text: # 4. FIRST-ORDER LOGIC
  
  Save this PDF as:
  
  Size: px
  Start display at page:
  
  ## Transcription
  
  1 4. FIRST-ORDER LOGIC Contents 4.1: Splitting the atom: names, predicates and relations 4.2: Structures 4.3: Quantification:, 4.4: Syntax 4.5: Semantics 4.6: De Morgan s laws for and 4.7: Truth trees for FOL L is useful, as we have seen, but also very limited. The goal of this final part of the course is to add features to L that will make it more powerful and more useful although there will be a price to pay in that the resulting system will be intrinsically harder to work with. We shall study what is known as first-order logic (FOL) and sometimes predicate logic. Essentially FOL = L + predicates + quantifiers. This logic is the basis of applications of logic to CS, such as ROLOG. For mathematics, we have the following Mathematics = Set theory + FOL. 1. Splitting the atom: names, predicates and relations A statement is a sentence which is capable of being either true or false. In L, statements can only be analysed further in terms of the usual L connectives until we get to atoms. The atoms cannot then be further analysed. We shall show how, in fact, atoms can be split by using some new ideas. A name is a word that picks out a specific individual. For example, 2, pı, Darth Vader are all names. Once something has been named, we can refer to it by that name. At its simplest, a sentence can be analysed into a subject and a predicate. For example, the sentence grass is green has grass as its subject and is green as its predicate. To make the nature of the predicate clearer, we might write is green to indicate the slot into which a name could be fitted. Or, more formally, we could use a variable and write x is green. We could symbolize this predicate thus G(x) = x is green. We may replace the variable x by names to get honest statements that may or may not be true. Thus G(grass) is the statement grass is green whereas G(cheese) is the statement cheese is green. This is an example of a 1-place predicate because there is exactly one slot into which a name can be placed to yield a statement. Other examples of 1-place predicates are: x is a prime and x likes honey. There are also 2-place predicates. For example, (x, y) = x is the father of y is such a predicate because there are two slots that can be replaced by names to 1
  
  2 2 4. FIRST-ORDER LOGIC yield statements. Thus (spoiler alert) (Darth Vader, Luke Skywalker) is true but (Winnie-the-ooh, Darth Vader) is false. More generally, (x 1,..., x n ) denotes an n-place predicate or an n-ary predicate. We say that the predicate has arity n. Here x 1,..., x n are n variables that mark the n slots into which names van be slotted. Most of the predicates we shall meet will have arities 1 or 2. But in theory there is no limit. Thus F (x 1, x 2, x 3 ) is the 3-place predicate x 1 fights against x 2 with a x 2. By inserting names, we get the statement F (Luke Skywalker, Darth Vader, banana) (a deleted scene). In FOL, names are called constants and are usually denoted by the rather more mundane a, b, c,... or a 1, a 2, a Variables are denoted by x, y, z... or x 1, x 2, x 3,.... They have no fixed meaning but serve as slots into which names can be slotted. An atomic formula is a predicate whose slots are filled with either variables or constants. We now turn to the question of what predicates do. 1-place predicates describe sets 1. Thus the 1-place predicate (x) describes the set = {a: (a) is true } although this is usually written simply as = {a: (a)}. For example, if (x) is the 1-place predicate x is a prime then the set it describes is the set of prime numbers. To deal with what 2-place predicates describe we need some new notation. An ordered pair is written (a, b) where a is the first component and b is the second component. Observe that we use round brackets. These always mean order matters. Let (x, y) be a 2-place predicate. Then it describes the set which is usually just written = {(a, b): (a, b) is true } = {(a, b): (a, b)}. A set of ordered pairs where the ordered pairs are taken from some set X is called a binary relation and sometimes just a relation. Thus 2-place predicates describe binary relations. We often denote binary relations by Greek letters. There is a nice graphical way to represent binary relations at least when the set they are defined on is not too big. Let ρ be a binary relation defined on the set X. We draw a directed graph or digraph of ρ. This consists of vertices labelled by the elements of X and arrows where we draw an arrow from a to a precisely when (a, a ) ρ. Example 1.1. The relation ρ = {(1, 1), (1, 2), (2, 3), (4, 1), (4, 3), (4, 5), (5, 3)} is defined on the set X = {1, 2, 3, 4, 5}. Its corresponding directed graph is 1 At least informally. There is the problem of Russell s paradox. How that can be dealt with, if indeed it can be, is left to a more advanced course.
  
