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Midterm Review
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Python class code for Aristotle example – animal, cat, human, and knowledge classes
class animal() : num_eyes = 2 num_teeth = 24 def __init__(self, e, t) : self.num_eyes = e self.num_teeth = t class cat(animal) : def __init__(self, c, l) : self.fur_color = c self.len_tail = l a1 = animal(2, 30)print a1.num_eyes, a1.num_teeth
my_cat = cat("yellow", 4)print my_cat.num_eyes, my_cat.num_teeth, my_cat.fur_color, my_cat.len_tail
class human(animal) : IQ = 100 sports_played = "Football" knowledge = [] def __init__(self, s, i = 100) : self.IQ = i self.sports_played = s
socrates = human(s="Golf")print socrates.num_eyes, socrates.num_teeth, socrates.IQ, socrates.sports_played
class knowledge() : grammatical = '' math = '' human = [] def __init__(self, g, m) : self.grammatical = g self.math = m
socrates_Knowledge = knowledge('Some Greek language knowledge', "Some math smarts")socrates.knowledge.append(socrates_Knowledge)socrates_Knowledge.human.append(socrates)print socrates.num_eyes, socrates.num_teeth, socrates.IQ, socrates.sports_played, socrates.knowledge[0].grammatical, socrates.knowledge[0].mathprint socrates.knowledgeprint socrates_Knowledge.grammatical, socrates_Knowledge.math, socrates_Knowledge.human[0].sports_played
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Code from September 17, 2013 Lectureclass Person() : name = None parents = [] kids = [] has = None def __init__(self, name) : self.name = name
p1 = Person("phil")p2 = Person("rita")p3 = Person("chris")p1.kids.append(p3)p3.parents.append(p1)
print "My name is ", p1.name, "My first kids name is ", p1.kids[0].name
class Job(): name = "" employees = [] def __init__(self, name) : self.name = name j1 = Job("professor")j1.employees = [] # This was added after class.p1.has = j1j1.employees.append(p1)print p1.has.name, j1.employees[0].name
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Code from September 17, 2013 Lectureclass Person(object): _registry = [] # _registry starts off as an empty list. name = "" amount = 0
def __init__(self, name, amount): self._registry.append(self) # Each instance of Person is added to _registry. self.name = name self.amount = amount def exchange(self, p, amnt) : # Behavior or Method self.amount -= amnt p.amount += amnt
p1, p2, p3, p4 = Person('tom', 1000), Person('jerry', 2000), Person('phineas', 3000), Person('ferb', 4000)for p in Person._registry: print p.name + ", " + str(p.amount) , # The comma at the end of the print statement causes all to print on one line.
def transfer(p1, p2, amnt) : p1.amount -= amnt # Fill in these 2 lines for Homework 1. Note, the “transfer” function p2.amount += amnt # requires no return statement.
transfer(p1, p2, 50)transfer(p3, p4, 50)
printfor p in Person._registry: print p.name + ", " + str(p.amount) , # More on Next Page
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Code from September 17, 2013 Lecture p1.exchange(p2, 50)
printfor p in Person._registry: print p.name + ", " + str(p.amount) , p10 = Person("me", "you")printprint p10._registryprintprint p1._registry
class MeekPerson(Person) : def __init__(self, name, amount): self._registry.append(self) # Each instance of Person is added to _registry. self.name = name self.amount = amount def exchange(self, p, amnt) : self.amount -= 2*amnt p.amount += 2*amnt
m1 = MeekPerson("snoopy", 10000)p6 = Person("charlie", 20000)m1.exchange(p6,50) # Dynamic Binding will be done here to choose the right method.printfor p in Person._registry: print p.name + ", " + str(p.amount) ,
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class LinkedList() : head = None tail = None size = 0 def cons(self, val) : n = Node(val) self.size+=1 if self.head == None : self.head = n self.tail = n else : n.next = self.head self.head = n return self def car(self) : return self.head.element def cdr(self) : l = self if self.size > 0 : l.head = l.head.next l.size -= 1 else : l.head, l.tail = None, None return l
def __str__(self): result = "'(" current = self.head for i in range(self.size+1): if current != None: result += str(current.element) if current.next != None : result += ", current = current.next else: result += ")" # Insert the closing ] in the string return result # Return an iterator for a linked list def __iter__(self): # To be discussed in Section 18.3 return LinkedListIterator(self.head)
# The Node classclass Node: def __init__(self, element): self.element = element self.next = None
class LinkedListIterator: def __init__(self, head): self.current = head
def next(self): if self.current == None: raise StopIteration else: element = self.current.element self.current = self.current.next return element
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Notables in LinkedList History
Aristotle Categories and Syllogism Object Oriented Concepts and Syllogistic Logic.
