Recursion and Induction
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CS311H: Discrete Mathematics Recursive Definitions Recursive
Recursive Definitions CS311H: Discrete Mathematics I Should be familiar with recursive functions from programming: Recursive Definitions public int fact(int n) { if(n <= 1) return 1; return n * fact(n - 1); Instructor: I¸sıl Dillig } I Recursive definitions are also used in math for defining sets, functions, sequences etc. Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Recursive Definitions 1/18 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Recursive Definitions 2/18 Recursive Definitions in Math Recursively Defined Functions I Consider the following sequence: I Just like sequences, functions can also be defined recursively 1, 3, 9, 27, 81,... I Example: I This sequence can be defined recursively as follows: f (0) = 3 f (n + 1) = 2f (n) + 3 (n 1) a0 = 1 ≥ an = 3 an 1 · − I What is f (1)? I First part called base case; second part called recursive step I What is f (2)? I Very similar to induction; in fact, recursive definitions I What is f (3)? sometimes also called inductive definitions Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Recursive Definitions 3/18 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Recursive Definitions 4/18 Recursive Definition Examples Recursive Definitions of Important Functions I Some important functions/sequences defined recursively I Consider f (n) = 2n + 1 where n is non-negative integer I Factorial function: I What’s a recursive definition for f ? f (1) = 1 f (n) = n f (n 1) (n 2) · − ≥ I Consider the sequence 1, 4, 9, 16,... I Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21,... I What is a recursive -
Chapter 3 Induction and Recursion
Chapter 3 Induction and Recursion 3.1 Induction: An informal introduction This section is intended as a somewhat informal introduction to The Principle of Mathematical Induction (PMI): a theorem that establishes the validity of the proof method which goes by the same name. There is a particular format for writing the proofs which makes it clear that PMI is being used. We will not explicitly use this format when introducing the method, but will do so for the large number of different examples given later. Suppose you are given a large supply of L-shaped tiles as shown on the left of the figure below. The question you are asked to answer is whether these tiles can be used to exactly cover the squares of an 2n × 2n punctured grid { a 2n × 2n grid that has had one square cut out { say the 8 × 8 example shown in the right of the figure. 1 2 CHAPTER 3. INDUCTION AND RECURSION In order for this to be possible at all, the number of squares in the punctured grid has to be a multiple of three. It is. The number of squares is 2n2n − 1 = 22n − 1 = 4n − 1 ≡ 1n − 1 ≡ 0 (mod 3): But that does not mean we can tile the punctured grid. In order to get some traction on what to do, let's try some small examples. The tiling is easy to find if n = 1 because 2 × 2 punctured grid is exactly covered by one tile. Let's try n = 2, so that our punctured grid is 4 × 4. -
Propositional Logic, Modal Logic, Propo- Sitional Dynamic Logic and first-Order Logic
A I L Madhavan Mukund Chennai Mathematical Institute E-mail: [email protected] Abstract ese are lecture notes for an introductory course on logic aimed at graduate students in Com- puter Science. e notes cover techniques and results from propositional logic, modal logic, propo- sitional dynamic logic and first-order logic. e notes are based on a course taught to first year PhD students at SPIC Mathematical Institute, Madras, during August–December, . Contents Propositional Logic . Syntax . . Semantics . . Axiomatisations . . Maximal Consistent Sets and Completeness . . Compactness and Strong Completeness . Modal Logic . Syntax . . Semantics . . Correspondence eory . . Axiomatising valid formulas . . Bisimulations and expressiveness . . Decidability: Filtrations and the finite model property . . Labelled transition systems and multi-modal logic . Dynamic Logic . Syntax . . Semantics . . Axiomatising valid formulas . First-Order Logic . Syntax . . Semantics . . Formalisations in first-order logic . . Satisfiability: Henkin’s reduction to propositional logic . . Compactness and the L¨owenheim-Skolem eorem . . A Complete Axiomatisation . . Variants of the L¨owenheim-Skolem eorem . . Elementary Classes . . Elementarily Equivalent Structures . . An Algebraic Characterisation of Elementary Equivalence . . Decidability . Propositional Logic . Syntax P = f g We begin with a countably infinite set of atomic propositions p0, p1,... and two logical con- nectives : (read as not) and _ (read as or). e set Φ of formulas of propositional logic is the smallest set satisfying the following conditions: • Every atomic proposition p is a member of Φ. • If α is a member of Φ, so is (:α). • If α and β are members of Φ, so is (α _ β). We shall normally omit parentheses unless we need to explicitly clarify the structure of a formula. -
