Theory of Computation

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Theory of Computation

• What types of things are computable?
• How can we demonstrate what things are computable?

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Foundations

• Cantors Set Theory - contradictions
• Multiple sizes to infinity.
• There exists at least one set bigger than the
  universal set.
• Hilberts rigor
• Find an algorithm that will generate proofs for
  all true statements
• Gödels Incompleteness Theorem
• No such algoritm exists.
• Even worse, there exist true statements for which
  no proof can ever be found.
• In any mathematical system there will either be
  true statements that cannot be proven or false
  statements that can.

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Computability

• The question then becomes, can we at least find
  an algorithm that can find any proof that does
  exist?
• Church Kleene and Post found no.
• Turing developed the universal algorithm
  machine, now called the Turing maching. This
  machine also had tasks it could not perform.

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Languages and grammars

• A language is a set of strings.
• A grammar is the set of rules that define what
  strings are valid members of a language.

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Rule structure

• Rules consist of three types of symbols
• Terminals are symbols that cannot be further
  expanded.
• Nonterminals are symbols that are not part of the
  languages alphabet, but can be expanded into
  larger substrings
• ?, which represents the empty string.
• Assume the alphabet is a, b. An example grammar
  might be
• S ? aSb
• S ? ba
• where a and b are terminals and S is a
  nonterminal.
• Note that rules can be defined recursively.z

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Formal grammars

• A grammar is thus the set of generation rules
  that define the valid strings of the language.
• Any string that can be generated by the grammar
  is a valid string in the language, and any string
  in the language that is valid can be generated by
  the grammar
• The type of grammar used by a language is
  determined by the restrictiveness of the rules.

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Regular grammars

• Regular grammars are ones where the the rules
  have a nonterminal on the left and on the right
  either
• The empty string
• A single terminal
• A single terminal and a single nonterminal
• Left regular grammars are ones where the
  nonterminal is to the left of the terminal in
  right regular grammars, the opposite is true.

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Example regular grammars

• S ? aB
• B ? bB
• B ? ?
• What language does this define?
• ab

S ? aB B ? bB B ? a What language does this
define? aba
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State machines

• Designed to allow for the modeling of transitions
  from one state to another.
• Consist of states representing the current state
  of the machine and transitions between those
  states.
• Entry Action - the action to perform when the
  state is entered
• Exit Action - the action to perform when the
  state is exited
• Input Action - an action to perform in a
  particular state with a a particular input
  (usually involves following a transition)
• Have a start state and an acceptor state
• Transitions define how the machine changes from
  one state to another. They usually define the
  condition(s) under which the machine changes
  states
• Transition Action - the action to perform when
  following a transition.

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Finite Automata

• Particular type of state machine that has no
  actions at all. Only states and transitions.

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Limits of Regular languages and finite automata

• What types of languages cant FAs accept? In
  other words, what limits are there on the
  complexity of regular languages?
• FAs lack memory, so that you cant have one part
  of a regular language dependent on another part.

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Context-free grammars

• In CFGs, the rules all take the form
• N ? w
• where N is a single non-terminal and w is some
  finite string of terminals and non-terminals, in
  any order.
• Whereas in regular languages, nonterminals were
  restricted as to where they could appear in the
  rules, now they can appear anywhere. Hence the
  term context-free.
• S ? aSb
• S ? ab

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CFG example

• S ? aB
• S ? bA
• A ? a
• A ? aS
• A ? bAA
• B ? b
• B ? bS
• B ? aBB

S ? U S ? V U ? TaU U ? TaT V ? TbV V ? TbT T
? aTbT T ? bTaT T ? ?
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Pushdown Automata

• Pushdown automata extend FAs in one very
  important way. We are now given a stack on which
  we can store information. This works like a
  standard LIFO stack, where information gets
  pushed onto the top and popped off the top.
• This means that we can now choose transitions
  based not just on the input, but also based on
  whats on the top of the stack.
• We also now have transition actions available to
  us. We can either push a specific element to the
  top of the stack, or pop the top element off the
  stack.

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Limits on PDAs and CFGs

• Adding memory is nice, but there are still
  significant limits on what a PDA can accomplish.
• Can a PDA be constructed that can do arithmetic?
• How, or why not?

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The Turing Machine

• Four components
• A tape of infinite length, divided into cells,
  which can contain one character of data
• A head which can read and write in the current
  cell, and which can move the tape one cell at a
  time left or right.
• A program, which is the set of rules that tells
  us what to do. Our transitions between states
  will now specify the letter we should read from
  the tape, the letter we should write to the tape,
  and the direction we should move the tape.