Lecture 2: Schreme
Available within the network will be functions and services to which you
subscribe on a regular basis and others that you call for when you need
them. In the former group will be investment guidance, tax counseling,
selective dissemination of information in your field of specialization,
announcement of cultural, sport, and entertainment events that fit your
interests, etc. In the latter group will be dictionaries, encyclopedias,
indexes, catalogues, editing programs, teaching programs, testing
programs, programming systems, data bases, and – most important –
communication, display, and modeling programs. All these will be – at
some late date in the history of networking - systematized and coherent;
you will be able to get along in one basic language up to the point at
which you choose a specialized language for its power or terseness.
J. C. R. Licklider and Robert W. Taylor,
The Computer as a Communication Device, April 1968
CS655: Programming Languages
David Evans
University of Virginia
http://www.cs.virginia.edu/~evans
Computer Science
Menu
• Registration Survey Results
– My Answers
•
•
•
•
Universal Languages
Stuff Languages are Made Of
Scheme
Infinitely many different functions
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Favorite Programming Languages
C
C++
Java
Perl
My answer:
41/3
21/3
21/3
1
– For getting work done: C (with LCLint)
– For reading other people’s code: CLU
– For fun: FL, Scheme
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Most Hated PL
COBOL
2
Perl
2 (also favorite)
Java
1
Lisp
1
None
4
My answer:
For asthetics, readability: APL
For teaching intro CS courses: C++
For writing big systems: Lisp
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Experience of Students in this Class
C++ (38 years), C (36), BASIC (29), Pascal (21),
various assemblies (12), Java (9), Perl (5),
FORTRAN (4), Python (2½), Ada (2),
Lisp dialects (1½), COBOL (1), Smalltalk (½)
• Algol60 direct successors = 106
• Pre-Algol languages = 46
• Completely object-oriented languages = ½
– But O-O doesn’t really mean anything
• Pure functional languages = 0
– Python is close (but has globals and state)
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Partner Preference
Assigned Randomly
Choose Your Own 4
No Clear Preference
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2
6
Can class go over time?
No problem 10
Problem
0
One said, “but if I am bored and you
keep me there I will be really annoyed
and glare at you”.
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Technical Questions
• Used higher-order procedures:
– No: 4 A little: 3 Yes: 3
• Wrote function that returned infinitely
many possible different functions
– No: 10 (some used tables of function
pointers)
• Fixed points: find x where f(x) = x
– Gets really interesting when x is a function!
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Why taking this class?
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Next: Computability and
Universal Programming
Languages
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Computability
• A programming language is universal if
it can express all computable functions.
• A function is “computable” if there exists
an algorithm that computes it.
• An algorithm is a computational
procedure that takes a finite number of
steps.
• What’s a computational procedure?
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Turing Machine
Tape head can:
- Read symbol from tape
- Write symbol on tape
- Move tape left or right one square
- Change its own state (finite state machine)


Infinitely long tape
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Universal Turing Machine
• Universal Turing Machine is a Turing
machine that can simulate any other
Turing machine
• Input (start state on tape) contains both
a description of a Turing Machine (i.e., a
program) and its input
• Yes, there really are such things
– Hopefully you will see it in CS660
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Computability
• A computational procedure is something
that can be described by a Turing
Machine
• A language is universal if it can express:
all algorithms
= all finite computational procedures
= all Turing machines
= one Universal Turing Machine
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Not everything is computable
• Halting Problem – will a Turing Machine
halt?
• Suppose we could compute it:
(define (halts? P) ...)
returns true if P halts, false otherwise.
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Based on Duane Boning,
http://sicp.ai.mit.edu/Fall-1999/lectures/lec25/6001-12-9-99-computability.ppt
(define contradicts-halts
(if (halts? contradict-halts)
(loop-forever)
#t))
If contradict-halts halts, it loops forever;
If contradict-halts doesn’t halt, value is true.
Yikes! halts? must not exisit!
Note: this is not a proof. (proof in 650)
Perhaps it is if that doesn’t exist.
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Is a language universal?
• How can we prove a language is
universal?
– Produce a program that implements a
Universal Turing Machine in the language
(usually this is pretty easy)
• How can we prove a language is not
universal?
– Prove that it cannot express the function of
some Turing Machine
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Elements of Language
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Elements of Language
• (Almost?) All Languages have:
– Primitives
– Means of combination
• English:
– Primitives = words (?)
