid’s algebraic properties. In particular, we characterize completely when one such monoid can be embedded into another as well as which trace monoids occur as submonoids. Surprisingly, these are precisely those trace monoids that embed into the direct product of two free monoids – which gives a partial answer to a question raised by Diekert et al. at STACS 1995.

Small Semi-Thue System Universal with Respect to the Termination Problem

Small Semi-Thue System Universal with Respect to the Termination Problem

The monoid of queue actions

We model the behavior of a fifo-queue as a monoid of transformations that are induced by sequences of writing and reading. We describe this monoid by means of a confluent and terminating semi-Thue system and study some of its basic algebraic properties such as conjugacy. Moreover, we show that while several properties concerning its rational subsets are undecidable, their uniform membership problem is $${{\mathsf {N}}}{{\mathsf {L}}}$$ -complete. Furthermore, we present an algebraic characterization of this monoid’s recognizable subsets. Finally, we prove that it is not Thurston-automatic.

Parikh Matrices and Parikh Rewriting Systems

Since the introduction of the Parikh matrix mapping, its injectivity problem is on top of the list of open problems in this topic. In 2010 Salomaa provided a solution for the ternary alphabet in terms of a Thue system with an additional feature called counter. This paper proposes the notion of a Pa rikh rewriting system as a generalization and systematization of Salomaa’s result. It will be shown that every Parikh rewriting system induces a Thue system without counters that serves as a feasible solution to the injectivity problem.

Efficient Completion of Weighted Automata

We consider directed graphs with edge labels from a semiring. We present an algorithm that allows efficient execution of queries for existence and weights of paths, and allows updates of the graph: adding nodes and edges, and changing weights of existing edges. We apply this method in the construction of matchbound certificates for automatically proving termination of string rewriting. We re-implement the decomposition/completion algorithm of Endrullis et al. (2006) in our framework, and achieve comparable performance.

On the Termination Problem for String Rewrite Systems

Based on some result given in 2 , concerning the termination of a semi-Thue systeme, we give some results illustrated by some examples to give some Noetherian semi-Thue systems.

On a conjecture about Parikh matrices

Based on Salomaa's characterization of M-equivalence, Atanasiu conjectured that a certain natural generalization of ME-equivalence solves the injectivity problem of Parikh matrices for the ternary alphabet. This paper refutes his conjecture but continues to study the interesting proposed Thue system. Characterization of certain irreducible elementary transformations under this system is obtained. Furthermore, these transformations are further scrutinized in terms of their replaceability by simpler ones.

A canonical automaton for one-rule length-preserving string rewrite systems

In this work, we use rearrangements in rewriting positions sequence in order to study precisely the structure of the derivations in one-rule length-preserving string rewrite systems. That yields to the definition of a letter-to-letter transducer that computes the relation induced by a one-rule length-preserving string rewrite system. This transducer can be seen as an automaton over an alphabet A×A. We prove that this automaton is finite if and only if the corresponding relation is rational. We also identify a sufficient condition for the context-freeness of the language L recognized by this automaton and, when this condition is satisfied, we construct a pushdown automaton that recognizes L.