MA/CSSE 474. Today's Agenda

Transcription

1 MA/CSSE 474 Theory of Computation Course Intro Today's Agenda Student questions Overview of yesterday's proof I placed online a "straight-line" writeup of the proof in detail, without the "here is how a proof works" commentary that was in the slides. See Session 1 resources on sched. page Introductions and Course overview Languages and Strings (if time) Operations on Languages 1

2 Overview of yesterday's proof S = L(M), language accepted by M T = {w {0,1}* : w does not have 11 as substring} Show that S = T. i.e., S T and T F S T: i.e. if wϵs, then wϵt By induction on w, showed If δ(q0, w) = q0, w has no 11 and does not end in 1. If δ(q0, w) = q1, w has no 11 and ends in 1. T S: i.e. if wϵt, then wϵs. We show the contrapositive: if w S, then w T. If w S then δ(q0, w) = q2 (the only non-accepting state). Show by induction that if δ(q0, w) = q2, then w contains 11. In the definition of M, I replaced A, B, C by q0, q1, q2. See Day 1 slides Roll Call Introductions If I mispronounce your name, or you want to be called by a nickname or different name than the Registrar lists for you, let me know. I have had most of you in class, but for some of you it has been a long time. Graders: Maisey Tucker, Devon Timaeus, Jonathan Taylor Instructor: Claude Anderson: F-210, x8331 Random Note: I often put more on my PowerPoint slides for a day than I expect we can actually cover that day, "just in case". 2

3 Instructor Professional Background Formal Education: BS Caltech, Mathematics 1975 Ph.D. Illinois, Mathematics 1981 MS Indiana, Computer Science 1987 Teaching: TA at Illinois, Indiana , Wilkes College (now Wilkes University) RHIT ? Major Consulting Gigs: Pennsylvania Funeral Directors Assn Navistar International Beckman Coulter ANGEL Learning Theory of Computation history Textbook Recent (as ToC books go) Thorough Literate Large (and larger!) Theory and Applications We ll focus more on theory; applications there for you to see 3

4 Online Materials Locations On the Schedule page public stuff Reading, HW, topics, resources, Suggestion: bookmark schedule page On Moodle personal stuff surveys, solutions, grades On piazza.com: Discussion forums and announcements Many things are under construction and subject to change, especially the course schedule. What we will focus on in 474 Definitions Theorems Examples Proofs A few applications, but mostly theory 4

5 Questions about course policies and procedures? From Syllabus? Schedule page? Said in class? Anything else? Why is attendance not mentioned in the syllabus? When do I plan to be in my office and available for you? Why you should go to faculty candidate talks! Learn things you might not hear form current faculty Help us evaluate candidate Help candidate evaluate Rose-Hulman The new faculty members are your future (even if you are a senior) And we are desperate! Greg Wilkin, O159, Wednesday 4:20 DISTRIBUTED SYSTEMS COMMUNICATION AND OVERCOMING THE IMPOSSIBLE IN COMPLEX EVENT PROCESSING 5

6 Languages and Strings Some Language-related Problems int alpha, beta; alpha = 3; beta = (2 + 5) / 10; (1) Lexical analysis: Scan the program and break it up into variable names, numbers, operators, punctuation, etc. (2) Parsing: Create a tree that corresponds to the sequence of operations that should be executed, e.g., / (3) Optimization: Realize that we can skip the first assignment since the value is never used, and that we can pre-compute the arithmetic expression, since it contains only constants. (4) Termination: Decide whether the program is guaranteed to halt. (5) Interpretation: Figure out what (if anything) useful it does. 6

7 A Framework for Analyzing Problems We need a single framework in which we can analyze a very diverse set of problems. The framework we will use is Language Recognition Most interesting problems can be restated as language recognition problems. Recap - Strings A string is a finite sequence (possibly empty) of symbols from some finite alphabet. is the empty string (some books/papers use instead) * is the set of all possible strings over an alphabet Counting: s is the number of symbols in s. = = 7 # c (s) is the number of times that c occurs in s. # a (abbaaa) = 4. 7

