STRUCTURES AND STRATEGIES FOR STATE SPACE SEARCH
|
|
- Lesley Preston
- 6 years ago
- Views:
Transcription
1 Slide STRUCTURES AND STRATEGIES FOR STATE SPACE SEARCH 3.0 Introduction 3.1 Graph Theory 3.2 Strategies for State Space Search 3.3 Using the State Space to Represent Reasoning with the Predicate Calculus 3.4 Epilogue and References 3.5 Exercises
2 Slide 3.2 Breadth-first search and shortest path because breadth-first search considers every node at each level of the graph before going deeper into the space, all states are first reached along the shortest path from the start state breadth-first search is therefore guaranteed to find the shortest path from the start state to the goal since all states are first found along the shortest path, any state encountered a second time are found along a path of equal or greater length since there is no chance that duplicate states were found along a better path, the algorithm simply discards any duplicate states
3 Slide 3.3 Figure 3.14: Graph of Figure 3.13 at iteration 6 of breadth-first search. States on open and closed are highlighted.
4 Slide 3.4 Depth-first search and shortest path not guaranteed to find the shortest path to a state the first time that state is encountered later in the search, a different path may be found to any state if path length matters in a problem solver, when the algorithm encounters a duplicate state, the algorithm should save the version reached along the shortest path this could be done by storing each state as a triple: (state,parent,length_of_path) when children are generated, the value of the path length is simply incremented by one and saved with the child if a child is reached along multiple paths, this information can be used to retain the best version
5 Function depth_first_search algorithm Slide 3.5
6 A trace of depth_first_search on the graph of Figure 3.13 Slide 3.6
7 Slide 3.7 Figure 3.16: Graph of Figure 3.13 at iteration 6 of depth-first search. States on open and closed are highlighted.
8 Slide 3.8 Figure 3.15: Breadth-first search of the 8-puzzle, showing order in which states were removed from open.
9 Figure 3.17: Depth-first search of the 8-puzzle with a depth bound of 5. Slide 3.9
10 Slide 3.10 Breadth-first and Depth-first search The choice of depth-first or breadth-first search depends on the specific problem being solved Significant features include the importance of finding the shortest path to a goal, the branching factor of the space, the available compute time and space resources, the average length of paths to a goal node and whether we want all solutions or only the first solution In making these decisions, there are advantages and disadvantages for each approach
11 Slide 3.11 Breadth-first It always examines all the nodes at level n before proceeding to level n+1, breadth-first search always finds the shortest path to a goal node In a problem where it is known that a simple solution exists, this solution will be found Unfortunately, if there is a bad branching factor, i.e. states have a high average number of descendants, the combinatorial explosion may prevent the algorithm from finding a solution using the available space This is due to the fact that all unexpected nodes for each level of the search must be kept on open For deep searches, or state spaces with a high branching factor, this can become quite cumbersome The space utilization of breadth-first search, measured in terms of the number of states in open, is an exponential function of the length of the path at any time If each state has an average of B children then there are B n states on level n
12 Slide 3.12 Depth-first Depth-first search gets quickly into a deep search space If it is know that the solution path will be long, depth-first search will not waste time searching a large number of shallow states in the graph Depth-first search can get lost deep in a graph, missing shorter paths to a goal or even becoming stuck in an infinitely long path that does not lead to a goal It is much more efficient for search spaces with many branches because it does not have to keep all the nodes at a given level on the open list The space usage of depth-first search is a linear function of the length of the path If each state has an average of B children per state, this requires a total space usage of B x n states to go to n levels deep into the space
13 Slide 3.13 State Space Description of a Logical System The soundness and completeness of predicate calculus inference rules guarantee the correctness of conclusions derived through this form of graph-based reasoning This ability to produce a formal proof of the integrity of a solution through the same algorithm that produces the solution is a unique attribute of much artificial intelligence and theorem proving based problem solving
14 Figure 3.18: State space graph of a set of implications in the propositional calculus. Slide 3.14
15 Slide 3.15 From the set of assertions and inference rule modus ponens, certain propositions (p,r, and u) may be inferred others ( such as v and q ) may not be inferred and do not logically follow from these assertions
16 Figure 3.19: And/or graph of the expression q r p. Slide 3.16
17 Slide 3.17
18 Figure 3.20: And/or graph of the expression q r p. Slide 3.18
19 Figure 3.21: And/or graph of a set of propositional calculus expressions. Slide 3.19
20 Slide 3.20 Example-- MACSYMA example of an and/or graph program for symbolically integrating mathematical functions well-known program that is used extensively by mathematicians breaking an expression into sub-expression that may be integrated independently of one another, with the result being combined algebraically into a solution expression this strategies include the rule for integration by parts and the rule for decomposing the integral of a sum into the sum of the integrals of the individual terms decomposition of a problem into independent subproblems can be represented by and nodes in the graph
