CSE 105: Introduction to Theory of Computability
Is there a problem that NO computer can ever solve?
What resources are needed to solve a problem?
Are some problems harder than others?
In this course, we will explore what it means to be "computable". We begin with a very simple model of computation, and work our way to the most powerful, the Turing machine, named after Alan Turing, who formalized the notion of "algorithm" even before there were any physical computers. You'll also learn about the relationship of these models to some essential tools in a computer scientist's toolkit, such as regular expressions and context-free grammars. Finally, you'll develop your technical communication skills in writing formal arguments and proofs.
Weekly Schedule (Lec A and Lec B)
| Week | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday |
|---|---|---|---|---|---|---|---|
| 1 | 1/8 Introduction Regular expressions Sec 1.3 (Read) Slides Annotated (LecA), Annotated (LecB) |
1/9 Discussion: Lecture review |
1/10 DFA Sec 1.1 (Read) Slides Annotated (LecA), Annotated (LecB) |
1/11 Discussion: Chapter 0 review |
1/12 DFA design Sec 1.1 (Read) Slides Annotated (LecA), Annotated (LecB) |
HW Indiv0 Due 1/13 11pm Pre-requisites PDF, source, style [Small typo fixed 1/8] Pre-class survey |
1/14 Review Quiz Link |
| 2 | 1/15 No lecture MLK Day |
HW Indiv1 Due 1/16 11pm RegExp, DFA PDF, source, hw1.png, hw1.jff, style Discussion: Lecture review |
1/17 Closure Sec 1.1 (Read) Slides Annotated (LecA), Annotated (LecB) |
1/18 Discussion: RegExp, DFA |
1/19 NFA Sec 1.2 (Read) Slides Annotated (LecA), Annotated (LecB) |
Group HW1 Due 1/20 11pm RegExp, DFA PDF, source, style [Additional Hints added 1/17] |
1/21 Review Quiz Link |
| 3 | 1/22 NFA to DFA Sec 1.2 (Read) Slides Annotated (LecA), Annotated (LecB) |
HW Indiv2 Due 1/23 11pm Closure, NFA PDF, source, hw2_1.png, hw2_2.png, hw2_3.png, hw2_4.png, hw2_1.jff, hw2_2.jff, hw2_3.jff, hw2_4.jff, style Discussion: Lecture review |
1/24 Equivalence with RegExp Sec 1.3 (Read) Slides, Annotated (LecA), Annotated (LecB) |
1/25 Discussion: Closure, NFA |
1/26 Non-regular sets Sec 1.4 (Read) Slides, Annotated (LecA), Annotated (LecB) |
Group HW2 Due 1/27 11pm Closure, NFA PDF, source, style hw2_5.png, hw2_5.jff, |
1/28 Review Quiz Link |
| 4 | 1/29 Pumping Lemma Sec 1.4 (Read) Slides, Annotated (LecA), Annotated (LecB) |
HW Indiv3 Due 1/30 11pm Regular languages, PL PDF, source, hw3_1.png, hw3_2.png, hw3_3.png, hw3_4.png, hw3_1.jff, hw3_2.jff, hw3_3.jff, hw3_4.jff, style Discussion: Lecture review |
1/31 PDA Sec 2.2 (Read) Slides, Annotated (LecA), Annotated (LecB) |
2/1 Discussion: Regular languages, PL |
2/2 PDA Design Sec 2.2 (Read) Slides, Annotated (LecA), Annotated (LecB) |
Group HW3 Due 2/3 11pm Regular languages, PL PDF, source, (No new jff or png files), style |
2/4 Review Quiz Link |
| 5 | 2/5 CFG Sec 2.1 (Read) Slides, Annotated (LecA), Annotated (LecB) |
2/6 Discussion: Lecture review |
2/7 Exam 1 Practice questions |
Discussion: PDA, CFG |
2/9 CFL Sec 2.1 (Read) Slides, Annotated (LecA), Annotated (LecB) |
2/11 Review Quiz Link |
|
| 6 | 2/12 TMs: Formal def Sec 3.1 (Read) Slides, Annotated (LecA), Annotated (LecB) |
HW Indiv4 Due 2/13 11pm PDA, CFG PDF, source, style 2/13 Discussion: Lecture review |
2/14 TMs: Implementation Sec 3.1 (Read) Slides, Annotated (LecA), Annotated (LecB) |
2/15 Discussion: Turing machines |
2/16 TMs: High-level Sec 3.1 (Read) Slides, Annotated (LecA), Annotated (LecB) |
Group HW4 Due 2/17 11pm PDA, CFG, CFL PDF, source, hw4_1.png, hw4_2.png, hw4_3.png, hw4_1.jff, hw4_2.jff, hw4_3.jff style |
2/18 Review Quiz Link |
| 7 | 2/19 No lecture Presidents' Day |
HW Indiv5 Due 2/20 11pm TM: formal, implementation PDF, source, style Discussion: Lecture review |
2/21 Church-Turing thesis Sec 3.2 (Read) Slides, Annotated (LecA), Annotated (LecB) |
