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::ICS 804::
Theory of Computation
- Ibrahim Otieno [email protected]
+254-0722-429297
SCI/ICT Building Rm. G15
ICS 804
ICS 804 - Delivery
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 4.5 hours a week
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Assessment
CAT
Assignment
Exams
 Theoretical course, but
 Interactivity
 Exercises (reflect exam questions)
 Laboratory sessions
ICS 804 - Delivery
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ICS 804 – Course Text
Introduction
 Computation from the point of view of
mathematics (discrete mathematics)
 Introduction to computability and
complexity theory
 What kind of problems are inherently
computable and which options are there?
 Regular languages and finite-state
automata
 Very recent field started in the 1930s
Course Outline

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

Mathematical Preliminaries
Turing Machines
Recursion Theory
Markov Algorithms
Register Machines
Regular Languages and finite-state automata
Aspects of Computability
Course Outline







Mathematical Preliminaries
Turing Machines
Recursion Theory
Markov Algorithms
Register Machines
Regular Languages and finite-state automata
Aspects of Computability
Discrete Mathematics



Theory of computability = new branch of mathematics
Presupposes background in discrete mathematics
Topics:
◦ Set-theoretic concepts
◦ Formal language theory
◦ Functions
◦ Big-O notation
◦ Propositional logic
◦ Proof Techniques
◦ Number-theoretic predicates
Sets
Sets








Elements of a set: may be physical objects
Set: never physical – is an abstract concept
a  S – a member of set S
Members of sets may be sets
S = {0, 1, {0, 1}}
Cardinality: card(S) = 3
Singleton: S = {a} – set containing single element
Sets  lists (unordered, elements are unique)
Set = family = class
Specifying (Naming) Sets
 Finite sets {a,e,i,o,u} – elements can be listed
 Infinite sets: come from mathematics
◦
enumerate elements in an extended sense
◦
◦
Natural numbers N = { 0,1,2,…}
Integers Z = {…,-2,-1,0,1,2,…}
 Set abstraction
◦
◦
◦
{n|n is prime} or {n:n is prime}
{n|…n…} or {n:…n…}: abstraction operator: the set of all n
such that n…
{n|n N & 2|n}: all even natural numbers
Specifying (Naming) Sets …
 By convention:
◦ Use variables n,m,k,j,…: natural numbers /
integers
◦ Use variables x,y,z,…: rational/real numbers
◦ Set variables: uppercase letters (A,B,C, …S)
◦ Universal set: U – all elements in current
domain
 U by set abstraction {x|x=x}
◦ Empty Set:  - with no element
  by set abstraction {x|xx}
Subsets
 A = {0,1,2,3,4} A is subset of N or AN
 A is a subset of B
◦
If for any x, x  A implies x  B
 Consequences:
◦
◦
Every set is a subset of itself i.e. A A, for any set A
  A, for any set A
 “proper subset”: non-identical subset
{1,2,3}  {1,2,3,4}
 Power set – all subsets of a set
◦
◦
◦
For a set S: P(S) = { A|A  S }
What is the power set of S = {1,2,3} ?
Card(P(S) = 2card(s) for a finite set S
Tuples
A tuple is an ordered list of elements
Ordered pair: <a,b> = <c,d> iff a = c and b = d
defined Ordered triple: <a,b,c>
0-tuple: A = {<>}
1-tuple: A = <a>
For any two sets A and B:






◦
◦
◦
◦
◦
◦
are said to be disjoint if AB = 
Card(AB)  Card(A) + Card(B)
Card(AB) = Card(A) + Card(B) iff AB = 
Compliment of B relative to A denoted A\B is defined as {x| x 
A and x  B}
We write BC and speak of compliment of B for the set U\B
Symmetric difference denoted by AB is as {x| (x  A and
x  B) or (x  B and x  A) }
Some set properties





