Giving a Step-By-Step Reduction from SAT to TSP and Giving Some Remarks on Neil Tennant's Changes of Mind
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NP-Completeness: Reductions Tue, Nov 21, 2017
CMSC 451 Dave Mount CMSC 451: Lecture 19 NP-Completeness: Reductions Tue, Nov 21, 2017 Reading: Chapt. 8 in KT and Chapt. 8 in DPV. Some of the reductions discussed here are not in either text. Recap: We have introduced a number of concepts on the way to defining NP-completeness: Decision Problems/Language recognition: are problems for which the answer is either yes or no. These can also be thought of as language recognition problems, assuming that the input has been encoded as a string. For example: HC = fG j G has a Hamiltonian cycleg MST = f(G; c) j G has a MST of cost at most cg: P: is the class of all decision problems which can be solved in polynomial time. While MST 2 P, we do not know whether HC 2 P (but we suspect not). Certificate: is a piece of evidence that allows us to verify in polynomial time that a string is in a given language. For example, the language HC above, a certificate could be a sequence of vertices along the cycle. (If the string is not in the language, the certificate can be anything.) NP: is defined to be the class of all languages that can be verified in polynomial time. (Formally, it stands for Nondeterministic Polynomial time.) Clearly, P ⊆ NP. It is widely believed that P 6= NP. To define NP-completeness, we need to introduce the concept of a reduction. Reductions: The class of NP-complete problems consists of a set of decision problems (languages) (a subset of the class NP) that no one knows how to solve efficiently, but if there were a polynomial time solution for even a single NP-complete problem, then every problem in NP would be solvable in polynomial time. -
Mapping Reducibility and Rice's Theorem
6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, 2010 Class 9 Nancy Lynch Today • Mapping reducibility and Rice’s Theorem • We’ve seen several undecidability proofs. • Today we’ll extract some of the key ideas of those proofs and present them as general, abstract definitions and theorems. • Two main ideas: – A formal definition of reducibility from one language to another. Captures many of the reduction arguments we have seen. – Rice’s Theorem, a general theorem about undecidability of properties of Turing machine behavior (or program behavior). Today • Mapping reducibility and Rice’s Theorem • Topics: – Computable functions. – Mapping reducibility, ≤m – Applications of ≤m to show undecidability and non- recognizability of languages. – Rice’s Theorem – Applications of Rice’s Theorem • Reading: – Sipser Section 5.3, Problems 5.28-5.30. Computable Functions Computable Functions • These are needed to define mapping reducibility, ≤m. • Definition: A function f: Σ1* →Σ2* is computable if there is a Turing machine (or program) such that, for every w in Σ1*, M on input w halts with just f(w) on its tape. • To be definite, use basic TM model, except replace qacc and qrej states with one qhalt state. • So far in this course, we’ve focused on accept/reject decisions, which let TMs decide language membership. • That’s the same as computing functions from Σ* to { accept, reject }. • Now generalize to compute functions that produce strings. Total vs. partial computability • We require f to be total = defined for every string. • Could also define partial computable (= partial recursive) functions, which are defined on some subset of Σ1*. -
The Satisfiability Problem
The Satisfiability Problem Cook’s Theorem: An NP-Complete Problem Restricted SAT: CSAT, 3SAT 1 Boolean Expressions ◆Boolean, or propositional-logic expressions are built from variables and constants using the operators AND, OR, and NOT. ◗ Constants are true and false, represented by 1 and 0, respectively. ◗ We’ll use concatenation (juxtaposition) for AND, + for OR, - for NOT, unlike the text. 2 Example: Boolean expression ◆(x+y)(-x + -y) is true only when variables x and y have opposite truth values. ◆Note: parentheses can be used at will, and are needed to modify the precedence order NOT (highest), AND, OR. 3 The Satisfiability Problem (SAT ) ◆Study of boolean functions generally is concerned with the set of truth assignments (assignments of 0 or 1 to each of the variables) that make the function true. ◆NP-completeness needs only a simpler Question (SAT): does there exist a truth assignment making the function true? 4 Example: SAT ◆ (x+y)(-x + -y) is satisfiable. ◆ There are, in fact, two satisfying truth assignments: 1. x=0; y=1. 2. x=1; y=0. ◆ x(-x) is not satisfiable. 5 SAT as a Language/Problem ◆An instance of SAT is a boolean function. ◆Must be coded in a finite alphabet. ◆Use special symbols (, ), +, - as themselves. ◆Represent the i-th variable by symbol x followed by integer i in binary. 6 SAT is in NP ◆There is a multitape NTM that can decide if a Boolean formula of length n is satisfiable. ◆The NTM takes O(n2) time along any path. ◆Use nondeterminism to guess a truth assignment on a second tape. -