21 Slide 3.21 Figure 3.22: And/or graph of part of the state space for integrating a function, from Nilsson (1971).
22 Slide 3.22 Example -- Goal-driven and/or search This example is taken from predicate calculus and represents a goal-driven graph where the goal is to be proved true in this situation is a predicate calculus expression containing variables the axioms are the logical descriptions of relationship between a dog, Fred, and his master, Sam assume that a cold day is not a warm day, bypassing issues such as the complexity added by equivalent expressions for predicates
23 Slide 3.23 The facts and rules of this example are given as English sentences followed by their predicate calculus equivalents:
24 Slide 3.24 The goal The goal is the expression X location( fred, X ) that means where is fred? from clause 7, a backward search algorithm examines this goal: if then fred is a good dog and fred has a master and fred s master is at a location that fred is at that location also
25 Figure 3.23: The solution subgraph showing that fred is at the museum. Slide 3.25
26 Slide 3.26 Example --The financial advisor revisited Assume that the individual has two dependents $20,000 in saving, a steady income of $30,000 goal is to find an investment, in predicate calculus X investment( X ) There are three rules (1,2 and 3) that conclude about investments
27 From Chapter 2 Slide 3.27
28 Figure 3.24: And/or graph searched by the financial advisor. Slide 3.28
29 Slide 3.29 Example -- An English Language Parser and Sentence Generator not from the predicate calculus consists of a set of rewrite rules for parsing sentences in a subset of English grammar rewrites rules take an expression and transform it into another by replacing the pattern on one side of arrow ( ) with the pattern on the other side rewrite rules transform a subset of English sentences into higher level grammatical constructs such as noun phrase, verb phrase, and sentence to determine whether they are well-formed sentences and to model the linguistics structure of the sentences
30 Five rules for a simple subset of English grammar are: Slide 3.30
31 Slide 3.31 Figure 3.25: And/or graph for the grammar of Example Some of the nodes (np, art, etc.) have been written more than once to simplify drawing the graph.
32 Figure 3.26: Parse tree for the sentence The dog bites the man. Note that this is a subtree of the graph of Figure Slide 3.32
33 Figure 3.27: A graph to be searched. Slide 3.33
State Space Search. Many problems can be represented as a set of states and a set of rules of how one state is transformed to another.
State Space Search Many problems can be represented as a set of states and a set of rules of how one state is transformed to another. The problem is how to reach a particular goal state, starting from
More informationSTRUCTURES AND STRATEGIES FOR STATE SPACE SEARCH
Slide 3.1 3 STRUCTURES AND STRATEGIES FOR STATE SPACE SEARCH 3.0 Introduction 3.1 Graph Theory 3.2 Strategies for State Space Search 3.3 Using the State Space to Represent Reasoning with the Predicate
More informationStructures and Strategies For Space State Search
Structures and Strategies For Space State Search 3.0 Introduction 3.1 Graph Theory 3.2 Strategies for Space State Search 3.3 Using Space State to Represent Reasoning with the Predicate Calculus AI Classnotes
More informationHEURISTIC SEARCH. 4.3 Using Heuristics in Games 4.4 Complexity Issues 4.5 Epilogue and References 4.6 Exercises
4 HEURISTIC SEARCH Slide 4.1 4.0 Introduction 4.1 An Algorithm for Heuristic Search 4.2 Admissibility, Monotonicity, and Informedness 4.3 Using Heuristics in Games 4.4 Complexity Issues 4.5 Epilogue and
More informationTHE PREDICATE CALCULUS
2 THE PREDICATE CALCULUS Slide 2.1 2.0 Introduction 2.1 The Propositional Calculus 2.2 The Predicate Calculus 2.3 Using Inference Rules to Produce Predicate Calculus Expressions 2.4 Application: A Logic-Based
More informationCHAPTER 10 GRAPHS AND TREES. Alessandro Artale UniBZ - artale/
CHAPTER 10 GRAPHS AND TREES Alessandro Artale UniBZ - http://www.inf.unibz.it/ artale/ SECTION 10.5 Trees Copyright Cengage Learning. All rights reserved. Trees In mathematics, a tree is a connected graph
More informationTrees Rooted Trees Spanning trees and Shortest Paths. 12. Graphs and Trees 2. Aaron Tan November 2017
12. Graphs and Trees 2 Aaron Tan 6 10 November 2017 1 10.5 Trees 2 Definition Definition Definition: Tree A graph is said to be circuit-free if, and only if, it has no circuits. A graph is called a tree
More informationCS 416, Artificial Intelligence Midterm Examination Fall 2004
CS 416, Artificial Intelligence Midterm Examination Fall 2004 Name: This is a closed book, closed note exam. All questions and subquestions are equally weighted. Introductory Material 1) True or False:
More informationAutomated Reasoning PROLOG and Automated Reasoning 13.4 Further Issues in Automated Reasoning 13.5 Epilogue and References 13.