Recursive Definitions and Structural Induction 1 Recursive Definitions
Massachusetts Institute of Technology Course Notes 6 6.042J/18.062J, Fall ’02: Mathematics for Computer Science Professor Albert Meyer and Dr. Radhika Nagpal Recursive Definitions and Structural Induction 1 Recursive Definitions Recursive definitions say how to build something from a simpler version of the same thing. They have two parts: • Base case(s) that don’t depend on anything else. • Combination case(s) that depend on simpler cases. Here are some examples of recursive definitions: Example 1.1. Define a set, E, recursively as follows: 1. 0 E, 2 2. if n E, then n + 2 E, 2 2 3. if n E, then n E. 2 − 2 Using this definition, we can see that since 0 E by 1., it follows from 2. that 0 + 2 = 2 E, and 2 2 so 2 + 2 = 4 E, 4 + 2 = 6 E, : : : , and in fact any nonegative even number is in E. Also, by 3., 2 2 2; 4; 6; E. − − − · · · 2 Is anything else in E? The definition doesn’t say so explicitly, but an implicit condition on a recursive definition is that the only way things get into E is as a consequence of 1., 2., and 3. So clearly E is exactly the set of even integers. Example 1.2. Define a set, S, of strings of a’s and b’s recursively as follows: 1. λ S, where λ is the empty string, 2 2. if x S, then axb S, 2 2 3. if x S, then bxa S, 2 2 4. if x; y S, then xy S. -
On Basic Probability Logic Inequalities †
mathematics Article On Basic Probability Logic Inequalities † Marija Boriˇci´cJoksimovi´c Faculty of Organizational Sciences, University of Belgrade, Jove Ili´ca154, 11000 Belgrade, Serbia; [email protected] † The conclusions given in this paper were partially presented at the European Summer Meetings of the Association for Symbolic Logic, Logic Colloquium 2012, held in Manchester on 12–18 July 2012. Abstract: We give some simple examples of applying some of the well-known elementary probability theory inequalities and properties in the field of logical argumentation. A probabilistic version of the hypothetical syllogism inference rule is as follows: if propositions A, B, C, A ! B, and B ! C have probabilities a, b, c, r, and s, respectively, then for probability p of A ! C, we have f (a, b, c, r, s) ≤ p ≤ g(a, b, c, r, s), for some functions f and g of given parameters. In this paper, after a short overview of known rules related to conjunction and disjunction, we proposed some probabilized forms of the hypothetical syllogism inference rule, with the best possible bounds for the probability of conclusion, covering simultaneously the probabilistic versions of both modus ponens and modus tollens rules, as already considered by Suppes, Hailperin, and Wagner. Keywords: inequality; probability logic; inference rule MSC: 03B48; 03B05; 60E15; 26D20; 60A05 1. Introduction The main part of probabilization of logical inference rules is defining the correspond- Citation: Boriˇci´cJoksimovi´c,M. On ing best possible bounds for probabilities of propositions. Some of them, connected with Basic Probability Logic Inequalities. conjunction and disjunction, can be obtained immediately from the well-known Boole’s Mathematics 2021, 9, 1409. -
Propositional Logic - Basics
Outline Propositional Logic - Basics K. Subramani1 1Lane Department of Computer Science and Electrical Engineering West Virginia University 14 January and 16 January, 2013 Subramani Propositonal Logic Outline Outline 1 Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Subramani Propositonal Logic (i) The Law! (ii) Mathematics. (iii) Computer Science. Definition Statement (or Atomic Proposition) - A sentence that is either true or false. Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox). Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation Why Logic? Subramani Propositonal Logic (ii) Mathematics. (iii) Computer Science. Definition Statement (or Atomic Proposition) - A sentence that is either true or false. Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox). Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation Why Logic? (i) The Law! Subramani Propositonal Logic (iii) Computer Science. Definition Statement (or Atomic Proposition) - A sentence that is either true or false. Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox). Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation Why Logic? (i) The Law! (ii) Mathematics. Subramani Propositonal Logic (iii) Computer Science. Definition Statement (or Atomic Proposition) - A sentence that is either true or false. Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. -
Predicate Logic. Formal and Informal Proofs
