Programming Languages

1
Programming Languages

• Syllogisms and Proof by Contradiction
• Midterm Review

2
Notions of Truth
Propositions Statements that can be either True
or False
Truth Are there well formed propositional
formulas (i.e., Statements) that return True when
their input is True truth1 (Bool -gt Bool) -gt
Bool truth1 wff (wff True) truth2 (Bool -gt
Bool -gt Bool) -gt Bool truth2 wff (wff True
True) ( \ p -gt not p) ( \ p q -gt (p q)
(not p gt q)) ( \ p q -gt not p gt q) ( \ p q -gt
(not p q) (not p gt q) )
If it was never possible for it not to be True
that something was going to exist, and it will
never be possible for it not to be True that
something existed in the past then it is
impossible for Truth ever to have had a beginning
or ever to have an end. That is, it was never
possible that Truth cannot be conceived not to
exist. If R is something that can be conceived
not to exist and T is something that cannot be
conceived not to exist and T is greater than R
and God is that, than which nothing greater can
be conceived, then God exists and is Truth.
3
Proof by Contradiction
Facts

• A is an Animal.
• A Barks.
• A is a Dog - A is an Animal, A Barks. (A is a
  Dog IF A is and Animal AND A Barks)
• 
  -
  ,
• -(A is a Dog)

Database
Rule
Query (If you want to know if A is a Dog based
upon the Facts and Rules in the Database try to
see if A is not a Dog.)
? Contradiction Therefore A is a Dog
Can this reasoning be automated?
4
Syllogisms
Note this is Prolog notation. In standard
notation, this would be P ? Q.
P implies Q (i.e., Q is True if P is True or If P
is True then Q is True)
Syllogism e.g., Modus Ponens 1). P is True
2). P ? Q is True, therefore Q is True.
Green assume 2 things the implied Result is in
Red if the result is just True, then the
syllogism is Valid, if the results are True and
False, then the syllogism is Invalid.
5
Proof by Contradiction
P1 P2
1). Let P Its raining, Im outside (comma
means ) 2). P1. (P1 is True, i.e., its
raining) 3). P2. (P2 is True, i.e., Im
outside) 4). Q - P Im wet - Its raining,
Im outside. (if its raining and Im


outside then Im wet) 5). Q (To answer the
Query Am I wet against the Database, assume Im
not wet) 6). (Its raining, Im outside)
( From 4 and 5 and Pattern 1 ) 7). Im
outside
( From 2 and 6 and Pattern 2 ) 8). Contradiction
Therefore Im wet ( From 3 and 7 and
Pattern 3 )
Facts
Rule
Pattern 1 (Modus Tollens) Q - (P1, P2). -Q ?
-(P1, P2) Pattern 2 (Affirming a
Conjunct) P1. -(P1, P2) ? -P2 Pattern
3 P2. -P2 ? Contradiction
Database
P
R
S
6
Proof by Contradiction

• A is an Animal.
• A Barks.
• A is a Dog - A is an Animal, A Barks.
• -(A is a Dog)
• 3 and 4 and Pattern 1 ? -(A is an Animal, A
  Barks)
• 1 and 5 and Pattern 2 ? -A Barks
• 2 and 6 and Pattern 3 ? Contradiction
• Therefore A is a Dog

Database
Query (If you want to know if A is a Dog based
upon the Facts and Rules in the Database try to
see if A is not a Dog.)
7
Midterm Review

• What we cannot speak about we must pass over in
  silence.
• The world consists of independent atomic
  factsexisting states of affairsout of which
  larger facts are built.
• Language consists of atomic, and then
  larger-scale, propositions that correspond to
  these facts by sharing the same "logical form".
• Thought, expressed in language, "pictures" these
  facts.
• Tractatus

8
Previous Midterm Exam
9
Study All the Homework and Quizzes
10
Functional Programming (Relation Programming)
Universe (Concepts for building a lisp
interpreter)
T. Skolem World
Relation Function Primitive Recursive Functions Primitive Recursive Arithmetic Primitive Recursive Relations Gödel's Incompleteness Theorem
A. Church and H. Curry World
? calculus Alpha-conversion Beta Reduction Combinator Logic ltSgt, ltKgt, ltBgt, ltCgt, ltYgt, and cond
J. McCarthy and G. Sussman/G. Steel World
Lisp / Scheme First-Class Functions Function Definition (lambda) e.g., (lambda (x) x) Function Application (f a) Substitution (let) e.g., (let ((x 3)) x) car, cdr, cons, s-expressions foldr, foldl, map Dynamic / Static Scoping
S. Krishnamurthi World
PLAI AE, WAE, FAE Function Definition (fun) e.g., (fun (x) x) Function Application (app) e.g., (app (id 'f) (id 'a)) Substitution (with) e.g., (with (x 3) x) Converting withs to (f a) (I.e., the FAE parser converts concrete syntax to abstract syntax) Environment / Closures Dynamic / Static Scoping Lazy / Eager Evaluation (interp (parse expr) Env.)
11
High Level Language Universe (Concepts for
building a HLL interpreter like ReL)
N. Chomsky Regular Grammar World
Regular Grammar Regular Expressions Finite State Machines Token (Non-Terminal) Lexical Analyzer (Tokenizer) Parser.jj ./grammar/Python.g
N. Chomsky Regular Context Free Grammar World
Context-free Grammar BNF EBNF Terminal Syntactic Analyzer (Parser) Internal Parse Tree Ambiguous Grammar Parser.jj ./grammar/Python.g
Abstract Syntax Tree World
Nodes on an AST ./src/org/python/antlr/ast ./dist/bin/jython ast/astview.py file.py
Interpreter World
Process an AST Visitor Pattern Symbol Table Type Checking Runtime Stack / Activation Record SQL, SIM RDF/RDFS/OWL/SPARQL ./src/org/python/antlr/ast/Tuple.java ./org/python/core/PyTuple.java
Tools
javacc / JJTree / JTB / JJDoc o Antlr ant SQLDeveloper jDeveloper or Eclipse jSQLParser
12
(No Transcript)
13
Relations and Functions
Relations A Relation is a subset of the
cross-product of a set of domains. Functions An
n-ary relation R is a function if the first n-1
elements of R are the functions arguments and
the last element is the functions results and
whenever R is given the same set of arguments, it
always returns the same results. Notice, this is
an unnamed function!.
14
Function Bodies
With the same goals as Alfred Whitehead and
Bertrand Russell in Principia Mathematica (i.e.,
to rid mathematics of the paradoxes of the
infinite and to show that mathematics is
consistent) but also to avoid the complexities of
mathematical logic - The foundations of
elementary arithmetic established by means of the
recursive mode of thought, without use of
apparent variables ranging over infinite domains
Thoralf Skolem, 1923
Thoralf Skolem
In a small town there is only one barber. This
man is defined to be the one who shaves all the
men who do not shave themselves. The question is
then asked, 'Who shaves the barber?' If the
barber doesn't shave himself, then -- by
definition -- he does. And, if the barber does
shave himself, then -- by definition -- he does
not. or Consider the statement 'This statement is
false. If the statement is false, then it is
true and if the statement is true, then it is
false.
This article can be found in From Frege to
Gödel A Source Book in Mathematical Logic,
1879-1931 (Source Books in the History of the
Sciences)
A function is called primitive recursive if there
is a finite sequence of functions ending with f
such that each function is a successor, constant
or identity function or is defined from preceding
functions in the sequence by substitution or
recursion.
15
Primitive Recursive Functions and Arithmetic (see
A Profile of Mathematical Logic by Howard
Delong pages 152 160)