  3 4. FIRST-ORDER LOGIC Example 1.2. Binary relations are very common in mathematics. Here are some examples. (1) The relation x y is defined on the set of natural numbers N = {0, 1, 2, 3,...} if x exactly divides y. (2) The relations (less than or equal to) and < (strictly less than) defined on Z = {..., 3, 2, 1, 0, 1, 2, 3,...}. (3) The relation is a subset of defined between sets. (4) the relation is an element of defined between sets. (5) The relation of logical equivalence defined on the set of wff. Just for convenience, I shall just use 1-place pedicates and and 2-places predicates and so also sets and binary relations only, but there is no limit on the arities of predicates and we can study ordered n-tuples just as well as ordered pairs. We begin with two examples. 2. Structures Example 2.1. Here is part of the family tree of various Anglo-Saxon kings and queens. Egbert = Redburga Ethelwulf = Osburga Ethelbald Alfred = Ealhswith Ethelfleda Edward Ælfthryth Ethelgiva We analyse the mathematics behind this family tree. First, we have a set D of kings and queens called the domain. This consists of eleven Tolkienesque elements { } Egbert, Redburga, Ethelwulf, Osburga, Ethelbald, Alfred, D =. Eahlswith, Ethelfleda, Edward, Ælfthryth, Ethelgiva But we have additional information. Beside each name is a symbl or which means that that person is respectively male or female. This is just a way of defining two subsets of D: M = {Egbert, Ethelwulf, Ethelbald, Alfred, Edward}
  
  4 4 4. FIRST-ORDER LOGIC and F = {Redburga, Osburga, Eahlswith, Ethelfleda, Ælfthryth, Ethelgiva}. There are also two other peices of information. The most obvious are the lines linking one generation to the next. This is the binary relation defined by the 2- place predicate x is the parent of y. It is the set of ordered pairs π. For example, (Ethelwulf, Alfred), (Osburga, Alfred) π. A little less obvious is the notation = which stands for the binary relation defined by the 2-place predicate x is married to y. It is the set of ordered pairs µ. For example, (Egbert, Redburga), (Redburga, Egbert), (Ethelwulf, Osburga) µ. It follows that the information contained in the family tree is also contained in the following package (D, M, F, π, µ). Example 2.2. This looks quite different at first from the previous example but is mathematically very closely related to it. Define (N, E, O,, ) where E and O are, respectively, the sets of odd and even natural numbers and and are binary relations. We define a structure to consist of a non-empty set D, called the domain, together with a finite slection of subsets, binary relations, etc. FOL will be a language that will enable us to talk about structures. 3. Quantification:, We begin with an example. Let A(x) be the 1-place predicate x is made of atoms. I want to say that everything is made from atoms. In L, I have no choice but to use infinitely many conjunctions A(jelly) A(ice-cream) A(blancmange)... This is inconvenient. We get around this by using what is called the universal quantifier. We write ( x)a(x) which should be read for all x, x is made from atoms or for each x, x is made from atoms. The variable does not have to be x, we have all of these ( y), ( z) and so on. As we shall see, you should think of ( x) and its ilk as being a new unary connective. x
  
