(Greek letters are written out so this can be read as plain text.)
A general term for an automata.
A machine could be a Turing Machine, a pushdown automata,
a finite state machine or any other restricted version of a Turing machine.
("state" think small and many, one bit of storage)
M = (Q, Sigma, delta, q0, F) is a definition of a Finite Automata.
To be fully defined the following must be provided:
Q a finite set of states often called q0, q1, ... , qn or s0, s1, ... , sn.
There is no requirement, in general, that every state be reachable.
sigma a finite input alphabet
delta a transition function mapping Q cross sigma to Q. delta is typically
given as a table with all states listed on the right, input symbols
listed across the top and next state in the table:
input
delta, the transition table
| 0 | 1 |
----+-----+-----+
s0 | s1 | s2 |
----+-----+-----+
state s1 | s2 | s0 |
----+-----+-----+
s2 | s2 | s1 |
----+-----+-----+
Example: This delta says that a machine in state s2
upon receiving an input of 1, will go to state s1.
This is down at the nano micro computer level.
When the transition table, delta, has all single entries, the machine
may be refereed to as a Deterministic Finite Automata, DFA.
The transition table is entered into a file, read by a computer
program, as three item lines (for above table):
s0 0 s1
s0 1 s2
s1 0 s2
s1 1 s0
s2 0 s2
s2 1 s1
There is no requirement, in general, that every entry in the table must
contain a state. If a machine tries to go to an empty table entry it
"crashes". The remainder of the input, if any, is unread. The machine
does not accept and can not possibly be in a final state.
When every state/input pair has at most one target state, the automata
is called deterministic. (See below for definition of nondeterministic.)
q0 the initial state from the set Q.
By definition this is the state the automata is in when it starts.
The automata gets the first symbol from the input, then goes from
the starting state to the state designated by the transition function.
F a set of final states from the set Q.
Final states are also known as accepting states.
The machine stops after the last input symbols is read and the
corresponding state transition occurs. If the machines state when
stopped is in F then the machine is said to accept the input string.
F can be a null set in which case only the empty language is accepted.
There is no requirement, in general, that any final state be reachable.
A machine defines a language, the set of all strings accepted by
the machine. This language is usually denoted L(M). The machine that
accepts a language L is usually denoted M(L).
There is a Finite Automata, as defined here, for every Regular Language and
a Regular Language for every Finite Automata.
Another common way to define a Finite Automata is via a diagram.
The states are shown as circles, often unlabeled, the initial state has
an arrow pointing to it, the final states have a double circle,
the transition function is shown as directed arcs with the input
symbol(s) on the arc.
Note: This has nothing to do with randomness!
M = (Q, sigma, delta, q0, F) is the same as for deterministic finite automata
above with the exception that delta can have sets of states.
delta for a nondeterministic machine looks like:
input
delta the transition table
| 0 | 1 |
----+---------+---------+
s0 | {s1,s2} | {s2} |
----+---------+---------+
state s1 | {s0,s2} | phi |
----+---------+---------+
s2 | phi | {s1} |
----+---------+---------+
A string is accepted if any sequence of transitions ends in a Final state.
There could be more than one sequence of transitions that end in a
Final state.
This would be called parallel processing, all transitions in a
set happen simultaneously.
Think of each transition that has more than one state as causing a tree
to branch. All branches are in some state and all branches transition on
every input. Any branch that reaches phi, the null or nonexistent state,
terminates.
Any NFA can be converted to a DFA but the DFA may require exponentially
more states than the NFA.
Push Down Automata, PDA, are a way to represent the language class called
Context Free Languages, CFL, covered above. By themselves PDA's are not
very important but the hierarchy of Finite State Machines with
corresponding Regular Languages, PDA's with corresponding CFL's and
Turing Machines with corresponding Recursively Enumerable Sets (Languages),
is an important concept.
The definition of a Push Down Automata is:
M = (Q, Sigma, Gamma, delta, q0, Z0, F) where
Q = a finite set of states including q0
Sigma = a finite alphabet of input symbols (on the input tape)
Gamma = a finite set of push down stack symbols including Z0
delta = a group of nondeterministic transitions mapping
Q x (Sigma union {epsilon}) x Gamma to finite sets of Q x Gamma star
q0 = the initial state, an element of Q, possibly the only state
Z0 = the initial stack contents, an element of gamma, possibly the only
stack symbol
F = the set of final, accepting, states but may be empty for a PDA
"accepting on an empty stack"
Unlike finite automata, the delta is not presented in tabular form.
The table would be too wide. Delta is a list of, nondeterministic,
transitions of the form:
delta(q,a,A) = { (qi,gammai), (qj,gammaj), ...} where
q is the current state,
a is the input tape symbol being read, an element of Sigma union {epsilon}
A is the top of the stack being read,
The ordered pairs (q sub i, gamma sub i) are respectively the next state
and the string of symbols to be written onto the stack. The machine
is nondeterministic, meaning that all the pairs are executed causing
a branching tree of PDA configurations. Just like the branching tree
for nondeterministic finite automata except additional copies of the
pushdown stack are also created at each branch.
