(*<*)

theory Paper

imports "../Myhill" "LaTeXsugar"

begin

declare [[show_question_marks = false]]

consts

 REL :: "(string \<times> string) \<Rightarrow> bool"

 UPLUS :: "'a set \<Rightarrow> 'a set \<Rightarrow> (nat \<times> 'a) set"

abbreviation

  "EClass x R \<equiv> R `` {x}"

notation (latex output)

  str_eq_rel ("\<approx>\<^bsub>_\<^esub>") and

  str_eq ("_ \<approx>\<^bsub>_\<^esub> _") and

  Seq (infixr "\<cdot>" 100) and

  Star ("_\<^bsup>\<star>\<^esup>") and

  pow ("_\<^bsup>_\<^esup>" [100, 100] 100) and

  Suc ("_+1" [100] 100) and

  quotient ("_ \<^raw:\ensuremath{\!\sslash\!}> _" [90, 90] 90) and

  REL ("\<approx>") and

  UPLUS ("_ \<^raw:\ensuremath{\uplus}> _" [90, 90] 90) and

  L ("\<^raw:\ensuremath{\cal{L}}>'(_')" [0] 101) and

  Lam ("\<lambda>'(_')" [100] 100) and 

  Trn ("'(_, _')" [100, 100] 100) and 

  EClass ("\<lbrakk>_\<rbrakk>\<^bsub>_\<^esub>" [100, 100] 100) and

  transition ("_ \<^raw:\ensuremath{\stackrel{\text{>_\<^raw:}}{\Longmapsto}}> _" [100, 100, 100] 100) and

(*>*)

section {* Introduction *}

text {*

  Regular languages are an important and well-understood subject in Computer

  Science, with many beautiful theorems and many useful algorithms. There is a

  wide range of textbooks on this subject, many of which are aimed at students

  and contain very detailed ``pencil-and-paper'' proofs

  (e.g.~\cite{Kozen97}). It seems natural to exercise theorem provers by

  formalising these theorems and by verifying formally the algorithms.

  There is however a problem: the typical approach to regular languages is to

  introduce finite automata and then define everything in terms of them.  For

  example, a regular language is normally defined as one whose strings are

  recognised by a finite deterministic automaton. This approach has many

  benefits. Among them is the fact that it is easy to convince oneself that

  regular languages are closed under complementation: one just has to exchange

  the accepting and non-accepting states in the corresponding automaton to

  obtain an automaton for the complement language.  The problem, however, lies with

  formalising such reasoning in a HOL-based theorem prover, in our case

  Isabelle/HOL. Automata are build up from states and transitions that 

  need to be represented as graphs, matrices or functions, none

  of which can be defined as inductive datatype. 

  In case of graphs and matrices, this means we have to build our own

  reasoning infrastructure for them, as neither Isabelle/HOL nor HOL4 nor

  HOLlight support them with libraries. Even worse, reasoning about graphs and

  matrices can be a real hassle in HOL-based theorem provers.  Consider for

  example the operation of sequencing two automata, say $A_1$ and $A_2$, by

  connecting the accepting states of $A_1$ to the initial state of $A_2$:  

  \begin{center}

  \begin{tabular}{ccc}

  \begin{tikzpicture}[scale=0.8]

  %\draw[step=2mm] (-1,-1) grid (1,1);

  \draw[rounded corners=1mm, very thick] (-1.0,-0.3) rectangle (-0.2,0.3);

  \draw[rounded corners=1mm, very thick] ( 0.2,-0.3) rectangle ( 1.0,0.3);

  \node (A) at (-1.0,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};

  \node (B) at ( 0.2,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};

  \node (C) at (-0.2, 0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};

  \node (D) at (-0.2,-0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};

  \node (E) at (1.0, 0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};

  \node (F) at (1.0,-0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};

  \node (G) at (1.0,-0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};

  \draw (-0.6,0.0) node {\footnotesize$A_1$};

  \draw ( 0.6,0.0) node {\footnotesize$A_2$};

  \end{tikzpicture}

  & 

  \raisebox{1.1mm}{\bf\Large$\;\;\;\Rightarrow\,\;\;$}

  &

  \begin{tikzpicture}[scale=0.8]

  %\draw[step=2mm] (-1,-1) grid (1,1);