Homework 4 Solutions Uploaded 4:00Pm on Dec 6, 2017 Due: Monday Dec 4, 2017
CS3510 Design & Analysis of Algorithms Section A Homework 4 Solutions Uploaded 4:00pm on Dec 6, 2017 Due: Monday Dec 4, 2017 This homework has a total of 3 problems on 4 pages. Solutions should be submitted to GradeScope before 3:00pm on Monday Dec 4. The problem set is marked out of 20, you can earn up to 21 = 1 + 8 + 7 + 5 points. If you choose not to submit a typed write-up, please write neat and legibly. Collaboration is allowed/encouraged on problems, however each student must independently com- plete their own write-up, and list all collaborators. No credit will be given to solutions obtained verbatim from the Internet or other sources. Modifications since version 0 1. 1c: (changed in version 2) rewored to showing NP-hardness. 2. 2b: (changed in version 1) added a hint. 3. 3b: (changed in version 1) added comment about the two loops' u variables being different due to scoping. 0. [1 point, only if all parts are completed] (a) Submit your homework to Gradescope. (b) Student id is the same as on T-square: it's the one with alphabet + digits, NOT the 9 digit number from student card. (c) Pages for each question are separated correctly. (d) Words on the scan are clearly readable. 1. (8 points) NP-Completeness Recall that the SAT problem, or the Boolean Satisfiability problem, is defined as follows: • Input: A CNF formula F having m clauses in n variables x1; x2; : : : ; xn. There is no restriction on the number of variables in each clause. -
11 Reductions We Mentioned Above the Idea of Solving Problems By
11 Reductions We mentioned above the idea of solving problems by translating them into known soluble problems, or showing that a new problem is insoluble by translating a known insoluble into it. Reductions are the formal tool to implement this idea. It turns out that there are several subtly different variations – we will mainly use one form of reduction, which lets us make a fine analysis of uncomputability. First, we introduce a technical term for a Register Machine whose purpose is to translate one problem into another. Definition 12 A Turing transducer is an RM that takes an instance d of a problem (D, Q) in R0 and halts with an instance d0 = f(d) of (D0,Q0) in R0. / Thus f must be a computable function D D0 that translates the problem, and the → transducer is the machine that computes f. Now we define: Definition 13 A mapping reduction or many–one reduction from Q to Q0 is a Turing transducer f as above such that d Q iff f(d) Q0. Iff there is an m-reduction from Q ∈ ∈ to Q0, we write Q Q0 (mnemonic: Q is less difficult than Q0). / ≤m The standard term for these is ‘many–one reductions’, because the function f can be many–one – it doesn’t have to be injective. Our textbook Sipser doesn’t like that term, and calls them ‘mapping reductions’. To hedge our bets, and for brevity, we’ll call them m-reductions. Note that the reduction is just a transducer (i.e. translates the problem) that translates correctly (so that the answer to the translated problem is the same as the answer to the original). -
Thy.1 Reducibility Cmp:Thy:Red: We Now Know That There Is at Least One Set, K0, That Is Computably Enumerable Explanation Sec but Not Computable
thy.1 Reducibility cmp:thy:red: We now know that there is at least one set, K0, that is computably enumerable explanation sec but not computable. It should be clear that there are others. The method of reducibility provides a powerful method of showing that other sets have these properties, without constantly having to return to first principles. Generally speaking, a \reduction" of a set A to a set B is a method of transforming answers to whether or not elements are in B into answers as to whether or not elements are in A. We will focus on a notion called \many- one reducibility," but there are many other notions of reducibility available, with varying properties. Notions of reducibility are also central to the study of computational complexity, where efficiency issues have to be considered as well. For example, a set is said to be \NP-complete" if it is in NP and every NP problem can be reduced to it, using a notion of reduction that is similar to the one described below, only with the added requirement that the reduction can be computed in polynomial time. We have already used this notion implicitly. Define the set K by K = fx : 'x(x) #g; i.e., K = fx : x 2 Wxg. Our proof that the halting problem in unsolvable, ??, shows most directly that K is not computable. Recall that K0 is the set K0 = fhe; xi : 'e(x) #g: i.e. K0 = fhx; ei : x 2 Weg. It is easy to extend any proof of the uncomputabil- ity of K to the uncomputability of K0: if K0 were computable, we could decide whether or not an element x is in K simply by asking whether or not the pair hx; xi is in K0. -
Redalyc.Evaluation of Consistency of the Knowledge in Agents