2/22 Discussion: Decidability, closure |
2/23 Algorithms Sec 3.3 (Read) Slides, Annotated (LecA), Annotated (LecB) |
Group HW5 Due 2/24 11pm TM: formal, implementation PDF, source, hw5TM.png, hw5TM.jff, style | 2/25 Review Quiz Link |
| 8 | 2/26 Decidable langs Sec 4.1 (Read) Slides, Annotated (LecA), Annotated (LecB) |
2/27 Discussion: Lecture review |
2/28 Undecidability Sec 4.2 (Read) Slides, Annotated (LecA), Annotated (LecB) |
3/1 Discussion: Lecture review |
3/2 Exam 2 Practice questions |
3/4 Review Quiz Link |
|
| 9 | 3/5 Reductions Sec 5.1 (Read) Slides, Annotated (LecA), Annotated (LecB) |
HW Indiv6 Due 3/6 11pm High-level descriptions, decidability PDF, source, style Discussion: Closure |
3/7 Reductions Sec 5.1 (Read) Slides, Annotated (LecA), Annotated (LecB) |
3/8 Discussion: Indiv6 review |
3/9 Reductions Sec 5.1 (Read) Slides, Annotated (LecA), Annotated (LecB) |
Group HW6 Due 3/10 11pm High-level descriptions, decidability PDF, source, style |
3/11 Review Quiz Link |
| 10 | 3/12 Time complexity Sec 7.1 (Read) Slides, Annotated (LecA), Annotated (LecB) |
HW Indiv7 Due 3/13 11pm Undecidability, reductions PDF, source, style Discussion: Lecture review |
3/14 P vs. NP Sec 7.1, 7.2 (Read) Slides, Annotated (LecA), Annotated (LecB) |
Group HW7 Due 3/15 11pm Undecidability, reductions PDF, source, style Discussion: Review |
3/16 Review Sec 5.1 (Read) Slides, Annotated (LecA), Annotated (LecB) Optional HW Indiv8 PDF, source, style Review Quiz Link (Complete by 3/17 11pm for credit) |
Final exam Saturday 3/17 11:30am-2:29pm Practice questions |
Syllabus
Upon successful completion of this course, you will be able to:
- Classify the computational complexity of a set of strings by determining whether it is regular, context-free, decidable, or undecidable.
- Give examples of sets in each of these classes (and prove them).
- Classify the computational complexity of a decision problem by translating it to a set of strings coding the problem.
- Use and design automata both formally and informally, including DFA, NFA, PDA.
- Deduce the language recognized by a machine or other formal description, including DFA, NFA, RegExp, PDA, CFG, TM.
- Prove invariants of computational classes, e.g. the class of regular languages is closed under ....
- Prove that certain models of computation are equivalent and translate between them algorithmically.
- Apply classical techniques including pumping lemma, determinization, diagonalization, and reduction to analyze the complexity of languages and problems.
CSE 105 instructional team
We are looking forward to working with you this quarter.
Announcements and Q&A are through Piazza (sign up link: piazza.com/ucsd/winter2018/cse105).
Our office hours and locations can be found in the calendar above.
For private questions to Prof. Minnes, you can reach me at minnes@eng.ucsd.edu
Textbook
The required textbook for this course is
This book is available at the Bookstore for under $20 and is also on reserve in the library. You will need the book to complete pre-class reading before each lecture period.
To brush up on proofs, use: Richard Hammack, Book of Proof, 2nd ed. (available for download here)
An iClicker2 is also required, and is available for purchase at the bookstore. Register your iClicker here.
Pre-class reading and post-class practice questions
Week 1
Pre-requisites
Reading Sec 0.2, 0.3, 0.4: In particular, review the definitions of set, element, subset,
infinite set, proper subset, natural numbers, integers, empty set, union, intersection, complement, sequence,
Cartesian product, function, domain, range, graph, self-loop, alphabet, symbol, string over an alphabet,
length, empty string, substring, concatenation, lexicographic order, shortlex/string order, prefix,
proof by contradiction.
