Complexity Theory Spring 2022


Charles E. Hughes
Computer Science
College of Engineering and Computer Science
University of Central Florida


Structure: TR 0900-1015 (9:00AM-10:15AM); 28 class periods, each 75 minutes long.
Room: HEC-103 when we meet fasce-to-face;

Class Zoom Link for when we meet virtually:

Go To Week
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14
, 15, 16

Instructor: Charles Hughes; Contact:; Use Subject COT6410
Office Hours: Tuesday/Thursday 10:45-12:00PM

OH Zoom Link:

GTA:  Parisa Darbar; Contact:; Use Subject COT6410

Office Hours: MW 2:00 PM-3:15 PM

OH Zoom Link:

Required Reading: All class notes linked from this site.
Recommended Reading:

Web Pages:
Base URL:
Notes URL: Introductory Notes; Formal Language and Automata Theory; Computability Theory; Complexity Theory
2021 Videos URL: OlderVideos/

Assignments: 6 for sure and maybe a seventh; Paper + Video Presentation with Slides

Exams: Midterm and Final.

Exam Dates (Tentative): Mid Term: Tuesday, March 15 in HEC-101; Withdraw Deadline: Friday, March 23; Spring Break: March 6-13; Final: Thurs., April 28, 7:00AM-9:50AM in HEC-101.

Evaluation (Tentative):

  1. Mid Term: 125 points; Final Exam: 125 points (balance between weights will be adjusted in your favor)
  2. Assignments: 75 points;
  3. Individual Paper Reviews and Presentations: 125 points
  4. Extra -- 50 points used to increase weight of best exam,  always to your benefit
  5. Total Available: 500
  6. Grading will be  A >= 90%, B+ >= 85%, B >= 80%, C+ >= 75%, C >= 70%, D >= 50%, F < 50%; minus grades might be used.


Weeks#1: (1/11, 1/13) -- Syllabus; About Me; Preliminaries; Introduction; Formal Languages and Automata Theory,
Videos from Last Year (2021) are here
The old videos for this week are: Preliminary.mp4, Intro.mp4, Day1_Trimmed.mp4, Day2_Trimmed.mp4
New Videos, as they are created will be in the Zoom Cloud Recording folder.
  1. Ground rules
  2. Decision problems (HMU 1.5)
  3. Solving vs checking
  4. Procedures vs algorithms
  5. Introduction to the theory of computation
  6. Terminology, goals
  7. Overview of automata (finite, pushdown, linear bounded, Turing machines)
  8. Overview of formal languages (regular, context free, context sensitive, phrase structured)
  9. Regular Languages (Sipser 1.1; HMU 2.1, 2.2)
  10. Closure Properties of Regular Languages (Sipser 1.2; HMU 4.2)
  11. Other Formal Models for Regular Languages
  12. Review material created by Prof. Jim Rogers, Earlam College
  13. Review material from Prof. David Workman, UCF Retired

Financial Aid (Assignment#1)

Survey at Webcourses
Due: Friday, January 14 at 11:59 PM

Week#2: (1/18, 1/20) -- Formal Languages and Automata Theory
  1. Continue Automata/Formal Languages Review
  2. Every Regular Language is a Regular Set (Sipser 1.3; HMU 3.1-3.3)
  3. State minimization using O(| Q | 2) table. Note alphabet is constant size. (Sipser p. 327; HMU 4.4)
  4. More closure properties (Sipser 1.1; HMU 4.2-4.3)
  5. Closure under min and max and discussions of various Reaching Algorithms (Depth-First Search)
  6. Pumping Lemma for Regular Languages (Pigeon Hole Principle) (Sipser 1.4; HMU 4.1)
  7. Myhill-Nerode Theorem (implicit in HMU 4.4)
  8. Myhill-Nerode as proof of min DFA uniqueness