A∪U=U
A∩=
C=U
UC = 
Idempotent Law: For any set A,
◦ A ∪A =A
◦ A ∩A =A
 Identity Law: For any set A,
◦ A ∪  =A
◦ A ∩ U =A
Some set properties
 Commutative Law: For any two sets A and B
◦ A ∪ B = B ∪A
◦ A ∩ B = B ∩A
 Associative Law: For any three sets A, B and C
◦ (A ∪ B) ∪ C = A ∪ (B ∪ C)
◦ A ∩ (B ∩ C) = (A ∩ B) ∩ C
 Distributive Law: For any three sets A, B and C
◦ A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
◦ A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
 De Morgan’s Law: For any two sets A and B
◦ (A ∪ B)' = A' ∩ B'
◦ (A ∩ B)' = A' ∪ B'
Special Sets
 N: set of natural numbers
closed under addition and multiplication
not closed under subtraction nor division
 Z: set of integers
closed under addition, subtraction and multiplication
 Q: set of rational numbers
Q = {x|x = p/q for some p,q  Z with q  0}
closed under addition, subtraction and multiplication
Q\{0} is closed under division
 R: set of real numbers
closed under addition, subtraction and multiplication
R\{0} closed under division

We define a partition of a set A to be a family of mutually
disjoint subsets such that A = A1 ∪ A2 ….An
Exercises
Exercise 1
 If A and B are two sets, then A ∩ (A ∪ B) equals
◦ (A) A
(B)
(C)

(D)
none of these
Exercise 2
 The set (A ∪ B ∪ C) ∩ (A ∩ B' ∩ C')' ∩ C' is equal to
◦ (A) B ∩ C'
(B) A ∩ C
(C)
B' ∩C'
(D) None of these
Formal Language Theory
Formal Language
 Formal language: specific kind of set
 Formal = not natural language, programming
language, …
but: NLP try to formalize natural language
 Ignoring semantics:
◦ possible to regard natural languages like Spanish
and programming languages such as PROLOG as
formal
Formal Language
 Alphabet : finite set of symbols, e.g.  =
{a,b}
 Word over alphabet : finite string of
elements of  allowing repetition
 Length of word w = |w| = number of symbol
occurrences in w
 Card() = m then there exist mn distinct
words over  of length n
◦
For alphabet = {a,b} we have 2n distinct words over  of length n
Formal Language
 Special superscript notation: anbm word
consisting of n times a and m times b
Example: a3b4a2 aaabbbbaa
 Distinct occurrences:
na(aba)=2
nb(aba)=1
Formal Language
 Null word: word of length 0
◦
◦
◦
Zero occurrences of a and b
 = a0
na() = 0 and nb() = 0
 Universal Language: * set of all words over 
 Language over : any subset of *
◦
◦
◦
Empty language:  - containing no word
Universal language: * - containing all words
Unit language L = {w} – containing word, w, only
 Alphabet  = {a,b}
◦
{w  *: |w| = 3} = aaa, aab, aba, abb, baa, bab, bba, bbb}
Operations on words
 Concatenation: ww’ of words w and w’
w is aba and w’ is bba
ww’ = ababba
 Reverse of a word: wR
abbR = bba
 Palindromes over  is the language
{w*|where w=wR}
Palindrome Examples
 Examples:
 Radar – level – mum – gig
 “Madam, in Eden, I’m Adam”
 “A man, a plan, a canal: Panama”
 “Doc, note. I dissent. A fast never prevents a fatness. I diet on cod.”
(Peter Hilton)
 Longest 1-word English palindrome:
 detartrated - (to remove tartrates)
 Longest 1-word palindrome in the world (Finnish)
 saippuakivikauppias - (lye dealer)
 Swahili palindromes: amesema, ataufuata, akatutaka
 Longest palindrome: 31358 words
Operations on words
 A word w’ is a prefix of a word w if w can be written w’w’’
for some word w’’
◦ ,a,ab,abb,abba are prefixes of the word abba
(first 4 are proper)
 A word w’’ is a suffix of a word w if w can be written w’w’’
for some word w’
◦ ,a,ba,bba,abba are suffixes of the word abba
(first 4 are proper)
Operations on words
 A word w’’ is a substring of word w if w can be written as
w’w’’w’’’ for some words w’ and w’’
abba:  a ab bb ba abb bba abba
Exercises

1 = {a,b,c}
2 = {a,b,ca}
3 = {a,b,Ab}
Determine to which * each word belongs
aba
bAb
cba
cab
caab
baAb
Solution




1 = {a,b,c}
2 = {a,b,ca}
3 = {a,b,Ab}
Determine to which * each word belongs
aba 1,2,3
bAb 3
cba 1
cab 1,2
caab1,2
baAb 3
Exercises
 Calculate how many words there are in the following languages.
List 3 elements in each of them. Which language contains 
◦
◦
◦
◦
*, where  = {a,b,c}
{w*| |w|  3}, where  = {a,b}
{w*| |w| =4}, where  = {a,b}
{anb|n is prime}
Solution

