State diagram
State diagrams are used to graphically represent
finite state machines.
State transition tables are another possible representation.
There are many forms of state diagrams that differ slightly and have different semantics.
Encyclopedia
State diagrams are used to graphically represent
finite state machines.
State transition tables are another possible representation.
There are many forms of state diagrams that differ slightly and have different semantics.
Directed graph
A classic form of a
state diagram for a
finite state machine is a
directed graph with the following elements:
States Q: a finite set of vertices normally represented by circles and labelled with unique designator symbols or words written inside them .
Input symbols S: a finite collection of input "symbols" or designators
S . For a
deterministic finite state machine ,
nondeterministic finite state machine , generalized nondeterministic finite state machine , or
Moore machine, input is signified on each edge, usually near the originating state. For a
Mealy machine, input and output are signified on each edge usually shown separated with a slash "/":
- Mealy input and output labels on an edge : "1/0" designates symbol "1" caused symbol "0" as output.
Output symbols Z: a finite collection of output "symbols" or designators . For a
Mealy machine, input and output are signified on each edge as shown above. For a
Moore machine the state's output is usually written inside the state's circle, separated from the state's designator with a slash "/".
- Example: If a state has a number of outputs the diagram should reflect this : e.g. "q5/1,0" designates state q5 with outputs a=1, b=0. This designator will be written inside the state's circle.
The "Output function ?" represents the mapping
? of input symbols
I x states
Q into output symbols
Z .
Edges d: represent the "transitions" between two states as caused by the input . An 'edge' is usually drawn as an arrow directed from the present-state toward the next-state.
d represents the mapping of input symbols
I x states
Q onto output symbols
Z .
Start state qo: . The start state qo is usually represented by an "arrow pointing at it from nowhere" . In older texts the start state is not shown and must be inferred from the text.
Accepting state F: If used -- a collection of double circles used to designate
accept states . Sometimes the accept state function as "
Final" states .
Examples
DFA, NFA, GNFA, or Moore machine
S1 and
S2 are states and
S1 is an accept state. Each edge is labeled with the input.
Mealy machine
S0,
S1, and
S2 are states. Each edge is labeled with "
j /
k" where
j is the input and
k is the output.
Harel statechart
Harel statecharts are gaining some more widespread usage since a variant has become part of
UML. The diagram type allows to model superstates, concurrent state diagrams and e.g. to model activities as part of a state.
Classic state diagrams are so called "or" diagrams, because the machine can only be in one state or the other. With Harel statecharts it is possible to model "and" machines, where a machine is in two or more states at the same time. This is due to the possibility of having superstates.
UML state diagram
The
Unified Modeling Language state diagram is essentially a state diagram with standardised notation that can describe a lot of things, from computer programs to business processes. The following tools can be used to make up a diagram:
- Filled circle, denoting START. Not absolutely necessary to use
- Hollow circle, denoting STOP. Not absolutely necessary to use
- Rectangle, denoting state. Top of the rectangle contains a name of the state. Can contain a horizontal line in the middle, below which the activity is written that is done in that state
- Arrow, denoting transition. An expression can be written on top of the line, enclosed in brackets denoting that this expression must be true for the transition to take place
- Thick horizontal line with either x>1 lines entering and 1 line leaving or 1 line entering and x>1 lines leaving. These denote join/fork, respectively.
Other extensions
An interesting extension is to allow arcs to flow from any number of states to any number of states. This only makes sense if the system is allowed to be in multiple states at once, which implies that an individual state only describes a condition or other partial aspect of the overall, global state. The resulting formalism is known as a
Petri net.
Another extension allows the integration of flowcharts within Harel statecharts. This extension supports the development of software that is both event driven and workflow driven.
References
-
-
- by Scott W. Ambler
- by Scott W. Ambler
- SCXML an XML language that provides a generic state-machine based execution environment based on Harel statecharts.
- Modelling and verification using UML statecharts, Drusinsky, D., Elsevier, 2006,
- Michael Sipser , Introduction to the Theory of Computation, Second Edition, Thomson Course Technology, Boston. ISBN-13: 978-0-534-95097-2, ISBN-10: 0-534-95097-3.
...
and Jeffrey Ullman
Introduction to Automata Theory, Languarges, and Computation, Addison-Wesley Publishing Company, Reading Mass, ISBN 0-201-02988-X.
- Taylor Booth Sequential Machines and Automata Theory, John Wiley and Sons, New York. Library of Congress Catalog Card Number: 67-25924.