Week#3: (1/25, 1/27) -- Formal Languages and Automata Theory
  1. Myhill-Nerode as proof of min DFA uniqueness
  2. Myhill-Nerode as a tool to show languages are not regular.
  3. The chosen language is
    L = { an bm | n is not equal to m}.
    This can be shown easily in an indirect manner by showing its complement is not regular. A direct approach is using right invariant equivalence classes as it is not amenable to the Pumping Lemma.
  4. Finite State Machines (Transducers): Mealy versus Moore Model (Sipser p. 87)
  5. Decision Problems for Regular Languages (HMU 3.4; HMU 4.3
  6. Introduction to Grammars (Regular and CFG mainly) (Sipser 2.1;
  7. Regular Grammars and Regular Languages (HMU p. 180)
  8. Context Free Grammars (A x where x (V ΣΣ)* ) (Sipser 2.1; HMU 5.1)
  9. Use of context free grammars in parsing and notion of Ambiguity (Sipser 2.1; HMU 5.2, 5.4)
  10. Review of reduced CFGs (HMU 7.1)
  11. Chomsky Normal Form (CNF) (Sipser 2.1; HMU 7.1)
  12. The use of CNF in the Cocke-Kasami-Younger O(N3) parsing of CFLs generated by CNFs (HMU 7.4)
  13. Pumping Lemma for CFLs (Sipser 2.3; HMU 7.2)
  14. Show L = { an bn cn | n i> 0 } and {L2 = { ww | w is in {a,b}+ } are not CFLs
  15. CSG for L
  16. Non-closure of CFLs under intersection and complement (Sipser 2.3)
  17. CFG for the complement of L
    { xy | |x| = |y| but x is not the same as y }
    can be first viewed as
    {x1 a x2 y1 b y2 | |x1|=|x2|, |y1|=|y2|} Union
      {x1 b x2 y1 a y2 | |x1|=|x2|, |y1|=|y2|}.
    But this can also be seen as
    {x1 a y1 x2 b y2 | |x1|=|x2|, |y1|=|y2|} Union
      {x1 b y1 x2 a y2 | |x1|=|x2|, |y1|=|y2|}.
    The above is easy to show as a CFL. We then union this with odd length strings and we have L1 complement.
  18. Decision problems for CFLs: is L(G) empty or finite/infinite are fine (Sipser 4.1; HMU 7.4)
Assignment #2

See Webcourses (Assignment # 2) for description
Sample of Similar Problems with Solutions

Due: 2/8 (Key)

Week#4: (2/1, 2/3) -- Computability Theory
  1. Show L = { an bn cn | n i> 0 } and {L2 = { ww | w is in {a,b}+ } are not CFLs
  2. CSG for L
  3. Non-closure of CFLs under intersection and complement (Sipser 2.3)
  4. CFG for the complement of L
    { xy | |x| = |y| but x is not the same as y }
    can be first viewed as
    {x1 a x2 y1 b y2 | |x1|=|x2|, |y1|=|y2|} Union
      {x1 b x2 y1 a y2 | |x1|=|x2|, |y1|=|y2|}.
    But this can also be seen as
    {x1 a y1 x2 b y2 | |x1|=|x2|, |y1|=|y2|} Union
      {x1 b y1 x2 a y2 | |x1|=|x2|, |y1|=|y2|}.
    The above is easy to show as a CFL. We then union this with odd length strings and we have L1 complement.
  5. Decision problems for CFLs: is L(G) empty or finite/infinite are fine (Sipser 4.1; HMU 7.4)
  6. Closure of CFLs under substitution and intersection with Regular (HMU 7.3)
  7. Checking ambiguity, equality to Sigma*,  equivalence and non-empty intersection with another CFL are nasty
  8. Very Basic Material on Context Sensitive Grammars (CSG) and Linear Bounded Automata (LBA)
  9. Show L1 = { an bn cn | n i> 0 } and L2 = { ww | w is in {a,b}+ } are CSLs
  10. Insights from intro to computability material
  11. Basic notions of computability and complexity
  12. Existence of unsolvable problems (counting and diagonalization)
  13. Solved, solvable (decidable, recursive), unsolved, unsolvable, re, non-re
  14. Hilbert's Tenth
  15. Undecidable problems made a bit more concrete
  16. Lots about problems and their complexity
  17. Halting Problem (HALT) is re, not decidable
  18. Set of algorithms (TOT) is non-re
  19. Halting Problem seen as fun
  20. Some consequences of non-re nature of algorithms

