Computation Binary Numbers Decimal numbers Binary numbers http

Computation

Binary Numbers • Decimal numbers • Binary numbers http: //faculty. mc 3. edu/pvetere/Applets/APPLETS/NUMSYS/applet_frame. htm

Representing Text • Decide how many characters we need to represent. • Determine the required number of bits. • Ascii: 7 bits. Can encode 27 = 128 different symbols.

Ascii http: //www. krisl. net/cgi-bin/ascbin. pl

Representing Text Four … 010001101111 01110101 01110010

When We Need More Characters What about things like: 简体字

When We Need More Characters What about things like: 简体字 Answer: Unicode: 32 bits. Over 4 million characters. http: //www. unicode. org/charts/

Digital Images

RGB The red channel

RGB The green channel

RGB Red Green Blue

Representing Pictures

Representing Sounds

Representing Programs public static Tree. Map<String, Integer> create() throws IOException { Integer freq; String word; Tree. Map<String, Integer> result = new Tree. Map<String, Integer>(); JFile. Chooser c = new JFile. Chooser(); int retval = c. show. Open. Dialog(null); if (retval == JFile. Chooser. APPROVE_OPTION) { Scanner s = new Scanner( c. get. Selected. File()); while( s. has. Next() ) { word = s. next(). to. Lower. Case(); freq = result. get(word); result. put(word, (freq == null ? 1 : freq + 1)); } } return result; } }

The Roots of Modern Technology 5 thc B. C. Aristotelian logic invented 1642 Pascal built an adding machine 1694 Leibnitz reckoning machine

The Roots, continued 1834 Charles Babbage’s Analytical Engine Ada writes of the engine, “The Analytical Engine has no pretensions whatever to originate anything. It can do whatever we know how to order it to perform. ” The picture is of a model built in the late 1800 s by Babbage’s son from Babbage’s drawings.

The Roots: Logic 1848 George Boole The Calculus of Logic chocolate and nuts and mint chocolate nuts mint

Boolean Logic P Q P P Q P Q True False True True False False False True True

Using Boolean Logic P Q P P Q (P Q) Q P ((P Q) Q) True False True False True True False True

How Big Are the Truth Tables? P Q R True True False True False False False

2 n

Increased Power of Quantified Logic • Tai. Shan. Has. Tail • Smoky. Has. Tail • Puffy. Has. Tail • Chumpy. Has. Tail • Snowflake. Has. Tail

Increased Power of Quantified Logic • Panda(Tai. Shan). • Bear(Smoky). • x (Panda(x) Bear(x, Tail)). • x (Bear(x) Has. Part(x)). • x (Bear(x) Animal(x)). • x (Animal(x) Bear(x)). • x (Animal(x) y (Mother-of(y, x))). • x ((Animal(x) Dead(x)) Alive(x)).

Using Quantified Logic (FOL) We can represent what we know as a set of logical formulas and then manipulate them.

Using Quantified Logic • Panda(Tai. Shan). • Bear(Smoky). • x (Panda(x) Bear(x, Tail)). • x (Bear(x) Has. Part(x, Tail)). • x (Bear(x) Animal(x)). • x (Animal(x) y (Mother-of(y, x))). • x ((Animal(x) Dead(x)) Alive(x)).

Search

Breadth-First Search Is this a good idea?

Depth-First Search

Scalability Solving hard problems requires search in a large space. To play master-level chess requires searching about 8 ply deep. So about 358 or 2 1012 nodes must be examined.

Growth Rates of Functions

Scalability To play one master-level game 2 1012 nodes Seconds since Big Bang 3 1017 Number of sequential games 150, 000 since Big Bang

The Advent of the Computer 1945 ENIAC The first electronic digital computer

1949 EDVAC The first stored program computer

Programs Are Just Strings (Bits) public static Tree. Map<String, Integer> create() throws IOException { Integer freq; String word; Tree. Map<String, Integer> result = new Tree. Map<String, Integer>(); JFile. Chooser c = new JFile. Chooser(); int retval = c. show. Open. Dialog(null); if (retval == JFile. Chooser. APPROVE_OPTION) { Scanner s = new Scanner( c. get. Selected. File()); while( s. has. Next() ) { word = s. next(). to. Lower. Case(); freq = result. get(word); result. put(word, (freq == null ? 1 : freq + 1)); } } return result; } }

Moore’s Law http: //www. intel. com/technology/mooreslaw/

How It Has Happened

How Much Computer Power Might It Take? http: //www. frc. ri. cmu. edu/~hpm/book 97/ch 3/index. html

How Much Compute Power is There? From Hans Moravec, Robot Mere Machine to Transcendent Mind 1998.

Kurweil’s Vision http: //www. pocket-lint. co. uk/news. phtml/12920/13944/Computersmatch-humans-by-2030. phtml

Limits to What We Can Compute Are there fundamentally uncomputable things? • Does God exist? • What’s the best way to run a country? • Does this puzzle have a solution?

Mathematics in the Early 20 th Century 1900 Hilbert’s program and the effort to formalize mathematics 1931 Kurt Gödel’s paper, On Formally Undecidable Propositions 1937 Alan Turing’s paper, On Computable Numbers with an application to the Entscheidungs problem

What Can We Do? 1. Can we make all true statements theorems? 2. Can we decide whether a statement is a theorem?