13 Automated Reasoning 13.0 Introduction to Weak Methods in Theorem Proving 13.1 The General Problem Solver and Difference Tables 13.2 Resolution Theorem Proving 13.3 PROLOG and Automated Reasoning 13.4
More informationLecture 14: March 9, 2015
324: tate-space, F, F, Uninformed earch. pring 2015 Lecture 14: March 9, 2015 Lecturer: K.R. howdhary : Professor of (VF) isclaimer: These notes have not been subjected to the usual scrutiny reserved for
More informationLOGIC AND DISCRETE MATHEMATICS
LOGIC AND DISCRETE MATHEMATICS A Computer Science Perspective WINFRIED KARL GRASSMANN Department of Computer Science University of Saskatchewan JEAN-PAUL TREMBLAY Department of Computer Science University
More informationArtificial Intelligence (part 4a) Problem Solving Using Search: Structures and Strategies for State Space Search
Artificial Intelligence (part 4a) Problem Solving Using Search: Structures and Strategies for State Space Search Course Contents Again..Selected topics for our course. Covering all of AI is impossible!
More informationFigure 4.1: The evolution of a rooted tree.
106 CHAPTER 4. INDUCTION, RECURSION AND RECURRENCES 4.6 Rooted Trees 4.6.1 The idea of a rooted tree We talked about how a tree diagram helps us visualize merge sort or other divide and conquer algorithms.
More informationOutline. Best-first search
Outline Best-first search Greedy best-first search A* search Heuristics Local search algorithms Hill-climbing search Beam search Simulated annealing search Genetic algorithms Constraint Satisfaction Problems
More informationConsider a description of arithmetic. It includes two equations that define the structural types of digit and operator:
Syntax A programming language consists of syntax, semantics, and pragmatics. We formalize syntax first, because only syntactically correct programs have semantics. A syntax definition of a language lists
More informationCS101 Introduction to Programming Languages and Compilers
CS101 Introduction to Programming Languages and Compilers In this handout we ll examine different types of programming languages and take a brief look at compilers. We ll only hit the major highlights
More informationTheorem proving. PVS theorem prover. Hoare style verification PVS. More on embeddings. What if. Abhik Roychoudhury CS 6214
Theorem proving PVS theorem prover Abhik Roychoudhury National University of Singapore Both specification and implementation can be formalized in a suitable logic. Proof rules for proving statements in
More informationOutline. Best-first search
Outline Best-first search Greedy best-first search A* search Heuristics Admissible Heuristics Graph Search Consistent Heuristics Local search algorithms Hill-climbing search Beam search Simulated annealing
More informationWhat is Parsing? NP Det N. Det. NP Papa N caviar NP NP PP S NP VP. N spoon VP V NP VP VP PP. V spoon V ate PP P NP. P with.
Parsing What is Parsing? S NP VP NP Det N S NP Papa N caviar NP NP PP N spoon VP V NP VP VP PP NP VP V spoon V ate PP P NP P with VP PP Det the Det a V NP P NP Det N Det N Papa ate the caviar with a spoon
More informationCOMP219: Artificial Intelligence. Lecture 7: Search Strategies
COMP219: Artificial Intelligence Lecture 7: Search Strategies 1 Overview Last time basic ideas about problem solving; state space; solutions as paths; the notion of solution cost; the importance of using
More informationParallel Programming. Parallel algorithms Combinatorial Search
Parallel Programming Parallel algorithms Combinatorial Search Some Combinatorial Search Methods Divide and conquer Backtrack search Branch and bound Game tree search (minimax, alpha-beta) 2010@FEUP Parallel
More informationSearch. (Textbook Chpt ) Computer Science cpsc322, Lecture 2. May, 10, CPSC 322, Lecture 2 Slide 1
Search Computer Science cpsc322, Lecture 2 (Textbook Chpt 3.0-3.4) May, 10, 2012 CPSC 322, Lecture 2 Slide 1 Colored Cards You need to have 4 colored index cards Come and get them from me if you still
More informationAn Evolution of Mathematical Tools
An Evolution of Mathematical Tools From Conceptualization to Formalization Here's what we do when we build a formal model (or do a computation): 0. Identify a collection of objects/events in the real world.