• Thoralf Skolem The foundations of elementary
  arithmetic by means of the recursive mode of
  thought, without the use of apparent variables
  ranging over infinite domains (1919, 1923).
• He assumed that the following notions are already
  understood
• natural number (0 is included in the book),
• successor of x (i.e., x),
• substitution of equals (if x y and y z, then
  x z),
• and the recursive mode of thought (see next
  page).

16
Primitive Recursive Functions and Arithmetic (see
A Profile of Mathematical Logic by Howard
Delong pages 152 160)

• The addition function
• Book notation Relation notation
  Arithmetic notation
• f(x, 0) x (x 0 x)
  x 0 x
• f(x, y) (f(x, y)) (x y ( x y
  b)) x y (x y)
• Example 2 3 (The similar example of 7 5 on
  pages 153-154 is actually a bit flawed as a
  comparison with the following will show)
• (0
  0 0) 2 0 2
• (0 0
  (0 0 b))
• gt (0 0 (0))
  gt (0 0 0) 2 1 3
• (0 0
  (0 0 b))
• gt (0 0 (0))
  gt (0 0 0) 2 2 4
• (0 0
  (0 0 b))
• gt (0 0 (0))
  gt (0 0 0) 2 3 5

17
Primitive Recursive Functions and Arithmetic (see
A Profile of Mathematical Logic by Howard
Delong pages 152 160)

• We may summarize the situation by saying that
  while the usual definition of a function defines
  it explicitly by giving an abbreviation of that
  expression, the recursive definition defines the
  function explicitly only for the first natural
  number, and then provides a rule whereby it can
  be defined for the second natural number, and
  then the third, and so on. The philosophical
  importance of a recursive function derives from
  its relation to what we mean by an effective
  finite procedure, and hence to what we mean by
  algorithm or decision procedure. DeLong, page
  156

18
Primitive Recursive Functions and Arithmetic (see
A Profile of Mathematical Logic by Howard
Delong pages 152 160)

• A Sequence of Functions from definitions on
  DeLong, page 157

Book notation Relation notation Relation notation Arithmetic notation
f1(x) x (x ( x 1)) (x ( x 1)) f1 is the successor function
f2(x) x (x x) (x x) f2 is the identity function with i 1
f3(y, z, x) z (y z x z) (y z x z) f3 is the identity function with i 2
f4(y, z, x) f1(f3(y,z,x)) (y z x ((x ( x 1)) (y z x z))) (y z x ((x ( x 1)) (y z x z))) f4 is defined by substitution for f1 and f3
This is how you would do this in lisp This is how you would do this in lisp This is how you would do this in lisp This is how you would do this in lisp
(let ((f1 (lambda (x) ( x 1))) (f3 (lambda (y z x) z))) (let ((f4 (lambda (y z x) (f1 (f3 y z x))))) (f4 2 4 6))) (let ((f1 (lambda (x) ( x 1))) (f3 (lambda (y z x) z))) (let ((f4 (lambda (y z x) (f1 (f3 y z x))))) (f4 2 4 6))) (let ((f1 (lambda (x) ( x 1))) (f3 (lambda (y z x) z))) (let ((f4 (lambda (y z x) (f1 (f3 y z x))))) (f4 2 4 6))) (let ((f1 (lambda (x) ( x 1))) (f3 (lambda (y z x) z))) (let ((f4 (lambda (y z x) (f1 (f3 y z x))))) (f4 2 4 6)))
f5(0, x) f2(x) (0 x ( x x)) f5(y, x) f4(y, f5(y,x), x) (not doable yet) f5(0, x) f2(x) (0 x ( x x)) f5(y, x) f4(y, f5(y,x), x) (not doable yet) f5 is defined by recursion and f2 and f4 f5 is defined by recursion and f2 and f4
f5 is primitive recursive addition f5 is primitive recursive addition f5 is primitive recursive addition f5 is primitive recursive addition
(let ((f1 (lambda (x) ( x 1))) (f2 (lambda (x) x)) (f3 (lambda (y z x) z))) (let ((f4 (lambda (y z x) (f1 (f3 y z x))))) (letrec ((f5 (lambda (a b) (if ( a 0) (f2 b) (f4 (- a 1) (f5 (- a 1) b) b))))) (f5 2 3)))) (let ((f1 (lambda (x) ( x 1))) (f2 (lambda (x) x)) (f3 (lambda (y z x) z))) (let ((f4 (lambda (y z x) (f1 (f3 y z x))))) (letrec ((f5 (lambda (a b) (if ( a 0) (f2 b) (f4 (- a 1) (f5 (- a 1) b) b))))) (f5 2 3)))) (let ((f1 (lambda (x) ( x 1))) (f2 (lambda (x) x)) (f3 (lambda (y z x) z))) (let ((f4 (lambda (y z x) (f1 (f3 y z x))))) (letrec ((f5 (lambda (a b) (if ( a 0) (f2 b) (f4 (- a 1) (f5 (- a 1) b) b))))) (f5 2 3)))) (let ((f1 (lambda (x) ( x 1))) (f2 (lambda (x) x)) (f3 (lambda (y z x) z))) (let ((f4 (lambda (y z x) (f1 (f3 y z x))))) (letrec ((f5 (lambda (a b) (if ( a 0) (f2 b) (f4 (- a 1) (f5 (- a 1) b) b))))) (f5 2 3))))
19
Primitive Recursive Functions and Arithmetic (see
A Profile of Mathematical Logic by Howard
Delong pages 152 160)