  5 4. FIRST-ORDER LOGIC 5 Will be the parse tree for ( x). The corresponding infinite disjunction (a) (b) (c)... is true when at least one of the terms of true. There is a corresponding existential quantifier. We write ( x)a(x) which should be read there exists an x such that (x) is true or there is at least one x such that (x) is true. The variable does not have to be x, we have all of these ( y), ( z) and so on. You should also think of ( x) and its ilk as being a new unary connective. x Will be the parse tree for ( x). 4. Syntax A first-order language consists of a choice of predicate letters with given arities. Which ones you choose, dpends on what problems you are interested in. In addition to our choice of predicate letters, we also have, for free, our logical symboms carried forward from L:,,,,, ; we also have variables x, y,..., x 1, x 2,...; constants (names) a, b... a 1, a 2,...; and quantifiers (forallx), ( y),... ( x 1 ), ( x 2 ),.... Recall that an atomic formula is a predicate letter with variables or constants inserted in all the available slots. We can now define a formula or wff in FOL. (1) All atomic formulae are wff. (2) If A and B are wff so too are ( A), (A B), (A B), (A B), (A B), (A B), and ( x)a, for any variable x and ( x)a for any variable x. (3) All wff arise by repeated application of steps (1) or (2) a finite number of times. I should add that I will carry forward to FOL from L the same conventions concerning the use of brackets without further comment. We may adapt parse trees to FOL. An expression such as (x 1, x 2, x 3 ), for example, gives rise to the parse tree x 1 x 2 x 3 The quantifiers ( x) and ( x) are treated as unary operators. The leaves of the parse tree are either variables or constants. Example 4.1. The formula ( x)[( (x) Q(x)) S(x, y)] has the parse tree
  
  6 6 4. FIRST-ORDER LOGIC x S Q x y x x We now come to a fundamental definition. Chhose any vertex v in a tree. The part of the tree, including v itself, that lies below v is clearly also a tree. It is called the subtree determined by v. In a parse tree, the subtree determined by an occurrence of ( x) is called the scope of this occurrence of the quantifier. Similarly for ( x). An occurrence of a variable x is called bound if it occurs with the scope of an occurrence of either ( x) and ( x). Otherwise, this occurrence is called free. Observe that the occurrence of x in both ( x) and ( x) is always bound. Example 4.2. Consider the formula ( (x) Q(y)) ( y)r(x, y). The scope of ( y) is R(x, y). Thus the x occurring in R(x, y) is free and the y occurring in R(x, y) is bound. A formula that has a free occurrence of some variable is called open otherwise it is called closed. A closed wff is called a sentence. FOL is about sentences. Example 4.3. Consider the wff ( x)(f (x) G(x)) (( x)f (x) ( x)g(x)). I claim this is a sentence. There are three occurrences of ( x). The first occurrence binds the x in F (x) G(x). The second occurrence binds the occurrence of x in the second occurrence of F (x). The third occurrence binds the occurreence of x in the second occurrence of G(x). Thus every occurrence of x in this wff is bound and there are no other variables. This the wff is closed as claimed. I shall explain why sentences are the natural things to study once I have defined the semantics of FOL. 5. Semantics Let L be a first-order language. For concreteness, suppose that it consists of two predicate letters where is a 1-place predicate letter and Q is a 2-place predicate letter. An interpretation I of L is a any structure (D, A, ρ) where D, the domain, is a non-empty set, A D is a subset and ρ is a binary relation on D. We interpret as A and Q as ρ. Thus the wff ( x)( (x) Q(x, x)) is interpreted as ( x D)((x A) ((x, x) ρ)). Under this interpretation, every sentence S in the language makes an assertion about the elements of D using A and ρ. We say that I is a model of S, written I S, if S is true when interpreted in this way
  
  7 4. FIRST-ORDER LOGIC 7 in I. A sentence S is said to be universally valid, written S is it is true in all interpretations. Whereas in L we studied tautologies, in FOL we study logically valid formulae. We write S 1 S 2 to mean that the sentences S 1 and S 2 are true in exactly the same interpretations. It is not hard to prove that this is equivalent to showing that S 1 S 2. The following example should help to clarify these definitions. Specifically, why it is sentences that we are interested in. Example 5.1. Interpret the 2-place predicate symbol (x, y) as the binary relation x is the father of y where the domain is the set of people. Observe that the phrase x is the father of y is neither true nor false since we know nothing about x and y. Consider now the wff ( y) (x, y). This says x is a father. This is still neither true nor false since x is not specified. Finally, consider the wff S 1 = ( x)( y) (x, y). This says everyone is a father. Observe that it is a sentence and that it is also a statement. In this case, it is false. I should add that in reading a sequence of quantifiers work your way in from left to right. The new sentence S 2 = ( x)( y) (x, y) is a different statement. It says that there is someone who is the father of everyone. This is also false. The new sentence S 3 = ( y)( x) (x, y) says that everyone has a father which is true. We now choose a new interpretation. This time we interpret (x, y) as x y and the domain as the set N = {0, 1, 2,...} as the set of natural numbers. The sentence S 1 says that for each natural number there is a natural number at least as big. This is true. The sentence S 2 says that there exists a natural number that is less than or equal to any natural number. This is true because 0 has exactly that property. Finally, sentence S 3 says that for each natural number there is a natural number that is no bigger. As we see in the above examples, sentences say something, or rather can be interpreted as saying something, that is making a statement that is therefore either true or false depending on the interpretation. I should add that sentences also arise (and this is a consequence of the definition but perhaps not obvious), when constants, that is names, are substituted for variables. Thus Darth Vader is the father of Winnie-the-ooh is a sentence, that happens to be false (and I believe has never been uttered by anyone before).
  