• e.g., “Floccipoccinihilipilification” (the act of
rendering useless)
– Means of combination
• e.g., Sentence = Subject Verb Object
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Means of Combination
• Allow us to say infinitely many things with
a finite number of primitives
– In English, words aren’t really primitives
(linguists call the primitives morphemes –
smallest units of meaning)
– Means of Combination work on word parts:
Antifloccipoccinihilipilification (the act of not
rendering useless)
Antifloccipoccinihilipilificationator (one who does the
act of not rendering useless)
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Describing Means of Combination
• BNF (Backus Naur Form)
<non-terminal> ::= <replacement>
Terminals are primitives in the language (don’t
appear on left-hand side.)
• Alternatives
<non-terminal> ::= <replacement1>
| <replacement2>
short for:
<non-terminal> ::= <replacement1>
<non-terminal> ::= <replacement2>
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BNF
• John Backus
– FORTRAN (manual described syntax using
verbose English, not a formal notation)
– Member of Algol60 committee, introduced
formal notation for syntax
– Influential in development of functional
languages: FP, FL, Turing Lecture
• Normal  Peter Naur (Donald Knuth’s
suggestion)
– Chair of Algol60 committee
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BNF Example
Sentence ::= NP VP
NP ::= Noun
VP ::= Verb
Noun ::= Dave | Scheme
Verb ::= rocks | sucks
Could this language be a universal programming language?
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BNF Example
Sentence ::= NP VP
NP ::= Noun | Noun and NP
VP ::= Verb
Noun ::= Dave | Scheme
Verb ::= rocks | sucks
We can express infinitely many things with a tiny language!
With the right meaning function, this could be a
universal programming language.
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Scheme
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Scheme
• Like all (?) languages, Scheme has:
– Primitives
– Means of Combination
• Also, like all (?) reasonable
programming languages it has:
– Means of Abstraction
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Mini-Scheme Primitives
• Numerals
• Functions
• Constants
0
+
#f
Examples
655 3.52
-58
#t
• You know what the primitives mean.
– Numeral  Number
– Function  Function
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+  math addition
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Means of Combination
• Application
Expression ::= (Expression)
Expression ::= (Expression Expression)
Expression ::= (Expression Expression Expression)
...
Expression ::= (Expression*)
(+ 655 58)
(* (+ 0 (+ 2 2)) 6)
“Evaluate all the expressions, then apply the first
expression (a procedure) to all the other values.”
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Means of Combination
• Special Forms
(if Expression1 Expression2 Expression3 )
If the value of Expression1 is #f, the value
of the if is the value of Expression3.
Otherwise, it is the value of Expression2 .
• A few others, but they are not necessary
(note: neither is if!)
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Means of Combination
• Make Procedure
Expression ::=
(lambda (Variable) Expression)
“A procedure that when applied to value,
evaluates to Expression with all instances
of Variable substituted for by value.”
Note: “substitution” is complicated.
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Mini-Scheme
Expression ::=
(Expression*)
| (lambda (Variable) Expression)
| (if Expression1 Expression2 Expression3 )
| numeral | #f | #t | +
Variable ::= identifier
Is mini-scheme a Universal Programming Language?
(Left as challenge problem.)
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Means of Abstraction
(define Variable Expression)
Substitute Expression for Variable.
(define two 2)
(+ two two)
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Abstracting Procedures
(define square (lambda (x) (* x x)) proc
(square two)
4
(define (Variable1 Variable2) Expression)
is just an abbreviation for:
(define Variable1
(lambda (Variable2) Expression)
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Multiple Arguments
(define (Variable1 Variable2 Variable3)
Expression)
is just an abbreviation for:
(define Variable1
(lambda (Variable2)
(lambda (Variable3)
Expression)))
This is called “currying”.
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Currying
(define add
(lambda (x)
(lambda (y)
(+ x y))))
(add 5)
proc
((add 5) 2)
7
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add
• add is a procedure that returns infinitely
many different procedures!
(define add
(lambda (x)
(lambda (y)
(+ x y))))
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triadd
(define triadd
(lambda (x) (lambda (y) (lambda (z)
(+ x y z)))))
• triadd is a procedure that returns
infinitely many different procedures, each
of which is a procedure that returns
infinitely many different procedures!
• So, after today your answer to survey
question: “Yes, infinitely many!”
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Charge
• Problem Set 1 out today
– Its not long but may be hard, start it soon
– Question 0 think about:
• Do we really need define?
• What scheme expressions do not have values?
• Next time: Higher-Order Functions, Intro to
Metacircular Evaluators
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