• Primitive Recursive Relations on DeLong, pages
  158-159
• Example of eq primitive recursive relation
• (letrec ((pd (lambda (x) (if ( x 0) 0 (- x 1))))
• (dm (lambda (x y) (if ( y 0) x (pd
  (dm x (pd y))))))
• (abs (lambda (x y) ( (dm x y)(dm y
  x))))
• (sg (lambda (x) (if ( x 0) 0 1)))
• (eq (lambda (x y) (sg (abs x y)))))
  (eq 1 1))

20
A little Bit of Lambda Calculus Lambda
Expressions

• The manipulation of meaningless symbols?
• ________________________________________________
• A lambda expression is a particular way to define
  a function
• LambdaExpression ? variable (M N) ( ?
  variable . M )
• M ? LambdaExpression
• N ? LambdaExpression
• E.g., ( ? x . (sq x) ) represents applying the
  square function to x.

21
A little Bit of Lambda Calculus Properties of
Lambda Expressions

• In (? x . M), x is bound. Other variables in M
  are free.
• A substitution of N for all occurrences of a
  variable x in M is written Mx ? N. Examples
• An alpha-conversion allows bound variable names
  to be changed. For example, alpha-conversion of
  ?x.x might yield ?y.y.
• A beta reduction ((? x . M)N) of the lambda
  expression (? x . M) is a substitution of all
  bound occurrences of x in M by N. E.g.,
• ((? x . x2) 5) 52

22
Lambda Calculus
Lambda Calculus
?x.x
?s.(s s)
?func.?arg.(func arg)
def identity ?x.x def self_apply ?s.(s s) def apply ?func.?arg.(func arg)
def select_first ?first.?second.first def select_second ?first.?second.second
def cond ?e1.?e2.?c.((c e1) e2)
def trueselect_first def falseselect_second def not ?x.(((cond false) true) x) Or def not ?x.((x false) true)
def and ?x.?y.(((cond y) false) x) Or def and ?x.?y.((x y) false)
def or ?x.?y.(((cond true) y) x) Or def or ?x.?y.((x true) y)
23
Lambda Calculus Substitution
In lambda calculus, if cond is defined as def
cond ?e1.?e2.?c.((c e1) e2), def and
?x.?y.(((cond y) false) x) is equivalent to def
and ?x.?y.((x y) false) because (((cond y)
false) x) (((?e1.?e2.?c.((c e1) e2) y) false)
x) ((?e2.?c.((c y) e2) false) x) (?c.((c y)
false) x) ((x y) false) In lambda calculus, if
cond is defined as def cond ?e1.?e2.?c.((c e1)
e2), def or ?x.?y.(((cond true) y) x) is
equivalent to def or ?x. ?y.((x true) y)
because (((cond true) y) x) (((?e1.?e2.?c.((c
e1) e2) true) y) x) ((?e2.?c.((c true) e2) y) x)
(?c.((c true) y) x) ((x true) y)
24
Simple Lisp in Scheme

• (letrec ( (first (lambda (List) (if (null? List)
  (list) (car List))))
• (sum-list (lambda (List) (if (null?
  List) 0 ( (car List) (sum-list (cdr List))))))
• (nth (lambda (N List) (if (not ( N
  0))(nth (- N 1) (cdr List))(car List))))
• (head (lambda (N List) (if ( N 0)
  (list) (cons (car List) (head (- N 1) (cdr
  List)))))) )
• (nth 1 (list 1 2 3)))
• ? 2
• (letrec ( (List (list 1 2 3 4 5 6))
• (first (lambda (List) (if (null?
  List) (list) (car List))))
• (sum-list (lambda (List) (if (null?
  List) 0 ( (car List) (sum-list (cdr List))))))
• (nth (lambda (N List) (if (not ( N
  0))(nth (- N 1) (cdr List))(car List))))
• (head (lambda (N List) (if ( N 0)
  (list) (cons (car List) (head (- N 1) (cdr
  List)))))) )
• (head (nth 1 List) List) )
• ? (list 1 2)
• Code for Chaitin page 43 - 44
• (letrec ( (map (lambda (Function List) (if (null?
  List) List (cons (Function (car List)) (map
  Function (cdr List))) )) )
• (factorial (lambda (N) (if ( N 0)
  1 ( N (factorial (- N 1)))))) )