  ### Predicate Calculus. There are certain arguments that seem to be perfectly logical, yet they cannot be expressed by using propositional calculus.
  
  Predicate Calculus (Alternative names: predicate logic, first order logic, elementary logic, restricted predicate calculus, restricted functional calculus, relational calculus, theory of quantification,
  
  ### 4 Domain Relational Calculus
  
  4 Domain Relational Calculus We now present two relational calculi that we will compare to RA. First, what is the difference between an algebra and a calculus? The usual story is that the algebra RA is
  
  ### Predicate Logic Review
  
  Predicate Logic Review UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane 1 Grammar A term is an individual constant or a variable. An individual constant is a lowercase letter from the beginning
  
  ### Invalidity in Predicate Logic
  
  Invalidity in Predicate Logic So far we ve got a method for establishing that a predicate logic argument is valid: do a derivation. But we ve got no method for establishing invalidity. In propositional
  
  ### Handout #1: Mathematical Reasoning
  
  Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or
  
  ### Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson
  
  Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement
  
  ### Semantics for the Predicate Calculus: Part I
  
  Semantics for the Predicate Calculus: Part I (Version 0.3, revised 6:15pm, April 14, 2005. Please report typos to hhalvors@princeton.edu.) The study of formal logic is based on the fact that the validity
  
  ### (LMCS, p. 317) V.1. First Order Logic. This is the most powerful, most expressive logic that we will examine.
  
  (LMCS, p. 317) V.1 First Order Logic This is the most powerful, most expressive logic that we will examine. Our version of first-order logic will use the following symbols: variables connectives (,,,,
  
  ### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
  
  Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
  
  ### CHAPTER 2. Logic. 1. Logic Definitions. Notation: Variables are used to represent propositions. The most common variables used are p, q, and r.
  
  CHAPTER 2 Logic 1. Logic Definitions 1.1. Propositions. Definition 1.1.1. A proposition is a declarative sentence that is either true (denoted either T or 1) or false (denoted either F or 0). Notation:
  
  ### Subjects, Predicates and The Universal Quantifier
  
  Introduction to Logic Week Sixteen: 14 Jan., 2008 Subjects, Predicates and The Universal Quantifier 000. Office Hours this term: Tuesdays and Wednesdays 1-2, or by appointment; 5B-120. 00. Results from
  
  ### 2. The Language of First-order Logic
  
  2. The Language of First-order Logic KR & R Brachman & Levesque 2005 17 Declarative language Before building system before there can be learning, reasoning, planning, explanation... need to be able to
  
  ### Mathematical Induction
  
  Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature. E.g., we would like to show: N 1 : is a number 1 : has the feature Φ ( x)(n 1 x! 1 x) How
  
  ### Predicate Logic. For example, consider the following argument:
  
  Predicate Logic The analysis of compound statements covers key aspects of human reasoning but does not capture many important, and common, instances of reasoning that are also logically valid. For example,
  
  ### Lecture 16 : Relations and Functions DRAFT
  
  CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence
  
  ### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2
  
  CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us
  
  ### Introduction to Predicate Logic. Ling324 Reading: Meaning and Grammar, pg
  
  Introduction to Predicate Logic Ling324 Reading: Meaning and Grammar, pg. 113-141 Usefulness of Predicate Logic for Natural Language Semantics While in propositional logic, we can only talk about sentences
  
  ### Propositional Logic. A proposition is a declarative sentence (a sentence that declares a fact) that is either true or false, but not both.
  