*, where  = {a,b,c}



{w*| |w|  3}, where  = {a,b}



15 words (23+22+21+20)
= {, a, b, ab, ba, bb, aa, aaa, baa, aba, aab, bba, bab, abb, bbb}
{w*| |w| =4}, where  = {a,b}



Infinite number of words
{, abba, ccccba, aaccccccccc, …}
16 words (24)
= {aaaa, aaab, aaba, abaa, baaa, aabb, abba, bbaa, baba, abab, baab, abbb, babb, bbab, bbba,
bbbb}
{anb|n is prime}


Infinite number of words
{aab, aaab, aaaaab, …}
Exercises
 Let  = {a,b,c,d}. List as many words as you can that are words
in the language
{w  *| |w| = 3 and w is a word in the English language}
Solution
 Let  = {a,b,c,d}. List as many words as you can that are words
in the language
{w  *| |w| = 3 and w is a word in the English language}
dad cab bad dab add cad baa
Exercises
 Suppose w is ab and w’ is bab. Identify each of the following
words:
ww’
wR
(ww’)R
ww’R
Solution
 Suppose w is ab and w’ is bab. Identify each of the following
words:
ww’ abbab
wR
ba
(ww’)R babba
ww’R abbab
Exercise
 Which of the following languages are palindrome languages
◦
◦
◦
◦
◦
wwR for any w  *
wR for any palindrome w  *
ww for any palindrome w  *
wawR for any w  *
ww’wR for any w  * and any palindrome w’  *
◦

Solution

Which of the following languages are palindrome languages
◦
yes
◦
yes
◦
yes
◦
yes
◦
yes
◦
yes
wwR for any w  *
wR for any palindrome w  *
ww for any palindrome w  *
wawR for any w  *
ww’wR for any w  * and any palindrome w’  *

Mapping and Functions
Mapping and Functions
 Mapping: association of members of one set with
members of another (not necessarily different) set
 First: domain
 Second: codomain or range
 Members of domain serve as: arguments
 Members of codomain serve as: values
 Mapping f between Dom(f)=A and Cod(f)=B
f: A  B
 A mapping f is a function if each member of Dom(f)
is mapped to one and only one member of Cod(f)
Function
 Function = single valued mapping
 Complete description of a mapping f consists of:
a) specification of Dom(f) and Cod(f)
b) description of values of f for any member of Dom(f)
taken as arguments
 Examples
f: N  N
f(n) = n + 1
Unary constant-7 function – returns 7 for all n
C17: N N
C17(n) = 7
f: * *
f(w) = wR
Functions
 Examples …continued
 Floor function
f: R  Z
f(x) = [x] = greatest integer ≤ x
 Ceiling function
f: R  Z
f(x) = [x] = least integer ≥ x
Multi-valued mapping
◦ f: Z  Z
f(n) = n ± 3
f(n) = n  3
Not a function
Functions
 Image of a function f defined as Image(f) is subset
of Cod(f) consisting precisely of the values of f for
members of Dom(f)
 Precisely in set language:
Image(f) = {yCod(f)|y=f(x) for some x  Dom(f)}
Number-theoretic Functions
 Number-theoretic functions: map natural numbers to
natural numbers: f: N  N
 Unary number-theoretic functions
f(n) = n + 1
 Binary number-theoretic functions
f: N 2 N
f(n,m) = n + m
 K-ary number-theoretic functions f: N k N
 0-ary number theoretic functions
C02: N 0  N
C02() = 2
Injective (1-to-1) Functions
 A function f is 1-to-1 or injective if no two distinct
elements of Dom(f) are mapped to one and the same
member of the Cod(f)
 Examples:
 Successor function
f: N  N
f(n) = n + 1
 Word reversal function
f: * *
f(w) = wR
 Ceiling, floor and constant-7 functions not injective
Surjective (onto) Functions
 A function f is onto or surjective provided that every
element of Cod(f) is a value of f for at least one element
of the Dom(f)
 Cod(f) = Image (f)
 Examples:
 Word reversal function
f: * *
f(w) = wR
 Ceiling function
f: R  Z
f(x) = [x] = least integer ≥ x
 Successor function not surjective – 0 not successor
 A function that is both injective and surjective is termed
as 1-1 correspondence or a bijective function
Functions forming operations
Inverse
 For two functions g and f:
g: N  N
g(n) = n/2
f: N  N
f(n) = 2n
 g is the inverse of f
 g = f-1
 The inverse operation (though not always defined) -1is;
Unary function forming operation
Composition
 For two functions g and f:
g: * N where ={a,b}
g(w) = na(w)
f: N  N
f(n) = 3n + 2
h: * N
h(w) = f(g(w))
h: composition of g and f (h = g  f)
 The composition operation is:
Binary function forming operation
Partial Functions
 For a function g:
g: N 2  N
g(n,m) = n – m
 The function g is undefined for pairs <n,m>
where n < m
 Number-theoretic functions that are undefined
for some n  N are Partial functions.
Partial Functions
 Let f(n) and g(n) be partial number-theoretic functions
 We can define a new partial number-theoretic function
h(n) = max{ f(n), g(n) }
 This means that