25
Develop Substitution for the Following
Expressions
Start with schema from Chapter 2 Get the
following expressions to work (subst (id 'x) 'x
(id 'y)) (subst (num 2) 'x (num 1)) (subst (id
'x) 'x (num 1)) (subst (id 'y) 'x (num
1)) (subst (add (id 'x) (id 'x)) 'x (num
1)) (subst (with 'y (num 2) (id 'x)) 'x (num
1)) (subst (with 'y (num 2) (add (id 'x) (id
'y))) 'x (num 1)) (subst (with 'y (id 'x) (add
(id 'x) (id 'y))) 'x (num 1)) (subst (with 'x
(id 'x) (id 'x)) 'x (num 1)) (calc (subst (with
'y (add (num 2) (id 'x)) (add (id 'y)(id 'x))) 'x
(num 1)))
26
Scheme for Textbook Chapter 3
lang plai (define-type WAE num (n number?)
add (lhs WAE?) (rhs WAE?) sub (lhs WAE?)
(rhs WAE?) with (name symbol?) (named-expr
WAE?) (body WAE?) id (name symbol?))
subst WAE symbol WAE!WAE (define (subst expr
sub-id val) (type-case WAE expr
num (n) expr add (l r) (add
(subst l sub-id val)
(subst r sub-id val)) sub (l r)
(sub (subst l sub-id val)
(subst r sub-id val)) with
(bound-id named-expr bound-body)
(if (symbol? bound-id sub-id)
(with bound-id
(subst named-expr sub-id val)
bound-body) (with
bound-id (subst
named-expr sub-id val)
(subst bound-body sub-id val)))
id (v) (if (symbol? v sub-id) val expr)))
calc WAE!number (define (calc expr)
(type-case WAE expr num (n) n
add (l r) ( (calc l) (calc r))
sub (l r) (- (calc l) (calc r))
with (bound-id named-expr bound-body)
(calc (subst bound-body
bound-id
(num (calc named-expr))))
id (v) (error 'calc "free identifier")))
27

• Functions and Recursion

28
let transformation, differed substitution and
closures, and interpretation in FAE let
transformation (let ((A B)) C) ((lambda (A)
C) B) A B ---------------- C
------------------- A
---------- B -------- C (let (( x 3)) (let ((f
(lambda (y) ( x y)))) (f 4)) ((lambda (x)
((lambda (f) (f 4)) (lambda (y) ( x y))))
3) (app (fun 'x app (fun 'f app (id 'f) (num
4)) (fun 'y (add (id 'x) (id 'y)))) (num 3))
(app ------- arg1
------------------ ------------arg2--------------
- (app ---------------------------------------
arg1-------------------------------------
--arg2 Differed substitution and closures (aSub
'f (closureV 'y (add (id 'x) (id 'y)) (aSub 'x
(numV 3) (mtSub))) (aSub 'x (numV 3)
(mtSub))) Interpretation (interp (app (id 'f)
(num 4)) (aSub 'f (closureV 'y (add (id 'x) (id
'y)) (aSub 'x (numV 3) (mtSub))) (aSub 'x (numV
3) (mtSub))))
(numV 7)
29
Static and Dynamic Scoping   Static
scoping (interp (parse 'with x 5 f 4)
(aSub 'f (closureV 'y (add (id 'x) (id 'y)) (aSub
'x (numV 3) (mtSub))) (aSub 'x (numV 3)
(mtSub)))) (numV 7) Dynamic Scoping (interp
(parse 'with x 5 f 4) (aSub 'f (closureV 'y
(add (id 'x) (id 'y)) (aSub 'x (numV 3) (mtSub)))
(aSub 'x (numV 3) (mtSub)))) (numV 9) Think
about this expression for both Static and Dynamic
Scoping (let ((z 3)) (let ((d 3) (f (lambda (x)
x))) (let ((z 27))
(let ((z 3) (a 5) (x (lambda (x y) ( x ( y
z))))) (let ((z 9) (a 7)) (x z a))))))
30
Recursive Functions Exemplified by foldr in lisp
(letrec ((f (lambda (l) (if (null? l) '()
(cons (car l) (f (cdr l))))))) (f '(1 2 3 4 5
6))) '(1 2 3 4 5 6) (letrec ((f (lambda (v l)
(if (null? l) v (cons (car l) (f v (cdr
l))))))) (f '() '(1 2 3 4 5 6))) '(1 2 3 4 5
6) (letrec ((f (lambda (f1 v l) (if (null? l)
v (f1 (car l) (f f1 v (cdr l))))))) (f
cons '() '(1 2 3 4 5 6))) '(1 2 3 4 5 6) f
foldr If f1 cons, foldr is the identity
function for a list. Its the same as (cons 1
(cons 2 (cons 3( cons 4 (cons 5 (cons 6 '()))))))
(cons 1 (cons 2 (cons 3 (cons 4 (cons
5 (cons 5 '()))))))
(cons 1 (cons 2 (cons 3 (cons 4 (cons
5 (cons 5 '()))))))
This can be thought of as a stack with conss on
it.
Here the stack is upside down
31
Recursive Functions Exemplified by foldl in lisp
(letrec ((f (lambda (f1 v l) (if (null? l)
v (f f1 (car l) (cdr l)))))) (f cons '() '(1
2 3 4 5 6))) 6 (letrec ((f (lambda (f1 v l) (if
(null? l) v (f f1 (f1 (car l) v) (cdr
l)))))) (f cons '() '(1 2 3 4 5 6))) '(6 5 4 3 2
1) f foldl If f1 cons, foldl reverses the
list. foldl is tail-recursive because nothing
goes on the stack. Its the same as (cons 6 (cons
5 (cons 4 ( cons 3 (cons 2 (cons 1 '()))))))
Nothing goes on the stack in this case.
32
PLAI Chapters 4, 5 and 6
Chapter 6, Pages 41 42 first-order Functions
are not values in the language. They can only be
defined in a designated portion of the program,
where they must be given names for use in the
remainder of the program. The functions in F1WAE
are of this nature, which explains the 1 in the
name of the language. higher-order Functions can
return other functions as values. first-class
Functions are values with all the rights of other
values. In particular, they can be supplied as
the value of arguments to functions, returned by
functions as answers, and stored in data
structures.
Page 27 - "
33
A Different Road to High-level Languages
34
Well be starting with javacc ? moving to ANTLR
later
Instance of a Programming Language
int main () return 0
Internal Parse Tree
Program (abstract syntax) Function main
Return type int params Block
Return Variable returnmain, LOCAL
addr0 IntValue 0
Abstract Syntax
35
Syntax and Grammar Parser.jj
PARSER_BEGIN(Parser) import java.io. import
java.util. public class Parser public
static void main(String args) throws
ParseException Parser parser new Parser
(System.in) parser.ae()
PARSER_END(Parser ) SKIP " "
"\t" "\n" "\r" lt"//" ("\n","\r")
("\n""\r")gt TOKEN lt LCURLY "" gt
lt RCURLY "" gt lt MINUS "-" gt lt
PLUS "" gt TOKEN / Literals /
lt INTEGER ("0"-"9") gt TOKEN
ltERROR gt
Parser
Grammar Production Rules
void ae() Token n n ltINTEGERgt
System.out.print("(num " n ")")
list() void list() LOOKAHEAD(2)
ltLCURLYgt ltPLUSgt System.out.print(" (add ") (
ae() ) ltRCURLYgt System.out.print(") ")
ltLCURLYgt ltMINUSgt
System.out.print(" (sub ") ( ae() ) ltRCURLYgt
System.out.print(") ")
Tokens, Terminals
36
Example Javacc Parser
void program() ( rels() ) ( comps()
) void rels() String id
ltRELATIONgt ( id identifier()
System.out.println(" Saw IDENTIFIER " id)
) System.out.println("Saw a RELATION")
String identifier() Token t t
ltIDENTIFIERgt return new String(t.image)
void comps() String id1, id2
ltCOMPOSEgt id1 identifier() ltWITHgt id2
identifier() System.out.println("Saw
Composiiton of " id1 " with " id2)
PARSER_BEGIN(Rel) import java.io. import
java.util. public class Rel public
static void main(String args) throws
ParseException Rel parser new
Rel(System.in) parser.program()
PARSER_END(Rel) SKIP " " "\t"
"\n" "\r" lt"//" ("\n","\r")
("\n""\r")gt TOKEN lt RELATION
"Relation" gt lt COMPOSE "Compose" gt lt
WITH "with" gt TOKEN / Literals /
lt IDENTIFIER "A"-"Z", "a"-"z"
( "A"-"Z", "a"-"z", "0"-"9", "_" )
gt TOKEN ltERROR gt
37
Chomsky Hierarchy