  irst Order Logic Propositional Logic A proposition is a declarative sentence (a sentence that declares a fact) that is either true or false, but not both. Are the following sentences propositions? oronto
  
  ### Foundations of mathematics
  
  Foundations of mathematics 1. First foundations of mathematics 1.1. Introduction to the foundations of mathematics Mathematics, theories and foundations Sylvain Poirier http://settheory.net/ Mathematics
  
  ### Lecture 17 : Equivalence and Order Relations DRAFT
  
  CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion
  
  ### Relations and Functions
  
  Section 5. Relations and Functions 5.1. Cartesian Product 5.1.1. Definition: Ordered Pair Let A and B be sets and let a A and b B. An ordered pair ( a, b) is a pair of elements with the property that:
  
  ### 4.1. Definitions. A set may be viewed as any well defined collection of objects, called elements or members of the set.
  
  Section 4. Set Theory 4.1. Definitions A set may be viewed as any well defined collection of objects, called elements or members of the set. Sets are usually denoted with upper case letters, A, B, X, Y,
  
  ### 2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.
  
  2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then
  
  ### The Syntax of Predicate Logic
  
  The Syntax of Predicate Logic LX 502 Semantics I October 11, 2008 1. Below the Sentence-Level In Propositional Logic, atomic propositions correspond to simple sentences in the object language. Since atomic
  
  ### Math 55: Discrete Mathematics
  
  Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 1, due Wedneday, January 25 1.1.10 Let p and q be the propositions The election is decided and The votes have been counted, respectively.
  
  ### 1 Propositional Logic
  
  1 Propositional Logic Propositions 1.1 Definition A declarative sentence is a sentence that declares a fact or facts. Example 1 The earth is spherical. 7 + 1 = 6 + 2 x 2 > 0 for all real numbers x. 1 =
  
  ### Problems on Discrete Mathematics 1
  
  Problems on Discrete Mathematics 1 Chung-Chih Li 2 Kishan Mehrotra 3 Syracuse University, New York L A TEX at January 11, 2007 (Part I) 1 No part of this book can be reproduced without permission from
  
  ### 6.3 Conditional Probability and Independence
  
  222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted
  
  ### CHAPTER 1. Logic, Proofs Propositions
  
  CHAPTER 1 Logic, Proofs 1.1. Propositions A proposition is a declarative sentence that is either true or false (but not both). For instance, the following are propositions: Paris is in France (true), London
  
  ### Applications of Methods of Proof
  
  CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The set-theoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are
  
  ### Predicate logic is an Extension of sentence logic. Several new vocabulary items are found in predicate logic.
  
  Atomic Sentences of Predicate Logic Predicate Logic and Sentence Logic Predicate logic is an Extension of sentence logic. All items of the vocabulary of sentence logic are items of the vocabulary of predicate
  
  ### The Foundations: Logic and Proofs. Chapter 1, Part III: Proofs
  
  The Foundations: Logic and Proofs Chapter 1, Part III: Proofs Rules of Inference Section 1.6 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build Arguments
  
  ### 1 if 1 x 0 1 if 0 x 1
  
  Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
  
  ### University of Ostrava. Reasoning in Description Logic with Semantic Tableau Binary Trees
  
  University of Ostrava Institute for Research and Applications of Fuzzy Modeling Reasoning in Description Logic with Semantic Tableau Binary Trees Alena Lukasová Research report No. 63 2005 Submitted/to
  
  ### Mathematical Induction
  
  Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,
  
  ### ! " # The Logic of Descriptions. Logics for Data and Knowledge Representation. Terminology. Overview. Three Basic Features. Some History on DLs
  
  ,!0((,.+#\$),%\$(-&.& *,2(-\$)%&2.'3&%!&, Logics for Data and Knowledge Representation Alessandro Agostini agostini@dit.unitn.it University of Trento Fausto Giunchiglia fausto@dit.unitn.it The Logic of Descriptions!\$%&'()*\$#)
  
  ### CHAPTER 3 Numbers and Numeral Systems
  
  CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,
  
  ### ON FUNCTIONAL SYMBOL-FREE LOGIC PROGRAMS
  
  PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical and Mathematical Sciences 2012 1 p. 43 48 ON FUNCTIONAL SYMBOL-FREE LOGIC PROGRAMS I nf or m at i cs L. A. HAYKAZYAN * Chair of Programming and Information
  
  ### CS 441 Discrete Mathematics for CS Lecture 5. Predicate logic. CS 441 Discrete mathematics for CS. Negation of quantifiers
  
  CS 441 Discrete Mathematics for CS Lecture 5 Predicate logic Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Negation of quantifiers English statement: Nothing is perfect. Translation: x Perfect(x)
  
  ### Proofs are short works of prose and need to be written in complete sentences, with mathematical symbols used where appropriate.
  
