• Recall:
  • A system is a set of entities, their attributes, and the relationships between them
  • Attributes are the external manifestations of the way the objects are known or observed
• System State:
  • The values of the attributes at a particular point (e.g., in time and/or space)

• A Simple System:
  • A light
• The Attributes:
  • A boolean named power
• The Possible States:
  • T (i.e., On)
  • F (i.e., Off)

• Abstraction Revisited:
  • Recall that abstraction is the process of considering what is important and ignoring what isn’t
• Unimportant Attributes in this Example:
  • Wattage
  • Voltage
  • Physical Dimensions
• Why Might Attributes be Unimportant?
  • They don’t change
  • They aren't relevant to the problem at hand

• Defined:
  • The set of all possible states of a system
• Kinds of State Spaces:
  • Discrete – each attribute is integer-valued
  • Continuous – each attribute is real-valued
  • Mixed/Hybrid – some attributes are integer-valued and some are real-valued

• The Computer Science Perspective:
  • We are (mostly) interested in digital computers in which everything is discrete
• The Implication:
  • We (almost) always work with discrete state spaces

• A Slightly More Complicated Example:
  • A light with one integer attribute, color, and one boolean attribute, power
• An Individual State:
  • Can be denoted by the ordered pair (power, color) where power can take on the values T and F, and color can take on the values R, G, and B
• The State Space:
  • {(T, R), (T, G), (T, B), (F, R), (F, G), (F, B)}

• An Observation:
  • Even discrete state spaces can be very large (sometimes called state space explosion)
• Some Discrete Math:
  • Letting \(x_i\) denote the number of values for attribute \(i\) and \(N\) denote the number of attributes, the (maximum) size of the state space, \(S\), is given by
  • \(S = \prod_{i=1}^{N} x_i\)

• An Example:
  • An HVAC system with two modes (heating and cooling), three levels (low, medium and high), and two zones (upstairs and downstairs) has a size of \(S = 2 \cdot 3 \cdot 2 = 12\)
• A More Realistic Example:
  • An HVAC system with two modes (heating and cooling), 70 levels (temperatures between 31 and 100), and two zones (upstairs and downstairs) has a size of \(S = 2 \cdot 70 \cdot 2 = 280\)

• Attributes of Interest:
  • Whether the door is opened or closed
  • Whether the overhead light is off, dim, or bright
• The State Space:
  • \(\{CLOSED, OPENED\} \times \{OFF, DIM, BRIGHT\}\)

• Recall:
  • A dynamic model describes how a system changes over time (which may be discrete or continuous)
• Time-Based Models:
  • Describe the system at every point in time regardless of whether the state of the system changes
• Event-Based Models:
  • Describe the system only at times when specific events (which cause a transition between states) occur

• States:
  • 
  • In general, the names of states should be adjectives
• Transitions:
  • 
  • In general, the names of transitions should be nouns or verbs

• The Simple Light Example:
  • 
• A Printer:
  • 

• Common Pseudo States:
  • Initial State
  • Final State
• An Example Revisited:
  • 

• Activities in States:
  • 
• Effects of Transitions:
  • 
  • The constraint in square brackets is called a guard

In general, one should use an entry action when the behavior should be executed regardless of how the state is entered. On the other hand, one should use an effect when the behavior is specific to the transition.

• An Observation:
  • Some states are realized for a predetermined amount of time
• An Example:
  • A Pre-Timed Traffic Light
  • 

• Specialization:
  • A specialization of a set (e.g., an extension of a class) is a subset in which each element has additional attributes in common (i.e., the specialization is less abstract)
• Generalization:
  • When one generalizes one ignores some attributes (i.e., the generalization is more abstract)

• The Process:
  • Create a disjunctive superstate from multiple substates
• Interpretation:
  • A system that is in a superstate is in exactly one of the substates that it can be decomposed into
• The Rationale:
  • Simplicity
  • Reduction of the state space

An Example

In the abstract model, this system has two states, Disabled and Enabled, and transitions between the two as a result of two events, Enable and Disable.

• The Process:
  • Create a conjunctive superstate from multiple substates
• Interpretation:
  • A system that is in a superstate must be in all of the states that it can be disaggregated into
• The Rationale:
  • Simplicity
• Example:
  • A GPS navigation system that, when in the Tracking state, is in both the Navigating state and the Recording state

• The Process:
  • Create an AND-composition in which the substates are independent
• Rationale:
  • Results in an additive increase in the size of the state space rather than a multiplicative increase
• Example:
  • A keyboard with a main keypad (that has a normal state and a caps lock state) and a numeric keypad (that has a normal state and a num lock state)

Example

Note that, because of the orthogonal regions, we do not have to define the states UnlockedPositioning, LockedPositioning, UnlockedNumeric, and UnlockedArrows.

• Transitions into Orthogonal States:
  • Fork vertices can be used to split an incoming transition
  • 
• Transitions out of Orthogonal States:
  • Join vertices can be used to merge transitions coming from orthogonal states
  • 

• Rationale:
  • A state can sometimes be viewed subsystem (i.e., has sufficient richness of relationships)
• UML Notation:
  • 

• The System:
  • A light with one integer attribute, color,and one boolean attribute, power
• The Obvious Model:
  • 

• State Diagrams:
  • Event-based
• Activity Diagrams:
  • Timing is incorporated using tokens
• Sequence Diagrams:
  • Time is incorporated using the vertical dimension
• Communication Diagrams:
  • Ordering over time is incorporated using sequence numbers