• Regular grammar used for tokenizing
• Context-free grammar (BNF) used for parsing
• Context-sensitive grammar not really used for
  programming languages

38
Regular Grammar

• Simplest least powerful
• Equivalent to
• Regular expression (think of perl)
• Finite-state automaton
• Right regular grammar
• ? Terminal,
• A and B ? Nonterminal
• A ? ? B
• A ? ?
• Example
• Integer ? 0 Integer 1 Integer ... 9 Integer
  
• 0 1 ... 9

39
Regular Grammar

• Less powerful than context-free grammars
• The following is not a regular language
• an bn n 1
• i.e., cannot balance ( ), , begin end

40
Regular Expressions
x a character x \x an escaped character,
e.g., \n name a reference to a name M
N M or N M N M followed by N M zero or
more occurrences of M M One or more
occurrences of M M? Zero or one occurrence of
M aeiou the set of vowels 0-9 the set of
digits . any single character
41

Finite State Automaton for Identifiers
(S, a2i) (I, 2i) (I, i)
(I, ) (F,
) Thus (S, a2i) (F, )
42
Context-Free Grammar

• Production
• a ? ß
• a ? Nonterminal
• ß ? (Nonterminal ? Terminal)
• ie, lefthand side is a single nonterminal, and
  righthand side is a string of nonterminals and/or
  terminals (possibly empty).

43
Context-Sensitive Grammar

• Production
• a ? ß a ß
• a, ß ? (Nonterminal ? Terminal)
• ie, lefthand side can be composed of strings of
  terminals and nonterminals, however, the number
  of items on the left must be smaller than the
  number of items on the right.

44
Syntax

• The syntax of a programming language is a precise
  description of all its grammatically correct
  programs.
• Precise syntax was first used with Algol 60, and
  has been used ever since.
• Three levels
• Lexical syntax - all the basic symbols of the
  language (names, values, operators, etc.)
• Concrete syntax - rules for writing expressions,
  statements and programs.
• Abstract syntax - internal representation of the
  program, favoring content over form.