CS 441 Discrete Mathematics for CS Lecture 5 Predicate logic Milos Hauskrecht [email protected] 5329 Sennott Square CS 441 Discrete mathematics for CS M. Hauskrecht Negation of quantifiers English statement: • Nothing is perfect. • Translation: ¬ x Perfect(x) Another way to express the same meaning: • Everything ... M. Hauskrecht 1 Negation of quantifiers English statement: • Nothing is perfect. • Translation: ¬ x Perfect(x) Another way to express the same meaning: • Everything is imperfect. • Translation: x ¬ Perfect(x) Conclusion: ¬ x P (x) is equivalent to x ¬ P(x) M. Hauskrecht Negation of quantifiers English statement: • It is not the case that all dogs are fleabags. • Translation: ¬ x Dog(x) Fleabag(x) Another way to express the same meaning: • There is a dog that … M. Hauskrecht 2 Negation of quantifiers English statement: • It is not the case that all dogs are fleabags. • Translation: ¬ x Dog(x) Fleabag(x) Another way to express the same meaning: • There is a dog that is not a fleabag. • Translation: x Dog(x) ¬ Fleabag(x) • Logically equivalent to: – x ¬ ( Dog(x) Fleabag(x) ) Conclusion: ¬ x P (x) is equivalent to x ¬ P(x) M. Hauskrecht Negation of quantified statements (aka DeMorgan Laws for quantifiers) Negation Equivalent ¬x P(x) x ¬P(x) ¬x P(x) x ¬P(x) M. Hauskrecht 3 Formal and informal proofs CS 441 Discrete mathematics for CS M. Hauskrecht Theorems and proofs • The truth value of some statement about the world is obvious and easy to assign • The truth of other statements may not be obvious, … …. But it may still follow (be derived) from known facts about the world To show the truth value of such a statement following from other statements we need to provide a correct supporting argument - a proof Important questions: – When is the argument correct? – How to construct a correct argument, what method to use? CS 441 Discrete mathematics for CS M. -
Logic in Computer Science
P. Madhusudan Logic in Computer Science Rough course notes November 5, 2020 Draft. Copyright P. Madhusudan Contents 1 Logic over Structures: A Single Known Structure, Classes of Structures, and All Structures ................................... 1 1.1 Logic on a Fixed Known Structure . .1 1.2 Logic on a Fixed Class of Structures . .3 1.3 Logic on All Structures . .5 1.4 Logics over structures: Theories and Questions . .7 2 Propositional Logic ............................................. 11 2.1 Propositional Logic . 11 2.1.1 Syntax . 11 2.1.2 What does the above mean with ::=, etc? . 11 2.2 Some definitions and theorems . 14 2.3 Compactness Theorem . 15 2.4 Resolution . 18 3 Quantifier Elimination and Decidability .......................... 23 3.1 Quantifiier Elimination . 23 3.2 Dense Linear Orders without Endpoints . 24 3.3 Quantifier Elimination for rationals with addition: ¹Q, 0, 1, ¸, −, <, =º ......................................... 27 3.4 The Theory of Reals with Addition . 30 3.4.1 Aside: Axiomatizations . 30 3.4.2 Other theories that admit quantifier elimination . 31 4 Validity of FOL is undecidable and is r.e.-hard .................... 33 4.1 Validity of FOL is r.e.-hard . 34 4.2 Trakhtenbrot’s theorem: Validity of FOL over finite models is undecidable, and co-r.e. hard . 39 5 Quantifier-free theory of equality ................................ 43 5.1 Decidability using Bounded Models . 43 v vi Contents 5.2 An Algorithm for Conjunctive Formulas . 44 5.2.1 Computing ¹º ................................... 47 5.3 Axioms for The Theory of Equality . 49 6 Completeness Theorem: FO Validity is r.e. ........................ 53 6.1 Prenex Normal Form . -
CS245 Logic and Computation
CS245 Logic and Computation Alice Gao December 9, 2019 Contents 1 Propositional Logic 3 1.1 Translations .................................... 3 1.2 Structural Induction ............................... 8 1.2.1 A template for structural induction on well-formed propositional for- mulas ................................... 8 1.3 The Semantics of an Implication ........................ 12 1.4 Tautology, Contradiction, and Satisfiable but Not a Tautology ........ 13 1.5 Logical Equivalence ................................ 14 1.6 Analyzing Conditional Code ........................... 16 1.7 Circuit Design ................................... 17 1.8 Tautological Consequence ............................ 18 1.9 Formal Deduction ................................. 21 1.9.1 Rules of Formal Deduction ........................ 21 1.9.2 Format of a Formal Deduction Proof .................. 23 1.9.3 Strategies for writing a formal deduction proof ............ 23 1.9.4 And elimination and introduction .................... 25 1.9.5 Implication introduction and elimination ................ 26 1.9.6 Or introduction and elimination ..................... 28 1.9.7 Negation introduction and elimination ................. 30 1.9.8 Putting them together! .......................... 33 1.9.9 Putting them together: Additional exercises .............. 37 1.9.10 Other problems .............................. 38 1.10 Soundness and Completeness of Formal Deduction ............... 39 1.10.1 The soundness of inference rules ..................... 39 1.10.2 Soundness and Completeness -
Arxiv:1410.2193V1 [Math.CO] 8 Oct 2014 There Are No Coincidences