Euclid 5 axioms of Geometry in the “The Elements”
“The Elements” was responsible for the notion of Certainty until the discovery of non-Euclidian Geometry in the 19th Century.
Gottlob Frege Modern mathematical logic Propositional Calculus and Predicate Calculus
Giuseppe Peano The principles of arithmetic, presented by a new method.
First attempt at an axiomatization of mathematics/arithmetic.
Georg Cantor Theory of sets, sizes of Infinity, and paradoxes.
Mathematics can be axiomatized using set theory.
David Hilbert, Alfred Whitehead, and Bertrand Russell
Principia Mathematica A 30 year attempt to restore Certainty to mathematics by formalizing it using set theory and logic and to prove that mathematics is “consistent”.
Thoralf Skolem Primitive Recursive Functions: A reaction to the works of Cantor, Hilbert, Whitehead and Russell in which the notion of Infinity is abandoned.
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Notables in LinkedList History
Kurt Gödel First Incompleteness Theorem
In any formal system capable of expressing primitive recursive functions/arithmetic the statement “I am a statement for which there is no proof” can be generated.
Second Incompleteness Theorem
No formal system capable of expressing primitive recursive functions/arithmetic can prove its own consistency.
Restored the notion of Certainty but in a manner very different from that envisioned by Hilbert, Whitehead and Russell.
Recursive Functions Functions which are defined for every input. Infinity returns.
Alonso Church
Lambda Calculus A language for defining functions and function application inspired by Gödel's recursive functions.
Undecidability of equivalence
There is no algorithm that takes as input two lambda expressions and determines if they are equivalent.
Alan Turing Turing Machine and Halting Problem
Created a theoretical model for a machine, (Turing machine), that could carry out calculations from inputs. There is in general no way to tell if it will halt (undecidability of halting.)
Church-Turing Thesis
Recursive Functions = Effectively Computable = Computational Completeness = calculable on a Turing machine = Turing-Complete (e.g., Lambda Calculus).
For more information see http://www.cs.utexas.edu/~ear/cs341/automatabook/
Haskell Currie Combinator Logic Developed a Turing-Complete language based solely upon function application of combinators.
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Notables in LinkedList History
John McCarthy The lisp programming language. 1960 paper: “Recursive Functions of Symbolic Expressions and Their Computation by Machine”, see class calendar for a copy.
McCarthy took concepts from Gödel's incompleteness proof (substitution), lambda calculus (function definition and function application) and combinator logic (car, cdr, and cons as primitive operations on
linked-lists)
(let ((l (cons 'a (cons 'b '())))) (let ((first (lambda (x) (car x)))) (first l)))
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Aristotle’s syllogistic logic
If it’s raining outside, then the grass is wet.Its raining outsideTherefore, the grass is wet.
If it’s raining outside, then the grass is wet.The grass is wet.Therefore, its raining outside.
