arXiv subjects

Hans Zantema

Publications and source records attributed to Hans Zantema.

At least 19 recordsLinked to original sources

arXiv2024

Equality of morphic sequences

Morphic sequences form a natural class of infinite sequences, typically defined as the coding of a fixed point of a morphism. Different morphisms and codings may yield the same morphic sequence. This paper investigates how to prove that two such representations of a morphic sequence by morphisms represent the same sequence. In particular, we focus on the smallest representations of the subsequences of the binary Fibonacci sequence obtained by only taking the even or odd elements. The proofs we give are induction proofs of several properties simultaneously, and are typically found fully automatically by a tool that we developed.

cs.SC↗

arXiv2023

Proving Termination of Graph Transformation Systems using Weighted Type Graphs over Semirings

We introduce techniques for proving uniform termination of graph transformation systems, based on matrix interpretations for string rewriting. We generalize this technique by adapting it to graph rewriting instead of string rewriting and by generalizing to ordered semirings. In this way we obtain a framework which is inspired by the tropical and arctic type graphs of [5] and introduces a new variant of arithmetic type graphs. These type graphs can be used to assign weights to graphs and to show that these weights decrease in every rewriting step in order to prove termination. We present an example involving counters and discuss the implementation in the tool Grez.

cs.LO↗

arXiv2023

Characterizing morphic sequences

Morphic sequences form a natural class of infinite sequences, extending the well-studied class of automatic sequences. Where automatic sequences are known to have several equivalent characterizations and the class of automatic sequences is known to have several closure properties, for the class of morphic sequences similar closure properties are known, but only limited equivalent characterizations. In this paper we extend the latter. We discuss a known characterization of morphic sequences based on automata and we give a characterization of morphic sequences by finiteness of a particular class of subsequences. Moreover, we relate morphic sequences to rationality of infinite terms and describe them by infinitary rewriting.

cs.FL↗

arXiv2022

Complexity of Simon's problem in classical sense

Simon's problem is a standard example of a problem that is exponential in classical sense, while it admits a polynomial solution in quantum computing. It is about a function $f$ for which it is given that a unique non-zero vector $s$ exists for which $f(x) = f(x \oplus s)$ for all $x$, where $\oplus$ is the exclusive or operator. The goal is to find $s$. The exponential lower bound for the classical sense assumes that $f$ only admits black box access. In this paper we investigate classical complexity when $f$ is given by a standard representation like a circuit. We focus on finding the vector space of all vectors $s$ for which $f(x) = f(x \oplus s)$ for all $x$, for any given $f$. Two main results are: (1) if $f$ is given by any circuit, then checking whether this vector space contains a non-zero element is NP-hard, and (2) if $f$ is given by any ordered BDD, then a basis of this vector space can be computed in polynomial time.

cs.CC↗

arXiv2021

Deadlock in packet switching networks

A deadlock in a packet switching network is a state in which one or more messages have not yet reached their target, yet cannot progress any further. We formalize three different notions of deadlock in the context of packet switching networks, to which we refer as global, local and weak deadlock. We establish the precise relations between these notions, and prove they characterize different sets of deadlocks. Moreover, we implement checking of deadlock freedom of packet switching networks using the symbolic model checker nuXmv. We show experimentally that the implementation is effective at finding subtle deadlock situations in packet switching networks.

cs.NI↗

arXiv2018

Counting symbol switches in synchronizing automata

Instead of looking at the lengths of synchronizing words as in Černý's conjecture, we look at the switch count of such words, that is, we only count the switches from one letter to another. Where the synchronizing words of the Černý automata $\mathcal{C}_n$ have switch count linear in $n$, we wonder whether synchronizing automata exist for which every synchronizing word has quadratic switch count. The answer is positive: we prove that switch count has the same complexity as synchronizing word length. We give some series of synchronizing automata yielding quadratic switch count, the best one reaching $\frac{2}{3} n^2 + O(n)$ as switch count. We investigate all binary automata on at most 9 states and determine the maximal possible switch count. For all $3\leq n\leq 9$, a strictly higher switch count can be reached by allowing more symbols. This behaviour differs from length, where for every $n$, no automata are known with higher synchronization length than $\mathcal{C}_n$, which has only two symbols. It is not clear if this pattern extends to larger $n$. For $n\geq 12$, our best construction only has two symbols.