45
Grammars

• Grammars Metalanguages used to define the
  concrete syntax of a language.
• Backus Normal Form Backus Naur Form (BNF)
• Stylized version of a context-free grammar (cf.
  Chomsky hierarchy)
• First used to define syntax of Algol 60
• Now used to define syntax of most major
  languages
• Production
• a ? ß
• a ? Nonterminal
• ß ? (Nonterminal ? Terminal)
• ie, lefthand side is a single nonterminal, and ß
  is a string of nonterminals and/or terminals
  (possibly empty).
• Example
• Integer ? Digit Integer Digit
• Digit ? 0 1 2 3 4 5 6 7 8 9

46
Extended BNF (EBNF)
Additional metacharacters a series of zero
or more ( ) must pick one from a list
pick none or one from a list Example Expression
-gt Term ( - ) Term IfStatement -gt if (
Expression ) Statement else Statement EBNF
is no more powerful than BNF, but its production
rules are often simpler and clearer. Javacc
EBNF ( ) a series of zero or more ( ) a
series of one or more optional
47
High Level Overview of Grammar Concepts
a series of zero or more ( ) must pick one
from a list pick none or one from a
list expression -gt term ( - ) term term
-gt factor ( / ) factor factor -gt (
expression ) number // the parenthesis
are part of the grammar not the EBNF number -gt
0 1 2 3 4 5 6 7 8 9
6 ( 11 7 ) / 3 100 ?
48
Associativity and Precedence

• A grammar can be used to define associativity and
  precedence among the operators in an expression.
• E.g., and - are left-associative operators in
  mathematics
• and / have higher precedence than and - .
• Consider the following grammar
• Expr -gt Expr Term Expr Term Term
• Term -gt Term Factor Term / Factor Term
  Factor Factor
• Factor -gt Primary Factor Primary
• Primary -gt 0 ... 9 ( Expr )

49
Associativity and Precedence
Parse of 423 5 6 7
50

• Functions and Recursion

51
Runtime Stack for Functions Program
int h, i void B(int w) int j, k
i 2w w w1 void A(int x, int y)
bool i, j B(h) int main()
int a, b h 5 a 3 b 2
A(a, b)

• parameters and local variables
• Return address
• Saved registers
• Temporary variables
• Return value

52
with ReL
53
ReL Components
CodeCompiler.java
Python.g
AST
Visitor
The Tuple entry in CodeCompiler was modified to
deal with putting SIM, SQL, Prolog, and ASP
information and expressions on the runtime stack
and to assure that a new instance of PyTuple is
created after the expressions are evaluated.
Interpreter
The Python grammar was changed to include SIM,
SQL, Prolog, and ASP statements.
Runtime Stack
PyTuple.java ---------------- parseSIM()
jSQLParser
jSIMParser
jSQLParser parses SQL statements from PyTuple and
produces an AST that can be visited.
There is no jSIMParser to parse SIM statements
like jSQLParser for SQL but I hope one will be
built as a project.
parseSIM parses SIM statements from PyTuple and
produces and passes AS information to SIMHelper.
SIMHelper.java
SQLVisitor.java
SIMHelper produces appropriate SPARQL statements
for the SIM statements which are sent to
SPARQLDoer.
SPARQLDoer.java
SQLVisitor visits the jSQLParser AST and produces
appropriate SPARQL statements for the SQL
statements which are sent to SPARQLDoer.
Oracle DBMS
54
What would you expect to find in Jython

• A grammar file for the python language
  ./grammar/python.g
• Java classes that get instantiated as nodes on
  the Abstract Syntax Tree
• ./src/org/python/antlr/ast
• A way of printing the Abstract Syntax Tree
• ./dist/bin/jython ast/astview.py file.py
• Visitor classes for the interpreter
• ./org/python/antlr/ast/VisitorBase.java
• ./org/python/antlr/ast/VisitorIF.java
• ./org/python/antlr/Visitor.java
• A way to hook into the visitors process of
  setting up the runtime stack
• ./org/python/compiler/Code.java
• ./org/python/compiler/CodeCompiler.java
• A way to work with the results from the jython
  interpreter
• an example that well use a lot is
  ./org/python/core/PyTuple.java