Valid argument
Fallacy
Aristotle
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Euclid’s “The Elements”
Five axioms (postulates)
Euclid
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P Q P && Q
False False False
False True False
True False False
True True True
P Q P || Q
False False False
False True True
True False True
True True True
P P
False True
True False
P Q PQFalse False TrueFalse True TrueTrue False FalseTrue True True
P Q P<=>Q
False False True
False True False
True False False
True True True
Propositional Logic
Frege
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Proposition: ((P Q)((P)Q))
P Q
False False
False True
True False
True True
False
True
True
True
(P Q) (P)
True
True
False
False
((P)Q)
False
True
True
True
((PQ)((P)Q))
False
True
True
True
Some True: prop is Satisfiable*
If they were all True: Valid / Tautology
All False: Contradiction (not satisfiable*)
*Satisfiability was the first known NP-complete (see next slide) problem
If prop is True when all variables are True:P, Q ((PQ)((P)Q))
A Truthdouble turnstile
Reasoning with Truth Tables
Frege
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Reasoning with Truth Tables
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Tautological Proof using Propositional Logic
It’s either summer or winter.If it’s summer, I’m happy.If it’s winter, I’m happy.
Is there anything you can uncompress from this?
The above statements can be written like this: ( (s w) ^ (s -> h) ^ (w -> h) ) -> h
This is a Haskell proof of this Tautologyvalid3 :: (Bool -> Bool -> Bool -> Bool) -> Boolvalid3 bf = and [ bf r s t| r <- [True,False], s <- [True,False], t <- [True,False]]
LOGIC> valid3 (\ s w h -> ((s || w) && (s ==> h) && (w ==> h)) ==> h)True
Another form of a un-compression (proof):
( (p q) ^ (¬p r) ^ (¬q r) ) -> r( (p ^ ¬p) (r ^ q) ^ (¬q r) ) -> r(F (q ^ ¬q) r) -> rr -> r¬r r.: T Frege
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Peano's Axioms
1. Zero is a number.
2. If a is a number, the successor of a is a number.
3. zero is not the successor of a number.
4. Two numbers of which the successors are equal are themselves equal.
5. (induction axiom.) If a set S of numbers contains zero and also the successor of every number in S, then every number is in S.
Peano's axioms are the basis for the version of number theory known as Peano arithmetic.
Peano
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Peano's Arithmetic
Peano
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Cantor Diagonalization
Create a new number from the diagonal by adding 1 and changing 10 to 0. The above example would give .960143…Now try to find a place for this number in the table above, it can’t be the first line because 8 != 9, it can’t be the second line because 5 != 6, etc. to infinity. So this line isn’t in the table above and therefore was not counted. The real numbers are not countable.
Cantor
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Set Theory ParadoxesSuppose there is a town with just one barber, who is male. In this town, every man keeps himself clean-shaven, and he does so by doing exactly one of two things:shaving himself; orgoing to the barber.Another way to state this is that "The barber is a man in town who shaves all those, and only those, men in town who do not shave themselves."From this, asking the question "Who shaves the barber?" results in a paradox because according to the statement above, he can either shave himself, or go to the barber (which happens to be himself). However, neither of these possibilities is valid: they both result in the barber shaving himself, but he cannot do this because he only shaves those men "who do not shave themselves".
Cantor and Russell
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A Sequence of Functions from definitions on DeLong, page 157:
Primitive Recursive Functions and Arithmetic(see “A Profile of Mathematical Logic” by Howard Delong – pages 152 – 160)
Skolem
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Gödel Numbering
1 = (0)’ = 211 x 31 x 513 x 73
Gödel
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Lambda Calculus
Lambda Calculus
x.x
s.(s s)
func.arg.(func arg)
def identity = x.xdef self_apply = s.(s s) def apply = func.arg.(func arg)
def select_first = first.second.firstdef select_second =first.second.second
def cond= e1.e2.c.((c e1) e2)
def true=select_first def false=select_seconddef 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)
Church
print (lambda x : x) (lambda s : s)
print (lambda s : (s) (s)) (lambda x : x)
print (lambda func, arg : (func) (arg)) ((lambda x : x + 3), 4)
Examples in python:
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Turing Machine
Turing
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Combinator Logic
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.
