Computer Science

In this dissertation, we study two of the global properties of 1-dimensional cellular automata (CAs) under periodic boundary condition, namely, reversibility and randomness. To address reversibility of finite CAs, we develop a mathematical tool, named reachability tree, which can efficiently characterize those CAs. A decision algorithm is proposed using minimized reachability tree which takes a CA rule and size n as input and verifies whether the CA is reversible for that n. To decide reversibility of a finite CA, we need to know both the rule and the CA size. However, for infinite CAs, reversibility is decided based on the local rule only. Therefore, apparently, these two cases seem to be divergent. This dissertation targets to construct a bridge between these two cases. To do so, reversibility of CAs is redefined and the notion of semi-reversible CAs is introduced. Hence, we propose a new classification of finite CAs -(1) reversible CAs, (2) semi-reversible CAs and (3) strictly irreversible CAs. Finally, relation between reversibility of finite and infinite CAs is established. This dissertation also explores CAs as source of randomness and build pseudo-random number generators (PRNGs) based on CAs. We identify a list of properties for a CA to be a good source of randomness. Two heuristic algorithms are proposed to synthesize candidate (decimal) CAs which have great potentiality as PRNGs. Two schemes tare developed o use these CAs as window-based PRNGs - (1) as decimal number generators and as (2) binary number generators. We empirically observe that in comparison to the best PRNG SFMT19937-64, average performance of our proposed PRNGs are slightly better. Hence, our decimal CAs based PRNGs are one of the best PRNGs today.

A word s of letters on edges of underlying graph Γ of deterministic finite automaton (DFA) is called synchronizing if s sends all states of the automaton to a unique state. J. Černy discovered in 1964 a sequence of n -state complete DFA possessing a minimal synchronizing word of length (n−1 ) 2 . The hypothesis, mostly known today as Černy conjecture, claims that (n−1 ) 2 is a precise upper bound on the length of such a word over alphabet Σ of letters on edges of Γ for every complete n -state DFA. The hypothesis was formulated in 1966 by Starke. Algebra with nonstandard operation over special class of matrices induced by words in the alphabet of labels on edges is used to prove the conjecture. The proof is based on the connection between length of words u and dimension of the space generated by solution L x of matrix equation M u L x = M s for synchronizing word s , as well as on relation between ranks of M u and L x . Important role below placed the notion of pseudo inverseL matrix, sometimes reversible.

Different classes of automata on infinite words have different expressive power. Deciding whether a given language L??Σ ? can be expressed by an automaton of a desired class can be reduced to deciding a game between Prover and Refuter: in each turn of the game, Refuter provides a letter in Σ , and Prover responds with an annotation of the current state of the run (for example, in the case of Büchi automata, whether the state is accepting or rejecting, and in the case of parity automata, what the color of the state is). Prover wins if the sequence of annotations she generates is correct: it is an accepting run iff the word generated by Refuter is in L . We show how a winning strategy for Refuter can serve as a simple and easy-to-understand certificate to inexpressibility, and how it induces additional forms of certificates. Our framework handles all classes of deterministic automata, including ones with structural restrictions like weak automata. In addition, it can be used for refuting separation of two languages by an automaton of the desired class, and for finding automata that approximate L and belong to the desired class.

Timed automata (TA) are a well-established formalism for specifying discrete-state/continuous-time behavior of time-critical reactive systems. Concerning the fundamental analysis problem of comparing a candidate implementation against a specification, both given as TA, it has been shown that timed trace equivalence is undecidable, whereas timed bisimulation equivalence is decidable. The corresponding proof utilizes region graphs, a finite, but generally very space-consuming characterization of TA semantics. Hence, most practical TA tools utilize zone graphs instead, a symbolic and generally more efficient representation of TA semantics, to automate analysis tasks. However, zone graphs only produce sound results for analysis tasks being reducible to plain reachability problems thus being too imprecise for checking timed bisimilarity. In this paper, we propose bounded zone-history graphs, a novel characterization of TA semantics facilitating an adjustable trade-off between precision and scalability of timed-bisimilarity checking. Our tool TimBrCheck is, to the best of our knowledge, the only currently available tool for effectively checking timed bisimilarity and even supports non-deterministic TA with silent moves. We further present experimental results gained from applying our tool to a collection of community benchmarks, providing insights into trade-offs between precision and efficiency, depending on the bound value.

Register automata are finite automata equipped with a finite set of registers in which they can store data, i.e. elements from an unbounded or infinite alphabet. They provide a simple formalism to specify the behaviour of reactive systems operating over data {\omega}-words. We study the synthesis problem for specifications given as register automata over a linearly ordered data domain (e.g. (N, {\leq}) or (Q, {\leq})), which allow for comparison of data with regards to the linear order. To that end, we extend the classical Church synthesis game to infinite alphabets: two players, Adam and Eve, alternately play some data, and Eve wins whenever their interaction complies with the specification, which is a language of {\omega}-words over ordered data. Such games are however undecidable, even when the specification is recognised by a deterministic register automaton. This is in contrast with the equality case, where the problem is only undecidable for nondeterministic and universal specifications. Thus, we study one-sided Church games, where Eve instead operates over a finite alphabet, while Adam still manipulates data. We show they are determined, and deciding the existence of a winning strategy is in ExpTime, both for Q and N. This follows from a study of constraint sequences, which abstract the behaviour of register automata, and allow us to reduce Church games to {\omega}-regular games. Lastly, we apply these results to the transducer synthesis problem for input-driven register automata, where each output data is restricted to be the content of some register, and show that if there exists an implementation, then there exists one which is a register transducer.

We prove various results about the largest exponent of a repetition in a factor of the Thue-Morse word, when that factor is considered as a circular word. Our results confirm and generalize previous results of Fitzpatrick and Aberkane & Currie.

Using a new approach based on automatic sequences, logic, and a decision procedure, we reprove some old theorems about circularly squarefree words and unbordered conjugates in a new and simpler way. Furthermore, we prove three new results about unbordered conjugates: we complete the classification, due to Harju and Nowotka, of binary words with the maximum number of unbordered conjugates; we prove that for every possible number, up to the maximum, there exists a word having that number of unbordered conjugates, and finally, we determine the expected number of unbordered conjugates in a random word.

Given, a sequence X of n variables, a time-series constraint ctr using the Sum aggregator, and a sliding time-series constraint enforcing the constraint ctr on each sliding window of X of m consecutive variables, we describe a Θ(n) time complexity checker, as well as a Θ(n) space complexity reformulation for such sliding constraint.

We make certain bounds in Krebs' proof of Cobham's theorem explicit and obtain corresponding upper bounds on the length of a common prefix of an aperiodic a -automatic sequence and an aperiodic b -automatic sequence, where a and b are multiplicatively independent. We also show that an automatic sequence cannot have arbitrarily large factors in common with a Sturmian sequence.

This paper studies a large class of two-player perfect-information turn-based parity games on infinite graphs, namely those generated by collapsible pushdown automata. The main motivation for studying these games comes from the connections from collapsible pushdown automata and higher-order recursion schemes, both models being equi-expressive for generating infinite trees. Our main result is to establish the decidability of such games and to provide an effective representation of the winning region as well as of a winning strategy. Thus, the results obtained here provide all necessary tools for an in-depth study of logical properties of trees generated by collapsible pushdown automata/recursion schemes.