1. The document defines a set of functions by specifying an initial set and rules for generating new functions from the initial set. The initial set includes identity and projection functions on integers. New functions can be generated by adding 1 or assigning a constant. 2. If a function F can be computed by a Turing machine, then F is recursive. Conversely, if a function F is recursive (part of the defined set), then a Turing machine can be defined to compute F. 3. A grammar G consists of symbols divided into final and auxiliary symbols, production rules that replace strings of symbols, a starting auxiliary symbol, and a language of final symbol strings generated by starting with the starting symbol and applying the production

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2. 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) = x+1
 Constant: Ck(x) = k
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3.  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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4.  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 => string of symbols
 final symbols have no production rules of the form
f => …
 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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5.  Think of auxiliaries as parts of speech and
final symbols as (whole) words
 Let
s be sentence
vp be verb phrase
np be noun phrase
n be noun
v be verb
do be direct object
adj be adjective
adv be adverb
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6.  Think of parts of speech
s => np vp
np => adj n | n
vp => v | v adv | v do | v do adv
do => n
n => boy | girl | dog | cheese
v => walks | runs | jumps | eats
adj => short | tall | spotted
adv => fast | slowly
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7.  To construct the parse tree for
tall girl walks
short boy eats cheese slowly
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8.  Production rules are of the form
auxiliary symbol => string of symbols
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9.  There also can be rules of the form
aPb => ….
 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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10.  S => T
T => a
T => a + T
T => (T)
 What language does this produce (list the
terms)?
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