Computability, etc.

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Computability, etc.

• Recap. Present homework. Grammar and machine
  equivalences. Turing history
• Homework Equivalence assignments. Presentation
  proposals, preparation. Keep up on postings.

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Recap

• Define different types of machines.
• abstract machines
• Machines are fed (my term) strings in a specified
  alphabet.
• Machines scan the strings and say yes or no
  (accept or reject)
• Each machine accepts strings in a certain
  pattern. We call the set of strings in that
  pattern a language.

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Recap

• Repeat a language is a SET of strings.
• This set may be finite or infinite.
• the empty set is a language, with no actual
  memberselements
• This set may or may not include the empty string

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Homework

• Define a FSA, alphabet A,blank, that accepts
  strings with even number of As. Define a second
  FSA that accepts strings with an odd number of
  As.
• Define a PDA, alphabet A, blank,(, ), that
  accepts A, AA, (AA), (A(AA)) etc. NOT A ,
  not unbalanced ( and ). HOLD OFF ON THIS ONE
  SEE HOMEWORK!
• Define TM that subtracts 1. If input is zero,
  then tape is cleared.
• Define Turing machine to subtract two numbers a
  and b, represented by a1 1s followed by a blank
  followed by b1 1s
• if (agtb), put (a-b) 1 1s on the tape
• if altb, leave zero 1s in the tape OR return
  something else if altb ???

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Turing machine

• can perform a function or
• define same type of machine plus accepting and
  rejecting states and say a TM defines a language
  L if strings in the language L make the machine
  stop in an accepting state and strings NOT in L
  stop in a rejecting state.

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Decidability

• A TM that always halts is called a decider.
• TM can keep going, that is, loop
• A language is Turing-decidable if a decider TM
  recognizes it.

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Univeral Turing Machine

• Consider the input ltM,wgt where M is some
  encoding for a Turing Machine and w is a string.
  
• Claim it is possible to define a TM U such that
  U simulates M on w! This is analogous to a
  general purpose computer.
• may have contributed to idea of stored program
  computing

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Halting problem

• Consider a universal Turing machine U.
• Now, if a M loops on a w, so would U.
• U is not a decider!
• Put another way the setATM ltM,wgt M is a TM
  and M accepts w is undecidable. It can be
  proven that there does not exist a TM that always
  halts, that says yes or no on all inputs ltM,wgt.

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Recap

• There are different definitions/constructions/conc
  epts that define the same set of functions or
  languages.
• So-called recursive functions (more next)
  correspond to computable (TM) functions.
• Languages defined by a Regular Expression (look
  it up) are languages recognized by a FSA.
• Languages defined by a Context Free Grammar (look
  it up) are languages recognized by a PDA.
• Some proofs of equivalence are easy and some more
  substantial.
• Will not focus on proofs, BUT these are possible
  topics for presentations.

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Recursive functions

• Define a set of functions by specifying a starter
  set and then ways of adding new functions.
  Functions can be on vectors. Stick to integers.
• Starter set
• identify function and projection I(x) x and
  Pi(x1,x2,xn) xi
• Add 1 A(x) x1
• Constant Ck(x) k

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Ways to add

• Composition if F and G are recursive (and F
  produces the right size vector, then the
  composition G F (first you do F and then G to the
  result) is recursive.
• Primitive recursion Given functions F and G,
  define HH(0,x1, x2, , xn) F(x1, x2, xn)
  andH(y1, x1, x2, xn) G(y,x1, ,xn,
  H(x1,x2,,xn))
• Minimal inverse Given a function F, then the
  function H defined by H(x) y such that F(y) x
  and y is the smallest integer such that that is
  true.

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Equivalence!

• If there exists a TM for a function F, then F is
  recursive.
• If a function F is recursive (that is, in this
  set), then you can define a TM that performs F.

