Language, machines and artificial intelligence

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• Language, machines and artificial intelligence

When two trains meet each other at a railroad
crossing, each shall come to a stop, and neither
shall proceed until the other has gone.
TEXAS state law
Source www.dumblaws.com
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Grammars

• A grammar is a set of rules used to define the
  structure of the strings in a language.
• If L is a language over an alphabet A, then a
  grammar for L consists of a set of grammar rules
  of the following form 
• ? ? ?
• where ? and ? denote strings of symbols taken
  from A and from a set of grammar symbols
  (non-terminals) that is disjoint from A

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Productions

• A grammar rule ? ? ? is often called a
  production, and it can be read in any of several
  ways as follows
• "replace ? by ?",
• ? produces ?,"
• "? rewrites to ? ,
• "? reduces to ?."

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Where to begin

• Every grammar has a special grammar symbol called
  a start symbol, and there must be at least one
  production with left side consisting of only the
  start symbol.
• For example, if S is the start symbol for a
  grammar, then there must be at least one
  production of the form
•   S? ?.

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The Four Parts of a Grammar

• An alphabet N of grammar symbols called
  non-terminals. (Usually upper case letters.)
•  2. An alphabet T of symbols called terminals.
  (Identical to the alphabet of the resulting
  language.)
•  3. A specific non-terminal called the start
  symbol. (Usually S. )
•  4. A finite set of productions of the form ? ?
  ?, where ? and ? are strings over the alphabet
  N ? T

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Example

• Let A a, b, c. Then a grammar for the
  language A can be described by the following
  four productions
• S ? ?
• S ? aS
• S ? bS
• S ? cS.
• Or in shorthand
• S ? ? aS bS cS,
• "S can be replaced by either ?, or aS, or bS, or
  cS."

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• A string made up of grammar symbols and language
  strings is called a sentential form.
• If x and y are sentential forms and ? ? ? is a
  production, then the replacement of ? by ? in x?y
  is called a derivation, and we denote it by
  writing
•   x?y ? x?y.

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Sample derivation.

• S ? ? aS bS cS,
• S ? aS
• S ? aS ? aaS.
• S ? aS ? aaS ? aacS ? aacbS..
• S ? aS ? aaS ? aacS ? aacbS ? aacb?
• aacb
• A short hand way of showing a derivation exists
• S ? aacb
• derives in zero or more steps

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Other shorthand

• The following three symbols with their associated
  meanings are used quite often in discussing
  derivations
•  ? derives in one step,
•  ? derives in one or more steps,
•  ? derives in zero or more steps.

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A more complex grammar

•   S ? AB
• A? ? aA
• B ? ? bB.
•  
• We can deduce that the grammar non-terminal
  symbols are S, A, and B, the start symbol is S,
  and the language alphabet includes ?, a, and b.

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Another derivation

• Let's consider the string aab.
• The statement S ? aab means that there exists a
  derivation of aab that takes one or more steps.
• For example, we have
•  S ? AB ? aAB ? aaAB ? aaB ? aabB ? aab.

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Grammar specifies a language

• If G is a grammar, then the language of G is the
  set of language strings derived from the start
  symbol of G. The language of G is denoted by
•  
• L(G)

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• If G is a grammar with start symbol S and set of
  language strings T, then the language of G is the
  following set
•  
• L(G) s s ? T and S ? s.

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Finite languages

• If the language is finite, then a grammar can
  consist of all productions of the form
• S ? w
• for each string w in the language.
• For example, the language a, ba can be
  described by the grammar S ? a ab.

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Infinite languages

• If the language is infinite, then some production
  or sequence of productions must be used
  repeatedly to construct the derivations.
• Notice that there is no bound on the length of
  strings in an infinite language.
• Therefore there is no bound on the number of
  derivation steps used to derive the strings.
• If the grammar has n productions, then any
  derivation consisting of n 1 steps must use
  some production twice

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• For example, the infinite language anb n ? 0
  can be described by the grammar,
• S ? b aS.
• To derive the string anb, use the production
• S ? aS
• repeatedly --n times to be exact-- and then stop
  the derivation by using the production S ? b.
• The production S ? aS allows us to say
• If S derives w, then it also derives aw,"

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Recursion

• A production is called recursive if its left side
  occurs on its right side.
• For example, the production S ? aS is recursive.
• A production S ? ? is indirectly recursive if S
  derives (in two or more steps) a sentential form
  that contains S.

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Indirect recursion

• For example, suppose we have the following
  grammar
•   S ? b aA
• A ? c bS.
•  The productions S ? aA and A ? bS are both
  indirectly recursive  
• S ? aA ?abS, 
• A ?bS ?baA.

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• A grammar is recursive if it contains either a
  recursive production or an indirectly recursive
  production.
• A grammar for an infinite language must be
  recursive!
• However, a given language can have many grammars
  which could produce it.

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Combining Grammars

• Suppose M and N are languages. We can describe
  them with grammars have disjoint sets of
  nonterminals.
• Assign the start symbols for the grammars of M
  and N to be A and B, respectively
• M A ? ? N B ? ?
• Then we have the following rules for creating new
  languages and grammars

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Union rule

• The union of the two languages, M ? N, starts
  with the two productions
• S ? A B
• followed by the grammars of M and N
• A ? ?
• B ? ?

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Product rule

• Similarly, the language M ? N starts with the
  production
• S ? AB
• followed by, as above,
• A ? ?
• B ? ?

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Closure rule

• Finally, the grammar for the closure of a
  language, M, starts with the production
• S ? AS ?
• followed by the grammar of M
• A ? ?

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Example union

• Suppose we want to write a grammar for the
  following language 
• L ?, a, b, aa, bb, ..., an, bn, ...
• L is the union of the two languages
• M an n ? N
• N bn n ? N

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• Thus we can write a grammar for L as follows
• S ? A B union rule,
• A ? ? aA grammar for M,
• B ? ? bB grammar for N.

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Example product

• Suppose we want to write a grammar for the
  following language 
• L ambn m,n ? N
• L is the product of the two languages
• M am m ? N
• N bn n ? N

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• Thus we can write a grammar for L as follows
•  
• S ? AB product rule,
• A ? ? aA grammar for M,
• B ? ? bB grammar for N.

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Example closure

• Suppose we want to construct the language L of
  all possible strings made up from zero or more
  occurrences of aa or bb.
• L aa, bb M
• M aa, bb

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• So we can write a grammar for L as follows
•  
• S ? AS ? closure rule,
• A ? aa bb grammar for aa, bb.

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An equivalent grammar

• We can simplify this grammar
• Replace the occurrence of A in S ? AS by the
  right side of A ? aa to obtain the production S ?
  aaS.
• Replace A in S ?AS by the right side of A ? bb to
  obtain the production S ? bbS.
• This allows us to write the the grammar in
  simplified form as
•   S ? aaS bbS ?

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Some simple grammars