CSCE 355 Foundations of Computation Lecture 1 Overview Topics Proof techniques: induction, contradiction Proof techniques June 1, 2015

Models of Computation

Course Outcomes Mathematical prerequisites: functions, relations, properties of relations, posets. Proof Techniques Finite automata: regular languages, regular expressions, DFAs, NFAs, equivalences. Limitations: pumping lemma Context free languages: grammars, push-down automata Turing machines: undecidability, the halting problem Intractability: NP, NP-Completeness

Prerequisites CSCE 211 Number systems, Boolean algebra, logic design, sequential machines Mealy machines Moore machines

Prerequisites CSCE 350 MATH 374 Techniques for representing and processing information, including the use of lists, trees, and graphs; analysis of algorithms; sorting, searching, and hashing techniques. MATH 374 Propositional and predicate logic; proof techniques; recursion and recurrence relations; sets, combinatorics, and probability; functions, relations, and matrices; algebraic structures.

Review of Relations on Sets Binary relations - (X, Y) ἐ R or X Rel Y < on integers likes (X,Y) Unary relation - properties boring(matthews) Ternary relation “X was introduced to Y by Z” -- ( X, Y, Z) Table in a relational database

Special types of Relations Injections Surjections Functions

Properties of Relations Property Def Example Neg-Example Reflexive Irreflexive symmetric antisymmetric asymmetric transitive Total Injection Surjection function

Posets Partially Ordered Sets (POSETS) Hasse Diagram Reflexive Antisymmetric Transitive Hasse Diagram Topological sorting

Equivalence relations

Proof Techniques 1.1 Direct proof 1.2 Proof by induction 1.3 Proof by transposition 1.4 Proof by contradiction 1.5 Proof by construction 1.6 Proof by exhaustion 1.7 Probabilistic proof 1.8 Combinatorial proof 1.9 Nonconstructive proof 1.10 Proof nor disproof 1.11 Elementary proof

Deductive Proofs the conclusion is established by logically combining the axioms, definitions, and earlier theorems Example: The sum of two even integers is even. Hypothesis

Theorem 1.3 used to prove Theorem 1.4 Theorem 1.3 If x >= 4 then 2x >= x2. Theorem 1.4 If x is the sum of the squares of 4 positive integers then 2x >= x2. Proof

Theorem 1.3 If x >= 4 then 2x >= x2. f(x) = x2 / 2x. Then what is the derivative f’ of f Derivative of quotient?? http://www.math.hmc.edu/calculus/tutorials/quotient_rule/ So f’(x) =

Proofs about Equality of Sets To prove S = T Show S is a subset of T, and T is a subset of S Commutative law of union Theorem 1.10 Distributive law of union over intersection Proof

Proof by Contradiction

If and only If statements IF H then C H = Hypothesis C = conclusion H implies C H only if C C if H A if and only if B If part : Only-if part Theorm 1.7 ceiling = floor  x is an integer

Induction Given a statement S(n) about an integer n that we want to prove. Basis Step: Show S(i) is true for a particular integer i Usually i = 0 or i = 1 Inductive Step: Assume S(n) is true for n >= i and then show S(n+1) is true Inductive Hypothesis: Assume S(n) is true

Example Induction Proof: Theorem 1.16

Number of leaves in complete tree of height h is 2h.

More general induction Basis step as before Assume S(k) for all k <= n then show S(n)

Recursive Def of Tree Basis: a single node is a tree. If T1, T2, … Tk are trees then a new tree can be formed by Add new node N, the root of the new tree Add copies of T1… Tk Add an edge from N to the root of each T1, T2, … Tk

Structural Induction For objects with recursive definitions consisting of base objects and then combining rules Basis step: show the proposition S(X) holds for every base object X. Inductive step: Given a recursive structure X formed from X1, X2, … Xn by the application of the def. then Assume S(X1) S(X2) …. S(Xn) are true and show that S(X) is true

Recursive Def of Arithmetic Expressions Basis: a number or a variable is an expression. If E and F are expressions then a new expression G can be formed by applying one of the three rules G = E + F G = E * F G = ( E )

Every Expression has equal number of left and right parenthses

Homework . Prove if a complete binary tree has n leaves then it has 2n-1 nodes.

References– Mathematical Foundations http://en.wikipedia.org/wiki/Binary_relation http://en.wikipedia.org/wiki/Relation_(mathematics) http://en.wikipedia.org/wiki/Mathematical_proof http://en.wikipedia.org/wiki/Proofs_from_THE_BOOK Extended “Proof” techniques http://www.maths.uwa.edu.au/~berwin/humour/invalid.proofs.html Fair Use Books Online http://fair-use.org/bertrand-russell/the-principles-of-mathematics/ Books Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills