1. Humans, Computers, and Computational Complexity J. Winters Brock Nathan Kaplan Jason Thompson

2. Can computer programs answer our questions? Can computer programs answer our questions? Can computer programs answer them quickly? Can computer programs answer them quickly? Theory of everything: “A finite set of principles from which one could mindlessly deduce all mathematical truths, by merely tediously following the rules of symbolic logic.”

3. Turing’s Halting Problem Will my computer program run into an infinite loop? Will my computer program run into an infinite loop? “Given a description of an algorithm and its initial input, determine whether the algorithm, when executed on this input, ever halts. The alternative is that it runs forever without halting.” 1936 – Alan Turing showed a general algorithm to solve the problem for all inputs cannot exist. 1936 – Alan Turing showed a general algorithm to solve the problem for all inputs cannot exist.

4. Elegant LISP Programs Turing’s proof was abstract, let’s take a real proof on a real machine. Turing’s proof was abstract, let’s take a real proof on a real machine. A program is “elegant” if no shorter program produces the same output. Chaitin shows with a similar paradox that it is impossible to prove that any particular large program is “elegant.” Chaitin shows with a similar paradox that it is impossible to prove that any particular large program is “elegant.”

5. Boolean Satisfiability Boolean Statement: A statement that is formed from variables, parentheses, and the logical connectors, AND, OR, NOT. (A V ¬B V ¬C) ^ (A V B V ¬C) ^ (¬A V B V C) Can you assign true and false values to A, B, and C in order to make the whole statement true?

6. Humans have problems with a problem of this nature, however computers can check all the possibilities in order to answer the question. Humans have problems with a problem of this nature, however computers can check all the possibilities in order to answer the question. Effectively checkable Effectively checkable Can we find a more efficient algorithm? Can we find a more efficient algorithm?

7. P vs. NP P: The set of all problems that can be solved by a computer program in polynomial time. P: The set of all problems that can be solved by a computer program in polynomial time. NP: The set of all problems that can be solved by a nondeterministic program in polynomial time. (effectively checkable) NP: The set of all problems that can be solved by a nondeterministic program in polynomial time. (effectively checkable)

8. NP-complete: A problem that is in NP that every other problem in NP can be reduced to. NP-complete: A problem that is in NP that every other problem in NP can be reduced to. Cook’s Theorem Cook’s Theorem Polynomial time algorithm for any NP-complete problem → P = NP Polynomial time algorithm for any NP-complete problem → P = NP Majority believe P ≠ NP Majority believe P ≠ NP

9. Are the algorithmic boundaries of computer computation applicable to human reasoning? Are the algorithmic boundaries of computer computation applicable to human reasoning? Computationalism: the view that all human activities are reducible to algorithms and therefore, could be implemented by a computer Computationalism: the view that all human activities are reducible to algorithms and therefore, could be implemented by a computer Anti-computationalism: cognition is not computational Anti-computationalism: cognition is not computational Pluralism: much of reasoning is computational, but some is not Pluralism: much of reasoning is computational, but some is not

10. Searle’s Chinese Room Thought Experiment “Imagine that you carry out the steps in a program for answering questions in a language you do not understand. I do not understand Chinese, so I imagine that I am locked in a room with a lot of boxes of Chinese symbols (the database), I get small bunches of Chinese symbols passed to me (questions in Chinese), and I look up in a rule book (the program) what I am supposed to do. I perform certain operations on the symbols in accordance with the rules (that is, I carry out the steps in the program) and give back small bunches of symbols (answers to the questions) to those outside the room. I am the computer implementing a program for answering questions in Chinese, but all the same I do not understand a word of Chinese. (Ftrain.com)” “Imagine that you carry out the steps in a program for answering questions in a language you do not understand. I do not understand Chinese, so I imagine that I am locked in a room with a lot of boxes of Chinese symbols (the database), I get small bunches of Chinese symbols passed to me (questions in Chinese), and I look up in a rule book (the program) what I am supposed to do. I perform certain operations on the symbols in accordance with the rules (that is, I carry out the steps in the program) and give back small bunches of symbols (answers to the questions) to those outside the room. I am the computer implementing a program for answering questions in Chinese, but all the same I do not understand a word of Chinese. (Ftrain.com)” Entire process is syntactical Entire process is syntactical Syntax does not guarantee understanding/semantics Syntax does not guarantee understanding/semantics No matter how seemingly intelligent you may be, the fact still remains that the process is meaningless to you and that you lack comprehension. This lack is what we know as the “intentional mental state.”

11. Language Processing For humans, it comes naturally. For humans, it comes naturally. Machines are not as efficient as humans in this area. Machines are not as efficient as humans in this area. “Buffalo buffalo buffalo buffalo.” Lexical Functional Grammar (LFG) recognition is NP-hard. Lexical Functional Grammar (LFG) recognition is NP-hard.