More informationLecture 5 - Axiomatic semantics
Program Verification March 2014 Lecture 5 - Axiomatic semantics Lecturer: Noam Rinetzky Scribes by: Nir Hemed 1.1 Axiomatic semantics The development of the theory is contributed to Robert Floyd, C.A.R
More informationAn Annotated Language
Hoare Logic An Annotated Language State and Semantics Expressions are interpreted as functions from states to the corresponding domain of interpretation Operators have the obvious interpretation Free of
More informationCS 4700: Foundations of Artificial Intelligence. Bart Selman. Search Techniques R&N: Chapter 3
CS 4700: Foundations of Artificial Intelligence Bart Selman Search Techniques R&N: Chapter 3 Outline Search: tree search and graph search Uninformed search: very briefly (covered before in other prerequisite
More informationLearning Objectives. c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 3.5, Page 1
Learning Objectives At the end of the class you should be able to: justify why depth-bounded search is useful demonstrate how iterative-deepening works for a particular problem demonstrate how depth-first
More informationGraphs Introduction and Depth first algorithm
Graphs Introduction and Depth first algorithm Carol Zander Introduction to graphs Graphs are extremely common in computer science applications because graphs are common in the physical world. Everywhere
More informationTyped Lambda Calculus for Syntacticians
Department of Linguistics Ohio State University January 12, 2012 The Two Sides of Typed Lambda Calculus A typed lambda calculus (TLC) can be viewed in two complementary ways: model-theoretically, as a
More informationCS 380/480 Foundations of Artificial Intelligence Winter 2007 Assignment 2 Solutions to Selected Problems
CS 380/480 Foundations of Artificial Intelligence Winter 2007 Assignment 2 Solutions to Selected Problems 1. Search trees for the state-space graph given below: We only show the search trees corresponding
More informationOn the Space-Time Trade-off in Solving Constraint Satisfaction Problems*
Appeared in Proc of the 14th Int l Joint Conf on Artificial Intelligence, 558-56, 1995 On the Space-Time Trade-off in Solving Constraint Satisfaction Problems* Roberto J Bayardo Jr and Daniel P Miranker
More informationApplication of recursion. Grammars and Parsing. Recursive grammar. Grammars
Application of recursion Grammars and Parsing So far, we have written recursive programs on integers. Let us now consider a new application, grammars and parsing, that shows off the full power of recursion.
More information1. true / false By a compiler we mean a program that translates to code that will run natively on some machine.
1. true / false By a compiler we mean a program that translates to code that will run natively on some machine. 2. true / false ML can be compiled. 3. true / false FORTRAN can reasonably be considered
More informationLing 571: Deep Processing for Natural Language Processing
Ling 571: Deep Processing for Natural Language Processing Julie Medero January 14, 2013 Today s Plan Motivation for Parsing Parsing as Search Search algorithms Search Strategies Issues Two Goal of Parsing
More informationLecture 17 of 41. Clausal (Conjunctive Normal) Form and Resolution Techniques
Lecture 17 of 41 Clausal (Conjunctive Normal) Form and Resolution Techniques Wednesday, 29 September 2004 William H. Hsu, KSU http://www.kddresearch.org http://www.cis.ksu.edu/~bhsu Reading: Chapter 9,
More informationDeduction Rule System vs Production Rule System. Prof. Bob Berwick. Rules Rule
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.034 Artificial Intelligence, Fall 2003 Recitation 2, September 11/12 Rules Rule Prof. Bob Berwick Agenda
More informationContext-Free Grammars