Acta Universitaria ISSN: 0188-6266 [email protected] Universidad de Guanajuato México Contreras, Meliza; Rodriguez, Miguel; Bello, Pedro; Aguirre, Rodolfo Evaluation of Consistency of the Knowledge in Agents Acta Universitaria, vol. 22, marzo, 2012, pp. 42-47 Universidad de Guanajuato Guanajuato, México Available in: http://www.redalyc.org/articulo.oa?id=41623190006 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Universidad de Guanajuato * ∗ ∗ ∗ In recent years, many formalisms have been proposed in the Artificial Intelligence literature to model common- sense reasoning. So, the revision and transformation of knowledge is widely recognized as a key problem in knowledge representation and reasoning. Reasons for the importance of this topic are the facts that intelligent systems are gradually developed and refined, and that often the environment of an intelligent system is not static but changes over time [4, 9]. Belief revision studies reasoning with changing information. Traditionally, belief revision techniques have been expressed using classical logic. Recently, the study of knowledge revision has increased since it can be applied to several areas of knowledge. Belief revision is a system that contains a corpus of beliefs which can be changed to accommodate new knowledge that may be inconsistent with any previous beliefs. Assuming the new belief is correct, as many of the previous ones should be removed as necessary so that the new belief can be in- corporated into a consistent corpus. -
CS21 Decidability and Tractability Outline Definition of Reduction
Outline • many-one reductions CS21 • undecidable problems Decidability and Tractability – computation histories – surprising contrasts between Lecture 13 decidable/undecidable • Rice’s Theorem February 3, 2021 February 3, 2021 CS21 Lecture 13 1 February 3, 2021 CS21 Lecture 13 2 Definition of reduction Definition of reduction • Can you reduce co-HALT to HALT? • More refined notion of reduction: – “many-one” reduction (commonly) • We know that HALT is RE – “mapping” reduction (book) • Does this show that co-HALT is RE? A f B yes yes – recall, we showed co-HALT is not RE reduction from f language A to no no • our current notion of reduction cannot language B distinguish complements February 3, 2021 CS21 Lecture 13 3 February 3, 2021 CS21 Lecture 13 4 Definition of reduction Definition of reduction A f B yes yes • Notation: “A many-one reduces to B” is written f no no A ≤m B – “yes maps to yes and no maps to no” means: • function f should be computable w ∈ A maps to f(w) ∈ B & w ∉ A maps to f(w) ∉ B Definition: f : Σ*→ Σ* is computable if there exists a TM Mf such that on every w ∈ Σ* • B is at least as “hard” as A Mf halts on w with f(w) written on its tape. – more accurate: B at least as “expressive” as A February 3, 2021 CS21 Lecture 13 5 February 3, 2021 CS21 Lecture 13 6 1 Using reductions Using reductions Definition: A ≤m B if there is a computable • Main use: given language NEW, prove it is function f such that for all w undecidable by showing OLD ≤m NEW, w ∈ A ⇔ f(w) ∈ B where OLD known to be undecidable – proof by contradiction Theorem: if A ≤m B and B is decidable then A is decidable – if NEW decidable, then OLD decidable Proof: – OLD undecidable. -
Recursion Theory and Undecidability
Recursion Theory and Undecidability Lecture Notes for CSCI 303 by David Kempe March 26, 2008 In the second half of the class, we will explore limits on computation. These are questions of the form “What types of things can be computed at all?” “What types of things can be computed efficiently?” “How fast can problem XYZ be solved?” We already saw some positive examples in class: problems we could solve, and solve efficiently. For instance, we saw that we can sort n numbers in time O(n log n). Or compute the edit distance of two strings in time O(nm). But we might wonder if that’s the best possible. Maybe, there are still faster algorithms out there. Could we prove that these problems cannot be solved faster? In order to answer these questions, we first need to be precise about what exactly we mean by “a problem” and by “to compute”. In the case of an algorithm, we can go with “we know it when we see it”. We can look at an algorithm, and agree that it only uses “reasonable” types of statements. When we try to prove lower bounds, or prove that a problem cannot be solved at all, we need to be more precise: after all, an algorithm could contain the statement “Miraculously and instantaneously guess the solution to the problem.” We would probably all agree that that’s cheating. But how about something like “Instantaneously find all the prime factors of the number n”? Would that be legal? This writeup focuses only on the question of what problems can be solved at all. -
Explanations, Belief Revision and Defeasible Reasoning