h(n) is defined just in case both f(n) and g(n) are defined

for any n  Dom(h) = Dom(f)  Dom(g), we have h(n) is
the maximum of f(n) and g(n)
Polynomial Functions
 We restrict ourselves to unary polynomial functions
 Polynomial in n of degree k is any expression of form:
aknk + ak-1nk-1 + ak-2nk-2 + ….. + a1n1 + a0n0
where k is any natural constant and coefficients ak , ak-1 ,
ak-2 , ….., a1 + a0 are integer constants with ak ≠ 0
 Polynomials of degree 0 are numerals
 Polynomials in n are not themselves natural numbers or
integers; they are expressions
Polynomial Functions
 For example:
2n5 + 2n3 + 4n2 +16n + 7
is a polynomial of degree 5
 The unary polynomial function is defined by:
f: N  N
f(n) = 2n5 + 2n3 + 4n2 +16n + 7
f(2) = ?
 Polynomial functions with negative coefficients may be partial
 Sometimes we write p(n) for a unary polynomial function of n
 For p1(n) and p2(n) => p2[p1(n)]) – result of replacing every
occurrence of n in p2(n) with p1(n)
Big-O Notation
Big-O
 Big-O (micron) notation provides standard means of
comparing growth of number-theoretic functions
 Analyze time and space requirements of algorithms
 Typical calculation: growth of a function as argument n
goes to infinity
 Monotone increasing: for all n1, n2  N, we have n1<
n2 implies f(n1) f(n2)
 Strictly monotone increasing: f(n1)<f(n2)
Big-O
 Four partial number-theoretic functions:
k(n) = 2n
h(n) = n2 + 1
g(n) = 3n
f(n) = 7[log2 n]
Big-O
Big-O
Which are (strictly) monotone increasing ?
Big-O
 Four partial number-theoretic functions
k(n) = 2n
h(n) = n2 + 1
g(n) = 3n
f(n) = 7[log2 n]
 Value 2:
k(2) = 4
h(2)=5
g(2) = 6
 Value 5:
k(5) = 32 h(5)=26 g(2) = 15
f(2)=7
f(2)=14
Big-O
Fastest growing function? Slowest?
Big-O
 For small numbers f is higher, but it is k that grows the fastest
for large numbers.
 Big-O expresses how fast a function grows relative to another
 Function f(n) is O(g(n))
◦ g(n) is at least as great as f(n); ultimately for all sufficiently
large n
 Similarly; g(n) is O(h(n)) and h(n) is O(k(n))
 Transitivity holds:
◦ If f(n) is O(g(n)) and g(n) is O(h(n)) then f(n) is O(h(n))
Big-O
 Definition:
 Let f(n) and g(n) be partial number-theoretic functions. Then
function f(n) is O(g(n)) if there exist natural number constants C
and n0 such that for any nn0 we have f(n)  C.g(n) whenever both
f(n) and g(n) are defined.
e.g. Function f(n) is O(g(n)): with C=1 and n0=5
4
5=n0
7
16
f(n) = 7[log2 n]
14
14
14
28
C.g(n) = 1.3n
12
15
21
48
Big-O
•
•
•
•
•
•
•
f1 is O((f2))
f3 is O((f4))
f5 is O((f6))
…..
…..
…..
f9 is O((f10))
•
Growth rate of a
function ftime is
O(f72(n)) for
instance
In other words
function ftime is
O(n2)
•
Big-O
Big-O
 Function f(0) is O(0)
 Function f(n) that is O(1) iff f(n) is bounded above by a constant for
sufficiently large arguments n
 Function f(n) that is O(n) has linear growth
 Function f(n) is O(n2) or O(n3): polynomial growth
 Function f(n) is O(2n) or O(3n): exponential growth
Exercises
(a)
(b)
 Characterize these functions as
◦ monotone increasing
◦ strictly monotone increasing
◦ neither
(c)
(e)
(d)
(f)
Solution
(a)
(c)
(e)
(b)
(d)
(f)
•
•
•
•
•
•
Strictly monotone
Neither
Strictly monotone
Monotone
Neither
Monotone
Exercises
 Which of the following functions are O(1)
◦
◦
◦
◦
◦
f(n) = [log2 n]