The operation of the PDA is to begin in state q0, read the symbol on the
input tape or read epsilon. If a symbol is read, the read head moves
to the right and can never reverse to read that symbol again.
The top of the stack is read by popping off the symbol.
Now, having a state, an input symbol and a stack symbol a delta transition
is performed. If there is no delta transition defined with these three
values the machine halts. If there is a delta transition with the (q,a,A)
then all pairs of (state,gamma) are performed. The gamma represents a
sequence of push down stack symbols and are pushed right to left onto
the stack. If gamma is epsilon, no symbols are pushed onto the stack. Then
the machine goes to the next state, q.
When the machine halts a decision is made to accept or reject the input.
If the last, rightmost, input symbol has not been read then reject.
If the machine is in a final state accept.
If the set of final states is empty, Phi, and the only symbol on the stack
is Z0, then accept. (This is the "accept on empty stack" case)
Now, using pictures we show the machines for DFA, PDA and TM
+-------------------------+----------------- DFA, NFA, NFA epsilon
| input string | accepts Regular Languages
+-------------------------+-----------------
^ read, move right
|
| +-----+
| | |--> accept
+--+ FSM | M = ( Q, Sigma, delta, q0, F)
| |--> reject
+-----+
+-------------------------+----------------- Push Down Automata
| input string |Z0 stack accepts Context Free Languages
+-------------------------+-----------------
^ read, move right ^ read and write (push and pop)
| |
+-----------------------+
| +-----+
| | |--> accept
+--+ FSM | M = ( Q, Sigma, Gamma, delta, q0, Z0, F)
| |--> reject
+-----+
+-------------------------+----------------- Turing Machine
| input string |BBBBBBBB ... accepts Recursively Enumerable
+-------------------------+----------------- Languages
^ read and write, move left and right
|
| +-----+
| | |--> accept
+--+ FSM | M = ( Q, Sigma, Gamma, delta, q0, B, F)
| |--> reject
+-----+
An example of a language that requires the PDA to be a NPDA,
Nondeterministic Push Down Automata,
is L = { w wr | w in Sigma and wr is w written backwards }
A Turing machine, named after A. Turing, is defined by
M = (Q, Sigma, Gamma, delta, q0, B, F)
where
Q is a finite set of states (same as DFA)
Gamma is a finite set of tape symbols
B is a symbol in Gamma is the blank #b, or space ## symbol
Sigma is a subset of Gamma not including B, the input tape symbols
delta is the transition function, next move table, that maps
Q cross Gamma to Q cross Gamma cross {Left, Right, Neither}
q0 is in Q and is the initial or starting state
F is a subset of Q, the final states. After the last input symbol has
been read and the tape is in the leftmost position,
a string is accepted if M is in a final state.
In this particular model, the tape is only unbounded on the right,
filled with B's and fixed on the left at the first input symbol.
Other models include a tape also unbounded on the left , symbols on the
tape for "left-end #[" and "right-end #]", ability to write no symbol,
ability to move {Left, Right or None}. These variations and many more are
provably equivalent. A Turing Machine that computes a value rather than
recognizing a language, leave the computed value on the tape and halts.
It is possible for a TM to never halt on some input strings that are not
in the language it accepts or on input values for which it is not
a complete function.
The set of all Turing machines is countable. Thus there exists real
numbers that can not be accepted by any Turing machine.
delta transition table might look like
input
delta, the transition table
| 0 | 1 | #b |
----+----------+----------+-----------+
s0 | (s1,1,R) | (s2,0,L) | Phi |
----+----------+----------+-----------+
state s1 | (s0,0,L) | Phi | (s0,#b,R) |
----+----------+----------+-----------+
s2 | Phi | (s1,1,R) | (s2,##,N) |
----+----------+----------+-----------+
A Turing machine may also have a nondeterministic transition table that
has sets of entries. Every member of the set of a transition then
continues operation in parallel, each with its own state and tape.
The transition table is entered into a file, read by a computer
program, as five item lines:
s0 0 s1 1 R write 1 to tape, move head one place Right
s0 1 s2 0 L write 0 to tape, move head one place Left
s0 #b Phi ## N no write, no movement
s1 0 s0 0 L
...
Phi is the null or nonexistent state. If a TM enters the phi state it is
"hung" and there is no indication to the outside world. There is no
accept and no reject output signal. No further tape head movement
occurs and no further symbols are read from the tape.
Each "step" executed by a Turing machine does the following:
The machine is in some state, q0 initially.
The read/write head is positioned on some tape cell, initially leftmost.
The pair (state q, input symbol a) select one cell in the delta
transition table. e.g. delta(q,a) = (qi, b , R)
The machine enters state qi, writes b onto the tape, moves right.
The running time of a Turing machine is measured in number of steps
and is usually related to the length of the input, the number of
symbols on the initial tape.
It is possible to code the definition of every Turing machine as a string of characters. A Universal Turing Machine is then a Turing machine that has on its input tape a definition of some Turing machine and a string. The Universal Turing Machine simulates the Turing machine defined on its input tape and accepts the string on its input tape if and only if the simulated Turing machine would accept the string. One UTM can simulate another UTM that is simulating a TM with an input string. Etc. Etc. Etc.