Artificial Intelligence 141 (2002) 1–28 www.elsevier.com/locate/artint Explanations, belief revision and defeasible reasoning Marcelo A. Falappa a, Gabriele Kern-Isberner b, Guillermo R. Simari a,∗ a Artificial Intelligence Research and Development Laboratory, Department of Computer Science and Engineering, Universidad Nacional del Sur Av. Alem 1253, (B8000CPB) Bahía Blanca, Argentina b Department of Computer Science, LG Praktische Informatik VIII, FernUniversitaet Hagen, D-58084 Hagen, Germany Received 15 August 1999; received in revised form 18 March 2002 Abstract We present different constructions for nonprioritized belief revision, that is, belief changes in which the input sentences are not always accepted. First, we present the concept of explanation in a deductive way. Second, we define multiple revision operators with respect to sets of sentences (representing explanations), giving representation theorems. Finally, we relate the formulated operators with argumentative systems and default reasoning frameworks. 2002 Elsevier Science B.V. All rights reserved. Keywords: Belief revision; Change theory; Knowledge representation; Explanations; Defeasible reasoning 1. Introduction Belief Revision systems are logical frameworks for modelling the dynamics of knowledge. That is, how we modify our beliefs when we receive new information. The main problem arises when that information is inconsistent with the beliefs that represent our epistemic state. For instance, suppose we believe that a Ferrari coupe is the fastest car and then we found out that some Porsche cars are faster than any Ferrari cars. Surely, we * Corresponding author. E-mail addresses: [email protected] (M.A. Falappa), [email protected] (G. Kern-Isberner), [email protected] (G.R. -
Limits to Parallel Computation: P-Completeness Theory
Limits to Parallel Computation: P-Completeness Theory RAYMOND GREENLAW University of New Hampshire H. JAMES HOOVER University of Alberta WALTER L. RUZZO University of Washington New York Oxford OXFORD UNIVERSITY PRESS 1995 This book is dedicated to our families, who already know that life is inherently sequential. Preface This book is an introduction to the rapidly growing theory of P- completeness — the branch of complexity theory that focuses on identifying the “hardest” problems in the class P of problems solv- able in polynomial time. P-complete problems are of interest because they all appear to lack highly parallel solutions. That is, algorithm designers have failed to find NC algorithms, feasible highly parallel solutions that take time polynomial in the logarithm of the problem size while using only a polynomial number of processors, for them. Consequently, the promise of parallel computation, namely that ap- plying more processors to a problem can greatly speed its solution, appears to be broken by the entire class of P-complete problems. This state of affairs is succinctly expressed as the following question: Does P equal NC ? Organization of the Material The book is organized into two parts: an introduction to P- completeness theory, and a catalog of P-complete and open prob- lems. The first part of the book is a thorough introduction to the theory of P-completeness. We begin with an informal introduction. Then we discuss the major parallel models of computation, describe the classes NC and P, and present the notions of reducibility and com- pleteness. We subsequently introduce some fundamental P-complete problems, followed by evidence suggesting why NC does not equal P. -
Redalyc.Abductive Reasoning: Challenges Ahead
THEORIA. Revista de Teoría, Historia y Fundamentos de la Ciencia ISSN: 0495-4548 [email protected] Universidad del País Vasco/Euskal Herriko Unibertsitatea España ALISEDA, Atocha Abductive Reasoning: Challenges Ahead THEORIA. Revista de Teoría, Historia y Fundamentos de la Ciencia, vol. 22, núm. 3, 2007, pp. 261- 270 Universidad del País Vasco/Euskal Herriko Unibertsitatea Donostia-San Sebastián, España Available in: http://www.redalyc.org/articulo.oa?id=339730804001 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Abductive Reasoning: Challenges Ahead Atocha ALISEDA BIBLID [0495-4548 (2007) 22: 60; pp. 261-270] ABSTRACT: The motivation behind the collection of papers presented in this THEORIA forum on Abductive reasoning is my book Abductive Reasoning: Logical Investigations into the Processes of Discovery and Explanation. These contributions raise fundamental questions. One of them concerns the conjectural character of abduction. The choice of a logical framework for abduction is also discussed in detail, both its inferential aspect and search strategies. Abduction is also analyzed as inference to the best explanation, as well as a process of epistemic change, both of which chal- lenge the argument-like format of abduction. Finally, the psychological question of whether humans reason abduc- tively according to the models proposed is also addressed. I offer a brief summary of my book and then comment on and respond to several challenges that were posed to my work by the contributors to this issue.