Mathematical Logic
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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 -
The Deduction Rule and Linear and Near-Linear Proof Simulations
The Deduction Rule and Linear and Near-linear Proof Simulations Maria Luisa Bonet¤ Department of Mathematics University of California, San Diego Samuel R. Buss¤ Department of Mathematics University of California, San Diego Abstract We introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (or formulas) in the proof. A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by \nearly linear" is meant the ratio of proof lengths is O(®(n)) where ® is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by O(n ¢ ®(n)). Also we show ¤Supported in part by NSF Grant DMS-8902480. 1 that a Frege proof of n lines can be transformed into a tree-like Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by O(n log n). -
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. -
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. -
Logic: Representation and Automated Reasoning
Logic Knowledge Representation & Reasoning Mechanisms Logic ● Logic as KR ■ Propositional Logic ■ Predicate Logic (predicate Calculus) ● Automated Reasoning ■ Logical inferences ■ Resolution and Theorem-proving Logic ● Logic as KR ■ Propositional Logic ■ Predicate Logic (predicate Calculus) ● Automated Reasoning ■ Logical inferences ■ Resolution and Theorem-proving Propositional Logic ● Symbols: ■ truth symbols: true, false ■ propositions: a statement that is “true” or “false” but not both E.g., P = “Two plus two equals four” Q = “It rained yesterday.” ■ connectives: ~, →, ∧, ∨, ≡ • Sentences - propositions or truth symbols • Well formed formulas (expressions) - sentences that are legally well-formed with connectives E.g., P ∧ R → and P ~ are not wff but P ∧ R → ~ Q is Examples P Q AI is hard but it is interesting P ∧ Q AI is neither hard nor interesting ~P ∧ ~ Q P Q If you don’t do assignments then you will fail P → Q ≡ Do assignments or fail (Prove by truth table) ~ P ∨ Q None or both of P and Q is true (~ P ∧ ~ Q) ∨ (P ∧ Q) ≡ T Exactly one of P and Q is true (~ P ∧ Q) ∨ (P ∧ ~ Q) ≡ T Predicate Logic ● Symbols: • truth symbols • constants: represents objects in the world • variables: represents ranging objects } Terms • functions: represent properties • Predicates: functions of terms with true/false values e.g., bill_residence_city (vancouver) or lives (bill, vancouver) ● Atomic sentences: true, false, or predicates ● Quantifiers: ∀, ∃ ● Sentences (expressions): sequences of legal applications of connectives and quantifiers to atomic -
Verifying the Unification Algorithm In
Verifying the Uni¯cation Algorithm in LCF Lawrence C. Paulson Computer Laboratory Corn Exchange Street Cambridge CB2 3QG England August 1984 Abstract. Manna and Waldinger's theory of substitutions and uni¯ca- tion has been veri¯ed using the Cambridge LCF theorem prover. A proof of the monotonicity of substitution is presented in detail, as an exam- ple of interaction with LCF. Translating the theory into LCF's domain- theoretic logic is largely straightforward. Well-founded induction on a complex ordering is translated into nested structural inductions. Cor- rectness of uni¯cation is expressed using predicates for such properties as idempotence and most-generality. The veri¯cation is presented as a series of lemmas. The LCF proofs are compared with the original ones, and with other approaches. It appears di±cult to ¯nd a logic that is both simple and exible, especially for proving termination. Contents 1 Introduction 2 2 Overview of uni¯cation 2 3 Overview of LCF 4 3.1 The logic PPLAMBDA :::::::::::::::::::::::::: 4 3.2 The meta-language ML :::::::::::::::::::::::::: 5 3.3 Goal-directed proof :::::::::::::::::::::::::::: 6 3.4 Recursive data structures ::::::::::::::::::::::::: 7 4 Di®erences between the formal and informal theories 7 4.1 Logical framework :::::::::::::::::::::::::::: 8 4.2 Data structure for expressions :::::::::::::::::::::: 8 4.3 Sets and substitutions :::::::::::::::::::::::::: 9 4.4 The induction principle :::::::::::::::::::::::::: 10 5 Constructing theories in LCF 10 5.1 Expressions :::::::::::::::::::::::::::::::: -
Chapter 9: Initial Theorems About Axiom System