  \draw[rounded corners=1mm, very thick] (-1.0,-0.3) rectangle (-0.2,0.3);

  \draw[rounded corners=1mm, very thick] ( 0.2,-0.3) rectangle ( 1.0,0.3);

  \node (A) at (-1.0,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};

  \node (B) at ( 0.2,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};

  \node (C) at (-0.2, 0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};

  \node (D) at (-0.2,-0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};

  \node (E) at (1.0, 0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};

  \node (F) at (1.0,-0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};

  \node (G) at (1.0,-0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};

  \draw (C) to [very thick, bend left=45] (B);

  \draw (D) to [very thick, bend right=45] (B);

  \draw (-0.6,0.0) node {\footnotesize$A_1$};

  \draw ( 0.6,0.0) node {\footnotesize$A_2$};

  \end{tikzpicture}

  \end{tabular}

  \end{center}

  \noindent

  On ``paper'' we can define the corresponding graph in terms of the disjoint 

  union of the state nodes. Unfortunately in HOL, the standard definition for disjoint 

  union, namely 

  %

  \begin{equation}\label{disjointunion}

  @{term "UPLUS A\<^isub>1 A\<^isub>2 \<equiv> {(1, x) | x. x \<in> A\<^isub>1} \<union> {(2, y) | y. y \<in> A\<^isub>2}"}

  \end{equation}

  \noindent

  changes the type---the disjoint union is not a set, but a set of pairs. 

  Using this definition for disjoint unions means we do not have a single type for automata

  automata, since there is no type quantification available in HOL. An

  alternative, which provides us with a single type for automata, is to give every 

  state node an identity, for example a natural

  number, and then be careful to rename these identities apart whenever

  connecting two automata. This results in clunky proofs

  establishing that properties are invariant under renaming. Similarly,

  connecting two automata represented as matrices results in very adhoc

  constructions, which are not pleasant to reason about.

  Functions are much better supported in Isabelle/HOL, but they still lead to similar

  problems as with graphs.  Composing, for example, two non-deterministic automata in parallel

  poses again the problem of how to implement disjoint unions. Nipkow \cite{Nipkow98} 

  dismisses the option of using identities, because it leads to ``messy proofs''. He

  \begin{quote}

  \it ``If the reader finds the above treatment in terms of bit lists revoltingly

  \end{quote}

  \noindent

  Moreover, it is not so clear how to conveniently impose a finiteness condition 

  upon functions in order to represent \emph{finite} automata. The best is

  Because of these problems to do with representing automata, there seems

  to be no substantial formalisation of automata theory and regular languages 

  carried out in a HOL-based theorem prover. Nipkow establishes in 

  \cite{Nipkow98} the link between regular expressions and automata in

  are carried out in Nuprl \cite{Constable00} and in Coq (for example 

  \cite{Filliatre97}).

  In this paper, we will not attempt to formalise automata theory in

  Isabelle/HOL, but take a completely different approach to regular

  languages. Instead of defining a regular language as one where there exists

  an automaton that recognises all strings of the language, we define a

  regular language as:

  \begin{definition}

  A language @{text A} is \emph{regular}, provided there is a regular expression that matches all

  strings of @{text "A"}.

  \end{definition}

  \noindent

  The reason is that regular expressions, unlike graphs, matrices and functons, can

  be easily defined as inductive datatype. Consequently a corresponding reasoning 

  infrastructure comes for free. This has recently been exploited in HOL4 with a formalisation

  of regular expression matching based on derivatives \cite{OwensSlind08}.  The purpose of this paper is to

  show that a central result about regular languages---the Myhill-Nerode theorem---can 

  be recreated by only using regular expressions. This theorem gives necessary

  and sufficient conditions for when a language is regular. As a corollary of this

  theorem we can easily establish the usual closure properties, including 

  complementation, for regular languages.\smallskip

  \noindent

  {\bf Contributions:} 

  There is an extensive literature on regular languages.

  To our knowledge, our proof of the Myhill-Nerode theorem is the

  first that is based on regular expressions, only. We prove the part of this theorem 

  stating that a regular expression has only finitely many partitions using certain 

  tagging-functions. Again to our best knowledge, these tagging functions have

  not been used before to establish the Myhill-Nerode theorem.

