CS 3512 Final Exam Topics Topics covered on Exam #1 Topic Text Chapter -------------------- ------------- Elementary Notions and Notations.. 1 A Proof Primer.................. 1.1 Sets............................ 1.2 skip 1.2.4 Ordered Structures.............. 1.3 Graphs and Trees (skip)......... 1.4 Facts About Functions............. 2 Definitions and Examples........ 2.1 2.1.1 only - skip the rest (but useful elsewhere in CS) Constructing Functions.......... 2.2 skip 2.2.2 (Map function) for now Properties of Functions......... 2.3 skip 2.3.4 and 2.3.5 Countability.................... 2.4 Topics covered on Exam #2 Topic Text Chapter -------------------- ------------- Construction Techniques........... 3 Inductively Defined Sets........ 3.1 skip 3.1.5 Recursive Functions & Procedures 3.2 skip 3.2.6 Grammars - skip for now ........ 3.3 Inductive Proof................... 4 skip 4.1, 4.2, 4.3 Inductive Proof................. 4.4 skip Example 4.47 for now Analysis Techniques............... 5 skip 5.1 through 5.5 Comparing Rates of Growth....... 5.6 skip "Divide & Conquer Recurrences (pages 381-389) Topics covered since Exam #2 Topic Text Chapter -------------------- ------------- Regular Languages & Finite Automata 11 Regular Languages................ 11.1 Review Properties of R.E.s p. 700 Finite Automata.................. 11.2 skip examples 11.5, 11.6, & 11.7 skip 11.2.3 (done better in 11.3.1) For 11.2.4, know the formula on page 713 for new(i,j) (in "Eliminate State k") skip 11.2.5 skip 11.2.6 but understand T and T* as in the lecture notes Constructing Efficient Finite Automata ..... 11.3 11.3.1 Know how to do the regular expression to NFA-Lambda algorithm 11.3.2 Know that the crucial step in going from an NFA to a DFA is that the states of the DFA are sets of states of the NFA - this is the "Subset Construction" of the Chapter 11 notes. Regular Languages Topics -Skip-.. 11.4 Also, from the Chapter 11 notes, know: 1) Regular languages are closed under union, concatenation, and star (by definition), and also complement, intersection, and set difference. 2) DFA's, NFA's, and NFA-Lambdas are all equivalent: language L is recognized by a DFA iff L is recognized by an NFA iff L is recognized by an NFA-Lambda (see 11.3.2 above) Distinguishability (only in the notes) Know how stings are distinguished by a language L. Know the "Distinguishability Theorem" If there are n strings that are distinguishable from one another, any DFA recognizing L has >= n states Know the "Distinguishability Corollary" If there is an infinite set of strings, each distinguished from the others by L, then L is not regular. Know the "Minimal DFA Theorem": the indistinguishability classes are the states - this is minimal by the Distinguishability Theorem Know what Kleene's Theorem says: a language is regular (represented by a regular expression) iff it is recognized by a DFA. Grammars, PDA's, Turing Machines Grammars ....................... 3.3 skip 3.3.1 (English grammar) Know 3.3.2, 3.3.3, 3.3.4 up to "Decimal Numerals, 3.3.5 Know the definition of a grammar Know that a grammar generates a language Know that grammars can be combined by the union, concatenation, & star constructions Know that some grammars are ambiguous Context-Free Languages/PDA's .... 12 Context-Free Languages ........ 12.1 Know the definition Know CFL's are closed under union, concatenation, and star Pushdown Automata ............. 12.2 Know they are non-deterministic Know Theorem 12.6 ......... 12.2.2 CFL's = langs. accepted by PDA's Turing Machines ................. 13 Turing Machine Definition ... 13.1 Church-Turing Thesis ........ 13.2 Know what this says (page 829) Computation ..................... 14 Computability ............... 14.1 Know that some problems have no algorithmic solution - e.g. the Halting Problem The Hierarchy of Languages .. 14.2 Know the contents of the table on the last page of the notes Exercises for Exam #1 The following is a list of Exercises from the text like the ones that may appear on exams, possibly in a slightly altered forms. Some have been assigned as homework. Also, you should know how to do (or at least understand) the "Additional Problems" from "Recommended Problems" 1 through 6. Chapter 1 1.1 Exercises (pages 12-13): All exercises, especially 2 c, d, and e, 3 b, and d, 4 c, 5, and 7 d, which were assigned as homework. 1.2 Exercises (pages 32-36): All exercises except 24-26 (not covered), but especially 3 b, d, f, and h, 5, 6 a, 7 a, 8, and 19, which were assigned as homework. Also, 13, 14, and 35 are of low priority. 1.3 Exercises (pages 53-56): All exercises except 19-21, but especially 2 f, 4 d, 8 b and e, 10 b and d, 11 b, 12 b, and 14 a, which were assigned as homework 1.4 - Skip this section. Chapter 2 2.1 Exercises (pages 92-96): Exercises 1, 2, 9, 15, 16, 27, and 28, but especially 2, 5 b and c, 27 b and c, and 28 c, d, and e, which were assigned as homework. 2.2 Exercises (pages 104-106): Skip all of these for now. 2.3 Exercises (pages 117-121): Exercises 1-3, 4 a, b, e, 5 c, f, g, 15-18. None of these were assigned as homework, but related questions appeared as "Additional Problems" in "Recommended Problems 5". 2.4 Exercises (pages 130-131): All exercises, but especially 1 a, 2 f, 3 a and d, and 9. Also, ""Additional Problems" 3 and 4 of "Recommended Problems 6" were assigned as homework. Exercises for Exam #2 The following is a list of Exercises from the text like the ones that may appear on exams, possibly in a slightly altered forms. Some have been assigned as homework. Also, you should know how to do (or at least understand) the "Additional Problems" from "Recommended Problems" 7 through 9 (it might be useful to review "Recommended Problems" 10 also in order to understand asymptotic notation better). Chapter 3 3.1 Exercises (pages 148-151): All exercises except 5, 11, 16 - 18, 20, and 21. 3.2 Exercises (pages 173-178): All exercises except 7, 13, and 20 - 26. Chapter 4 - just Section 4.4 4.4 Exercises (pages 274-278): All exercises except 18, and 22 - 25. Chapter 5 - just Section 5.6 5.6 Exercises (pages 389-392): All exercises except 10 - 13. Exercises since Exam #2 The following is a list of Exercises from the text like the ones that may appear on exams, possibly in a slightly altered forms. Some have been assigned as homework. Also, you should know how to do (or at least understand) the "Additional Problems" from "Recommended Problems" 10 through 12 (though 10 was partly covered on Exam #2, it might be useful to review "Recommended Problems" 10 also in order to understand asymptotic notation better). Chapter 11 11.1 Exercises (pages 702-704) Exercises 1-4, 8, 9 11.2 Exercises (pages 726-728) Exercises 1-6 11.3 Exercises (pages 743-745) Exercises 1, 3b, 12 Distinguishability There are no exercises on this from the text, but "Recommended Problems" 12 has some. Grammars, PDA's, Turing Machines You don't have to know how to do any exercises from the sections that cover this "Survey" material, but you should know the definitons and results mentioned above, including the Church-Turing Thesis