State Machines Lecture

State Machines Overview

• State machines model systems by representing states and transitions between them.
• Essential components include states, actions, initial states, and state transitions based on actions.
• State machines simplify the abstraction of complex systems, making them easier to reason about.

Importance of State Machines

• All computer systems operate as state machines, involving states of registers and memory.
• Transitions between states reflect the program's operations leading from initial to final states.
• A programming language defines permissible operations, shaping the state machine.

Applications of State Machines

• Useful when abstracting irrelevant details reduces complexity to a manageable number of states.
• Employed in model checking to explore every possibility in system behavior.
• An essential tool for designing communication protocols and managing complex distributed algorithms.

Modeling Foundations

• State machines form the basis for advanced modeling techniques like Z, CSP, and Temporal Logic.
• There's no universal notation for state machines, with concepts broadly agreed upon but varying rigor based on specific scenarios.

Components of a State Machine

• States: Snapshot of system conditions, including variable values and execution locations.
• Actions: Events that trigger transitions between states.
• A complete definition specifies the possible states, initial states, actions, and transition behaviors.

Simple State Machine Examples

• A very basic state machine can consist of a single state, such as {stop}, with no actions associated.
• More dynamic examples can include states like {alarm} with actions such as {beep}.

Advanced State Machine Characteristics

• A more complex state machine may include multiple states like {off, idle, moving, crashed} with various actions.
• Non-determinism arises when a single action can lead to multiple outcomes or transitions from the same state.
• Machines can exhibit potentially infinite steps due to non-deterministic transitions.

Formal Definition

• A state machine is formally defined as a 4-tuple (S, A, I, T) where:
  • S represents the set of possible states.
  • A is the set of actions that can induce state transitions.
  • I denotes the set of initial states from which the machine can begin.
  • T defines the transition relation indicating how states change in response to actions.