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)

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(x 3 )?  x 3  x 3 + 2X + 3  x 2 log x  7  6x  … Functions not in O(x 3 )?  x 4  x X + 3  x 3 log x  7 x  …

Complexity: O-notation Search (e: element, A[]: array) i  1 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))

P all the other sorts: Comparison of Problems / Solutions by Their Complexity Simple instruction O(1) Binary search ordered array Search in complete Binary Search Trees O(log N) Search in unordered arrayO(N ) HeapsortQuicksort Shortest pathMST O(N log N ) O(N 2 ) All the other sorts

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 “current state” Input data “transition” “new state”

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

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 Most books Homework: why 1 implies 2 and why 3 implies 2?

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

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

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: prob nprob Polynomial transformation solution

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 Cook Theorem (1971):The CNF-SAT is NP-complete `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-SAT): Given a CNF form, is there an assignment of the variables that makes the formula true? Problem (CNF-problem): Given a CNF form obtain Boolean assignments that make form true Homework: Proof that CNF-SAT is in NP (use definition 3 of Slide 11)

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) Computer Memory Program … …. Program … …. State1: S1State2: S2 S1 S2 A computation: S1, S2, S3, …, Sm

(Vague) Idea of The Proof (II) Computer Memory Program … …. Program … …. State j SjSj SjSj S j can be represented as a logic formula F j 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 NP- Complete We want to proof that prob is NP complete. This is done in two steps: 1.Show that prob is in NP 2. Show that a known NP-complete (e.g., CNF-sat) problem can be reduced (polynomial) into prob nprob Polynomial transformation solution CNF- sat prob Polynomial transformation solution

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 y x  y x ¬x¬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

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: 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