55
cat tests/demo.py dist/bin/jython
ast/astview.py tests/demo.py MAKECONNECT URL
jdbcoraclethin_at_rising-sun.microlab.cs.utexas.ed
u1521orcl UNAME cs345_50 PWORD cs345_50p DROP
TABLE NEWTEST1 CREATE TABLE NEWTEST1 (VAL1
NUMBER, VAL2 NUMBER, VAL3 NUMBER) x3 for i in
2, 4, 7         INSERT INTO NEWTEST1 (VAL1,
VAL2, VAL3) VALUES ((lambda xx1) (4), i, x)
print (SELECT FROM NEWTEST1) print (SELECT
VAL2 FROM NEWTEST1) print (SELECT VAL1, VAL2
FROM NEWTEST1 WHERE VAL3 x) ('Module',
 ('body',   ('Expr (1,0)',    ('value',
('Connection (1,0)', ('elts',), ('ctx',
('Load',))))),   ('Expr (2,0)', ('value',
('Tuple (2,0)', ('elts',), ('ctx', ('Load',))))),
  ('Assign (4,0)',    ('targets', ('Name
(4,0)', ('id', 'x'), ('ctx', ('Store',)))),   
('value', ('Num (4,2)', ('n', 3)))),   ('For
(5,0)',    ('target', ('Name (5,4)', ('id',
'i'), ('ctx', ('Store',)))),    ('iter',    
('List (5,9)',      ('elts',       ('Num
(5,10)', ('n', 2)),       ('Num (5,13)', ('n',
4)),       ('Num (5,16)', ('n', 7))),     
('ctx', ('Load',)))),    ('body',     ('Expr
(6,1)',      ('value',       ('Tuple (6,1)',
       ('elts',         ('Call (6,49)',
         ('func',           ('Lambda (6,50)',
           ('args',             ('arguments',
             ('args',               ('Name
(6,57)', ('id', 'x'), ('ctx', ('Param',)))),
             ('vararg', None),
           ('body',             ('BinOp
(6,59)',              ('left',              
('Name (6,59)', ('id', 'x'), ('ctx',
('Load',)))),              ('op', ('Add',)),
             ('right', ('Num (6,61)', ('n',
1))))))),          ('args', ('Num (6,65)', ('n',
4))),          ('keywords',),         
('starargs', None),          ('kwargs', None)),
        ('Name (6,69)', ('id', 'i'), ('ctx',
('Load',))),         ('Name (6,72)', ('id',
'x'), ('ctx', ('Load',)))),        ('ctx',
('Load',)))))),    ('orelse',)),   ('Print
(7,0)',    ('dest', None),    ('values',
('Tuple (7,7)', ('elts',), ('ctx', ('Load',)))),
   ('nl', True)),   ('Print (8,0)',   
('dest', None),    ('values', ('Tuple (8,7)',
('elts',), ('ctx', ('Load',)))),    ('nl',
True)),   ('Print (9,0)',    ('dest', None),
   ('values',     ('Tuple (9,7)',     
('elts', ('Name (9,52)', ('id', 'x'), ('ctx',
('Load',)))),      ('ctx', ('Load',)))),   
('nl', True))))
AST
56
MAKECONNECT Python.g
57
OWL Inferencing A short Primer
owlFunctionalProperty owlInverseFunctionalProper
ty owlSymmetricProperty owlTransitiveProperty ow
linverseOf
rdfssubClassOf rdfssubPropertyOf rdfsdomain rdf
srange
owlequivalentClass owlequivalentProperty owldis
jointWith owlcomplementOf
owlsameAs owldifferentFrom
owlsomeValuesFrom owlallValuesFrom owlhasValue
58
Inference Examples
owlFunctionalProperty owlInverseFunctionalProper
ty owlSymmetricProperty owlTransitiveProperty ow
linverseOf
hasMother rdftype owlFunctionalProperty
John hasMother Mary John
hasMother Maria gt Mary owlsameAs
Maria Maria owlsameAs Mary
59
Inference Examples
owlFunctionalProperty owlInverseFunctionalProper
ty owlSymmetricProperty owlTransitiveProperty ow
linverseOf
hasSSN rdftype owlInverseFunctionalProperty
John hasSSN 123-45-6789 Johny
hasSSN 123-45-6789 gt John
owlsameAs Johny Johny owlsameAs John
60
Inference Examples
owlFunctionalProperty owlInverseFunctionalProper
ty owlSymmetricProperty owlTransitiveProperty ow
linverseOf
hasSibling rdftype owlSymmetricProperty
John hasSibling Mary gt Mary
hasSibling John
61
Inference Examples
owlFunctionalProperty owlInverseFunctionalProper
ty owlSymmetricProperty owlTransitiveProperty ow
linverseOf
hasAncestor rdftype owlTransitiveProperty
John hasAncestor Mary Mary
hasAncestor Tom gt John
hasAncestor Tom
62
Inference Examples
owlFunctionalProperty owlInverseFunctionalProper
ty owlSymmetricProperty owlTransitiveProperty ow
linverseOf
hasParent owlinverseOf hasChild
John hasParent Mary gt Mary
hasChild John
63
Inference Examples
owlequivalentClass owlequivalentProperty owldis
jointWith owlcomplementOf
Male owldisjointWith Female
John rdftype Male Mary
rdftype Female gt John
owldifferentFrom Mary Mary
owldifferentFrom John
64
Inference Examples
owlequivalentClass owlequivalentProperty owldis
jointWith owlcomplementOf
NonHuman owlcomplementOf Human
Fish rdfssubClassOf
NonHuman gt Fish owldisjointWith
Human Human owldisjointWith
Fish
65
Semantic Web RDF and RDFS Example
66
Semantic Web RDFS and OWL Example
67
Semantic Web SQL Type Query
USA
68
Semantic Web SPARQL
69
OWL Inferencing dont copy this demo from a
PDF, it wont work use the PPT file instead
-- If you dont have an RDS_DATA_TABLE in your
user account, uncomment the following two lines
the first time you run this. -- EXECUTE IMMEDIATE
'CREATE TABLE RDF_DATA_TABLE( id NUMBER, triple
SDO_RDF_TRIPLE_S)' -- SEM_APIS.CREATE_RDF_MODEL('
RDF_MODEL_CS345_PROF1', 'RDF_DATA_TABLE',
'triple') TRUNCATE TABLE "RDF_DATA_TABLE" drop
storage DROP SEQUENCE RDF_DATA_TABLE_SQNC CREATE
SEQUENCE RDF_DATA_TABLE_SQNC START WITH 1
NOCACHE INSERT INTO RDF_DATA_TABLE VALUES (