  Advice for homework: Proofs are short works of prose and need to be written in complete sentences, with mathematical symbols used where appropriate. Even if a problem is a simple exercise that doesn t
  
  ### CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs
  
  CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce
  
  ### Introduction to Proofs
  
  Chapter 1 Introduction to Proofs 1.1 Preview of Proof This section previews many of the key ideas of proof and cites [in brackets] the sections where they are discussed thoroughly. All of these ideas are
  
  ### Math 3000 Section 003 Intro to Abstract Math Homework 2
  
  Math 3000 Section 003 Intro to Abstract Math Homework 2 Department of Mathematical and Statistical Sciences University of Colorado Denver, Spring 2012 Solutions (February 13, 2012) Please note that these
  
  ### TU e. Advanced Algorithms: experimentation project. The problem: load balancing with bounded look-ahead. Input: integer m 2: number of machines
  
  The problem: load balancing with bounded look-ahead Input: integer m 2: number of machines integer k 0: the look-ahead numbers t 1,..., t n : the job sizes Problem: assign jobs to machines machine to which
  
  ### WRITING PROOFS. Christopher Heil Georgia Institute of Technology
  
  WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this
  
  ### POWER SETS AND RELATIONS
  
  POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty
  
  ### CHAPTER 7 GENERAL PROOF SYSTEMS
  
  CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimes
  
  ### Discrete Mathematics: Solutions to Homework (12%) For each of the following sets, determine whether {2} is an element of that set.
  
  Discrete Mathematics: Solutions to Homework 2 1. (12%) For each of the following sets, determine whether {2} is an element of that set. (a) {x R x is an integer greater than 1} (b) {x R x is the square
  
  ### Predicate Logic. Example: All men are mortal. Socrates is a man. Socrates is mortal.
  
  Predicate Logic Example: All men are mortal. Socrates is a man. Socrates is mortal. Note: We need logic laws that work for statements involving quantities like some and all. In English, the predicate is
  
  ### Mathematics for Computer Science
  
  Mathematics for Computer Science Lecture 2: Functions and equinumerous sets Areces, Blackburn and Figueira TALARIS team INRIA Nancy Grand Est Contact: patrick.blackburn@loria.fr Course website: http://www.loria.fr/~blackbur/courses/math
  
  ### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11
  
  CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According
  
  ### SJÄLVSTÄNDIGA ARBETEN I MATEMATIK
  
  SJÄLVSTÄNDIGA ARBETEN I MATEMATIK MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET Automated Theorem Proving av Tom Everitt 2010 - No 8 MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET, 106 91 STOCKHOLM
  
  ### Consistency, completeness of undecidable preposition of Principia Mathematica. Tanmay Jaipurkar
  
  Consistency, completeness of undecidable preposition of Principia Mathematica Tanmay Jaipurkar October 21, 2013 Abstract The fallowing paper discusses the inconsistency and undecidable preposition of Principia
  
  ### 7 Relations and Functions
  
  7 Relations and Functions In this section, we introduce the concept of relations and functions. Relations A relation R from a set A to a set B is a set of ordered pairs (a, b), where a is a member of A,
  
  ### 3 Some Integer Functions
  
  3 Some Integer Functions A Pair of Fundamental Integer Functions The integer function that is the heart of this section is the modulo function. However, before getting to it, let us look at some very simple
  
  ### Expressive powver of logical languages
  
  July 18, 2012 Expressive power distinguishability The expressive power of any language can be measured through its power of distinction or equivalently, by the situations it considers indistinguishable.
  
