94 lines
4.1 KiB
Plaintext
94 lines
4.1 KiB
Plaintext
theory RegExp
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imports "Functional-Automata.Execute"
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begin
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term Atom
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value "Star (Times(Plus (Atom(CHR ''a'')) (Atom(CHR ''b''))) (Atom(CHR ''c'')))"
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notation Star ("\<lbrace>(_)\<rbrace>\<^sup>*" [0]100)
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notation Plus (infixr "||" 55)
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notation Times (infixr "~~" 60)
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notation Atom ("\<lfloor>_\<rfloor>" 65)
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(*
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datatype 'a rexp = Empty ("<>")
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| Atom 'a ("\<lfloor>_\<rfloor>" 65)
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| Alt "('a rexp)" "('a rexp)" (infixr "||" 55)
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| Conc "('a rexp)" "('a rexp)" (infixr "~~" 60)
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| Star "('a rexp)" ("\<lbrace>(_)\<rbrace>\<^sup>*" [0]100)
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*)
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definition rep1 :: "'a rexp \<Rightarrow> 'a rexp" ("\<lbrace>(_)\<rbrace>\<^sup>+")
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where "\<lbrace>A\<rbrace>\<^sup>+ \<equiv> A ~~ \<lbrace>A\<rbrace>\<^sup>*"
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definition opt :: "'a rexp \<Rightarrow> 'a rexp" ("\<lbrakk>(_)\<rbrakk>")
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where "\<lbrakk>A\<rbrakk> \<equiv> A || One"
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value "Star (Conc(Alt (Atom(CHR ''a'')) (Atom(CHR ''b''))) (Atom(CHR ''c'')))"
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text\<open>or better equivalently:\<close>
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value "\<lbrace>(\<lfloor>CHR ''a''\<rfloor> || \<lfloor>CHR ''b''\<rfloor>) ~~ \<lfloor>CHR ''c''\<rfloor>\<rbrace>\<^sup>*"
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section\<open>Definition of a semantic function: the ``language'' of the regular expression\<close>
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text\<open> This is just a reminder - already defined in @{theory Regular_Exp} as @{term lang}.\<close>
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text\<open>In the following, we give a semantics for our regular expressions, which so far have
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just been a term language (i.e. abstract syntax). The semantics is a ``denotational semantics'',
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i.e. we give a direct meaning for regular expressions in some universe of ``denotations''.
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This universe of denotations is in our concrete case:\<close>
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definition enabled :: "('a,'\<sigma> set)da \<Rightarrow> '\<sigma> set \<Rightarrow> 'a list \<Rightarrow> 'a list"
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where "enabled A \<sigma> = filter (\<lambda>x. next A x \<sigma> \<noteq> {}) "
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text\<open>Now the denotational semantics for regular expression can be defined on a post-card:\<close>
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fun L :: "'a rexp => 'a lang"
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where L_Emp : "L Zero = {}"
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|L_One: "L One = {[]}"
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|L_Atom: "L (\<lfloor>a\<rfloor>) = {[a]}"
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|L_Un: "L (el || er) = (L el) \<union> (L er)"
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|L_Conc: "L (el ~~ er) = {xs@ys | xs ys. xs \<in> L el \<and> ys \<in> L er}"
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|L_Star: "L (Star e) = Regular_Set.star(L e)"
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text\<open>A more useful definition is the \<close>
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fun L\<^sub>s\<^sub>u\<^sub>b :: "'a::order rexp => 'a lang"
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where L\<^sub>s\<^sub>u\<^sub>b_Emp: "L\<^sub>s\<^sub>u\<^sub>b Zero = {}"
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|L\<^sub>s\<^sub>u\<^sub>b_One: "L\<^sub>s\<^sub>u\<^sub>b One = {[]}"
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|L\<^sub>s\<^sub>u\<^sub>b_Atom: "L\<^sub>s\<^sub>u\<^sub>b (\<lfloor>a\<rfloor>) = {z . \<forall>x. x \<le> a \<and> z=[x]}"
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|L\<^sub>s\<^sub>u\<^sub>b_Un: "L\<^sub>s\<^sub>u\<^sub>b (el || er) = (L\<^sub>s\<^sub>u\<^sub>b el) \<union> (L\<^sub>s\<^sub>u\<^sub>b er)"
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|L\<^sub>s\<^sub>u\<^sub>b_Conc: "L\<^sub>s\<^sub>u\<^sub>b (el ~~ er) = {xs@ys | xs ys. xs \<in> L\<^sub>s\<^sub>u\<^sub>b el \<and> ys \<in> L\<^sub>s\<^sub>u\<^sub>b er}"
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|L\<^sub>s\<^sub>u\<^sub>b_Star: "L\<^sub>s\<^sub>u\<^sub>b (Star e) = Regular_Set.star(L\<^sub>s\<^sub>u\<^sub>b e)"
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definition XX where "XX = (rexp2na example_expression)"
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definition YY where "YY = na2da(rexp2na example_expression)"
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(* reminder from execute *)
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value "NA.accepts (rexp2na example_expression) [0,1,1,0,0,1]"
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value "DA.accepts (na2da (rexp2na example_expression)) [0,1,1,0,0,1]"
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definition zero where "zero = (0::nat)"
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definition one where "one = (1::nat)"
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typ "'a set"
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export_code zero one Suc Int.nat nat_of_integer int_of_integer
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Zero One Atom Plus Times Star
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rexp2na na2da enabled
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NA.accepts DA.accepts
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example_expression
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in SML
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module_name RegExpChecker file "RegExpChecker.sml"
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SML_file "RegExpChecker.sml"
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no_notation Atom ("\<lfloor>_\<rfloor>")
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end
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