Optional extra practice: Chapter 0 Exercise # 1, 2, 3, 4, 5, 6.
Regular expressions
Reading Sec 1.3: Def 1.52 of regular expressions (p. 64), example 1.53 (p. 65)
Optional extra practice: Chapter 1 Exercise # 20
DFA
Reading Sec 1.1: Figure 1.4 (p. 34), Definition 1.5 (p. 35)
Optional extra practice: Chapter 1 Exercise # 1, 2, 3
Week 2
Closure operations (Wednesday)
Reading Pages 45-47 (paragraph before Theorem 1.25 and Theorem 1.25 and its proof).
Optional extra practice Chapter 1 Exercise # 4, 5
Nondeterminism (Friday)
Reading Page 48 (Figure 1.27 and description below it), Example 1.35 on page 52.
Optional extra practice Chapter 1 Exercise # 7, 9
Week 3
NFA to DFA (Monday)
Reading Pages 55 paragraph that starts "Recall the "reader as automaton"..." and Example 1.41 on pages 56-58..
Optional extra practice Chapter 1 Exercise # 13, 14, 16
NFA to RegExp (Wednesday)
Reading Example 1.56 on page 68.
Optional extra practice Chapter 1 Exercise # 7, 17
Nonregular sets (Friday)
Reading Page 77.
Optional extra practice Chapter 1 Exercise # 30
Week 4
Pumping Lemma (Monday)
Reading Example 1.75 (page 81) and Example 1.77 (page 82)
Optional extra practice Chapter 1 Exercise # 29, 30
PDA (Wednesday)
Reading Informal description of PDA on pages 111-112.
PDA Design (Friday)
Reading Example 2.16 (page 115) and Example 2.18 (page 116)
Optional extra practice Chapter 2 Exercise # 5
Week 5
Context-free grammars> (Monday)
Reading Intro to section 2.1 (pp. 102-103)
Optional extra practice Chapter 2 Exercise # 2,4
Context-free languages (Friday)
Reading Designing CFGs (pp. 106-107), Ambiguity Definition 2.7 (p. 108)
Optional extra practice Chapter 2 Exercise # 15
Week 6
Turing Machines (Monday)
Reading Pages 166-167 on differences between finite automata and Turing machines.
Turing machines (Wednesday)
Reading Example 3.7 on page 171, Example 3.9 page 173.<
Optional extra practice Chapter 3 Exercise # 2
Turing machines (Friday)
Reading Bottom of page 166 and top of page 167 (high-level and implementation level definitions of M1),
then terminology for describing Turing machines pages 184-185
Optional extra practice Chapter 3 Exercise # 7,8
Week 7
Variants of Turing machines (Wednesday)
Reading Section 3.2, especially "Equivalence with other models" on page 181.
Optional extra practice Chapter 3 Exercise # 18
Computational problems (Friday)
Reading Format and notation for describing Turing machines, middle of page 185
Optional extra practice Chapter 3 Exercise # 15, 16; Chapter 4 Exercise # 1
Week 8
Decidable problems (Monday)
Reading Section 4.1, Theorem 4.5 (page 197) and Theorem 4.8 (page 199)
Optional extra practice Chapter 4 Exercise # 3,5
Undecidability (Wednesday)
Reading Section 4.2 page 207-209
Optional extra practice Chapter 4 Exercise # 7,8
Week 9
Reductions (Monday)
Reading Section 4.2 Theorem 4.22, Section 5.1 Theorem 5.1
Reductions (Wednesday)
Reading Section 5.1 Theorem 5.2
Reductions (Friday)
Reading Section 5.1 Theorem 5.3, 5.4
Optional extra practice Chapter 5 Exercise # 18 (without Rice's theorem)
Week 10
Complexity: P and NP (Monday)
Reading Section 7.1 example at the top of page 279, Theorem 7.8, Theorem 7.11; Section 7.2 Definition 7.12, Theorem 7.14;
Section 7.3 Theorem 7.24
Optional extra practice Chapter 7 Exercise # 6, 7, 8, 10
NP-completeness (Wednesday)
Reading Section 7.1 example at the top of page 279, Theorem 7.8, Theorem 7.11; Section 7.2 Definition 7.12, Theorem 7.14;
Section 7.3 Theorem 7.24
Optional extra practice Chapter 7 Exercise # 5, read through sample solution of 28