g(n) = [log10 n]
h(n) = min(n,100)
j(n) = 4n2 + 7n + 3
k(n) = 5n5 + n3 + n + 68
Solution
 Which of the following functions are O(1)
◦ f(n) = [log2 n]
◦ g(n) = [log10 n]
◦ h(n) = min(n,100)
only this one is O(1)
◦ j(n) = 4n2 + 7n + 3
◦ k(n) = 5n5 + n3 + n + 68
Propositional Logic
Proposition
 Proposition: any statement with a truth value
 Any statement that is either true or false
 Examples:
◦ For all n in P, n3 – n is divisible by 6
◦ N2 is O(n3)
◦ 6 is a prime number
◦ Nairobi is located in Kenya
◦ Goldbach’s Conjecture: every even integer strictly greater than 4 is
the sum of two primes
◦ Don’t forget to vote
◦ Is Russian beer the best beer in the world?
◦ We define the degree of a vertex v within an undirected graph G
to be the number of edges in G incident upon v.
Notation
 PL: infinite symbol set of
◦ lower-case letters of the Roman alphabet,
p,q,r,p’,q’,r’,p’’,q’’,r’’,… which stand for propositions:
sentence letters
◦ Propositional connectives:   &  
◦ Aggregation signs - (), [], {}
 Language of propositional logic- set of strings over
alphabet PL
Negation
 Symbol: 
p
p
T
F
F
T
 p: John is tall
 p: John is not tall
 Unary connective
Conjunction
 Binary connective: symbol - & 
p
q
p&q
T
T
T
F
T
F
T
F
F
F
F
F
For conjuncts p and q
p: John is tall
q: Andrew is short
p&q: John is tall and Andrew is short
Disjunction
 Binary connective: symbol - 
p
q
pq
T
T
T
F
T
T
T
F
T
F
F
F
 Inclusive or
 For disjuncts p and q
p: John is tall
q: Andrew is short
p  q: Either John is tall or Andrew is short
Conditional
 Used to express conditional statements
 Binary connective: symbol 
p
q
pq
T
T
T
F
T
T
T
F
F
F
F
T
p: It is raining
q: The streets are wet
 Antecedent p and consequent q
 False wherever antecedent is true and consequent is false
Conditional
 Binary connective: 
 Not quite the same semantics as in natural language
 If Kenya is in Europe, then 2+2=4
= false
Biconditional
 Binary connective: symbol
p
q
pq
T
T
T
F
T
F
T
F
F
F
F
T
 If and only if
p: 2+2 = 4
q: 1+1 = 2
True if component sentence letters have same truth values
Semantic Issues
 Sentences of propositional logic classified as:
 Tautologous
◦ True under all assignments of component sentence letters
◦ pp
 Contradictory
◦ False under all assignments of component sentence letters
◦ p&p
 Contingent
◦ Sentences that are neither tautologous or contradictory
◦ p & q and p  q
Semantic Issues
 Satisfiable
◦ A sentence is satisfiable if for at least one assignment of the
component letters the sentence holds true
◦ Not contradictory
◦ Truth tables: decision procedure for satisfiability
◦ The satisfiability problem for propositional logic is decidable
 Equivalence
Two sentences S1 and s2 are (logically) equivalent if for any
given assignment of truth values to component sentence
letters, the resulting truth value of S1 is identical with value of
S2
Idempotence, Associativity, Commutativity,
Distributivity and DeMorgans Laws also hold
PROOF TECHNIQUES
Proof Techniques
Conditional Proof
pq
assume truth of the antecedent
reason to the truth of consequent q
Example 1
If I drink too much alcohol, I can’t drive a car
• When I drink alcohol, the alcohol gets absorbed in my blood. My
blood reaches my brain, where it will distort its functionality,
