* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download A Quick Overview of Computational Complexity
Exact cover wikipedia , lookup
Algorithm characterizations wikipedia , lookup
Artificial intelligence wikipedia , lookup
A New Kind of Science wikipedia , lookup
Natural computing wikipedia , lookup
Lateral computing wikipedia , lookup
Genetic algorithm wikipedia , lookup
Mathematical optimization wikipedia , lookup
K-nearest neighbors algorithm wikipedia , lookup
Computational electromagnetics wikipedia , lookup
Simulated annealing wikipedia , lookup
Pattern recognition wikipedia , lookup
Travelling salesman problem wikipedia , lookup
Inverse problem wikipedia , lookup
Knapsack problem wikipedia , lookup
Multiple-criteria decision analysis wikipedia , lookup
Halting problem wikipedia , lookup
Factorization of polynomials over finite fields wikipedia , lookup
Theoretical computer science wikipedia , lookup
Themes of Presentations • • • • • • • • • • • • Rule-based systems/expert systems (Ulit) MS Clippy (Justin) Fuzzy Logic (Andrew) Configuration Systems () Tutoring and Help Systems () Design () Help-Desk Systems (David) (*): Intelligent Sales Support with CBR Experience/case Maintenance (Sean) e-commerce () Recommender systems (Alexandra) Semantic web and CBR () Physically-Grounded CBR (Konstantin) Sources: (*) Case-Based Reasoning Technology: From Foundations to Applications Homework (next class) Read Chapter 2 of the Experience Management book and answer the following questions: • Provide an example of something that is data but not information, something that is information but not knowledge, and something that is knowledge • Give an example of experience. Why can’t experience be general knowledge? • What is the relation between experience management and CBR? What is/are the difference(s)? • Provide an example for each of the 4 phases of the CBR cycle for a domain of your own (can’t be the restaurant example). First you would need to think what is the task that you are trying to solve. Please specify. Is this a classification or a synthesis task? Please specify From Data to Knowledge Abstract Experience Knowledge Information Data Clauses or meta-relations: GrandParent(X,Z) if Parent(X,Y) and Parent(Y,Z) Relations: parent(john, Sebastian) Simple objects: john, Sebastian Concrete Experience Management vs CBR (Organization) Problem acquisition Experience base Reuserelated knowledge Experience presentation Experience adaptation BOOK CBR Experience evaluation and retrieval Development and Management Methodologies Experience Management Complex problem solving Case Library 1. Retrieve 4. Retain Background Knowledge (IDSS) 3. Revise 2. Reuse Computational Complexity CSE 335/435 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. 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? Definition O(g) = { f : lim n f(n)/g(n) is a real number} For example: what functions are in O(x3)? x3 x3 + 2X + 3 x2 log x 7 6x3 - 1000 … Functions not in O(x3)? x4 x10 + 2X + 3 x3 log x 7x … Complexity: O-notation Search (e: element, A[]: array) i1 While (A [i] e and i < N+1) i i +1 Return i Worst case: k(N+1), where k = time for making the comparison A [i] e This algorithm’s complexity is lineal (i.e., O(N)) Comparison of Problems / Solutions by Their Complexity •Quicksort O(N2) •MST •Shortest path O(N log N ) •Search for similar case O(N ) (assuming constant similarity) O(log N) •Binary search ordered array • all the other sorts: O(1) •Simple instruction •Search in complete Binary Search Trees •Heapsort P 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? Input data “current state” “transition” “new state” If the answer to both of these questions is “yes”, the computation is said to be deterministic 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 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 O(N) DeterministicSearch (e: element, A[]: array) i1 While (A [i] e and i < N+1) i i +1 Return i NP is the class of problems that can be solved in O(Nk), where k is some constant by a nondeterministic computer O(1) Non-determinisitcSearch(e: element, A[]: Array) i Oracle(e, A) return i NP Complexity (I) Homework: why 1 implies 2 and why 3 implies 2? 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 Standard Definition (and the one we use) 3. Write programs that: (1) generate a possible solution S and (2) tests if S is a solution to prob. Both programs need to be in P. Solution: Why (1) implies (2) • Let prob be a problem in P • There is a deterministic algorithm alg that solves prob in polynomial time O(nk), for some constant k • That same algorithm alg runs in a nondeterminsitc machine (it just do not use the oracle) • Thus, alg has the same polynomial complexity, O(nk) • Thus, prob is in NP Solution: Why (3) implies (2) • If a problem prob satisfies Guess and Check solution prob …. polynomial • The nondeterministic version solution prob …. 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 Some Problems Seem Too Hard (NP-Complete) •SAT •Circuit-SAT P •Vertex Cover •TSP NP 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. Reduction: nprob Polynomial transformation solution prob 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 Decision problem Homework: Proof that CNF-SAT is in NP (use definition 3 of Slide 11) `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 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) (Vague) Idea of The Proof (I) State1: S1 State2: S2 Computer Memory S2 S1 S1 S2 Program … <instruction> …. A computation: S1, S2, S3, …, Sm (Vague) Idea of The Proof (II) Statej Computer Memory Sj Sj 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 How to proof that A Problem is NPComplete We want to proof that nprob is NP complete. This is done in two steps: 1. Show that nprob is in NP 2. Show that a known NP-complete (e.g., CNF-sat) problem can be reduced (polynomial) into nprob nprob Polynomial transformation solution CNFsat Polynomial transformation solution prob 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: x y x y x xy xy ¬x 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 x y z Circuit-SAT: Given a Boolean combinatorial circuit, is there a Boolean assignment of the circuit’s input such that the output is true Readings • http://users.forthnet.gr/ath/kimon/CC/CCC1b.htm • This is part A, from there follow to Parts B, C and D • Introduction to Algorithms, Cormen, Chapter 34 “NPCompleteness” Homework 1. (CSE 335) Obtain an algorithm (pseudo-code) solving the Circuit-SAT 2. (CSE 335) Explain why your solution is not polynomial 3. Prove that Circuit-Sat is NP complete: a) Show that Circuit-SAT is in NP b) 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