Gödel’s Incompleteness Theorem Kurt Gödel showed, in the proof of his Incompleteness Theorem [Gödel 1931], that the answer to question 1 is no. In particular, he showed that there exists no decidable axiomatization of Peano arithmetic that is both consistent and complete.

The Entscheidungsproblem Does there exist an algorithm to decide, given an arbitrary sentence w in first order logic, whether w is valid (i. e. , must be true)?

Turing Machines At each step, the machine must: ● choose its next state, ● write on the current square, ● move left or right. and

An Example

The Church-Turing Thesis All formalisms powerful enough to describe everything we think of as a computational algorithm are equivalent. • Turing machines • Lambda calculus • Standard programming languages like Java • Conway’s game of life • DNA computing

The Game of Life Playing the game The rules: ● A dead cell with exactly three live neighbors becomes a live cell (birth). ● A live cell with two or three live neighbors stays alive (survival). ● In all other cases, a cell dies or remains dead (overcrowding or loneliness). A game halts iff it reaches some stable configuration.

Back to the Entscheidungsproblem Theorem: The Entscheidungsproblem is unsolvable. Proof: (A variant of Turing’s proof) 1. 2. 3. 4. Given a Turing machine M, we can construct a logical formula F that is true iff M ever halts. If there were a solution to the Entscheidungsproblem, then we could determine the truth of F and thus be able to decide whether M ever halts. But there is no procedure for determining (in the general case) whether M halts. So there is no solution to the Entscheidungsproblem.

The Halting Problem Turing machine, M input string, w Does M halt on w? Accept Reject

An Example of a Similar Question Does this program halt on all inputs? times 3(x: positive integer) = While x 1 do: If x is even then x = x/2. Else x = 3 x + 1. Let’s try it.

The Halting Problem Is Undecidable Proof: If it were decidable, then some TM MH would decide it. MH would implement the specification: halts(<M: string, w: string>) = If <M> is a Turing machine description and M halts on input w then accept. Else reject.

Trouble(x: string) = if halts(x, x) then loop forever, else halt. What is Trouble(<Trouble>)? What is halts(<Trouble, Trouble>)? ● If halts reports that Trouble(<Trouble>) halts, Trouble ____. ● If halts reports that Trouble(<Trouble>) does not halt, Trouble _____.

Another Undecidable Problem The Post Correspondence Problem

A PCP Instance With a Simple Solution i X 1 b 2 abb Y aab b a 3 aba 4 baaa baba

A PCP Instance With a Simple Solution i X 1 b 2 abb Y aab b a 3 aba 4 baaa baba Solution: 3, 4, 1

Another PCP Instance i X 1 11 2 01 Y 011 3 001 110 0

A PCP Instance With No Simple Solution i X Y 1 1101 1 2 0110 11 3 1 110

A PCP Instance With No Simple Solution i X Y 1 1101 1 2 0110 11 3 1 110 Shortest solution has length 252.

A Tiling Problem Given a finite set T of tiles of the form: Is it possible to tile an arbitrary surface in the plane?

A Set of Tiles That Cannot Tile the Plane

Is the Tiling Problem Decidable? Wang’s conjecture: If a given set of tiles can be used to tile an arbitrary surface, then it can always do so periodically. In other words, there must exist a finite area that can be tiled and then repeated infinitely often to cover any desired surface. But Wang’s conjecture is false.

The Implications • The Entscheidungs problem is undecidable. • There’s no black box reasoning engine for FOL. • Would quantum computing change the picture? • Does undecidability doom our attempt to make artificial copies of ourselves?

Is Decidability Enough?

Boolean Logic, Again P Q P P Q (P Q) Q P ((P Q) Q) True False True False True True False True How many steps would it take a deterministic Turing machine to examine this table?

Nondeterminism P is True Q is False P is False Q is True Q is False

Nondeterministically Deciding SAT P Q P P Q (P Q) Q P ((P Q) Q) True False True False True True False True How many steps in a single path of the process of examining this table?

P and NP • P – problems that are solvable deterministically in polynomial time. • NP – problems that are solvable nondeterministically in polynomial time.

The Traveling Salesman Problem 15 10 20 4 25 28 8 40 9 7 3 23 Given n cities and the distances between each pair of them, find the shortest tour that returns to its starting point and visits each other city exactly once along the way.

The Traveling Salesman Problem 15 10 20 4 25 28 8 40 9 7 3 23 Given n cities: Choose a first city Choose a second Choose a third … n n-1 n-2 n!

The Traveling Salesman Problem Can we do better than n! ● First city doesn’t matter. ● Order doesn’t matter. So we get (n-1!)/2.

The Growth Rate of n! 2 2 11 479001600 3 6 12 6227020800 4 24 13 87178291200 5 120 14 1307674368000 6 720 15 20922789888000 7 5040 16 355687428096000 8 40320 17 6402373705728000 9 362880 18 121645100408832000 10 3628800 19 2432902008176640000 11 39916800 36 3. 6 1041

The Traveling Salesman Problem

Solving it Nondeterministically First, we change the problem slightly. New problem: Given a value n, is there a tour of length less than n? 15 10 20 4 25 28 8 40 9 7 3 23

SAT and Traveling Salesman D ND 2 n n Traveling Salesman n! n SAT

Growth Rates of Functions, Again

P and NP • P – problems that are solvable deterministically in polynomial time. • NP – problems that are solvable nondeterministically in polynomial time. • NP-complete – NP problems that are at least as hard as every other NP-complete problem.

Does Complexity Doom AI?