  ### 8 Divisibility and prime numbers
  
  8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express
  
  ### . 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9
  
  Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a
  
  ### Basic Set Theory. 1. Motivation. Fido Sue. Fred Aristotle Bob. LX 502 - Semantics I September 11, 2008
  
  Basic Set Theory LX 502 - Semantics I September 11, 2008 1. Motivation When you start reading these notes, the first thing you should be asking yourselves is What is Set Theory and why is it relevant?
  
  ### Quotient Rings and Field Extensions
  
  Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.
  
  ### The Graphical Method: An Example
  
  The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,
  
  ### INTRODUCTORY SET THEORY
  
  M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,
  
  ### 5544 = 2 2772 = 2 2 1386 = 2 2 2 693. Now we have to find a divisor of 693. We can try 3, and 693 = 3 231,and we keep dividing by 3 to get: 1
  
  MATH 13150: Freshman Seminar Unit 8 1. Prime numbers 1.1. Primes. A number bigger than 1 is called prime if its only divisors are 1 and itself. For example, 3 is prime because the only numbers dividing
  
  ### 3. Mathematical Induction
  
  3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
  
  ### 3. Equivalence Relations. Discussion
  
  3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,
  
  ### If an English sentence is ambiguous, it may allow for more than one adequate transcription.
  
  Transcription from English to Predicate Logic General Principles of Transcription In transcribing an English sentence into Predicate Logic, some general principles apply. A transcription guide must be
  
  ### Mathematical Induction. Lecture 10-11
  
  Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach
  
  ### Chapter 4: The Logic of Boolean Connectives
  
  Chapter 4: The Logic of Boolean Connectives 4.1 Tautologies and logical truth Logical truth We already have the notion of logical consequence. A sentence is a logical consequence of a set of sentences
  
  ### Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition
  
  Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing
  
  ### This asserts two sets are equal iff they have the same elements, that is, a set is determined by its elements.
  
  3. Axioms of Set theory Before presenting the axioms of set theory, we first make a few basic comments about the relevant first order logic. We will give a somewhat more detailed discussion later, but
  
  ### Just the Factors, Ma am
  
  1 Introduction Just the Factors, Ma am The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive
  
  ### Mathematical Induction. Mary Barnes Sue Gordon
  
  Mathematics Learning Centre Mathematical Induction Mary Barnes Sue Gordon c 1987 University of Sydney Contents 1 Mathematical Induction 1 1.1 Why do we need proof by induction?.... 1 1. What is proof by
  
  ### So let us begin our quest to find the holy grail of real analysis.
  
  1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers
  
  ### 6.080/6.089 GITCS Feb 12, 2008. Lecture 3
  
  6.8/6.89 GITCS Feb 2, 28 Lecturer: Scott Aaronson Lecture 3 Scribe: Adam Rogal Administrivia. Scribe notes The purpose of scribe notes is to transcribe our lectures. Although I have formal notes of my
  
  ### Predicate logic. Logic in computer science. Logic in Computer Science (lecture) PART II. first order logic
  
  PART II. Predicate logic first order logic Logic in computer science Seminar: INGK401-K5; INHK401; INJK401-K4 University of Debrecen, Faculty of Informatics kadek.tamas@inf.unideb.hu 1 / 19 Alphabets Logical
  
  ### CmSc 175 Discrete Mathematics Lesson 10: SETS A B, A B
  
  CmSc 175 Discrete Mathematics Lesson 10: SETS Sets: finite, infinite, : empty set, U : universal set Describing a set: Enumeration = {a, b, c} Predicates = {x P(x)} Recursive definition, e.g. sequences
  
  ### Propositional Logic. Definition: A proposition or statement is a sentence which is either true or false.
  
  Propositional Logic Definition: A proposition or statement is a sentence which is either true or false. Definition:If a proposition is true, then we say its truth value is true, and if a proposition is
  
  ### Sets and set operations
  
  CS 441 Discrete Mathematics for CS Lecture 7 Sets and set operations Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square asic discrete structures Discrete math = study of the discrete structures used
  
  ### 13 Infinite Sets. 13.1 Injections, Surjections, and Bijections. mcs-ftl 2010/9/8 0:40 page 379 #385
  
  mcs-ftl 2010/9/8 0:40 page 379 #385 13 Infinite Sets So you might be wondering how much is there to say about an infinite set other than, well, it has an infinite number of elements. Of course, an infinite
  