*}

section {* Preliminaries *}

text {*

  Strings in Isabelle/HOL are lists of characters with the \emph{empty string}

  are sets of strings. The language containing all strings is written in

  Isabelle/HOL as @{term "UNIV::string set"}. The concatenation of two languages 

  is written @{term "A ;; B"} and a language raised to the power @{text n} is written 

  @{term "A \<up> n"}. Their definitions are

  \begin{center}

  @{thm Seq_def[THEN eq_reflection, where A1="A" and B1="B"]}

  \hspace{7mm}

  @{thm pow.simps(1)[THEN eq_reflection, where A1="A"]}

  \hspace{7mm}

  @{thm pow.simps(2)[THEN eq_reflection, where A1="A" and n1="n"]}

  \end{center}

  \noindent

  where @{text "@"} is the usual list-append operation. The Kleene-star of a language @{text A}

  is defined as the union over all powers, namely @{thm Star_def}. In the paper

  we will make use of the following properties of these constructions.

  \begin{proposition}\label{langprops}\mbox{}\\

  \end{tabular}

  \end{proposition}

  \noindent

  our formalisation.\footnote{Available at ???}

  The notation in Isabelle/HOL for the quotient of a language @{text A} according to an 

  equivalence relation @{term REL} is @{term "A // REL"}. We will write 

  @{text "\<lbrakk>x\<rbrakk>\<^isub>\<approx>"} for the equivalence class defined 

  as @{text "{y | y \<approx> x}"}.

  Central to our proof will be the solution of equational systems

  involving sets of languages. For this we will use Arden's lemma \cite{Brzozowski64}

  which solves equations of the form @{term "X = A ;; X \<union> B"} in case

  @{term "[] \<notin> A"}. However we will need the following ``reverse'' 

  version of Arden's lemma.

  \begin{lemma}[Reverse Arden's Lemma]\label{arden}\mbox{}\\

  If @{thm (prem 1) arden} then

  @{thm (lhs) arden} has the unique solution

  @{thm (rhs) arden}.

  \end{lemma}

  \begin{proof}

  For the right-to-left direction we assume @{thm (rhs) arden} and show

  that @{thm (lhs) arden} holds. From Prop.~\ref{langprops}@{text "(i)"} 

  we have @{term "A\<star> = {[]} \<union> A ;; A\<star>"},

  which is equal to @{term "A\<star> = {[]} \<union> A\<star> ;; A"}. Adding @{text B} to both 

  sides gives @{term "B ;; A\<star> = B ;; ({[]} \<union> A\<star> ;; A)"}, whose right-hand side

  is equal to @{term "(B ;; A\<star>) ;; A \<union> B"}. This completes this direction. 

  For the other direction we assume @{thm (lhs) arden}. By a simple induction

  on @{text n}, we can establish the property

  \begin{center}

  @{text "(*)"}\hspace{5mm} @{thm (concl) arden_helper}

  \end{center}

  \noindent

  Using this property we can show that @{term "B ;; (A \<up> n) \<subseteq> X"} holds for

  all @{text n}. From this we can infer @{term "B ;; A\<star> \<subseteq> X"} using the definition

  of @{text "\<star>"}.

  For the inclusion in the other direction we assume a string @{text s}

  with length @{text k} is element in @{text X}. Since @{thm (prem 1) arden}

  we know by Prop.~\ref{langprops}@{text "(ii)"} that 

  @{term "s \<notin> X ;; (A \<up> Suc k)"} since its length is only @{text k}

  (the strings in @{term "X ;; (A \<up> Suc k)"} are all longer). 

  From @{text "(*)"} it follows then that

  @{term s} must be element in @{term "(\<Union>m\<in>{0..k}. B ;; (A \<up> m))"}. This in turn

  implies that @{term s} is in @{term "(\<Union>n. B ;; (A \<up> n))"}. Using Prop.~\ref{langprops}@{text "(iii)"} 

