Sequentiable structures are a subclass of monoids that generalise the free monoids and the monoid of non-negative real numbers with addition. In this paper we consider functions f : Σ∗ → M and define the Myhill-Nerode relation for these functions. We prove that a function of finite index, n, can be represented with a subsequential transducer with n states.

The DFA model of computation has explicit state names for every possible state that a machine might be in. Nevertheless, when writing programs, although we do think about the different states that a program might be in, we do not explicitly label them. We now consider how to identify machine states associated with a language L merely by identifying a particular relationship between strings in L...

The paper introduces recognizable languages as inverse images of sets of arrows from finite categories internal to monoids. The first result is the Myhill-Nerode Theorem as a conservative extension of the classic result for tree languages. The second result shows that a language of planar acyclic circuit diagrams whose gates have non-empty lists of input and output ports is recognizable if, and...

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...

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.

By the Myhill-Nerode Theorem, a tree language L is recognisable if and only if ≡L has finite index. The minimal bottom-up DFTA of L is then defined using the congruence classes of ≡L as states. Exercise 1 (Bottom-up Residuals). Let L ⊆ T (F) and t ∈ T (F). The bottom-up residual of L by t is the set of all contexts C such that C[t] ∈ L: t−1L def = {C ∈ C(F) | C[t] ∈ L} . 1. Show that L is recog...

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...

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...