Week#5: (2/8, 2/10) -- Computability Theory
  1. Models of computation
  2. Systems related to FRS (Petri Nets, Vector Addition, Abelian Semi-Groups)
  3. RM simulated by Ordered FRS
  4. Primitive Recursive Functions (prf)
  5. Initial functions
  6. Closure under composition and recursion
  7. Primitive recursive functions SNAP, TERM, STP and VALUE
  8. Primitive Recursive Function in detail
  9. Addition and multiplication examples
  10. Sample functions and predicates
  11. Closure under cases
  12. Bounded minimization
  13. Arithmetic fuctions that use bounded search
  14. Pairing functions
  15. Limitations of primitive recursive
  16. mu-recursion and the partial recursive functions
  17. Notions of instantaneous descriptions
  18. Encodings
  19. Equivalence of models
  20. TMs to Register Machines
  21. RM to Factor Replacement Systems
  22. Factor Replacement to Recursive Functions
  23. Gory details on FRS to REC
  24. Universal machines
  25. Recursive Functions to TMs
  26. Consequences of equivalence
    Assignment #3

    See Webcourses (Assignment # 3) for description
    Sample of Similar Problems with Solutions

    Due: 2/22 (Key)

Week#6: (2/15, 2/17) -- Computability Theory
  1. Recursive Functions to TMs (completes equivalence)
  2. Consequences of equivalence
  3. The primitive recursive predicate STP and function VALUE
  4. Review Undecidability (Halting Problem, shown by diagonalization)
  5. RE sets and semidecidability
  6. Enumeration Theorem: The set of all re sets W0, W1, W2, ...
  7. The set K = { n | n is in the n-th re set } = { n | n is in Wn } is re, non-recursive
  8. The set K0 = HALT = { <n,x> | x is in the n-th re set }
  9. Alternative characterizations of re sets
  10. Parameter Theorem (aka Sm,n Theorem)
  11. Quantification and re sets
  12. Quantification and co-re sets
  13. Quantification and non-re / non-co-re sets
  14. Reduction
  15. Classic sets Halt = Ko = Lu, NON-EMPTY (Lne), EMPTY (Le)
  16. Reduction from Ko that shows undecidability of K, HasZero, IsNonEmpty
  17. Complete re set (K and K0 as examples)
  18. Note: To be re-complete a set must be re
  19. Reduction from Ko to HasIndentity, TOTAL, IsZero, IsEmpty, IsIdentity
  20. Equivalence of certain re sets (K, HasZero, IsNonEmpty, HasIndentity) to Ko
  21. Equivalence of certain non-re sets (IsZero, IsEmpty, IsIdentity) to TOTAL
  22. Reducibility and degrees (many-one, one-one, Turing)
  23. Hierarchy or RE equivalence class (m-1 and 1-1 degrees)
  24. Rice's Theorem

        Assignment #4

    See Webcourses (Assignment # 4) for description
    Sample with key

        Due: 2/28 (Key)

Week#7: (2/22, 2/24) -- Computability Theory
  1. "Picture" proofs for Rice's Theorem
  2. Constant Time and Mortal Machines
  3. Introduction to rewriting systems (Thue, Post)
  4. Union, intersection, complement for recursive, re and non-re sets (can be, cannot be)
  5. Rewriting systems
  6. Post Canonical Forms
  7. Thue and Semi-Thue systems (relation to group theory)
  8. Word problems (Semi-Thue and Thue) and equivalence problems (Thue)
  9. Simulating Turing Machine by Semi-Thue System
  10. Simulating Turing Machines by Thue Systems
  11. Grammars and re sets
  12. Brief introduction to Post Correspondence Systems and Relation to Semi-Thue Systems
  13. Post Correspondence Problem (in detail)
  14. Unsolvable problems related to context-free grammars/languages
  15. Ambiguity of CFGs
  16. Non-Emptiness of CFL Intersections
  17. Context-Sensitive Grammars and Unsolvability Results
  18. Valid (CSL) and Invalid Traces (CFL)
  19. Details on Valid (CSL) and Invalid Traces (CFL)
  20. Intersection of CFLs revisited
  21. Quotients of CFLs
  22. Type 0 grammars and Traces
  23. L =  Sigma*  for L a Regular or CFL
  24. L = L^2 for L a CFL
  25. Summary of Grammar Results


