Studying the internal structure of REC, the set of solvable problems

1
Lecture 14

• Studying the internal structure of REC, the set
  of solvable problems
• Complexity theory overview
• Automata theory preview
• Motivating Problem
• string searching

2
Studying REC

• Complexity Theory
• Automata Theory

3
Current picture of all languages
All Languages
REC Solvable
REC Solvable
RE-REC
All languages - RE
Half Solvable
Unsolvable
Which language class should be studied further?
4
Complexity Theory
REC
RE - REC
All languages - RE

• In complexity theory, we differentiate problems
  by how hard a problem is to solve
• Remember, all problems in REC are solvable
• Which problem is harder and why?
• Max
• Input list of n numbers
• Task return largest of the n numbers
• Element
• Input list of n numbers
• Task return any of the n numbers

5
Resource Usage

• How do we formally measure the hardness of a
  problem?
• We measure the resources required to solve input
  instances of the problem
• typical resources are
• Running time
• Memory
• Need a notion of size of an input instance
• Obviously larger input instances require more
  resources to solve

6
Poly Language Class
REC
RE - REC
All languages - RE
Poly
Rest of REC
Informal Definition A problem L1 is easier than
problem L2 if problem L1 can be solved in less
time than problem L2. Poly the set of problems
which can be solved in polynomial time (typically
referred to as P, not Poly) Major goal Identify
whether or not a problem belongs to Poly
7
Working with Poly
Poly
Rest of REC

• How do you prove a problem L is in Poly?
• Show there exists a program P which solves L in
  polynomial time
• How do you prove a problem L is not in Poly?
• You must prove that no program P solves L in
  polynomial time
• We are not very good at this.
• For a large class of interesting problems, we
  have techniques (polynomial-time
  answer-preserving input transformations) which
  show a problem L probably is not in Poly, but
  few which prove it.

8
Automata Theory
REC
RE - REC
All languages - RE

• In automata theory, we will define new models of
  computation which we call automata
• Finite State Automata (FSA)
• Pushdown Automata (PDA)
• Key concept
• FSAs and PDAs are restricted models of
  computation
• We take away capabilities and resources
• We then identify which problems can be solved
  using FSAs and PDAs

9
New language classes
REC
RE - REC
All languages - RE

• REC is the set of solvable languages when we
  start with a general model of computation like
  C programs
• We want to identify which problems in REC can be
  solved when using these restricted automata

Rest of REC
10
Recap

• Complexity Theory
• Studies structure of the set of solvable problems
• Method analyze resources (processing time) used
  to solve a problem
• Automata Theory
• Studies structure of the set of solvable problems
• Method define automata with restricted
  capabilities and resources and see what they can
  solve (and what they cannot solve)
• This theory also has important implications in
  the development of programming languages and
  compilers

11
Motivating Problem

• String Searching

12
String Searching

• Input
• String x
• String y
• Tasks
• Return location of y in string x
• Does string y occur in string x?

• Can you identify applications of this type of
  problem in real life?
• Try and develop a solution to this problem.

13
String Searching II

• Input
• String x
• pattern y
• Tasks
• Return location of y in string x
• Does pattern y occur in string x?

• Pattern
• anything.html
• EN4
• Do you see applications for this in real life?
• Try and develop a solution to this problem.

14
String Searching

• We will show an easy way to solve these string
  searching problems
• In particular, we will show that we can solve
  these problems in the following manner
• write down the pattern
• the computer automatically turns this into a
  program which performs the actual string search