  this is equal to @{term "B ;; A\<star>"}, as we needed to show.\qed

  \end{proof}

  \noindent

  Regular expressions are defined as the inductive datatype

  \begin{center}

  @{text r} @{text "::="}

  @{term NULL}\hspace{1.5mm}@{text"|"}\hspace{1.5mm} 

  @{term EMPTY}\hspace{1.5mm}@{text"|"}\hspace{1.5mm} 

  @{term "CHAR c"}\hspace{1.5mm}@{text"|"}\hspace{1.5mm} 

  @{term "SEQ r r"}\hspace{1.5mm}@{text"|"}\hspace{1.5mm} 

  @{term "ALT r r"}\hspace{1.5mm}@{text"|"}\hspace{1.5mm} 

  @{term "STAR r"}

  \end{center}

  \noindent

  and the language matched by a regular expression is defined as

  \begin{center}

  \begin{tabular}{c@ {\hspace{10mm}}c}

  \begin{tabular}{rcl}

  @{thm (lhs) L_rexp.simps(1)} & @{text "\<equiv>"} & @{thm (rhs) L_rexp.simps(1)}\\

  @{thm (lhs) L_rexp.simps(2)} & @{text "\<equiv>"} & @{thm (rhs) L_rexp.simps(2)}\\

  @{thm (lhs) L_rexp.simps(3)[where c="c"]} & @{text "\<equiv>"} & @{thm (rhs) L_rexp.simps(3)[where c="c"]}\\

  \end{tabular}

  &

  \begin{tabular}{rcl}

  @{thm (lhs) L_rexp.simps(4)[where ?r1.0="r\<^isub>1" and ?r2.0="r\<^isub>2"]} & @{text "\<equiv>"} &

       @{thm (rhs) L_rexp.simps(4)[where ?r1.0="r\<^isub>1" and ?r2.0="r\<^isub>2"]}\\

  @{thm (lhs) L_rexp.simps(5)[where ?r1.0="r\<^isub>1" and ?r2.0="r\<^isub>2"]} & @{text "\<equiv>"} &

       @{thm (rhs) L_rexp.simps(5)[where ?r1.0="r\<^isub>1" and ?r2.0="r\<^isub>2"]}\\

  @{thm (lhs) L_rexp.simps(6)[where r="r"]} & @{text "\<equiv>"} &

      @{thm (rhs) L_rexp.simps(6)[where r="r"]}\\

  \end{tabular}

  \end{tabular}

  \end{center}

  @{text "\<epsilon>"} to define @{term "\<Uplus>rs"}. This function, roughly speaking, folds @{const ALT} over the 

  set @{text rs} with @{const NULL} for the empty set. We can prove that for finite sets @{text rs}

  \begin{center}

  @{thm (lhs) folds_alt_simp}@{text "= \<Union> (\<calL> ` rs)"}

  \end{center}

  \noindent

  holds, whereby @{text "\<calL> ` rs"} stands for the 

  image of the set @{text rs} under function @{text "\<calL>"}.

*}

section {* Finite Partitions Imply Regularity of a Language *}

text {*

  The key definition in the Myhill-Nerode theorem is the

  \emph{Myhill-Nerode relation}, which states that w.r.t.~a language two 

  strings are related, provided there is no distinguishing extension in this

  language. This can be defined as:

  \begin{definition}[Myhill-Nerode Relation]\mbox{}\\

  @{thm str_eq_def[simplified str_eq_rel_def Pair_Collect]}

  \end{definition}

  \noindent

  It is easy to see that @{term "\<approx>A"} is an equivalence relation, which

  partitions the set of all strings, @{text "UNIV"}, into a set of disjoint

  equivalence classes. An example is the regular language containing just

  into the three equivalence classes @{text "X\<^isub>1"}, @{text "X\<^isub>2"} and  @{text "X\<^isub>3"}

  as follows

  \begin{center}

  @{text "X\<^isub>1 = {[]}"}\hspace{5mm}

  @{text "X\<^isub>2 = {[c]}"}\hspace{5mm}

  @{text "X\<^isub>3 = UNIV - {[], [c]}"}

  \end{center}

  One direction of the Myhill-Nerode theorem establishes 

  that if there are finitely many equivalence classes, then the language is 

  regular. In our setting we therefore have to show:

  \begin{theorem}\label{myhillnerodeone}

  @{thm[mode=IfThen] hard_direction}

  \end{theorem}

  \noindent

  To prove this theorem, we first define the set @{term "finals A"} as those equivalence

  classes that contain strings of @{text A}, namely

  %

  \begin{equation} 

  @{thm finals_def}

  \end{equation}

  \noindent

  It is straightforward to show that in general @{thm lang_is_union_of_finals} and 

  @{thm finals_in_partitions} hold. 