Week#8: (3/1, 3/3) -- Review and Complexity Theory
  1. Review session and sample exams
  2. Exam Topics
  3. Sample exam1; Sample exam1 key
  4. Sample exam2; Sample exam2 key
  5. Key to Samples from Notes
  6. Yet Other Examples (Focused on Formal Languages and Automata Theory)
  7. Yet Other Examples key (Focused on Formal Languages and Automata Theory)
  8. Midterm Legal Cheat Sheet
  9. Basics of Complexity Theory
  10. Decision vs Optimization Problems (achieving a goal vs achieving min cost)
  11. Polynomial == Easy; Exponential == Hard
  12. Polynomial reducibility
  13. Verifiers versus solvers
  14. P as solvable in deterministic polynomial time
  15. NP as solvable in non-deterministic polynomial time
  16. NP as verifiable in deterministic polynomial time
  17. Concepts of NP-Complete and NP-Hard
  18. Canonical NP-Complete problem: SAT (Satisfiability)
  19. Some NP problems that do not appear to be in P: Graph Coloring, Vertex Cover, SubsetSum
  20. Million dollar question: P = NP ?
  21. Boolean expressions: Tautologies, Satisfiability, Truth Tables
  22. Axiomatic Systems for propositional logic: Substitution and modus ponens versus refutation/resolution
  23. Unsolvability of deducibility in fragments of propositional calculus

Week#9 Spring Break

Week#10 (3/15, 3/17: Midterm Exam and Complexity Theory

  1. Midterm (Tuesday, March 15 in HEC-101)
  2. Midterm Key
  3. More Comments on unsolvability of deducibility in fragments of propositional calculus
  4. Construction that maps every problem solvable in non-deterministic polynomial time on TM to SAT
  5. SAT is polynomial reducible to (<=P) 3SAT
  6. 3SAT as a second NP-Complete problem
  7. Integer Linear Programming
  8. 3SAT <=P SubsetSum
  9. SubsetSum <=P Partition
  10. Partition equivalence to SubsetSum

Week#11: (3/22, 3/24, (3/25 is Withdrawal Deadline)) -- Complexity Theory
  1. Discussion of presentations
  2. Reduction of 3SAT to k-Vertex Cover
  3. 3-SAT to 3-Coloring
  4. Isomophism of k-Coloring with k-Register Allocation of live variables
  5. Scheduling problems introduced 
  6. Scheduling on multiprocessor systems
  7. Scheduling problems (fixed number of processors, minimize final finishing time)
  8. N processors, M tasks, no constraints
  9. Partition and scheduling problems
  10. Greedy heuristics
  11. 2-processor scheduling -- greedy based on list, sorted long to short, sorted short to long, optimal. Tradeoffs.
  12. Precedences (lists, delays, preemption)
  13. Anomalies (reducing precedence, increasing processors, reducing times)
  14. Unit Execution Time: Trees, forest, anti forests
  15. UET: DAGs and m=2
  16. Hamiltonian Path
  17. Traveling Salesman
  18. Knapsack (relation to SubsetSum), Dynamic Programming Pseudo-polynomial Solution
  19. Bin packing (fixed capacity, minimize number of bins)
  20. Pseudo polynomial-time solution for Knapsack using dynamic programming with changed parameters,
    n*W versus 2^n but W = 2^log(W). This kind of problem is called Weak NP Complete. I'll chat about that later.
  21. You will each, individually, develop a paper and a presentation, based on an existing research paper, just as if you had to present to your peers and advisors. In addition to presenting the results and the way in which these were proven, you should comment on the paper's importance, readability, and replicability, and even its validity if you question that, just as if you were a journal or conference reviewer. Your report must be a tutorial on the topic of the paper so those with less time to delve into the paper can get a strong sense of the results and the context of those results. Specifying new open problems that you see as interesting is also a goal, but may not be attainable for some of these. The length of your paper is 6 to 12 pages double-spaced, 1" margins all around, using either Times Roman or Calibri 11 point, or Arial 10-point. The references are in addition to the narrative and appear on separate pages that do not count against the 6 to 12-page limit. Images may be used but if the total number of images exceeds a page, then all past that one-page aggregate will lead to a requirement for additional text. The presentation slides must be designed to support your 8 to 10-minute video. This means that there are likely 10 to 15 slides.
  22. Towards the end of the semester you give a presentation to a small group of other students and they to you. This will be done in a breakout Zoom session where your presentation and any questions/answers are captured. You may, if you wish, create the presentation in advance and share a video, but you need to be there to participate in the Q&A and to hear other students' presentations. To do this, I assume you all have access to a webcam and use Zoom or a similar tool that captures your slides and you in one extended screen. Let me know if you lack a webcam. Note that you will be asked to provide some brief feedback to me on the papers of your fellow students taking part in teh same breakout.
  23. I have placed about 55 sample papers at Sample Topics. You may choose from any of these except for ones that have already been taken. You may also choose your own separate from these with permission from me. In general, the minimum page length in IEEE 2-column format is 8 pages or 10 pages in single column, single-spaced layout. The prefix CLAIMED_NAME means NAME has already chosen that. Please send me an email to assure you succeed in making your "claim."
  24. Realize that many of these suggested papers will be from arXiv and, while arXiv is a fantastic source, it is not refereed so if you question the validity of some result, that is actually a reasonable outcome so long as you present their approach and your counterarguments just as if you were a reviewer.
  25. I also put out a folder that contains some examples (papers and presentations) from past semesters.