cs.FL↗

arXiv2018

Lower Bounds for Synchronizing Word Lengths in Partial Automata

It was conjectured by Černý in 1964, that a synchronizing DFA on $n$ states always has a synchronizing word of length at most $(n-1)^2$, 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 $n \leq 5$, and with bounds on the number of symbols for $n \leq 12$. Here we give the full analysis for $n \leq 7$, without bounds on the number of symbols. For PFAs (partial automata) on $\leq 7$ states we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding $(n-1)^2$ for $n \geq 4$. 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 $\leq 10$ 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 $n$. For $n=6,7,8,9$, the construction on two symbols is the unique one reaching the maximal length. For both series the growth is faster than $(n-1)^2$, 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.

cs.FL↗

arXiv2018

The k-cube is k-representable

A graph is called $k$-representable if there exists a word $w$ over the nodes of the graph, each node occurring exactly $k$ times, such that there is an edge between two nodes $x,y$ if and only after removing all letters distinct from $x,y$, from $w$, a word remains in which $x,y$ alternate. We prove that if $G$ is $k$-representable for $k>1$, then the Cartesian product of $G$ and the complete graph on $n$ nodes is $(k+n-1)$-representable. As a direct consequence, the $k$-cube is $k$-representable for every $k \geq 1$. Our main technique consists of exploring occurrence based functions that replace every $i$th occurrence of a symbol $x$ in a word $w$ by a string $h(x,i)$. The representing word we construct to achieve our main theorem is purely composed from concatenation and occurrence based functions.

math.CO↗

arXiv2018

Solving computational problems in the theory of word-representable graphs

A simple graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ iff $xy\in E$. Word-representable graphs generalize several important classes of graphs. A graph is word-representable iff it admits a semi-transitive orientation. We use semi-transitive orientations to enumerate connected non-word-representable graphs up to the size of 11 vertices, which led to a correction of a published result. Obtaining the enumeration results took 3 CPU years of computation. Also, a graph is word-representable iff it is $k$-representable for some $k$, that is, if it can be represented using $k$ copies of each letter. The minimum such $k$ for a given graph is called graph's representation number. Our computational results in this paper not only include distribution of $k$-representable graphs on at most 9 vertices, but also have relevance to a known conjecture on these graphs. In particular, we find a new graph on 9 vertices with high representation number. Finally, we introduce the notion of a $k$-semi-transitive orientation refining the notion of a semi-transitive orientation, and show computationally that the refinement is not equivalent to the original definition unlike the equivalence of $k$-representability and word-representability.

math.CO↗

arXiv2017

Slowly synchronizing automata with fixed alphabet size

It was conjectured by Černý in 1964 that a synchronizing DFA on $n$ states always has a shortest synchronizing word of length at most $(n-1)^2$, and he gave a sequence of DFAs for which this bound is reached. In this paper, we investigate the role of the alphabet size. For each possible alphabet size, we count DFAs on $n \le 6$ states which synchronize in $(n-1)^2 - e$ steps, for all $e < 2\lceil n/2 \rceil$. Furthermore, we give constructions of automata with any number of states, and $3$, $4$, or $5$ symbols, which synchronize slowly, namely in $n^2 - 3n + O(1)$ steps. In addition, our results prove Černý's conjecture for $n \le 6$. Our computation has led to $27$ DFAs on $3$, $4$, $5$ or $6$ states, which synchronize in $(n-1)^2$ steps, but do not belong to Černý's sequence. Of these $27$ DFA's, $19$ are new, and the remaining $8$ which were already known are exactly the \emph{minimal} ones: they will not synchronize any more after removing a symbol. So the $19$ new DFAs are extensions of automata which were already known, including the Černý automaton on $3$ states. But for $n > 3$, we prove that the Černý automaton on $n$ states does not admit non-trivial extensions with the same smallest synchronizing word length $(n-1)^2$.

cs.FL↗

arXiv2017

DFAs and PFAs with Long Shortest Synchronizing Word Length

It was conjectured by Černý in 1964, that a synchronizing DFA on $n$ states always has a shortest synchronizing word of length at most $(n-1)^2$, 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 $n \leq 4$, and with bounds on the number of symbols for $n \leq 10$. Here we give the full analysis for $n \leq 6$, without bounds on the number of symbols. For PFAs the bound is much higher. For $n \leq 6$ we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding $(n-1)^2$ for $n =4,5,6$. For arbitrary n we give a construction of 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.

