Hereditarily finite (HF) set theory provides a standard universe of sets, but with no infinite sets. Its utility is demonstrated through a formalisation of the theory of regular languages and finite automata, including the Myhill-Nerode theorem and Brzozowski’s minimisation algorithm. The states of an automaton are HF sets, possibly constructed by product, sum, powerset and similar operations.

We study the problem of learning sequential top-down tree-toword transducers (stws). First, we present a Myhill-Nerode characterization of the corresponding class of sequential tree-to-word transformations (ST W). Next, we investigate what learning of stws means, identify fundamental obstacles, and propose a learning model with abstain. Finally, we present a polynomial learning algorithm.

Motivated by a conjecture of Ellentuck concerning fibers f^x(C), f recursive and C an element of one of Barback's "tame models" (Tame models in the isols, Houston J. Math. 12 (1986), 163-175), we study such fibers in the more general context of Nerode semirings. The principal results are that (1) all existentially complete Nerode semirings meet all of their recursive fibers, and (2) not all Ner...

This paper gives a new presentation of Kozen’s proof of Kleene algebra completeness featured in his article A completeness theorem for Kleene algebras and the algebra of regular events. A few new variants are introduced, shortening the proof. Specifically, we directly construct an ε-free automaton to prove an equivalent to Kleene’s representation theorem (implementing Glushkov’s instead of Thom...

Cassel, S. 2015. Learning Component Behavior from Tests. Theory and Algorithms for Automata with Data. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1311. 70 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-9395-0. Formal models are often used to describe the behavior of a computer program or component. Behavioral models have man...

The distinguishability language of a regular language L is the set of words distinguishing between pairs of words under the Myhill-Nerode equivalence induced by L, i.e., between pairs of distinct left quotients of L. The similarity relation induced by a language L is a similarity relation inspired by the Myhill-Nerode equivalence and it was used to obtain compact representation of automata for ...

Consider an extension of the classical model of finite automata operating on finite strings in which the machine simultaneously reads a fixed advice tape — an infinite string A. A deterministic finite-string automaton with advice A is like a deterministic finite-string automaton, except that at step n the next state depends on the current state, the nth symbol of the input w ∈ Σ∗, and the nth s...

We generalize the classical Myhill-Nerode theorem for finite automata to the setting of sequential transducers over unique GCDmonoids, which are cancellative monoids in which every two non-zero elements admit a unique greatest common (left) divisor. We prove that a given formal power series is sequential, if and only if it is directed and our Myhill-Nerode equivalence relation has finite index....

There are many proofs of the Myhill-Nerode theorem using automata. In this library we give a proof entirely based on regular expressions, since regularity of languages can be conveniently defined using regular expressions (it is more painful in HOL to define regularity in terms of automata). We prove the first direction of the MyhillNerode theorem by solving equational systems that involve regu...

We present a formal constructive theory of regular languages consisting of about 1400 lines of Coq/Ssreflect. As representations we consider regular expressions, deterministic and nondeterministic automata, and Myhill and Nerode partitions. We construct computable functions translating between these representations and show that equivalence of representations is decidable. We also establish the...