There are no Coincidences Tanya Khovanova October 21, 2018 Abstract This paper is inspired by a seqfan post by Jeremy Gardiner. The post listed nine sequences with similar parities. In this paper I prove that the similarities are not a coincidence but a mathematical fact. 1 Introduction Do you use your Inbox as your implicit to-do list? I do. I delete spam and cute cats. I reply, if I can. I deal with emails related to my job. But sometimes I get a message that requires some thought or some serious time to process. It sits in my Inbox. There are 200 messages there now. This method creates a lot of stress. But this paper is not about stress management. Let me get back on track. I was trying to shrink this list and discovered an email I had completely forgotten about. The email was sent in December 2008 to the seqfan list by Jeremy Gardiner [1]. The email starts: arXiv:1410.2193v1 [math.CO] 8 Oct 2014 It strikes me as an interesting coincidence that the following se- quences appear to have the same underlying parity. Then he lists six sequences with the same parity: • A128975 The number of unordered three-heap P-positions in Nim. • A102393, A wicked evil sequence. • A029886, Convolution of the Thue-Morse sequence with itself. 1 • A092524, Binary representation of n interpreted in base p, where p is the smallest prime factor of n. • A104258, Replace 2i with ni in the binary representation of n. n • A061297, a(n)= Pr=0 lcm(n, n−1, n−2,...,n−r+1)/ lcm(1, 2, 3,...,r). -
Lecture 1: Propositional Logic
Lecture 1: Propositional Logic Syntax Semantics Truth tables Implications and Equivalences Valid and Invalid arguments Normal forms Davis-Putnam Algorithm 1 Atomic propositions and logical connectives An atomic proposition is a statement or assertion that must be true or false. Examples of atomic propositions are: “5 is a prime” and “program terminates”. Propositional formulas are constructed from atomic propositions by using logical connectives. Connectives false true not and or conditional (implies) biconditional (equivalent) A typical propositional formula is The truth value of a propositional formula can be calculated from the truth values of the atomic propositions it contains. 2 Well-formed propositional formulas The well-formed formulas of propositional logic are obtained by using the construction rules below: An atomic proposition is a well-formed formula. If is a well-formed formula, then so is . If and are well-formed formulas, then so are , , , and . If is a well-formed formula, then so is . Alternatively, can use Backus-Naur Form (BNF) : formula ::= Atomic Proposition formula formula formula formula formula formula formula formula formula formula 3 Truth functions The truth of a propositional formula is a function of the truth values of the atomic propositions it contains. A truth assignment is a mapping that associates a truth value with each of the atomic propositions . Let be a truth assignment for . If we identify with false and with true, we can easily determine the truth value of under . The other logical connectives can be handled in a similar manner. Truth functions are sometimes called Boolean functions. 4 Truth tables for basic logical connectives A truth table shows whether a propositional formula is true or false for each possible truth assignment. -
The Peripatetic Program in Categorical Logic: Leibniz on Propositional Terms
THE REVIEW OF SYMBOLIC LOGIC, Page 1 of 65 THE PERIPATETIC PROGRAM IN CATEGORICAL LOGIC: LEIBNIZ ON PROPOSITIONAL TERMS MARKO MALINK Department of Philosophy, New York University and ANUBAV VASUDEVAN Department of Philosophy, University of Chicago Abstract. Greek antiquity saw the development of two distinct systems of logic: Aristotle’s theory of the categorical syllogism and the Stoic theory of the hypothetical syllogism. Some ancient logicians argued that hypothetical syllogistic is more fundamental than categorical syllogistic on the grounds that the latter relies on modes of propositional reasoning such as reductio ad absurdum. Peripatetic logicians, by contrast, sought to establish the priority of categorical over hypothetical syllogistic by reducing various modes of propositional reasoning to categorical form. In the 17th century, this Peripatetic program of reducing hypothetical to categorical logic was championed by Gottfried Wilhelm Leibniz. In an essay titled Specimina calculi rationalis, Leibniz develops a theory of propositional terms that allows him to derive the rule of reductio ad absurdum in a purely categor- ical calculus in which every proposition is of the form AisB. We reconstruct Leibniz’s categorical calculus and show that it is strong enough to establish not only the rule of reductio ad absurdum,but all the laws of classical propositional logic. Moreover, we show that the propositional logic generated by the nonmonotonic variant of Leibniz’s categorical calculus is a natural system of relevance logic ¬ known as RMI→ . From its beginnings in antiquity up until the late 19th century, the study of formal logic was shaped by two distinct approaches to the subject. The first approach was primarily concerned with simple propositions expressing predicative relations between terms.