State transition table
In
Automata Theory, a state transition table is a table describing the transition function T of a finite automaton. This function governs what state the automaton will move to, given an input to the machine. Given a
state diagram of a finite automaton, a state transition table can be derived from it and vice versa.
Encyclopedia
In
Automata Theory, a
state transition table is a table describing the transition function
T of a finite automaton. This function governs what state the automaton will move to, given an input to the machine. Given a
state diagram of a finite automaton, a state transition table can be derived from it and vice versa.
Common forms
State transition tables are typically two-dimensional tables. There are two common forms for arranging them.
- The vertical dimension indicates current states, the horizontal dimension indicates events, and the cells in the table contain the next state if an event happens .
State Transition Table Events
State | E1 | E2 | ... | En |
| S1 | - | Ay/Sj | ... | - |
| S2 | - | - | ... | Ax/Si |
| ... | ... | ... | ... | ... |
| Sm | Az/Sk | - | ... | - |
- The vertical dimension indicates current states, the horizontal dimension indicates next states, and the row/column intersections contain the event which will lead to a particular next state.
State Transition Table next
current | S1 | S2 | ... | Sm |
| S1 | Ay/Ej | - | ... | - |
| S2 | - | - | ... | Ax/Ei |
| ... | ... | ... | ... | ... |
| Sm | - | Az/Ek | ... | - |
Example
An example of a state transition table for a machine
M together with the corresponding state diagram is given below.
State Transition Table Input
State | 1 | 0 | | S1 | S1 | S2 | | S2 | S2 | S1 |
| | State Diagram
|
All the possible inputs to the machine are enumerated across the columns of the table. All the possible states are enumerated across the rows. From the state transition table given above, it is easy to see that if the machine is in
S1 , and the next input is character
1, the machine will stay in
S1. If a character
0 arrives, the machine will transition to
S2 as can be seen from the second column. In the diagram this is denoted by the arrow from
S1 to
S2 labeled with a
0.
For a
nondeterministic finite automaton , a new input may cause the machine to be in more than one state, hence its non-determinism. This is denoted in a state transition table by a pair of curly braces
with the set of all target states between them. An example is given below.
State Transition Table for an NFA Input
State | 1 | 0 | e |
| S1 | S1 | | F |
| S2 | S2 | S1 | F |
| S3 | S2 | S1 | S1 |
Here, a nondeterministic machine in the state
S1 reading an input of
0 will cause it to be in two states at the same time, the states
S2 and
S3. The last column defines the legal transition of states of the special character, e. This special character allows the NFA to move to a different state when given no input. In state
S3, the NFA may move to
S1 without consuming an input character. The two cases above make the finite automaton described non-deterministic.
Transformations from/to state diagram
It is possible to draw a
state diagram from the table. A sequence of easy to follow steps is given below:
- Draw the circles to represent the states given.
- For each of the states, scan across the corresponding row and draw an arrow to the destination state. There can be multiple arrows for an input character if the automaton is an NFA.
- Designate a state as the start state. The start state is given in the formal definition of the automaton.
- Designate one or more states as accept state. This is also given in the formal definition.
References
- Michael Sipser: Introduction to the Theory of Computation. PWS Publishing Co., Boston 1997 ISBN 0-534-94728-X