math.CO↗

arXiv2017

Synchronizing non-deterministic finite automata

In this paper, we show that every D3-directing CNFA can be mapped uniquely to a DFA with the same synchronizing word length. This implies that Černý's conjecture generalizes to CNFAs and that the general upper bound for the length of a shortest D3-directing word is equal to the Pin-Frankl bound for DFAs. As a second consequence, for several classes of CNFAs sharper bounds are established. Finally, our results allow us to detect all critical CNFAs on at most 6 states. It turns out that only very few critical CNFAs exist.

math.CO↗

arXiv2017

Termination of Cycle Rewriting by Transformation and Matrix Interpretation

We present techniques to prove termination of cycle rewriting, that is, string rewriting on cycles, which are strings in which the start and end are connected. Our main technique is to transform cycle rewriting into string rewriting and then apply state of the art techniques to prove termination of the string rewrite system. We present three such transformations, and prove for all of them that they are sound and complete. In this way not only termination of string rewriting of the transformed system implies termination of the original cycle rewrite system, a similar conclusion can be drawn for non-termination. Apart from this transformational approach, we present a uniform framework of matrix interpretations, covering most of the earlier approaches to automatically proving termination of cycle rewriting. All our techniques serve both for proving termination and relative termination. We present several experiments showing the power of our techniques.

cs.LO↗

arXiv2016

Proceedings 9th International Workshop on Computing with Terms and Graphs

This volume contains the proceedings of TERMGRAPH 2016, the Ninth International Workshop on Computing with Terms and Graphs which was held on April 8, 2016 in Eindhoven, The Netherlands, as a satellite event of the European Joint Conferences on Theory and Practice of Software (ETAPS 2016).

cs.LO↗

arXiv2015

The Degree of Squares is an Atom (Extended Version)

We answer an open question in the theory of degrees of infinite sequences with respect to transducibility by finite-state transducers. An initial study of this partial order of degrees was carried out in (Endrullis, Hendriks, Klop, 2011), but many basic questions remain unanswered. One of the central questions concerns the existence of atom degrees, other than the degree of the `identity sequence' 1 0^0 1 0^1 1 0^2 1 0^3 .... A degree is called an `atom' if below it there is only the bottom degree 0, which consists of the ultimately periodic sequences. We show that also the degree of the `squares sequence' 1 0^0 1 0^1 1 0^4 1 0^9 1 0^{16} ... is an atom. As the main tool for this result we characterise the transducts of `spiralling' sequences and their degrees. We use this to show that every transduct of a `polynomial sequence' either is in 0 or can be transduced back to a polynomial sequence for a polynomial of the same order.

cs.FL↗

arXiv2015

Proving Looping and Non-Looping Non-Termination by Finite Automata

A new technique is presented to prove non-termination of term rewriting. The basic idea is to find a non-empty regular language of terms that is closed under rewriting and does not contain normal forms. It is automated by representing the language by a tree automaton with a fixed number of states, and expressing the mentioned requirements in a SAT formula. Satisfiability of this formula implies non-termination. Our approach succeeds for many examples where all earlier techniques fail, for instance for the S-rule from combinatory logic.

cs.LO↗

arXiv2014

Non-termination using Regular Languages

We describe a method for proving non-looping non-termination, that is, of term rewriting systems that do not admit looping reductions. As certificates of non-termination, we employ regular (tree) automata.

cs.LO↗

arXiv2013

A Rewriting View of Simple Typing

This paper shows how a recently developed view of typing as small-step abstract reduction, due to Kuan, MacQueen, and Findler, can be used to recast the development of simple type theory from a rewriting perspective. We show how standard meta-theoretic results can be proved in a completely new way, using the rewriting view of simple typing. These meta-theoretic results include standard type preservation and progress properties for simply typed lambda calculus, as well as generalized versions where typing is taken to include both abstract and concrete reduction. We show how automated analysis tools developed in the term-rewriting community can be used to help automate the proofs for this meta-theory. Finally, we show how to adapt a standard proof of normalization of simply typed lambda calculus, for the rewriting approach to typing.

cs.PL↗