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### mindfa -- minimize a DFA
record dfa(Q,S,d,q0,F) # a DFA
procedure main()
x := getdfa()
every 1 to 10 do
showdfa("Reduced",minimize(showdfa("Original",x)))
end
## - getdfa() -- accept a dfa from input, return it
##
procedure getdfa()
local Q,S,d,q0,F
local q,a
Q := readset("Enter states (1 character names): ")
S := readset("Enter input alphabet: ")
F := readset("Enter Final states (subset of states): ")
writes("What is the start state? ")
q0 := read()
d := table()
every q := !Q & a := !S do {
writes("enter delta(",q,",",a,") = ")
d[q||":"||a] := read()
}
return dfa(Q,S,d,q0,F)
end
## readset(s) - get a set
#
procedure readset(s)
local t1
writes(s)
t1 := []
every put(t1,!cset(read())) # the cset removes duplicates
return t1
end
## showdfa(msg,D) -- show a dfa
#
procedure showdfa(msg,D)
local q,a
every 1 to 3 do write()
write(msg," Deterministic Finite Automaton is:")
write()
write("\t(Q,S,delta,q0,F)")
write()
write("where:")
write()
writeset("Q",D.Q)
writeset("S",D.S)
writeset("F",D.F)
write("\tStart state is ",D.q0)
write("\tDelta: ")
every q := !D.Q do {
every writes("\td(",q,",",a := !D.S,") = ",D.d[q||":"||a])
write()
}
return D
end
## writeset(msg,s) -- display a set
#
procedure writeset(msg,s)
local tmp
tmp := ""
every tmp ||:= !s || ","
write("\t",msg," = {",tmp[1:-1],"}")
return
end
## minimize(D) -- minimize a dfa
#
global distab, dlists
procedure minimize(D)
local F,QF
local p,q,a,cs
distab := table()
dlists := table()
F := D.F
QF := diff(D.Q,D.F)
every p := !F & q := !QF do
distab[cset(p||q)] := "X"
every ((p := !F & q := !F) |
(p := !QF & q := !QF)) & p ~== q do
if \distab[cset(D.d[p||":"||(a:=!D.S)]||D.d[q||":"||a])] then {
distab[cset(p||q)] := "X"
marklists(dlists[cset(p||q)])
}
else
every a := !D.S do
if D.d[p||":"||a] ~== D.d[q||":"||a] then {
cs := cset(D.d[p||":"||a]||D.d[q||":"||a])
if cs == cset(p||q) then next
/dlists[cs] := []
put(dlists[cs],cset(p||q))
}
return makemdfa(D,distab)
end
## marklists(l) -- recursively mark the pair of nodes
# on list l.
procedure marklists(l)
local e
if /l then return
every e := !l do {
distab[e] := "X"
marklists(dlists[e])
}
return
end
## makemdfa(D,DT) -- Use the table from the minimization
# to construct the minimal dfa
procedure makemdfa(D,DT)
local elist, etab, qset, tlist, echeck
local p, q, Delta, q0
etab := table() # table of new states
qset := ''
every p := !D.Q do {
qset ++:= p
plike := equiv(p,etab) | cset(p)
every q := !diff(D.Q,qset) & p ~== q do
if /distab[cset(p||q)] then {
plike ++:= equiv(q,etab) | q
}
etab[plike] := plike
}
tlist := []
elist := []
Delta := table()
q0 := equiv(D.q0,etab) # start state of reduced machine
put(tlist,q0)
put(elist,q0) # only worry about states reachable
# from [q0]
echeck := table() # keep track of states
echeck[q0] := q0
while q := get(tlist) do
every a := !D.S do {
Delta[q||":"||a] := equivdelta(q,a,D,etab)
if /echeck[Delta[q||":"||a]] then {
echeck[Delta[q||":"||a]] := Delta[q||":"||a]
put(tlist,Delta[q||":"||a])
put(elist,Delta[q||":"||a])
}
}
return dfa(elist,D.S,Delta,q0,finalstates(D,elist))
end
## equiv(q,el) -- return the equivalence class in el containing q
#
procedure equiv(q,el)
every p := !el do
if p++q == p then return p
end
## equivdelta(p,a,D,el) -- apply delta to equiv. classes
#
procedure equivdelta(p,a,D,el)
local q, r
q := !p # any state in equiv. class p
r := D.d[q||":"||a] # find state in original dfa
return equiv(r,el) # return its equivalence class
end
## finalstates(D,el) -- build the set of final states
#
procedure finalstates(D,el)
local flist, p, q
ftab := table()
every p := !D.F do
ftab[q := equiv(p,el)] := q
flist := []
every put(flist,(!sort(ftab))[1])
return flist
end
## diff(l1,l2) -- return the difference of two sets
#
procedure diff(l1,l2)
local l,t1,t2
t1 := ''
every t1 ++:= !l1
t2 := ''
every t2 ++:= !l2
l := []
every put(l,!(t1--t2))
if *l = 0 then fail
return l
end
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