s f g x = f x (g x) k x y = x b f g x = f (g x) c f g x = f x g y f = f (y f) cond p f g x = if p x then f x else g x
-- Some Primitive Recursive Functions on Natural Numbers
addition x z = y (b (cond ((==) 0) (k z)) (b (s (b (+) (k 1)) ) (c b pred))) x multiplication x z = y (b (cond ((==) 0) (k 0)) (b (s (b (pradd1) (k z)) ) (c b pred))) x exponentiation x z = y (b (cond ((==) 0) (k 1)) (b (s (b (prmul1) (k x)) ) (c b pred))) z factorial x = y (b (cond ((==) 0) (k 1)) (b (s (prmul1) ) (c b pred))) x
No halting problem here but not Turing complete either
Implies recursion or bounded loops, if-then-else constructs and run-time stack.
see the BlooP language in
Combinators
Haskell Currie
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John McCarthy’s Takeaways
-- Primitive Recursive Functions on Lists are more interesting than Primitive Recursive Functions on Numbers
length x = y (b (cond ((==) []) (k 0)) (b (s (b (+) (k 1)) ) (c b cdr))) x sum x = y (b (cond ((==) []) (k 0)) (b (s (b (+) (car)) ) (c b cdr))) x product x = y (b (cond ((==) []) (k 1)) (b (s (b (*) (car)) ) (c b cdr))) x map f x = y (b (cond ((==) []) (k [])) (b (s (b (:) (f) ) ) (c b cdr))) x -- map (\ x -> (car x) + 2) [1,2,3] or -- map (\ x -> add (car x) 2) [1,2,3]
-- A programming language should have first-class functions as (b p1 p2 . . . Pn), substitution, lists with car, cdr and cons operations and recursion.car (f:r) = f cdr (f:r) = r cons is : op
John’s 1960 paper: “Recursive Functions of Symbolic Expressions and Their Computation by Machine” – see copy on class calendar.
McCarthy
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class Stack(LinkedList): def push(self, e): self.addFirst(e) def pop(self): return self.removeFirst() def peek(self): return self.car() class Queue(LinkedList): def enqueue(self, e): self.cons(e) def dequeue(self): return self.removeLast() print "Stack test"
Stack and Queue
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Stack Exampledef calc(expr): s1 =Stack() for s in expr: if s == ")": e = "" for i in range(s1.getSize()): v = s1.pop() if v == "(": break else: e += str(v) # print e, e[::-1], eval(e[::-1]) e = eval(e[::-1]) # Comment for no extra credit s1.push(str(e)[::-1]) # Comment for no extra credit # s1.push(eval(e)) # Uncomment for no extra credit # print s1 else: s1.push(s) # return s1.pop() # Uncomment for no extra credit return s1.pop()[::-1] # Comment for no extra credit print eval("(((12+33.3)*(1.4*701))/(2.2-1.1))")print calc("(((12+33.3)*(1.4*701))/(2.2-1.1))")
print eval("(((2+3)*(4*7))*2)")print calc("(((2+3)*(4*7))*2)"
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Bubble Sort
l = [i for i in range(10, 0, -1)]# or [i for i in range(10)][::-1]cnt = len(l) - 1total = 0while cnt > 0: swap = 0 i = 0 for e in l: if i + 1 < len(l) and l[i] > l[i+1]: l[i], l[i+1] = l[i+1], l[i] swap += 1 i += 1 total += swap cnt -= 1print total
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Merge Sort
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Merge Sort
