Language: Set of Strings Preliminaries Language: Set of Strings Examples: Names, dates, email addresses, Java, C++, Patterns in PERL, Python, … CSC 361 Preliminaries

Specification and recognition of languages Sets Operations Defining sets (inductively) Proving properties about sets (using induction) Defining functions on sets (using recursion) Proving properties about functions (using induction) Strings Languages : set of strings CSC 361 Preliminaries

Representation of Sets Set: collection of objects Finite : can use enumeration E.g., {a,b,c} Infinite : requires describing an infinite characteristic of members using finite terms E.g., { n e N | even(n) /\ square(n) } CSC 361 Preliminaries

Sets: Examples Sets are denoted by { <collection of elements> } {} {a,b} {{}} {1, 2, …, 100} {0, 1, 2, …} {0,2,4, …} {2n | n  } “the empty set” “the set consisting of the elements a and b” “the set consisting of the empty set” “the set consisting of the first 100 natural numbers” “the set consisting of all natural numbers” “the set of all natural pair numbers” “the set of all natural pair numbers” CSC 361 Preliminaries

Set Inclusion and Set Equality Operations on Sets Set Inclusion and Set Equality Definition: Given 2 sets, A and B, A is contained in B, denoted by A  B, if every element in A is also an element in B True or false: {e,i,t,c} {a, b, …, z} for any set A, A  A for any set A, A  {A} true true false Definition: Given 2 sets A and B, A is equal to B, denoted by A = B, if A  B and B  A CSC 361 Preliminaries

Operations on Sets Review: Member Union Intersection Subset Powerset Cartesian product Definition of an op – its properties CSC 361 Preliminaries

Operations on Sets Set Difference DeMorgan’s Laws CSC 361 Definition of an op – its properties CSC 361 Preliminaries

Operations on Sets Cartesian Product Definition: Given two sets, A and B, the Cartesian product of A and B, denoted by A  B, is the following set: {(a,b) | a A and b  B} pair or 2-tuple Examples: What is: {1, 2 , 3}  {a,b} = True or false: {(1,a), (3,b)}  {1, 2 , 3}  {a,b} True or false: {1,2,3}  {1, 2 , 3}  {a,b} {(1,a), (1,b), (2,a), (2,b), (3,a), (3,b)} true false CSC 361 Preliminaries

Operations on Sets Cartesian Product of More Than 2 Sets Definition: Given three sets, A, B and C, the Cartesian product of A, B, and C denoted by A  B  C, is the following set: {(a,b,c) | a A, b  B, c  C} Triple or 3-tuple Definition. (x,y,z) = (x’,y’, z’) only if x = x’, y = y’ and z = z’ These definitions can be extended to define the Cartesian product: A1  A2  …  An and equality between n-tuples CSC 361 Preliminaries

Operations on Sets Some Cartesian product problems to try out: What is: {1, 2 , 3}  {a,b}  {,} = What is the form of the set A  B  C  D What is the form of the set A  B  (C  D) What is the form of the set (A  B )  (C  D) CSC 361 Preliminaries

Partition of a Set X The set of subsets of X is a partition of X iff “covers all subsets” “pair-wise disjoint” Collectively Exhaustive / Mutually Exclusive Combinatorics: list partition, number partition, … CSC 361 Preliminaries

Examples Set of Natural numbers is partitioned by “mod 5” relation into five “equivalence classes”: { {0,5,10,…}, {1,6,11,…}, {2,7,12,…}, {3,8,13,…}, {4,9,14,…} } “String length” can be used to partition the set of all bit strings. { {},{0,1},{00,01,10,11},{000,…,111},… } Equivalence = Similar for a purpose (approximation) CSC 361 Preliminaries

Relations Definition: Given two sets, A and B, A relation R is any subset of A  B. In other words, R  A  B Relations are used to describe how elements of different sets interact Question: What does the relation A  B indicate? Examples of relations Simple: {(c,s) | c is a class at KU, s is a student at KU, AND s is in C } CSC 361 Preliminaries

Relations Example: Let I be the set of all classroom instructors at Kutztown Let S be the set of all students at Kutztown Then, we can define the relation Students of an Instructor, over I x S, as follows: Students of an Instructor= {(i,s) | i I, s S, s is in a course of instructor i} Questions: Does this relation represent an equivalence relation? Suppose this instructor teaches courses c1, c2, and c3. Would the sets Students in instructor i’s course, for courses c1, c2, and c3, be disjoint? CSC 361 Preliminaries

Equivalence Relations A relation R is an equivalence relation if R is reflexive, symmetric, and transitive Relation R is: reflexive if (a,a)  R for each a in the language symmetric if when (a,b)  R then (b,a)  R transitive if (a,b)  R and (b,c)  R, then (a,c)  R Equivalence relations are generalizations of the equality relation x {(1,2),(1,3),…, (2,3),(2,4),…} {(a,b) | a and b are males between 35 and 40 years old} “the relation x < y”  CSC 361 Preliminaries

Functions Definition: A function f from a set A to a set B, denoted by f: A  B , is rule of correspondence between A and B such that there is a unique element in B assigned to each element in A i.e. for each a A there is one and only one b B such that (a,b) f Note: A is the domain of f B is the range of f Question: Is the following relation also a function from the set of KU classes to the set of KU students? {(c,s) | c is a class at KU, s is a student at KU, AND s is in C } CSC 361 Preliminaries

Functions Kinds of Functions Total One to one (injection) All domain elements map to an element in the range One to one (injection) Total, and each mapping from domain to range is unique Onto (surjection) All elements of the range are mapped to from a domain element 1-1 & Onto (bijection) CSC 361 Preliminaries

(Total) Function Domain Co-domain ( Range) A B f CSC 361 Preliminaries

One to One Function (injection) Domain Co-domain ( Range) A B f CSC 361 Preliminaries

Onto Function (surjection) Domain Co-domain ( Range) A B f CSC 361 Preliminaries

1-1 Onto Function (bijection) Domain Co-domain ( Range) A B f CSC 361 Preliminaries

Inductive Definitions Constructive Example Set of natural numbers N = {0,1,2,3,…} Finite representation in terms of Seed element: zero Closure Operation: successor function {0,s(0),s(s(0)),s(s(s(0))),…} Imposes additional structure on the domain. CSC 361 Preliminaries

(* Minimality condition to uniquely determine N *) Basis case: Recursive step: Closure: only if it can be obtained from 0 by a finite number of applications of the operation s. (* Minimality condition to uniquely determine N *) CSC 361 Preliminaries

Languages A language can be specified as a set of strings. Languages are classified by how they can be formed Regular languages are the most constrained Others Context-free Context-sensitive Recursively enumerable CSC 361 Preliminaries

Language Semantics - Syntax Regular sets/languages Generator Regular Expressions, Regular Grammars Recognizer Finite State Automata Context-free languages Context-free Grammar Push-down Automata Identifiers, numbers, etc in a PL; Tokenizer BNF; Parser DTD; XML Parser CSC 361 Preliminaries