  Therefore if we know that there exists a regular expression for every

  equivalence class in @{term "finals A"} (which by assumption must be

  Our proof of Thm.~\ref{myhillnerodeone} relies on a method that can calculate a

  regular expression for \emph{every} equivalence class, not just the ones 

  in @{term "finals A"}. We

  %

  \begin{equation} 

  @{thm transition_def}

  \end{equation}

  \noindent

  @{text X}. Note that we do not define an automaton here, we merely relate two sets

  (with the help of a regular expression). In our concrete example we have 

  Next we build an equational system that

  contains an equation for each equivalence class. Suppose we have 

  the equivalence classes @{text "X\<^isub>1,\<dots>,X\<^isub>n"}, there must be one and only one that

  contains the empty string @{text "[]"} (since equivalence classes are disjoint).

  Let us assume @{text "[] \<in> X\<^isub>1"}. We build the following equational system

  \begin{center}

  \begin{tabular}{rcl}

  @{text "X\<^isub>1"} & @{text "="} & @{text "(Y\<^isub>1\<^isub>1, CHAR c\<^isub>1\<^isub>1) + \<dots> + (Y\<^isub>1\<^isub>p, CHAR c\<^isub>1\<^isub>p) + \<lambda>(EMPTY)"} \\

  @{text "X\<^isub>2"} & @{text "="} & @{text "(Y\<^isub>2\<^isub>1, CHAR c\<^isub>2\<^isub>1) + \<dots> + (Y\<^isub>2\<^isub>o, CHAR c\<^isub>2\<^isub>o)"} \\

  & $\vdots$ \\

  @{text "X\<^isub>n"} & @{text "="} & @{text "(Y\<^isub>n\<^isub>1, CHAR c\<^isub>n\<^isub>1) + \<dots> + (Y\<^isub>n\<^isub>q, CHAR c\<^isub>n\<^isub>q)"}\\

  \end{tabular}

  \end{center}

  \noindent

  where the pairs @{text "(Y\<^isub>i\<^isub>j, CHAR c\<^isub>i\<^isub>j)"} stand for all transitions 

  single ``initial'' state in the equational system, which is different from

  states. We are forced to set up the equational system in this way, because 

  the Myhill-Nerode relation determines the ``direction'' of the transitions. 

  The successor ``state'' of an equivalence class @{text Y} can be reached by adding 

  characters to the end of @{text Y}. This is also the reason why we have to use 

  our reverse version of Arden's lemma.) 

  Overloading the function @{text L} for the two kinds of terms in the 

  \begin{center}

  @{thm L_rhs_item.simps(1)[where r="r", THEN eq_reflection]}

  \end{center}

  \noindent

  we can prove for @{text "X\<^isub>2\<^isub>.\<^isub>.\<^isub>n"} that the following equations

  %

  \begin{equation}\label{inv1}

  @{text "X\<^isub>i = \<calL>(Y\<^isub>i\<^isub>1, CHAR c\<^isub>i\<^isub>1) \<union> \<dots> \<union> \<calL>(Y\<^isub>i\<^isub>q, CHAR c\<^isub>i\<^isub>q)"}.

  \end{equation}

  \noindent

  hold. Similarly for @{text "X\<^isub>1"} we can show the following equation

  %

  \begin{equation}\label{inv2}

  @{text "X\<^isub>1 = \<calL>(Y\<^isub>i\<^isub>1, CHAR c\<^isub>i\<^isub>1) \<union> \<dots> \<union> \<calL>(Y\<^isub>i\<^isub>p, CHAR c\<^isub>i\<^isub>p) \<union> \<calL>(\<lambda>(EMPTY))"}.

  \end{equation}

  \noindent

  The reason for adding the @{text \<lambda>}-marker to our equational system is 

  the other terms contain the empty string. 

  equational system into a \emph{solved form} maintaining the invariants

  \eqref{inv1} and \eqref{inv2}. From the solved form we will be able to read

  off the regular expressions. 

  In order to transform an equational system into solved form, we have two main 

  operations: one that takes an equation of the form @{text "X = rhs"} and removes

  the recursive occurences of @{text X} in @{text rhs} using our variant of Arden's 

  and substitutes @{text X} throughout the rest of the equational system

  \begin{center}