Department of Linguistics Ohio State University Syntax 2 (Linguistics 602.02) January 3, 2012 (CFGs) A CFG is an ordered quadruple T, N, D, P where a. T is a finite set called the terminals; b. N is a
More informationKnowledge Representation
Knowledge Representation References Rich and Knight, Artificial Intelligence, 2nd ed. McGraw-Hill, 1991 Russell and Norvig, Artificial Intelligence: A modern approach, 2nd ed. Prentice Hall, 2003 Outline
More informationAn Appropriate Search Algorithm for Finding Grid Resources
An Appropriate Search Algorithm for Finding Grid Resources Olusegun O. A. 1, Babatunde A. N. 2, Omotehinwa T. O. 3,Aremu D. R. 4, Balogun B. F. 5 1,4 Department of Computer Science University of Ilorin,
More informationCS 188 Introduction to Artificial Intelligence Fall 2018 Note 1
CS 188 Introduction to Artificial Intelligence Fall 2018 Note 1 These lecture notes are heavily based on notes originally written by Nikhil Sharma. Agents In artificial intelligence, the central problem
More informationMonte Carlo Methods; Combinatorial Search
Monte Carlo Methods; Combinatorial Search Parallel and Distributed Computing Department of Computer Science and Engineering (DEI) Instituto Superior Técnico November 22, 2012 CPD (DEI / IST) Parallel and
More informationA4B36ZUI - Introduction ARTIFICIAL INTELLIGENCE
A4B36ZUI - Introduction to ARTIFICIAL INTELLIGENCE https://cw.fel.cvut.cz/wiki/courses/a4b33zui/start Michal Pechoucek, Branislav Bosansky, Jiri Klema & Olga Stepankova Department of Computer Science Czech
More informationOctober 19, 2004 Chapter Parsing
October 19, 2004 Chapter 10.3 10.6 Parsing 1 Overview Review: CFGs, basic top-down parser Dynamic programming Earley algorithm (how it works, how it solves the problems) Finite-state parsing 2 Last time
More informationSystem Analysis and Design. Data Flow Diagram. System Analysis and Design
Data Flow Diagram 1 Data Flow diagram The dataflow diagram is a modeling tool that allows us to picture a system as a network of functional processes, connected to one another by pipelines and holding
More informationHomework & NLTK. CS 181: Natural Language Processing Lecture 9: Context Free Grammars. Motivation. Formal Def of CFG. Uses of CFG.
C 181: Natural Language Processing Lecture 9: Context Free Grammars Kim Bruce Pomona College pring 2008 Homework & NLTK Review MLE, Laplace, and Good-Turing in smoothing.py Disclaimer: lide contents borrowed
More informationLogic and Computation
Logic and Computation From Conceptualization to Formalization Here's what we do when we build a formal model (or do a computation): 0. Identify a collection of objects/events in the real world. This is
More informationHEURISTIC SEARCH. 4.3 Using Heuristics in Games 4.4 Complexity Issues 4.5 Epilogue and References 4.6 Exercises
4 HEURISTIC SEARCH Slide 4.1 4.0 Introduction 4.1 An Algorithm for Heuristic Search 4.2 Admissibility, Monotonicity, and Informedness 4.3 Using Heuristics in Games 4.4 Complexity Issues 4.5 Epilogue and
More informationCombinatorial Search; Monte Carlo Methods
Combinatorial Search; Monte Carlo Methods Parallel and Distributed Computing Department of Computer Science and Engineering (DEI) Instituto Superior Técnico May 02, 2016 CPD (DEI / IST) Parallel and Distributed
More informationContext-Free Grammars
Context-Free Grammars Carl Pollard yntax 2 (Linguistics 602.02) January 3, 2012 Context-Free Grammars (CFGs) A CFG is an ordered quadruple T, N, D, P where a. T is a finite set called the terminals; b.
More informationCMSC Introduction to Algorithms Spring 2012 Lecture 16
CMSC 351 - Introduction to Algorithms Spring 2012 Lecture 16 Instructor: MohammadTaghi Hajiaghayi Scribe: Rajesh Chitnis 1 Introduction In this lecture we will look at Greedy Algorithms and Dynamic Programming.
