State-transition table
In automata theory and sequential logic, a state-transition table is a table showing what state (or states in the case of a nondeterministic finite automaton) a finite-state machine will move to, based on the current state and other inputs. It is essentially a truth table in which the inputs include the current state along with other inputs, and the outputs include the next state along with other outputs.
A state-transition table is one of many ways to specify a finite-state machine. Other ways include a state diagram.
Common forms
One-dimension
State-transition tables are sometimes one-dimensional tables, also called characteristic tables. They are much more like truth tables than their two-dimensional form. The single dimension indicates inputs, current states, next states and (optionally) outputs associated with the state transitions.
| Input | Current state | Next state | Output |
|---|---|---|---|
| I1 | S1 | Si | Ox |
| I2 | S1 | Sj | Oy |
| … | … | … | … |
| In | S1 | Sk | Oz |
| I1 | S2 | Si′ | Ox′ |
| I2 | S2 | Sj′ | Oy′ |
| … | … | … | … |
| In | S2 | Sk′ | Oz′ |
| … | … | … | … |
| I1 | Sm | Si″ | Ox″ |
| I2 | Sm | Sj″ | Oy″ |
| … | … | … | … |
| In | Sm | Sk″ | Oz″ |
Two-dimensions
State-transition tables are typically two-dimensional tables. There are two common ways for arranging them.
In the first way, one of the dimensions indicates current states, while the other indicates inputs. The row/column intersections indicate next states and (optionally) outputs associated with the state transitions.
Input Current state |
I1 | I2 | … | In |
|---|---|---|---|---|
| S1 | Si/Ox | Sj/Oy | … | Sk/Oz |
| S2 | Si′/Ox′ | Sj′/Oy′ | … | Sk′/Oz′ |
| … | … | … | … | … |
| Sm | Si″/Ox″ | Sj″/Oz″ | … | Sk″/Oz″ |
In the second way, one of the dimensions indicates current states, while the other indicates next states. The row/column intersections indicate inputs and (optionally) outputs associated with the state transitions.
Next state Current state |
S1 | S2 | … | Sm |
|---|---|---|---|---|
| S1 | Ii/Ox | — | … | — |
| S2 | — | — | … | Ij/Oy |
| … | — | — | — | — |
| Sm | — | Ik/Oz | … | — |
Other forms
Simultaneous transitions in multiple finite-state machines can be shown in what is effectively an n-dimensional state-transition table in which pairs of rows map (sets of) current states to next states.[1] This is an alternative to representing communication between separate, interdependent finite-state machines.
At the other extreme, separate tables have been used for each of the transitions within a single finite-state machine: "AND/OR tables"[2] are similar to incomplete decision tables in which the decision for the rules which are present is implicitly the activation of the associated transition.
Example
An example of a state-transition table together with the corresponding state diagram for a finite-state machine is given below:
Input Current state
|
1 | 0 |
|---|---|---|
| S1 | S1 | S2 |
| S2 | S2 | S1 |
|
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 (the first row), 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. This process can be described statistically using Markov Chains.
For a nondeterministic finite-state machine, an 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 of a state-transition table for a nondeterministic finite-state machine is given below:
Input Current state
|
1 | 0 | ε |
|---|---|---|---|
| S1 | S1 | {S2, S3} | Φ |
| S2 | S2 | S1 | Φ |
| S3 | S2 | S1 | S1 |
Here, a nondeterministic finite-state 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 state transitions of the special character, ε. This special character allows the machine to move to a different state when given no input. In state S3, the machine may move to S1 without consuming an input character. The two cases above make the finite-state machine described nondeterministic.
Transformations from/to state diagram
It is possible to draw a state diagram from a state-transition 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(s). There can be multiple arrows for an input character if the finite-state machine is nondeterministic.
- Designate a state as the start state. The start state is given in the formal definition of a finite-state machine.
- Designate one or more states as accept state. This is also given in the formal definition of a finite-state machine.
See also
References
- ^ Breen, Michael (2005), "Experience of using a lightweight formal specification method for a commercial embedded system product line" (PDF), Requirements Engineering Journal, 10 (2): 161–172, CiteSeerX 10.1.1.60.5228, doi:10.1007/s00766-004-0209-1
- ^ Leveson, Nancy; Heimdahl, Mats Per Erik; Hildreth, Holly; Reese, Jon Damon (1994), "Requirements Specification for Process-Control Systems" (PDF), IEEE Transactions on Software Engineering, 20 (9): 684–707, CiteSeerX 10.1.1.72.8657, doi:10.1109/32.317428
Further reading
- Michael Sipser: Introduction to the Theory of Computation. PWS Publishing Co., Boston 1997 ISBN 0-534-94728-X