Exploring the Limits of Computation and Modeling

1. Computability andModeling Computation • What are some really impressive things that computers can do? • Land the space shuttle (and other aircraft) from the ground • Automatically track the location of a space or land vehicle • Beat a grandmaster at chess • Are there any things that computers can't do? • Yes, and these are referred to as “undecidable problems,” “unsolvable problems,” or “non-computable functions.”

2. Some Difficult Problems • For example, could a computer tell us what the meaning of life is? • Suppose a computer outputs the string "You are born, live your life, and then you die.” • Has that computer given us the meaning of life? • The statement may or may not be true • Doesn't tell one how to live a happy life • Doesn't tell one how to live a successful life • Doesn't tell if there is an afterlife • The program doesn’t arrive at the solution by itself • Point #1: Problems need to be precisely defined, before we can determine if they are solvable.

3. Some Difficult Problems, Cont. • Similarly, could a computer tell us if there is life after death? • Consider the following two programs – the first simply outputs the string “There is life after death,” and the second simply outputs “There is no life after death.” • Note that one of the two programs answers the question correctly, but which one? Also note that neither justifies or develops its’ own answer. • Point #2: Before we can determine if something can be solved or computed, we have to define what we mean by “compute.”

4. A Formal Model of Computation:Turing Machines • Before determining what computers are ultimately capable of, one must answer the following questions: • What is computation? • What are “reasonable” steps that can be performed in a computation? • Formally, computation is represented by a mathematical model called a Turing Machine. • Proposed in 1926 by Alan Turing to model “any possible computation” • Church’s hypothesis is that any general and reasonable way to compute, or any type of general purpose computer can be modeled/represented by a Turing machine. • Detailed discussion of Turing machines is beyond the scope of this course. • Other, less computer-like models of computation include recursive function theory and predicate calculus.

5. A Less Formal Model of Computation: Algorithms • Informally, an algorithm is a finite length sequence of operations, each of which is well-defined and finite. In addition, an algorithm must terminate after a finite number of operations. • Well-defined means it should be clear at each step what is to be done. • “Compare x and y" doesn't say how x any are to be compared • Finite means each step must involve a finite amount of information (i.e., data), which is manipulated to only a finite degree. • Adding 1 to a number is finite • Repeatedly dividing a number by 2 forever is not.

6. Algorithms, Cont. • Intuitively, a well-defined and finite step could be performed by a person by pencil and paper in a fixed amount of time. • Algorithms can be described by a computer program, flowchart, pseudo-code, or by some other similar method. • Example: • Set x =0 reasonable • Let x = x +1 reasonable • Set y = x/0 unreasonable • Place an arbitrary question in a memory location, wait 2 seconds for the answer, and then process the answer. unreasonable

7. Problems to be Solved • What kinds of problems are there? • Mathematical problems (simple and complex) • Logical problems • Data processing problems, etc. • Consider just mathematical decision (I.e., yes/no) problems: • Let x be an integer, is x even? • Let x and y be integers, is x<y? • Are there integers x, y and z such that x*2+y = zy?

8. Unsolvable Problems • Can all mathematical decision problems be solved with algorithms? • Answer: No! (by “solved,” we mean in all cases) • Example of unsolvable problems: • Let P be a program. Does P contain an infinite loop, i.e., will P terminate eventually or loop forever? • Let P be a program, and let x be a variable in P. Is x ever assigned a value when P is executed? • Let P be a program, and let s be a statement in P. Is s ever executed when P is executed?

9. Problems vs. Computational Steps • In general, problems fall into one of three categories: • Those which are provably solvable • Those which are provable unsolvable • Those whose status is unknown; currently there is no algorithm to solve the problem, but it has not been proven that none exists • Among this last category of problems, there is considerable debate as to their ultimate status, e.g., is there a program that can “understand” a natural language such as English? • Note however, that this is an entirely different question from what constitutes a “reasonable” step in a computation, and on that issue there is no debate. • A “reasonable” step in a computation is well-defined and finite, and can be performed by pencil and paper in a fixed amount of time.

10. Efficiency vs. Inefficiency • In addition, solvable problems fall into one of three categories: • Solvable problems which have been proven to be solvable efficiently, i.e., solvable in (deterministic) polynomial time. • Solvable problems which have been proven not to be solvable efficiently, i.e., NP-hard or NP-complete (NP stands for Nondeterministic Polynomial). • Solvable problems for which there is no efficient solution, but which have not been proven not be be solvable efficiently.