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Grammars

• A grammar G is a
• set of symbols, divided in final symbols and
  auxiliary symbols
• (finite) set of production rules of the form
  string of symbols gt string of symbols
• final symbols have no production rules of the
  formf gt
• In this exposition, we view certain strings of
  letters as single symbols.
• one auxiliary is called the started symbol s
• The language generated (defined by) G is all the
  strings of final symbols generated by a sequence
  starting with s

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Example

• Think of auxiliaries as parts of speech and final
  symbols as (whole) words
• Let s be sentencevp be verb phrasenp be noun
  phrasen be nounv be verbdo be direct
  objectadj be adjectiveadv be adverb

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Example

• Think of parts of speechs gt np vpnp gt adj n
  nvp gt v v adv v do v do advdo gt nn gt
  boy girl dog cheesev gt walks runs
  jumps eatsadj gt short tall spottedadv gt
  fast slowly

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Try

• To construct the parse tree fortall girl
  walksshort boy eats cheese slowly

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Context free grammar

• Production rules are of the formauxiliary
  symbol gt string of symbols

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So what isn't CFG?

• There also can be rules of the formaPb gt .
• where P is an auxiliary symbol. Think of it as P
  in the context of a and b produces something. So
  this is NOT context-free.

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Example CFG

• S gt TT gt aT gt a TT gt (T)
• What language does this produce (list the terms)?

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Regular grammar

• All rules are of the form
• auxiliary symbol gt final symbol or
• auxiliary symbol gt single final symbol followed
  by an auxiliary symbol

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• So a regular grammar is CFG!

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Regular grammars FSA

• Any language accepted by a regular grammar is
  accepted by a FSA and
• Any language accepted by a FSA is accepted by a
  regular grammar.
• Said a different way
• If a language L is accepted by a regular grammar
  G, we can define a FSA that accepts it AND if a
  language L is accepted by a FSA, we can define a
  regular grammar that accepts it.

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Informal proof

• Have a Grammar G with production rules of the
  accepted form.
• Define a FSA with one state for each auxiliary
  symbol, plus one more state. Make the initial
  state the one corresponding to the initial
  symbol. Add one more state, F and let it be the
  final state.
• If PgtaQ, where P and Q are auxiliaries and a is
  a symbol in the final alphabet, make an edge from
  P to Q, with label a.
• If Pgta, make an edge going from P to F with
  label a.

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Cont.

• Given a FSA, define a grammar with auxiliary
  symbols corresponding to the states. The initial
  symbol is the one corresponding to the initial
  state.
• Add production rules PgtaQ for all edges from P
  to Q.
• Change all production rules PgtaF to Pgta.

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Closure

• Regular languages are closed with respect to
  various operations, including concatenation.
• If M is a regular language and N is a regular
  language, then we can define MN is the language
  made up of all the strings of the form sm sn
  where sn where sm is in M and sn is in N. Then MN
  also is regular.

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Closure

• If M and N are regular languages then so is the
  union of M and N.
• This is defined as the set of strings that are in
  M or are in N (or both).
• This also is a regular language.

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Regular expressions

• Yet another equivalent definition!
• A regular expression over an alphabet is a
  pattern defined using letters from the alphabet
  combined using notation for union, concatenation,
  wild card, counts.
• The regular expession defines a set of strings,
  that is, a language

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CFG andPDA

• Same equivalence here.

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Turing history

• Invented what came to be called Turing
  machines. Proved various things about
  them, including Halting problem
• Worked in Bletchley Park during WWII built a
  machine to help decode German codes (codes
  changed). Given medal (in secret)
• Worked in general area, including spending time
  at Princeton with Van Neumann, others. Wrote
  papers on computer chess, notion of Turing test
• Arrested convicted for homosexual acts. Made to
  take hormone drugs. Appeared to be okay
• Committed suicide

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Homework

• Pick 1 Please think about it first instead of or
  before trying to look it up. Informal proofs are
  okay.
• CFG and PDA equivalence
• If you believe this, then there does exist a PDA
  for the homework problem.
• Regular expression and (either) FSA or Regular
  grammar equivalence
• Possibilities for presentation topics in Turing
  work, history
• Keep up postings
• Shai videos