More informationIntroductory logic and sets for Computer scientists
Introductory logic and sets for Computer scientists Nimal Nissanke University of Reading ADDISON WESLEY LONGMAN Harlow, England II Reading, Massachusetts Menlo Park, California New York Don Mills, Ontario
More informationAssignment 4 CSE 517: Natural Language Processing
Assignment 4 CSE 517: Natural Language Processing University of Washington Winter 2016 Due: March 2, 2016, 1:30 pm 1 HMMs and PCFGs Here s the definition of a PCFG given in class on 2/17: A finite set
More informationGuidelines for Writing Mathematical Proofs
Appendix A Guidelines for Writing Mathematical Proofs One of the most important forms of mathematical writing is writing mathematical proofs. The writing of mathematical proofs is an acquired skill and
More informationInteger Programming ISE 418. Lecture 7. Dr. Ted Ralphs
Integer Programming ISE 418 Lecture 7 Dr. Ted Ralphs ISE 418 Lecture 7 1 Reading for This Lecture Nemhauser and Wolsey Sections II.3.1, II.3.6, II.4.1, II.4.2, II.5.4 Wolsey Chapter 7 CCZ Chapter 1 Constraint
More informationSURIEM 2016 Final Report: Games on Graphs
SURIEM 2016 Final Report: Games on Graphs Julie Anne Bowman, Arthur Diep-Nguyen, Rashmika Goswami, Dylan King, Nicholas Lindell, Emily Olson, Robert W. Bell July 14, 2016 1 Introduction The game of Cops
More informationRelated Course Objec6ves
Syntax 9/18/17 1 Related Course Objec6ves Develop grammars and parsers of programming languages 9/18/17 2 Syntax And Seman6cs Programming language syntax: how programs look, their form and structure Syntax
More informationMATHEMATICAL STRUCTURES FOR COMPUTER SCIENCE
MATHEMATICAL STRUCTURES FOR COMPUTER SCIENCE A Modern Approach to Discrete Mathematics SIXTH EDITION Judith L. Gersting University of Hawaii at Hilo W. H. Freeman and Company New York Preface Note to the
More informationCSE 473. Chapter 4 Informed Search. CSE AI Faculty. Last Time. Blind Search BFS UC-BFS DFS DLS Iterative Deepening Bidirectional Search
CSE 473 Chapter 4 Informed Search CSE AI Faculty Blind Search BFS UC-BFS DFS DLS Iterative Deepening Bidirectional Search Last Time 2 1 Repeated States Failure to detect repeated states can turn a linear
More informationEquivalence of NTMs and TMs
Equivalence of NTMs and TMs What is a Turing Machine? Similar to a finite automaton, but with unlimited and unrestricted memory. It uses an infinitely long tape as its memory which can be read from and
More informationComputational Linguistics: Syntax-Semantics Interface
Computational Linguistics: Syntax-Semantics Interface Raffaella Bernardi KRDB, Free University of Bozen-Bolzano P.zza Domenicani, Room: 2.28, e-mail: bernardi@inf.unibz.it Contents 1 Lambda terms and DCG...................................
More informationAXIOMS OF AN IMPERATIVE LANGUAGE PARTIAL CORRECTNESS WEAK AND STRONG CONDITIONS. THE AXIOM FOR nop
AXIOMS OF AN IMPERATIVE LANGUAGE We will use the same language, with the same abstract syntax that we used for operational semantics. However, we will only be concerned with the commands, since the language
More information6.034 Notes: Section 2.1
6.034 Notes: Section 2.1 Slide 2.1.1 Search plays a key role in many parts of AI. These algorithms provide the conceptual backbone of almost every approach to the systematic exploration of alternatives.
More informationAlgorithm Design Techniques (III)
Algorithm Design Techniques (III) Minimax. Alpha-Beta Pruning. Search Tree Strategies (backtracking revisited, branch and bound). Local Search. DSA - lecture 10 - T.U.Cluj-Napoca - M. Joldos 1 Tic-Tac-Toe
More informationScribe: Virginia Williams, Sam Kim (2016), Mary Wootters (2017) Date: May 22, 2017
CS6 Lecture 4 Greedy Algorithms Scribe: Virginia Williams, Sam Kim (26), Mary Wootters (27) Date: May 22, 27 Greedy Algorithms Suppose we want to solve a problem, and we re able to come up with some recursive
More informationIn Our Last Exciting Episode
In Our Last Exciting Episode #1 Lessons From Model Checking To find bugs, we need specifications What are some good specifications? To convert a program into a model, we need predicates/invariants and
More informationNatural Language Processing
Natural Language Processing Classification III Dan Klein UC Berkeley 1 Classification 2 Linear Models: Perceptron The perceptron algorithm Iteratively processes the training set, reacting to training errors
More informationSymbolic and Concolic Execution of Programs
Symbolic and Concolic Execution of Programs Information Security, CS 526 Omar Chowdhury 10/7/2015 Information Security, CS 526 1 Reading for this lecture Symbolic execution and program testing - James
More informationContext-free Grammars
1 contents of Context-free Grammars Phrase Structure Everyday Grammars for Programming Language Formal Definition of Context-free Grammars Definition of Language Left-to-right Application cfg ects Transforming
