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Themes of Presentations

Themes of Presentations. Rule-based systems/expert systems (Catie) Software Engineering (Khansiri) Fuzzy Logic (Mark) Configuration Systems (Sudhan) * Tutoring and Help Systems (Nicolas) Design (Liam) * Help-Desk Systems (Denis) * Experience/case Maintenance (Fabiana)

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Themes of Presentations

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  1. Themes of Presentations • Rule-based systems/expert systems (Catie) • Software Engineering (Khansiri) • Fuzzy Logic (Mark) • Configuration Systems (Sudhan) • * Tutoring and Help Systems (Nicolas) • Design (Liam) • * Help-Desk Systems (Denis) • * Experience/case Maintenance (Fabiana) • Markov Decision Processes (Megan) • Reinforcement learning (Megan) • e-commerce (Joe) • * Recommender systems (Chad) • * Conversational case-based reasoning (Shruti) • * Semantic web and CBR (Steve)

  2. Computational Complexity CSE 335/435

  3. Why Studying Computational Complexity in IDSS? • We will observe that some techniques seem ideal to provide decision support • We will formulate those techniques as computational problems • Many of these problems will turn out to be intractable (NP-complete or worse) • Thus, we will study relaxations that approximate solutions. These relaxations are in P.

  4. A Quick Overview of Computational Complexity • What does the notation O(f) indicates • When do we say that a program has polynomial complexity • What does it mean that a problem is P?, in NP? • What does it mean that a problem is NP-complete?

  5. Definition O(g) = { f : lim n  f(n)/g(n) is a real number} For example: what functions are in O(x3)? Functions not in O(x3)? • x3 • x3 +2X + 3 • x2 log x • 7 • 6x3 - 1000 • … • x4 • x10 +2X + 3 • x3 log x • 7x • …

  6. Complexity: O-notation Search (e: element, A[]: array) i  1 While (A [i]  e and i < N+1) i  i +1 Return i k(N+1), where k = time for making the comparison A [i]  e Worst case: This algorithm’s complexity is lineal (i.e., O(N))

  7. O(N log N ) • MST • Shortest path • Binary search ordered array • Search in complete Binary Search Trees • Heapsort • Quicksort P O(N2) • all the other sorts: • All the other sorts Comparison of Problems / Solutions by Their Complexity • Search in unordered array O(N ) O(log N) O(1) • Simple instruction

  8. Input data “new state” “current state” “transition” Deterministic Computation (Informal) • Key questions: if a computer is confronted with a certain state of the computation where a choice must be made, • 1. are all the alternatives transitions known?, and • 2. given some input data, is it known which transition the machine will make? If the answer to both of these questions is “yes”, the computation is said to be deterministic

  9. Nondeterministic Computation If the answer to any of these questions is “no”, the computation is said to be nondeterministic • That is, either • some transitions are unknown, or • given some input data, the machine can make more than one transition

  10. P versus NP P is the class of problems that can be solved in O(Nk), where k is some constant by a deterministic computer DeterministicSearch (e: element, A[]: array) i  1 While (A [i]  e and i < N+1) i  i +1 Return i O(N) NP is the class of problems that can be solved in O(Nk), where k is some constant by a nondeterministic computer Non-determinisitcSearch(e: element, A[]: Array) i  Oracle(e, A) return i O(1)

  11. Most books Homework: why 1 implies 2 and why 3 implies 2? NP Complexity (I) How to proof that a problem prob is in NP: • 1. Show that prob is in P, or • 2. Write a program solving prob using the oracle that runs in polynomial time, or • 3. Write 2 polynomial programs that: (1) generate a possible solution S and (2) tests if S is a solution to prob

  12. NP Complexity (II) The class NP consists of all problems that can be solved in polynomial time by nondeterministic computers NP Include all problems in P The key question is are there problems in NP that are not in P or is P = NP? We don’t know the answer to the previous question But there are a particular kind of problems, the NP-complete problems, for which all known deterministic algorithms have an exponential complexity

  13. SAT • Circuit-SAT • Vertex Cover • TSP NP Some Problems Seem Too Hard (NP-Complete) P

  14. Reduction: Polynomial transformation nprob prob solution NP-Complete • A problem prob is NP-complete if: • prob is in NP • Every other problem nprob in NP can be reduced in polynomial time into prob.

  15. Conjunctive Normal Form A conjunctive normal form (CNF) is a Boolean expression consisting of one or more disjunctive formulas connected by an AND symbol (). A disjunctive formula is a collection of one or more (positive and negative) literals connected by an OR symbol (). Example: (a)  (¬ a  ¬b  c  d)  (¬c  ¬d)  (¬d) Problem (CNF-problem): Given a CNF form obtain Boolean assignments that make form true Example (above): a  true, b  false, c  true, d  false

  16. Homework: Proof that CNF-SAT is in NP (use definition 3 of Slide 11) Decision problem • `Decision problem: problem with YES/NO answer • Decision problems can be easier than the standard variant • But for proving NP-completeness they facilitate the proofs Problem (CNF-problem): Given a CNF form obtain Boolean assignments that make form true Problem (CNF-SAT): Given a CNF form, is there an assignment of the variables that makes the formula true? Cook Theorem (1971):The CNF-SAT is NP-complete

  17. Illustration of NP-Completeness of CNF-SAT We will show that the problem of determining if an element e is contained in an array A can be reduced to CNF-sat Solution: The following CNF formula is true if and only if e is in A: (A[1] = e  A[2] = e  …  A[n] = e) Traversing A to obtain this formula can be done in O(N)

  18. S1 S1 S2 S2 (Vague) Idea of The Proof (I) State1: S1 State2: S2 Computer Memory Program … <instruction> …. A computation: S1, S2, S3, …, Sm

  19. Sj Sj (Vague) Idea of The Proof (II) Statej Computer Memory Program … <instruction> …. Sj can be represented as a logic formula Fj The computation can be represented S1, S2, S3, …, Sm as (F1  F2  …  Fm), which is transformed into a CNF

  20. Polynomial transformation Polynomial transformation nprob CNF- sat prob solution solution How to proof that A Problem is NP-Complete • We want to proof that prob is NP complete. • This is done in two steps: • Show that prob is in NP • 2. Show that a known NP-complete (e.g., CNF-sat) problem can be reduced (polynomial) into prob

  21. x x  y y x x  y y x ¬x Circuit-sat (I) A Boolean combinatorial circuit consists of one or more Boolean components connected by wires such that there is one connected component (i.e., there are no separate parts) and the circuit has only one output. Boolean components:

  22. x y z Circuit-sat (II) Circuit-problem: Given a Boolean combinatorial circuit, find a Boolean assignment of the circuit’s input such that the output is true Circuit-SAT: Given a Boolean combinatorial circuit, is there a Boolean assignment of the circuit’s input such that the output is true

  23. Homework • 1. Obtain an algorithm (pseudo-code) solving the Circuit-SAT • 2. Explain why your solution is not polynomial • 3. Prove that Circuit-Sat is NP complete: • Show that Circuit-SAT is in NP • Prove that CNF-SAT can be reduced into Circuit-SAT: (a)  (¬a  ¬b  c  d)  (¬c  ¬d)  (¬d) • Show a circuit representing the above formula • Describe an algorithm for this transformation • Explain why this algorithm is in P

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