global swapsswaps = 0def mergeSort(list): if len(list) > 1: # Merge sort the first half # firstHalf = list[ : len(list) // 2] This is what the book does. firstHalf = [i for i in list[: len(list) // 2]] # This uses list comprehension. mergeSort(firstHalf)
# Merge sort the second half # secondHalf = list[len(list) // 2 : ]This is what the book does. secondHalf = [i for i in list[len(list) // 2 :]] # This uses list comprehension. mergeSort(secondHalf)
# Merge firstHalf with secondHalf into list merge(firstHalf, secondHalf, list)
# Merge two sorted lists */def merge(list1, list2, temp): global swaps current1 = 0 # Current index in list1 current2 = 0 # Current index in list2 current3 = 0 # Current index in temp
while current1 < len(list1) and current2 < len(list2): if list1[current1] < list2[current2]: temp[current3] = list1[current1] current1 += 1 current3 += 1 swaps += 1 else: temp[current3] = list2[current2] current2 += 1 current3 += 1 swaps += 1
while current1 < len(list1): temp[current3] = list1[current1] current1 += 1 current3 += 1 swaps += 1
while current2 < len(list2): temp[current3] = list2[current2] current2 += 1 current3 += 1 swaps += 1
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Bubble and Merge Sort Efficiency
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List Comprehension and Lambda Expressions for Propositional Logic
# m stands for "it's Monday", t stands for "the next day is Tuesday", and d stands for "I will win a thousand dollars"print [(lambda (m, t, d) : not (m and (not m or t) and (not t or d)) or d) ((i, j, k)) for i in (False, True) for j in (False, True) for k in (False, True)]
It's Monday.If it's Monday, the next day is Tuesday.If the next day is Tuesday, then I will win a thousand dollars.
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List Comprehension and Lambda Expressions and SQLemp = (('EMPNO', 'ENAME', 'JOB', 'MGR', 'HIREDATE', 'SAL', 'COMM', 'DEPTNO'), (7369, 'SMITH', 'CLERK', 7902, '1980-12-17 00:00:00', 800.0, 0.0, 20), (7499, 'ALLEN', 'SALESMAN', 7698, '1981-02-20 00:00:00', 1600.0, 300.0, 30), (7521, 'WARD', 'SALESMAN', 7698, '1981-02-22 00:00:00', 1250.0, 500.0, 30), (7566, 'JONES', 'MANAGER', 7839, '1981-04-02 00:00:00', 2975.0, 0.0, 20), (7654, 'MARTIN', 'SALESMAN', 7698, '1981-09-28 00:00:00', 1250.0, 1400.0, 30), (7698, 'BLAKE', 'MANAGER', 7839, '1981-05-01 00:00:00', 2850.0, 0.0, 30), (7782, 'CLARK', 'MANAGER', 7839, '1981-06-09 00:00:00', 2450.0, 0.0, 10), (7788, 'SCOTT', 'ANALYST', 7566, '1982-12-09 00:00:00', 3000.0, 0.0, 20), (7839, 'KING', 'PRESIDENT', 0, '1981-11-17 00:00:00', 5000.0, 0.0, 10), (7844, 'TURNER', 'SALESMAN', 7698, '1981-09-08 00:00:00', 1500.0, 0.0, 30), (7876, 'ADAMS', 'CLERK', 7788, '1983-01-12 00:00:00', 1100.0, 0.0, 20), (7900, 'JAMES', 'CLERK', 7698, '1981-12-03 00:00:00', 950.0, 0.0, 30), (7902, 'FORD', 'ANALYST', 7566, '1981-12-03 00:00:00', 3000.0, 0.0, 20), (7934, 'MILLER', 'CLERK', 7782, '1982-01-23 00:00:00', 1300.0, 0.0, 10))
dept = (('DEPTNO', 'DNAME', 'LOC'), (10, 'ACCOUNTING', 'NEW YORK'), (20, 'RESEARCH', 'DALLAS'), (30, 'SALES', 'CHICAGO'), (40, 'OPERATIONS', 'BOSTON'))
print[ (i[1], j[1]) for i in emp for j in dept if i[7] == j[0] ]
İs similar to “select ename, dname from emp join dept on( emp.deptno = dept.deptno)” in SQL