More informationArbori Starter Manual Eugene Perkov
Arbori Starter Manual Eugene Perkov What is Arbori? Arbori is a query language that takes a parse tree as an input and builds a result set 1 per specifications defined in a query. What is Parse Tree? A
More information1.3. Conditional expressions To express case distinctions like
Introduction Much of the theory developed in the underlying course Logic II can be implemented in a proof assistant. In the present setting this is interesting, since we can then machine extract from a
More informationCOURSE: DATA STRUCTURES USING C & C++ CODE: 05BMCAR17161 CREDITS: 05
COURSE: DATA STRUCTURES USING C & C++ CODE: 05BMCAR17161 CREDITS: 05 Unit 1 : LINEAR DATA STRUCTURES Introduction - Abstract Data Types (ADT), Arrays and its representation Structures, Stack, Queue, Circular
More informationArtificial Intelligence
Artificial Intelligence Informed Search and Exploration Chapter 4 (4.1 4.2) A General Search algorithm: Chapter 3: Search Strategies Task : Find a sequence of actions leading from the initial state to
More informationCSE 311: Foundations of Computing. Lecture 8: Predicate Logic Proofs
CSE 311: Foundations of Computing Lecture 8: Predicate Logic Proofs Last class: Propositional Inference Rules Two inference rules per binary connective, one to eliminate it and one to introduce it Elim
More informationAnalysis of Algorithms - Greedy algorithms -
Analysis of Algorithms - Greedy algorithms - Andreas Ermedahl MRTC (Mälardalens Real-Time Reseach Center) andreas.ermedahl@mdh.se Autumn 2003 Greedy Algorithms Another paradigm for designing algorithms
More information6.001 Notes: Section 4.1
6.001 Notes: Section 4.1 Slide 4.1.1 In this lecture, we are going to take a careful look at the kinds of procedures we can build. We will first go back to look very carefully at the substitution model,
More informationFormal Predicate Calculus. Michael Meyling
Formal Predicate Calculus Michael Meyling May 24, 2013 2 The source for this document can be found here: http://www.qedeq.org/0_04_07/doc/math/qedeq_formal_logic_v1.xml Copyright by the authors. All rights
More informationComputer Science and Software Engineering University of Wisconsin - Platteville. 3. Search (Part 1) CS 3030 Lecture Notes Yan Shi UW-Platteville
Computer Science and Software Engineering University of Wisconsin - Platteville 3. Search (Part 1) CS 3030 Lecture Notes Yan Shi UW-Platteville Read: Textbook Chapter 3.7-3.9,3.12, 4. Problem Solving as
More informationPRESENTATION SLIDES FOR PUN-LOSP William Bricken July Interface as Overview
PRESENTATION SLIDES FOR PUN-LOSP William Bricken July 1997 Interface as Overview Input Logic directory of files (edif, pun) filename logic specification string (symbolic logic notations) table propositional
More informationCS 771 Artificial Intelligence. Informed Search
CS 771 Artificial Intelligence Informed Search Outline Review limitations of uninformed search methods Informed (or heuristic) search Uses problem-specific heuristics to improve efficiency Best-first,
More informationCS510 \ Lecture Ariel Stolerman
CS510 \ Lecture02 2012-10-03 1 Ariel Stolerman Midterm Evan will email about that after the lecture, at least 2 lectures from now. The exam will be given in a regular PDF (not an online form). We will
More informationfor all x, the assertion P(x) is false. there exists x, for which the assertion P(x) is true.
You can t prove a predicate is true because a predicate is not an assertion, you can t prove it is valid as it is not a deduction! If someone asks you to prove P(x), it is not totally clear what they mean.
More informationLambda Calculus and Type Inference
Lambda Calculus and Type Inference Björn Lisper Dept. of Computer Science and Engineering Mälardalen University bjorn.lisper@mdh.se http://www.idt.mdh.se/ blr/ October 13, 2004 Lambda Calculus and Type
More informationCSE 417 Branch & Bound (pt 4) Branch & Bound
CSE 417 Branch & Bound (pt 4) Branch & Bound Reminders > HW8 due today > HW9 will be posted tomorrow start early program will be slow, so debugging will be slow... Review of previous lectures > Complexity
More information7.1 Introduction. A (free) tree T is A simple graph such that for every pair of vertices v and w there is a unique path from v to w
Chapter 7 Trees 7.1 Introduction A (free) tree T is A simple graph such that for every pair of vertices v and w there is a unique path from v to w Tree Terminology Parent Ancestor Child Descendant Siblings
More informationIntroduction to dependent types in Coq
October 24, 2008 basic use of the Coq system In Coq, you can play with simple values and functions. The basic command is called Check, to verify if an expression is well-formed and learn what is its type.