Initial Theorems about Axiom 9 System AS1 1. Theorems in Axiom Systems versus Theorems about Axiom Systems ..................................2 2. Proofs about Axiom Systems ................................................................................................3 3. Initial Examples of Proofs in the Metalanguage about AS1 ..................................................4 4. The Deduction Theorem.......................................................................................................7 5. Using Mathematical Induction to do Proofs about Derivations .............................................8 6. Setting up the Proof of the Deduction Theorem.....................................................................9 7. Informal Proof of the Deduction Theorem..........................................................................10 8. The Lemmas Supporting the Deduction Theorem................................................................11 9. Rules R1 and R2 are Required for any DT-MP-Logic........................................................12 10. The Converse of the Deduction Theorem and Modus Ponens .............................................14 11. Some General Theorems About ......................................................................................15 12. Further Theorems About AS1.............................................................................................16 13. Appendix: Summary of Theorems about AS1.....................................................................18 2 Hardegree, -
Logic- Sentence and Propositions
Sentence and Proposition Sentence Sentence= वा啍य, Proposition= त셍कवा啍य/ प्रततऻप्तत=Statement Sentence is the unit or part of some language. Sentences can be expressed in all tense (present, past or future) Sentence is not able to explain quantity and quality. A sentence may be interrogative (प्रश्नवाच셍), simple or declarative (घोषड़ा配म셍), command or imperative (आदेशा配म셍), exclamatory (ववमयतबोध셍), indicative (तनदेशा配म셍), affirmative (भावा配म셍), negative (तनषेधा配म셍), skeptical (संदेहा配म셍) etc. Note : Only indicative sentences (सं셍ेता配म셍तवा啍य) are proposition. Philosophy Dept, E- contents-01( Logic-B.A Sem.III & IV) by Dr Rajendra Kr.Verma 1 All kind of sentences are not proposition, only those sentences are called proposition while they will determine or evaluate in terms of truthfulness or falsity. Sentences are governed by its own grammar. (for exp- sentence of hindi language govern by hindi grammar) Sentences are correct or incorrect / pure or impure. Sentences are may be either true or false. Philosophy Dept, E- contents-01( Logic-B.A Sem.III & IV) by Dr Rajendra Kr.Verma 2 Proposition Proposition are regarded as the material of our reasoning and we also say that proposition and statements are regarded as same. Proposition is the unit of logic. Proposition always comes in present tense. (sentences - all tenses) Proposition can explain quantity and quality. (sentences- cannot) Meaning of sentence is called proposition. Sometime more then one sentences can expressed only one proposition. Example : 1. ऩानीतबरसतरहातहैत.(Hindi) 2. ऩावुष तऩड़तोत (Sanskrit) 3. It is raining (English) All above sentences have only one meaning or one proposition. -
Mathematical Induction
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 the first rung of the ladder. 2. If we can reach a particular rung of the ladder, then we can reach the next rung. From (1), we can reach the first rung. Then by applying (2), we can reach the second rung. Applying (2) again, the third rung. And so on. We can apply (2) any number of times to reach any particular rung, no matter how high up. This example motivates proof by mathematical induction. Principle of Mathematical Induction Principle of Mathematical Induction: To prove that P(n) is true for all positive integers n, we complete these steps: Basis Step: Show that P(1) is true. Inductive Step: Show that P(k) → P(k + 1) is true for all positive integers k. To complete the inductive step, assuming the inductive hypothesis that P(k) holds for an arbitrary integer k, show that must P(k + 1) be true. Climbing an Infinite Ladder Example: Basis Step: By (1), we can reach rung 1. Inductive Step: Assume the inductive hypothesis that we can reach rung k. Then by (2), we can reach rung k + 1. Hence, P(k) → P(k + 1) is true for all positive integers k. We can reach every rung on the ladder. Logic and Mathematical Induction • Mathematical induction can be expressed as the rule of inference (P(1) ∧ ