6001 structure

1
6001structure interpretation of computer
programsrecitation 9/ october 15, 1997
2
overview

• todays ideas
• dictionaries
• pattern matching
• cross products

3
dictionaries

• model
• a dictionary is a table that associates keys with
  values
• each key is associated with at most one value
• constructors
• new gt Dict
• add Dict x Key x Val gt Dict
• observers
• containsKey Dict x Key gt Bool
• lookup Dict x Key gt Val
• contract
• (containsKey (new) k) gt f
• (containsKey (add d k1 v) k2) gt t when k1
  k2, or whatever (containsKey d k2 v) evals to
  otherwise
• (lookup (add d k1 v) k2) gt v when k1 k2, or
  whatever (lookup d k2 v) evals to otherwise
• assume lookup is only called when dict contains
  keyand add is only called with keys not
  contained in dict

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implementing dictionaries

• simple representation association list
• (define (lookup d k) (cadr (assq k d)))
• orders of growth (in n, length of assoc list)
• time to lookup n
• time to insert new entry 1
• if list is sorted?
• worst case is no better
• but on average, only search half the list
• better representations
• can learn about these in algorithms course
• tree entries are sorted by key, and arranged on
  a tree
• time is order (log n) to lookup
• hash table
• entries are stored in lists of buckets
• each list is associated with an array element
• hash function applied to key gives array
  index,designed to spread entries evenly so that
  lists are short
• time is order (1) to lookup

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dictionary merge

• a new operation
• merge Dict x Dict --gt Dict
• resulting dictionary contains the entries of both
  dictionaries
• if the argument dictionaries have any common
  keys, they must agree on their values
• otherwise merge fails
• implementation
• note use of special failure value
• (define (merge d1 d2)(if (null? d1) d2
  (let ((b (car d1)) (k1 (car b)) (v1 (cdr
  b))) (if (containsKey d2 k1) (if (equal? v1
  (lookup d2 k1)) (merge (cdr d1)
  d2) fail) (cons b (merge (cdr d1)
  d2)))))))

6
using dictionaries

• examples
• browser cache URLs to cached web pages
• email client aliases to email addresses
• file directory file name to file contents
• dictionary packages
• many programming languages provide built-in
  support for dictionary-like objects
• Javas util package has a dictionary class with
  several implementations, such as hash table
• Scheme has some built-in procedures for handling
  association lists
• assq Key x List (Pair (Key, Val)) gt Pair (Key,
  Val)
• (define al (a.1 b.2 c.3))
• (assq b al) gt a.1

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patterns

• definition
• a pattern is a symbol or a list of symbols
• each symbol is either an atom or a variable
• a variable is prefixed with a question mark ?v
• instantiation
• given a pattern and a dictionary from vars to
  atoms,we can generate a symbol or a list of
  symbols
• example
• pattern is (?greeting there ?name )
• dictionary is ((?greeting hello) (?name alice))
• result is (hello there alice)
• matching
• given some result and a pattern, can we find a
  dictionary? these are called matches
• may be none or one
• examples for datum (hello there alice)
• pattern (?greeting there ?firstname) one match
• pattern (?greeting there ?firstname ?lastname)
  no matches
• pattern (?greeting ?name) no matches

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tilde variables

• new kind of variable
• v is instantiated with a list of symbols
• the elements of this list (not the list itself)
  are inserted when instantiated
• example
• pattern is (greeting ?name )
• dictionary is ((greeting (hello there)) (?name
  alice))
• result is (hello there alice)
• v can be instantiated by an empty list
• matching
• how many matches now?
• can have several matches for same pattern and
  datum
• example datum (hello friend alice)
• pattern (greeting name) produces three matches,
  these
• ((greeting (hello) (name (friend alice)))
• ((greeting (hello friend) (name (alice)))
• and also these
• ((greeting ()) (name (hello friend alice)))
• ((greeting (hello friend alice) (name ()))

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rules

• a rule consists of
• two patterns
• one is used to generate matches (a list of
  dictionaries) called the matching pattern
• the other is used as a template to generate
  results called the template
• examples from toy Eliza program
• (a mother b) (tell me more about your mother)
• ?x (why are you being short with me ?)
• (phrase) (what makes you say phrase ?)
• yes (you sound confident)
• example dotted tail notation
• (define (sum x . t) (if (null? t) x ( x (sum
  t)))
• how many matches for this kind of pattern?
• nameless vars
• no point giving names to vars not used in the
  template
• so just use ? and by themselves
• note that template is not exactly a pattern it
  can use ? and as atoms

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how many?

• how many results of applying a rule?
• matching can produce 0, 1 or several dictionaries
• instantiation always produces exactly one result
  from each pattern and dictionary
• but two dictionaries may yield the same result
• example
• datum is (1 2)
• pattern is (a b)
• template (b a)
• 3 matches (a associated with (), (1), (1 2))
• but only 2 results (1 2) and (2 1)

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multiply-occurring vars

• what happens when a variable occurs twice?
• given our definition of matching, answer is
  clearmust be matched to the same value in both
  places
• implementation in problem set uses dictionary
  merging
• example
• ( a a ) (youre repeating yourself)
• ( ?a ?a ) (youre stammering)
• (?x ?x) (2 ?x)
• (?x / ?x) (1)

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more elaborate rules

• in the problem set, youll encounter
• restriction rules
• apply an arbitrary test to the dictionary before
  allowing match
• user provides code for test as a procedure Dict
  gt Bool
• general rules
• get around simple instantiation mechanism
• in problem set, user-supplied procedure
  constructs result from dictionary directly

13
sequencing the application of rules

• a rule-based system has
• a set of rules, often very large
• heuristics that determine which order to apply
  the rules in
• which rule is applied first?
• are rules applied repeatedly?
• partial matches
• rules may be applied against part of the input
• for example, in arithmetic simplification want to
  reduce (x x) to 2xwhatever context it appears
  in
• so (y x x) should be simplified to (y 2x)

14
forming cross products

• problem
• given two lists, p and q, form the list
  consisting of all pairs pi.qjin order, where pi
  is the ith element of p and qj is the jth element
  of q
• prelude apply
• (apply op l) applies the procedure op to the
  arguments given by the list l
• (apply (1 2 3)) gt 6
• a neat solution
• first make a list from p by pairing each of its
  elements with all the elements of q(define
  (tensor p q) (map (lambda (pi) (map (lambda (qj)
  (cons pi qj)) q) p)
• now append all the resulting lists
  together(define (cross p q) (apply append
  (tensor p q)))
• puzzle
• can you generalize this so that rather than just
  pairing pi and qj it applies someprocedure proc,
  giving a list of the results of evaluating (proc
  pi qj)?
• can you eliminate all pairings that give a result
  satisfying some property? (example proc is ,
  drop all zeros)

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student puzzle

• for Wednesday
• implement a procedure
• remdups List(t) x (t x t gt bool) gt List(t)
• which given a list, returns a list containing the
  same elementsin the same order, but with
  duplicates removed
• equality of elements is determined using the
  procedure argument
• hint use filter