reducing my coordination abilities. To drive a car I need
coordination.
Proof Techniques
Conditional Proof
Example 2
If Jones is elected, then the treasury will be bankrupted
• Given that Jones is lavish and a spendthrift
• He will spend vast amounts on annual banquets
• Given his slackness in supervising his subordinates, there will be
widespread corruption
Therefore, if elected then the treasury will be bankrupted
Proof Techniques
Indirect Proof
pq
Proof by contradiction
• 11 is not a prime number
if it is not a prime number, it must have a non-trivial factor. But it
is not 2,3,4,5,6,7,8,9,10 and it can’t be larger than n. This is a
contradiction, so 11 is a prime after all
•
•
•
•
Appropriate when p is unstructured.
Derive a contradiction of the form p & p.
Having assumed p is true, continue to show that p & p is true.
Conclude then that p is not true, thereby showing p is true.
Number-theoretic
Predicates
Number-theoretic Predicates
• Properties of objects, relations, holding between pairs
of objects, relations, holding among triples of objects
are expressed using predicates.
•If the objects are members of N, then one speaks
of number-theoretic predicates
•e.g. prime(n) is a unary number-theoretic
predicate that is satisfied by 2 and 3, but not by 4
or 6
•e.g. predicate n|m hold just in case m is a multiple
of n so that 2|6 and 2|0 hold
Number-theoretic Predicates
•
The extension of a k-ary number-theoretic predicate is the
set of all and only those k-tuples of natural numbers
satisfying that predicate
extension of the predicate prime(n) is the set
{n  N|prime(n)}
extension of cube_of(n,m) is the infinite set of ordered pairs
{<0,0>,<1,1>,<8,2>,<27,3>,<64,4>,…}
Number-theoretic Predicates
• Predicates can be combined using the five propositional
connectives:   &  
C1(n1,n2,…nk) & C2(n1,n2,…nk)
is true if k-tuple <n1,n2,…,nk> satisfies both
C1(n1,n2,…nk) and C2(n1,n2,…nk)
•
•
A binary predicate C(n,n) is said to be reflexive if, for any and
all n  N, we have C(n,n)
A binary predicate C(n1,n2) is said to be symmetric if, for any n,
m  N; C(n,m) implies C(m,n)
Notation
Quantifier: may play a role in specification of predicates
◦ Universal quantifier (n)(C1(n) C1(n))
 C1(n): prime(n)
 Given any natural number n, either n is prime or it is not
 Given any natural number n is prime and n> 2 is odd ?
◦ Existential quantifier (n)(C1(n) & C2(n))
 C1(n): prime(n)
 C2(n): odd(n)
 There exists an n such that n is prime and n is not odd
 There is no prime greater than 2 that is even?
Exercise
Given: the following predicates C1(n) C2(n,m,k) C3(n,m,k) with n,m,k  N
 C1(n1)  n1 is prime
 C2(n1,n2,n3)  n1 = n2 + n3
 C3(n1,n2,n3)  n1 = n2 . n3
For each of the predicates below, determine whether it is satisfied by the pair
<4,12> n=4 m=12
(k)C2(n,m,k)
(k)[C2(k,n,m) & C1(k)]
(k)C3(n,m,k)
(k)C3(k,n,m)
(k)[C3(k,n,m)&C1(k)]
(kn)[C3(m,k,n) & C1(k)]
Exercise
Given: the following predicates C1(n) C2(n,m,k) C3(n,m,k) with n,m,k  N
 C1(n1)  n1 is prime
 C2(n1,n2,n3)  n1 = n2 + n3
 C3(n1,n2,n3)  n1 = n2 . n3
For each of the predicates below, determine whether it is satisfied by the pair
<4,12> n=4 m=12
(k)C2(n,m,k)
no
(k)[C2(k,n,m) & C1(k)]
no
(k)C3(n,m,k)
yes
(k)C3(k,n,m)
yes
(k)[C3(k,n,m)&C1(k)]
no
(kn)[C3(m,k,n) & C1(k)] yes