Review of Basic Computer Science Background

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Review of Basic Computer Science Background
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Computer Science Youll Need

• Some models of computation
• Finite state machines
• Recursive functions / Turing machines
• Logic circuit models
• Cellular automata
• Computability theory
• Universal vs. non-universal models
• Computational complexity theory
• Polynomial vs. exponential problem classes
• Computer architecture/engineering concepts
• functional units, CPUs, instruction sets, etc...

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Models of Computation

• Non-universal models
• Finite state machines (w. read-only inputs)
• Related to regular expression languages
• Pushdown automata (FSM w. a stack)
• Relation to context-free grammar languages
• Universal models
• Recursive function theory (Church)
• File clerk model / Turing machines (Turing)
• FSM with unbounded-size read-write external
  memory
• Logic circuits (Shannon)
• RAM machines (von Neumann)
• Cellular automata (von Neumann)

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Church-Turing thesis

• All the universal models were proved early on
  to be able to simulate each other
• Straightforward to write emulator programs
• Church-Turing thesis
• Anything that can be computed physically (in any
  way whatsoever) can be computed by machines in
  these models
• Hence the term universal
• This is in the form of a physical postulate.
• Note that universality the Church-Turing thesis
  say nothing about efficiency.

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Computational Complexity Theory

• Characterizes the complexity (defined in terms
  of resource usage) of computations
• This complexity can be viewed as essentially
  the reciprocal of computational efficiency
• efficiency would be amount of computation
  completed per unit resources consumed
• Traditional focus in complexity theory only on
  either of two resources, in isolation
• Time (most popular)
• Number of primitive ops. or parallel steps
  performed
• Space
• Max. number of bits of storage needed at any time

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Cost-Efficiency

• Resources (time, space, hardware, energy, etc.)
  have some associated cost for their expenditure
• Resource usage should be minimized
• A 100 efficient process is one that wastes no
  resources. Want to maximize efficiency.
• Let min be the minimum cost required to carry
  out a process, the actual cost.
• So min/, or cost-efficiency, is the
  fraction of cost that is well spent.
• Note that is inversely proportional to cost
  (or complexity), even if min is unknown.

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Complexity Classes

• In complexity theory, it is useful to classify
  computational problems or tasks on a given
  machine in terms of the asymptotic order of
  growth of the complexity.
• Allows ignoring the effects of minor
  implementation details
• Further, similar orders of growth are often
  grouped together, e.g.
• P - The class of problems solvable in polynomial
  (O(nk) for some k) time
• This class was thought to be independent of the
  machine model used, until the discovery of
  quantum computing!

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Order of Growth Notation