RDF_DATA_TABLE_SQNC.nextval, SDO_RDF_TRIPLE_S('RDF
_MODEL_CS345_PROF1lthttp//www.example.org/people.
owlgt', 'lthttp//www.example.org/people.owlOBJECT
gt', 'rdftype', 'rdfsClass')) INSERT INTO
RDF_DATA_TABLE VALUES ( RDF_DATA_TABLE_SQNC.nextva
l, SDO_RDF_TRIPLE_S('RDF_MODEL_CS345_PROF1lthttp/
/www.example.org/people.owlgt', 'lthttp//www.examp
le.org/people.owlANIMALgt', 'rdftype',
'rdfsClass'))
70
OWL Inferencing
INSERT INTO RDF_DATA_TABLE VALUES (
RDF_DATA_TABLE_SQNC.nextval, SDO_RDF_TRIPLE_S('RDF
_MODEL_CS345_PROF1lthttp//www.example.org/people.
owlgt', 'lthttp//www.example.org/people.owlANIMAL
gt', 'rdfssubClassOf', 'lthttp//www.example.org/pe
ople.owlOBJECTgt')) INSERT INTO RDF_DATA_TABLE
VALUES ( RDF_DATA_TABLE_SQNC.nextval,
SDO_RDF_TRIPLE_S('RDF_MODEL_CS345_PROF1lthttp//ww
w.example.org/people.owlgt', 'lthttp//www.example.
org/people.owlCATgt', 'rdftype',
'rdfsClass')) INSERT INTO RDF_DATA_TABLE
VALUES ( RDF_DATA_TABLE_SQNC.nextval,
SDO_RDF_TRIPLE_S('RDF_MODEL_CS345_PROF1lthttp//ww
w.example.org/people.owlgt', 'lthttp//www.example.
org/people.owlCATgt', 'rdfssubClassOf',
'lthttp//www.example.org/people.owlANIMALgt')) I
NSERT INTO RDF_DATA_TABLE VALUES (
RDF_DATA_TABLE_SQNC.nextval, SDO_RDF_TRIPLE_S('RDF
_MODEL_CS345_PROF1lthttp//www.example.org/people.
owlgt', 'lthttp//www.example.org/people.owli5gt',
'rdftype', 'lthttp//www.example.org/people.owlCA
Tgt')) INSERT INTO RDF_DATA_TABLE VALUES (
RDF_DATA_TABLE_SQNC.nextval, SDO_RDF_TRIPLE_S('RDF
_MODEL_CS345_PROF1lthttp//www.example.org/people.
owlgt', 'lthttp//www.example.org/people.owli5gt',
'lthttp//www.example.org/people.owlvaluegt',
'"101"xsdinteger')) Commit
71
OWL Inferencing
-- named graph local inference (NGLI) BEGIN sem_ap
is.create_entailment( 'RDF_ENT_CS345_PROF1',
models_in gt sem_models('RDF_MODEL_CS345_P
ROF1'), rulebases_in gt sem_rulebases('owl2
rl'), passes gt SEM_APIS.REACH_CLOSUR
E, inf_components_in gt null, options
gt 'LOCAL_NG_INFT' ) END /
72
OWL Inferencing
select g, s, p, o from table(sem_match('graph
?g ?s ?p ?o', sem_models('RDF_MODEL_CS345_PROF1
'),sem_rulebases('owl2rl'),null,null)) MINUS selec
t g, s, p, o from table(sem_match('graph ?g ?s
?p ?o', sem_models('RDF_MODEL_CS345_PROF1'),null
,null,null)) Returns http//www.example.org/peo
ple.owl http//www.example.org/people.owlCAT http
//www.w3.org/2000/01/rdf-schemasubClassOf
http//www.example.org/people.owlOBJECT
http//www.example.org/people.owl http//www.exam
ple.org/people.owli5 http//www.w3.org/1999/02/22
-rdf-syntax-nstype
http//www.example.org/people.owlANIMAL http//ww
w.example.org/people.owl http//www.example.org/pe
ople.owli5 http//www.w3.org/1999/02/22-rdf-synta
x-nstype http//www.example.org/p
eople.owlOBJECT
73
OWL Inferencing
SEM_MODELS('RDF_MODEL_CS345_prof1'),
null, SEM_ALIASES( SEM_ALIAS('',
'http//www.example.org/people.owl')), null)
) SELECT value from table( sem_match('select
where ?indiv rdftype CAT. ?indiv value
?value . ', SEM_MODELS('RDF_MODEL_CS345_prof1'),
sem_rulebases('owl2rl'), SEM_ALIASES(
SEM_ALIAS('', 'http//www.example.org/people.owl'
)), null) ) Returns Value 101
74
OWL Inferencing
SEM_MODELS('RDF_MODEL_CS345_prof1'),
null, SEM_ALIASES( SEM_ALIAS('',
'http//www.example.org/people.owl')), null)
) SELECT value from table( sem_match('select
where ?indiv rdftype ANIMAL. ?indiv
value ?value . ', SEM_MODELS('RDF_MODEL_CS345_pr
of1'), sem_rulebases('owl2rl'), SEM_ALIASES(
SEM_ALIAS('', 'http//www.example.org/people.owl'
)), null) ) Returns Value 101
75
OWL Inferencing
SEM_MODELS('RDF_MODEL_CS345_prof1'),
null, SEM_ALIASES( SEM_ALIAS('',
'http//www.example.org/people.owl')), null)
) SELECT value from table( sem_match('select
where ?indiv rdftype OBJECT. ?indiv
value ?value . ', SEM_MODELS('RDF_MODEL_CS345_pr
of1'), sem_rulebases('owl2rl'), SEM_ALIASES(
SEM_ALIAS('', 'http//www.example.org/people.owl'
)), null) ) Returns Value 101
76
Initial Concepts for
77
Proof by Contradiction
P1 P2
1). Let P Its raining, Im outside (comma
means ) 2). P1. (P1 is True, i.e., its
raining) 3). P2. (P2 is True, i.e., Im
outside) 4). Q - P Im wet - Its raining,
Im outside. (if its raining and Im


outside then Im wet) 5). Q (To answer the
Query Am I wet against the Database, assume Im
not wet) 6). (Its raining, Im outside)
( From 4 and 5 and Pattern 1 ) 7). Im
outside
( From 2 and 6 and Pattern 2 ) 8). Contradiction
Therefore Im wet ( From 3 and 7 and
Pattern 3 )
Facts
Rule
Pattern 1 (Modus Tollens) Q - (P1, P2). -Q ?
-(P1, P2) Pattern 2 (Affirming a
Conjunct) P1. -(P1, P2) ? -P2 Pattern
3 P2. -P2 ? Contradiction
Database
P
R
S
78
Proof by Contradiction

• A is an Animal.
• A Barks.
• A is a Dog - A is an Animal, A Barks.
• -(A is a Dog)
• 3 and 4 and Pattern 1 ? -(A is an Animal, A
  Barks)
• 1 and 5 and Pattern 2 ? -A Barks
• 2 and 6 and Pattern 3 ? Contradiction
• Therefore A is a Dog

Database
Query (If you want to know if A is a Dog based
upon the Facts and Rules in the Database try to
see if A is not a Dog.)