8 More Functions on Strings Concatenation: st is the concatenation of s and t. If x = good and y = bye, then xy = goodbye. Note that xy = x + y. is the identity for concatenation of strings. So: x (x = x = x). Concatenation is associative. So: s, t, w ((st)w = s(tw)). More Functions on Strings Replication: For each string w and each natural number i, the string w i is: w 0 =, w i+1 = w i w Examples: a 3 = aaa (bye) 2 = byebye a 0 b 3 = bbb Reverse: For each string w, w R is defined as: if w = 0 then w R = w = if w 1 then: a ( u * (w = ua)). So define w R = a u R. 8

9 Concatenation and Reverse of Strings Theorem: If w and x are strings, then (w x) R = x R w R. Example: (nametag) R = (tag) R (name) R = gateman Do the proof on the board Relations on Strings: aaa is a substring of aaabbbaaa aaaaaa is not a substring of aaabbbaaa aaa is a proper substring of aaabbbaaa Every string is a substring of itself. is a substring of every string. s is a prefix of t iff: x *(t = sx). s is a proper prefix of t iff: s is a prefix of t and s t. Examples: The prefixes of abba are:, a, ab, abb, abba. The proper prefixes of abba are:, a, ab, abb. Every string is a prefix of itself. is a prefix of every string. s is a suffix of t iff: x *(t = xs) then continue as above: proper suffix, self, 9

10 Defining a Language A language is a (finite or infinite) set of strings over a finite alphabet. Examples: Let = {a, b} Some languages over :, { }, {a, b}, {, a, aa, aaa, aaaa, aaaaa} If is nonempty, the language * contains an infinite number of strings, including:, a, b, ab, ababaa. Example Language Definitions 1. L = {x {a, b}* : all a s precede all b s}, a, aa, aabbb, and bb are in L. aba, ba, and abc are not in L. 2. L = {x : y {a, b}* : x = ya} Simple English description: 3. L = {x#y: x, y {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}* and, when x and y are viewed as the decimal representations of natural numbers, square(x) = y}. Examples: 3#9, 12#144, 3#8, 12, 12#12#12, # 4. L = {a n : n 0} Simpler symbolic description of this language: 10

11 English is hard to describe L = {w: w is a sentence in English}. Examples: Bill hit the ball. Colorless green ideas sleep furiously. The window needs fixed. Ball the Stacy hit blue. A Halting Problem Language L = {w: w is a Java program that, when given any finite input string, is guaranteed to halt}. Is this language well specified? Can we decide what strings it contains? 11

12 Languages and Prefixes What are the following languages? L = {w {a, b}*: no prefix of w contains b} L = {w {a, b}*: no prefix of w starts with a} L = {w {a, b}*: every prefix of w starts with a} Operations on Languages 12

13 Concatenation of Languages If L 1 and L 2 are languages over : Examples: L 1 L 2 = {w * : s L 1 ( t L 2 (w = st))} L 1 = {cat, dog} L 2 = {apple, pear} L 1 L 2 ={catapple, catpear, dogapple, dogpear} Concatenation of Languages { } is the identity for concatenation: L{ } = { }L = L is a zero for concatenation: L = L = L 1 = {a n : n 0} L 2 = {b n : n 0} L 1 L 2 = {a n b m : n, m 0} L 1 L 2 {a n b n : n 0} δ(q0, w) = q2 13

14 Kleene Star L* = { } {w * : k 1 ( w 1, w 2, w k L (w = w 1 w 2 w k ))} Example: L = {dog, cat, fish} L* = {, dog, cat, fish, dogdog, dogcat, fishcatfish, fishdogdogfishcat, } The + Operator L + = L L* L + = L* - { } iff L L + is the closure of L under concatenation. 14

15 Concatenation and Reverse of Languages Theorem: (L 1 L 2 ) R = L 2R L 1R. Proof: x ( y ((xy) R = y R x R )) Theorem 2.1 (L 1 L 2 ) R = {(xy) R : x L 1 and y L 2 } Definition of concatenation of languages = {y R x R : x L 1 and y L 2 } Lines 1 and 2 = L 2R L R 1 Definition of concatenation of languages 15