              Final Paper and PowerPoint

                   Read above. Its weight is 125 points. The split is nominally 75 for the paper and 50 for your presentation
                   and feedback on other presentations you provide.

              Due: 4/22

    Assignment #5

            See Webcourses (Assignment # 5) for description
            Sample with Key

  26.  Due: 4/5 (Key)

Week#12: (3/29, 3/31) -- Complexity Theory
  1. Tiling the plane and Bounded Tiling
  2. Bounded PCP
  3. Co-NP
  4. Reduction techniques
  5. P = co-P ; P contained in intersection of NP and co-NP
  6. NP-Hard
  7. QSAT as an example of NP-Hard, possibly not NP
  8. NP-Hard problems are in general functional not necessarily decision problems
  9. NP-Complete are decision problems as they are in NP
  10. NP-Easy -- these are problems that are polynomial when using an NP oracle
  11. NP-Equivalent is the class of NP-Easy and NP-Hard problems
  12. Optimization versions of SubsetSum and K-Color
  13. Validity of SubsetSum optimization reduction to SubsetSum Decision Problem Oracle
  14. 2SAT (linear time complexity)
  15. Uniform Min-1 is NP Equivalent
  16. Triangle Strip versus Triangle List
  17. Weakly versus Strongly NP-Hard/NP-Complete

            Assignment #6

                 See Webcourses (Assignment # 6) for description
                 Sample with Key

            Due: 4/19 (Key)

Week#13: (4/5, 4/7) -- Complexity Theory
  3. ATM (Alternating NDTM)
  4. QSAT, Petri Nets, Presburger
  5. Why does PSPACE = NPSPACE? Key is Savitch's Theorem
  6. PSPACE-Complete problems (CSL membership, QSAT)
  7. FP is functional equivalent to P; R(x,y) in FP if can provide value y for input x via deterministic polynomial time algorithm
  8. FNP is functional equivalent to NP; R(x,y) in FNP if can verify any pair (x,y) via deterministic polynomial time algorithm
  9. TFNP is the subset of FNP where a solution always exists, i.e., there is a y for each x such that R(x,y).
  10. Factoring is in TFNP, but is it in FP?
  11. P = (NP intersect Co-NP) is interesting analogue to intersection of RE and co-RE but may not hold here
  12. It appears that TFNP does not have any complete problems!!!
  13. Khot's Conjecture and its implications

Week#14: (4/12, 4/14) More Computability Notes

  1. Cleanups (topics that were left hanging earlier)
  2. Final Exam Topics
  3. More stuff for final
  4. Sample Final Exam (Key)
  5. Exam Review (emphasis on Formal Languages and Computability topics)


Week#15: (4/19, 4/21)

  1. Exam Review (emphasis on complexity topics and sample exam questions)
  2. On 4/21 we will have the breakout sessions in which you give your presentations and hear those of a few of your fellow students (typically five per breakout room). These will be set up prior to the 21st.
  3. Rules of the game and group composition for Breakout Rooms on the 21st.
  4. Breakout Room Videos

            Assignment #7

                 Just turn in your discussion of someone's or several someone's presentation(s)

            Due: 4/23


Week#16:  (4/28 -- Final Exam in HEC-101)

  1. I will run  a help session on Tuesday, 4/26 at the normal class time
  2. All Project Videos, Papers, and Presentations are Due on Friday, 4/22 by midnight
  3. Final Exam is Thursday April 28; 7:00AM to 9:50AM


UCF (Charles E. Hughes)  -- Last Modified  4/23/2022