  ### 1 Mathematical Models of Cost, Revenue and Profit
  
  Section 1.: Mathematical Modeling Math 14 Business Mathematics II Minh Kha Goals: to understand what a mathematical model is, and some of its examples in business. Definition 0.1. Mathematical Modeling
  
  ### Pythagorean Triples. Chapter 2. a 2 + b 2 = c 2
  
  Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the
  
  ### First-Order Logics and Truth Degrees
  
  First-Order Logics and Truth Degrees George Metcalfe Mathematics Institute University of Bern LATD 2014, Vienna Summer of Logic, 15-19 July 2014 George Metcalfe (University of Bern) First-Order Logics
  
  ### Announcements. CompSci 230 Discrete Math for Computer Science Sets. Introduction to Sets. Sets
  
  CompSci 230 Discrete Math for Computer Science Sets September 12, 2013 Prof. Rodger Slides modified from Rosen 1 nnouncements Read for next time Chap. 2.3-2.6 Homework 2 due Tuesday Recitation 3 on Friday
  
  ### CS 3719 (Theory of Computation and Algorithms) Lecture 4
  
  CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 Church-Turing thesis Let s recap how it all started. In 1990, Hilbert stated a
  
  ### Open and Closed Sets
  
  Open and Closed Sets Definition: A subset S of a metric space (X, d) is open if it contains an open ball about each of its points i.e., if x S : ɛ > 0 : B(x, ɛ) S. (1) Theorem: (O1) and X are open sets.
  
  ### Working with whole numbers
  
  1 CHAPTER 1 Working with whole numbers In this chapter you will revise earlier work on: addition and subtraction without a calculator multiplication and division without a calculator using positive and
  
  ### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.
  
  Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the
  
  ### Definition 14 A set is an unordered collection of elements or objects.
  
  Chapter 4 Set Theory Definition 14 A set is an unordered collection of elements or objects. Primitive Notation EXAMPLE {1, 2, 3} is a set containing 3 elements: 1, 2, and 3. EXAMPLE {1, 2, 3} = {3, 2,
  
  ### Taylor Polynomials and Taylor Series Math 126
  
  Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will
  
  ### Tastes and Indifference Curves
  
  C H A P T E R 4 Tastes and Indifference Curves This chapter begins a 2-chapter treatment of tastes or what we also call preferences. In the first of these chapters, we simply investigate the basic logic
  
  ### Computability Theory
  
  CSC 438F/2404F Notes (S. Cook and T. Pitassi) Fall, 2014 Computability Theory This section is partly inspired by the material in A Course in Mathematical Logic by Bell and Machover, Chap 6, sections 1-10.
  
  ### Week 5: Binary Relations
  
  1 Binary Relations Week 5: Binary Relations The concept of relation is common in daily life and seems intuitively clear. For instance, let X be the set of all living human females and Y the set of all
  
  ### Chapter 9. Systems of Linear Equations
  
  Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables
  
  ### Examples of Functions
  
  Examples of Functions In this document is provided examples of a variety of functions. The purpose is to convince the beginning student that functions are something quite different than polynomial equations.
  
  ### Fundamentele Informatica II
  
  Fundamentele Informatica II Answer to selected exercises 1 John C Martin: Introduction to Languages and the Theory of Computation M.M. Bonsangue (and J. Kleijn) Fall 2011 Let L be a language. It is clear
  
  ### Permutation Groups. Rubik s Cube
  
  Permutation Groups and Rubik s Cube Tom Davis tomrdavis@earthlink.net May 6, 2000 Abstract In this paper we ll discuss permutations (rearrangements of objects), how to combine them, and how to construct
  
  ### Regular Languages and Finite Automata
  
  Regular Languages and Finite Automata 1 Introduction Hing Leung Department of Computer Science New Mexico State University Sep 16, 2010 In 1943, McCulloch and Pitts [4] published a pioneering work on a
  
  ### Type Theory & Functional Programming
  
  Type Theory & Functional Programming Simon Thompson Computing Laboratory, University of Kent March 1999 c Simon Thompson, 1999 Not to be reproduced i ii To my parents Preface Constructive Type theory has