See Computability Definitions
The Turing machine restricted to a number of moves bounded by a fixed polynomial in n, p(n), where n = |x|, n is the length of the input string. The machine must accept all strings in a language in polynomial time in order to be in P. P is a set of languages, also called a class of languages.
Change the deterministic delta transition table to a nondeterministic delta transition table and the TM represents a class of languages that are believed to be different from the language class P. NP is a set of languages, also called a class of languages.
A Moore machine is defined as M = (Q, sigma, Delta, delta, gamma, q0)
Where Q, Sigma, delta and q0 are as defined for finite automata.
Delta is the output alphabet.
gamma is a mapping from Q to Delta.
+-------------------------+
| input string | Moore machine, no more powerful than DFA
+-------------------------+
^ read, move right
|
| +-----+
| | |
+--+ FSM | M = ( Q, Sigma, Delta, delta, gamma, q0)
| |
+-----+
|
v
+-----------------------------
| output tape alphabet Delta (May be same or vary from Sigma)
+-----------------------------
Example: Q= {q0, q1} Sigma= {a, b} Delta= {A, B} gamma= q0->A, q1->B
delta | a | b | gamma
------+----+----+------
q0 | q1 | q0 | A
q1 | q0 | q1 | B
A B <--shows symbol written to output when entering state
_____ a _____
--> / q0 \-------->/ q1 \ and other connecting transitions like a DFA
\_____/<------- \_____/
^ | a ^ |
\-b-/ \-b-/
This is a finite state machine that outputs a symbol on every state
transition starting with gamma(q0) which is always output.
Note there is no set F of final states. A Moore machine is equivalent to
a DFA when Delta = {0, 1} and a zero output is interpreted as reject,
while a 1 output is interpreted as accept and gamma maps final states
to 1 and all other states to 0.
A finite automata can only accept (last input symbol put machine in a final
state) or reject an input string. A Moore machine is like a simple
function where an input is processed and a result is produced.
Moore machines are usually not based on NFA's because this would be like
a multi valued function. Some input could produce more than one output.
A Mealy machine is defined as M = (Q, sigma, Delta, delta, gamma, q0)
using the definitions from the Moore machine with the exception that
gamma maps Q cross Sigma to Delta.
Delta is the output alphabet.
Thus the output is a function of the state and the input symbol,
written during the state transition.
The difference from a Moore machine is the output being a function
of the transition rather than just a function of the sequence of states.
The empty string as input produces an empty output.
+-------------------------+
| input string | Mealy machine, no more powerful than DFA
+-------------------------+
^ read, move right
|
| +-----+
| | |
+--+ FSM | M = ( Q, Sigma, Delta, delta, gamma, q0)
| |
+-----+
|
v
+-----------------------------
| output tape alphabet Delta (May be same or vary from Sigma)
+-----------------------------
Example: Q= {q0, q1} Sigma= {a, b} Delta= {A, B, C, D} gamma is in delta
delta | a | b
------+------+------
q0 | q1A | q0C
q1 | q0D | q1B
/--shows symbol written to output upon transition
_____ a/A _____
--> / q0 \------->/ q1 \ and other connecting transitions state/output
\_____/<------ \_____/
^ | a/D ^ |
\-b-/ \-b-/
C B symbol written to output upon transition
Mealy machines are usually not based on NFA's because this would be like
a multi valued function. Some input could produce more than one output.
The proof that a Mealy machine and a DFA are equivalent, using gamma being
a one for accept on transition to a final state, and gamma being zero
otherwise, also needs a little effort to have a Mealy machine accept
the null string, epsilon.
The following machines are provably equivalent
FA = DFA = NFA = NFA epsilon
FA = two way FA (tape can move left, right, or not move)
FA = Moore machine with output alphabet {0, 1} and state q accepting if and
only if gamma(q)=1
FA = Mealy machine with some technicalities
TM = unrestricted grammar
TM = lambda calculus
TM = Post formal systems
TM = NTM
TM = TM with more tapes, write or not write, move or not move, etc.
PDA not equal NPDA
FA = regular languages PDA(NPDA) = context free languages TM = recursively enumerable languages
The following group can have equivalent computational power
partial recursive functions
r.e. languages = recursively enumerable languages
Turing machines that may not halt when input is not accepted
unrestricted grammars
lambda calculus
Post Formal Systems
computer programs in a reasonable language that may not halt on some input
Chomsky type 0 languages
The following group can have equivalent computational power
total recursive functions
recursive languages = rec
Turing machines that halt on all inputs
grammars that stop on all inputs
lambda calculus with finite recursion
computer programs that halt on all inputs
The following group can have equivalent computational power
context sensitive languages, CSL's
context sensitive grammars, CSG's
Chomsky type 1 languages
The following group can have equivalent computational power
context free languages, CFL's
context free grammars, CFG's
NPDA nondeterministic pushdown automata
Chomsky type 2 languages
The following group can have equivalent computational power
regular expressions
regular languages
regular grammars
deterministic and nondeterministic finite automata, DFA, NFA, NFA epsilon
Chomsky type 3 languages
Last updated 5/1/2021