Models of Computation: Automata and Processes

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Models of Computation Automata and Processes

• Jos Baeten

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Models of computation

• Automata theory and formal languages
• Turing machine
• Computability
• Algorithm
• Complexity

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Models of computation

• Automaton is model of computer
• input/output function
• stand alone
• batch process
• Add notion of interaction -gt
• concurrency theory, process theory

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Thesis

• Process theory is just as important to every
  computer science student as automata theory
• Can be taught together in every undergraduate
  curriculum
• Analogies help, standardization can ensue

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A Cooks tour of automata and processes

• Some answers, some questions

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From automata theory

• A regular language
• deterministic finite automaton
• non-deterministic finite automaton
• right-linear grammar
• left-linear grammar
• regular expression

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From process theory

• A regular process
• deterministic finite automaton
• non-deterministic finite automaton
• right-linear grammar
• left-linear grammar
• regular expression

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A regular process

• A finite transition system modulo bisimulation
• edge-labeled, initial state, set of final
  states
• A finite guarded linear recursive specification
• action prefix
• recursion
• 0
• 1

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A regular expression

• Not all regular processes can be described by a
  regular expression (w.r.t. bisimulation)

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Example
a
b

• X a Y 1
• Y b X 1
• X aY 1 a(bX 1) 1 abX a 1
• X (ab)(a 1)
• Y b(ab)(a 1) 1
• left distributivity of over holds
• in language equivalence

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Milner 1984

• Not for bisimulation equivalence.
• Question which finite behaviours are
  bisimulation equivalent to a regular expression?
• Answer well-behaved ones
• B, Corradini, Grabmayer. JACM 2007.

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Consequence

• Iteration not basic programming construct for
  parallel programming.

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Regular expressions

• choice
• sequential composition
• 0 inaction, d
• 1 skip, e
• a action
• iteration

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Reg. exp. to finite automata
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Algebra for regular expressions
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Recursive specifications (grammars)

• X a Y 1 Y b X 1
• RSP x yx z ? x yz
• guarded, i.e.
• y does not have the empty word property,
• y ? y 1

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Open question

• The algebra with RSP is ground-complete.

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Context-free language

• Context-free grammar
• Push-down automaton

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What is a context-free process?

• A guarded recursive specification over TSP
• choice
• sequential composition
• 0 inaction
• 1 skip
• a action (prefix)

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Context-free process

• B, Bergstra, Klop. JACM 1993.
• Restricted Greibach normal form.

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A push-down automaton?

• Theorem any context-free process is bisimulation
  equivalent to a regular process communicating
  with a stack.

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Stack

• S T S
• T ? ?d U !d
• U 1 T U

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Different classes of processes

• Regular action, , 0, 1
• Context-free add seq. comp.
• Basic parallel add , no communication
• Basic communicating add with comm.
• Parallel add seq.comp. to basic parallel
• Communicating add seq.comp. to basic comm.

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Bag

• B ? ?d . (B !d)

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A non-theorem

• Theorem any basic parallel process is
  bisimulation equivalent to a regular process
  communicating with a bag.
• Problem need to detect termination.
• Bag with termination is not basic parallel.

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More questions

• What set of processes do you get as a regular
  process communicating with a queue
  (with termination)?
• A queue is a communicating process, only if
  arbitrary communication is allowed (or binary
  communication plus renaming).