Monadic Second-Order Logic with Arbitrary Monadic Predicates

We study Monadic Second-Order Logic (MSO) over finite words, extended with (non-uniform arbitrary) monadic predicates. We show that it defines a class of languages that has algebraic, automata-theoretic and machine-independent characterizations. We consider the regularity question: given a language in this class, when is it regular? To answer this, we show a substitution property and the existence of a syntactical predicate. We give three applications. The first two are to give very simple proofs that the Straubing Conjecture holds for all fragments of MSO with monadic predicates, and that the Crane Beach Conjecture holds for MSO with monadic predicates. The third is to show that it is decidable whether a language defined by an MSO formula with morphic predicates is regular.Comment: Conference version: MFCS'14, Mathematical Foundations of Computer Science Journal version: ToCL'17, Transactions on Computational Logi

Uniformly automatic classes of finite structures

We investigate the recently introduced concept of uniformly tree-automatic classes in the realm of parameterized complexity theory. Roughly speaking, a class of finite structures is uniformly tree-automatic if it can be presented by a set of finite trees and a tuple of automata. A tree t encodes a structure and an element of this structure is encoded by a labeling of t. The automata are used to present the relations of the structure. We use this formalism to obtain algorithmic meta-theorems for first-order logic and in some cases also monadic second-order logic on classes of finite Boolean algebras, finite groups, and graphs of bounded tree-depth. Our main concern is the efficiency of this approach with respect to the hidden parameter dependence (size of the formula). We develop a method to analyze the complexity of uniformly tree-automatic presentations, which allows us to give upper bounds for the runtime of the automata-based model checking algorithm on the presented class. It turns out that the parameter dependence is elementary for all the above mentioned classes. Additionally we show that one can lift the FPT results, which are obtained by our method, from a class C to the closure of C under direct products with only a singly exponential blow-up in the parameter dependence

Inkdots as advice for finite automata

We examine inkdots placed on the input string as a way of providing advice to finite automata, and establish the relations between this model and the previously studied models of advised finite automata. The existence of an infinite hierarchy of classes of languages that can be recognized with the help of increasing numbers of inkdots as advice is shown. The effects of different forms of advice on the succinctness of the advised machines are examined. We also study randomly placed inkdots as advice to probabilistic finite automata, and demonstrate the superiority of this model over its deterministic version. Even very slowly growing amounts of space can become a resource of meaningful use if the underlying advised model is extended with access to secondary memory, while it is famously known that such small amounts of space are not useful for unadvised one-way Turing machines.Comment: 14 page

On the Complexity of Infinite Advice Strings

We investigate in this paper a notion of comparison between infinite strings. In a general way, if M is a computation model (e.g. Turing machines) and C a class of objects (e.g. languages), the complexity of an infinite word alpha can be measured with respect to the amount of objects from C that are presentable with machines from M using alpha as an oracle. In our case, the model M is finite automata and the objects C are either recognized languages or presentable structures, known respectively as advice regular languages and advice automatic structures. This leads to several different classifications of infinite words that are studied in detail; we also derive logical and computational equivalent measures. Our main results explore the connections between classes of advice automatic structures, MSO-transductions and two-way transducers. They suggest a closer study of the resulting hierarchy over infinite words

NFA reduction via hypergraph vertex cover approximation

In this thesis, we study in minimum vertex cover problem on the class of k-partite k-uniform hypergraphs. This problem arises when reducing the size of nondeterministic finite automata (NFA) using preorders, as suggested by Champarnaud and Coulon. It has been shown that reducing NFAs using preorders is at least as hard as computing a minimal vertex cover on 3-partite 3-uniform hypergraphs, which is NP-hard. We present several classes of regular languages for which NFAs that recognize them can be optimally reduced via preorders. We introduce an algorithm for approximating vertex cover on k-partite k-uniform hypergraphs based on a theorem by Lovász and explore the use of fractional cover algorithms to improve the running time at the expense of a small increase in the approximation ratio

Canonical Algebraic Generators in Automata Learning

Many methods for the verification of complex computer systems require the existence of a tractable mathematical abstraction of the system, often in the form of an automaton. In reality, however, such a model is hard to come up with, in particular manually. Automata learning is a technique that can automatically infer an automaton model from a system -- by observing its behaviour. The majority of automata learning algorithms is based on the so-called L* algorithm. The acceptor learned by L* has an important property: it is canonical, in the sense that, it is, up to isomorphism, the unique deterministic finite automaton of minimal size accepting a given regular language. Establishing a similar result for other classes of acceptors, often with side-effects, is of great practical importance. Non-deterministic finite automata, for instance, can be exponentially more succinct than deterministic ones, allowing verification to scale. Unfortunately, identifying a canonical size-minimal non-deterministic acceptor of a given regular language is in general not possible: it can happen that a regular language is accepted by two non-isomorphic non-deterministic finite automata of minimal size. In particular, it thus is unclear which one of the automata should be targeted by a learning algorithm. In this thesis, we further explore the issue and identify (sub-)classes of acceptors that admit canonical size-minimal representatives.Comment: PhD thesi