More informationModule 8. Other representation formalisms. Version 2 CSE IIT, Kharagpur
Module 8 Other representation formalisms 8.1 Instructional Objective The students should understand the syntax and semantic of semantic networks Students should learn about different constructs and relations
More informationTyped Lambda Calculus
Department of Linguistics Ohio State University Sept. 8, 2016 The Two Sides of A typed lambda calculus (TLC) can be viewed in two complementary ways: model-theoretically, as a system of notation for functions
More informationChapter 3. Describing Syntax and Semantics
Chapter 3 Describing Syntax and Semantics Chapter 3 Topics Introduction The General Problem of Describing Syntax Formal Methods of Describing Syntax Attribute Grammars Describing the Meanings of Programs:
More informationCOMP219: Artificial Intelligence. Lecture 14: Knowledge Representation
COMP219: Artificial Intelligence Lecture 14: Knowledge Representation 1 Overview Last time Game playing Minimax decisions Alpha-beta pruning Today Introduce the need for explicit knowledge representation
More informationSearch and Optimization
Search and Optimization Search, Optimization and Game-Playing The goal is to find one or more optimal or sub-optimal solutions in a given search space. We can either be interested in finding any one solution
More informationParameterized Complexity - an Overview
Parameterized Complexity - an Overview 1 / 30 Parameterized Complexity - an Overview Ue Flarup 1 flarup@imada.sdu.dk 1 Department of Mathematics and Computer Science University of Southern Denmark, Odense,
More informationInformed search strategies (Section ) Source: Fotolia
Informed search strategies (Section 3.5-3.6) Source: Fotolia Review: Tree search Initialize the frontier using the starting state While the frontier is not empty Choose a frontier node to expand according
More information15.4 Longest common subsequence
15.4 Longest common subsequence Biological applications often need to compare the DNA of two (or more) different organisms A strand of DNA consists of a string of molecules called bases, where the possible
More informationA MECHANIZATION OF TYPE THEORY. Gerard P. HUBT IRIA - LABORIA Rocquencourt FRANCE
Session 6 Logic: II Theorem Proving and A MECHANIZATION OF TYPE THEORY Gerard P. HUBT IRIA - LABORIA Rocquencourt FRANCE ABSTRACT A refutational system of logic for a language of order w ia presented.
More informationFormal Systems and their Applications
Formal Systems and their Applications Dave Clarke (Dave.Clarke@cs.kuleuven.be) Acknowledgment: these slides are based in part on slides from Benjamin Pierce and Frank Piessens 1 Course Overview Introduction
More informationPropositional Logic. Part I
Part I Propositional Logic 1 Classical Logic and the Material Conditional 1.1 Introduction 1.1.1 The first purpose of this chapter is to review classical propositional logic, including semantic tableaux.
More informationThis is already grossly inconvenient in present formalisms. Why do we want to make this convenient? GENERAL GOALS
1 THE FORMALIZATION OF MATHEMATICS by Harvey M. Friedman Ohio State University Department of Mathematics friedman@math.ohio-state.edu www.math.ohio-state.edu/~friedman/ May 21, 1997 Can mathematics be
More informationDISCRETE MATHEMATICS
DISCRETE MATHEMATICS WITH APPLICATIONS THIRD EDITION SUSANNA S. EPP DePaul University THOIVISON * BROOKS/COLE Australia Canada Mexico Singapore Spain United Kingdom United States CONTENTS Chapter 1 The
More informationUniversity of Nevada, Las Vegas Computer Science 456/656 Fall 2016
University of Nevada, Las Vegas Computer Science 456/656 Fall 2016 The entire examination is 925 points. The real final will be much shorter. Name: No books, notes, scratch paper, or calculators. Use pen
More informationArtificial Intelligence (part 4c) Strategies for State Space Search. (Informed..Heuristic search)
Artificial Intelligence (part 4c) Strategies for State Space Search (Informed..Heuristic search) Search Strategies (The Order..) Uninformed Search breadth-first depth-first iterative deepening uniform-cost
More information