1. What would be our focus ?
Geometry deals with Declarative or “What is” knowledge.
Computer Science deals with Imperative or “How to” knowledge
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2. Declarative Knowledge
“What is true” knowledge
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3. Imperative Knowledge “How to” knowledge
To find an approximation of square root of x:
Make a guess G
Improve the guess by averaging G and x/G
Keep improving the guess until it is good enough
X = 2
G = 1
X/G = 2
G = ½ (1+ 2) = 1.5
X/G = 4/3
G = ½ (3/2 + 4/3) = 17/12 =
X/G = 24/17
G = ½ (17/ /17) = 577/408 =
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4. “How to” knowledge
Process – series of specific, mechanical steps for deducing information, based on simpler data and set of operations
Procedure – particular way of describing the steps a process will evolve through
Need a language for description:
vocabulary
rules for connecting elements – syntax
rules for assigning meaning to constructs – semantics
Using language of procedures to solve problems
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5. Scheme Basics Rules for Scheme
Legal expressions have rules for constructing from simpler pieces
(Almost) every expression has a value, which is “returned” when an expression is “evaluated”.
Every value has a type.
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6. Language Elements Primitives Means of combination Means of abstraction
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7. Language elements -- primitives
Numbers – self-evaluating rule
23  23
-36  -36
Names for built-in procedures
+, *, /, -, =, …
What is the value of such an expression?
+  [#procedure …]
Evaluating by looking up value associated with name
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8. Language elements – combinations
How do we create expressions using these procedures?
(+ 2 3)
Close paren
Open paren
Expression whose value is a procedure
Other expressions
Evaluate by getting values of subexpressions, then apply operator to values of arguments
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9. Language elements - combinations
Can use nested combinations – just apply rules recursively
(+ (* 2 3) 4) 10
(* (+ 3 4) (- 8 2)) 42
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10. Language elements -- abstractions
In order to abstract an expression, need way to give it a name
(define score 23)
This is a special form
Does not evaluate second expression
Rather, it pairs that name with the value of the third expression in an environment
Return value is unspecified
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11. Language elements -- abstractions
To get the value of a name, just look up pairing in environment
score  23
Note that we already did this for +, *, …
(define total ( ))
(* 100 (/ score total))  92
can create a complex thing, name it, treat it as primitive
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12. Scheme Basics Rules for evaluation If self-evaluating, return value.
If a name, return value associated with name in environment.
If a special form, do something special.
If a combination, then
a. Evaluate all of the subexpressions of combination (in any order)
b. apply the operator to the values of the operands (arguments) and return result
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13. Capturing common patterns
Here are some common patterns
(* 5 5)
(* 23 23)
(* score score)
(* x x)
How do we generalize (e.g. the last expression)?
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14. Language elements -- abstractions
Need to capture ways of doing things – use procedures
(lambda (x) (* x x))
parameters
body
To process
something
multiply it by itself
Special form – creates a procedure and returns it as value
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15. Language elements -- abstractions
Use this anywhere you would use a proceduure
((lambda (x) (* x x)) 5)
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16. Scheme Basics Rules for evaluation Rules for application
If self-evaluating, return value.
If a name, return value associated with name in environment.
If a special form, do something special.
If a combination, then
a. Evaluate all of the subexpressionss of combination (in any order)
b. apply the operator to the values of the operands (arguments) and return result
Rules for application
If procedure is primitive procedure, just do it.
If procedure is a compound procedure, then:
evaluate the body of the procedure with each formal parameter replaced by the corresponding actual argument value.
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17. Language elements -- abstractions
Use this anywhere you would use a proceduure
((lambda (x) (* x x)) 5)
(* 5 5) 25
Can give it a name
(define square (lambda (x) (* x x)))
(square 5)  25
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18. Taxonomy of Expressions
Primitive Expressions
Constants (self-evaluating)
Numerals, e.g.17 == Sch-Num 7
Strings, e.g. “hello” == Sch-String hello
Booleans, e.g. #f, #t == Sch-Bool
Names
E.g. + == primitive procedure
E.g. x == variable created by define or procedure application
Compound Expressions
Combinations: (<operator> <operand> <operand> …)
Special forms:
Define: (define <name> <exp>)
Lambda: (lambda (<formal1> <formal2> …) <body>)
If: (if <predicate> <consequent> <alternative>)
…. More to come
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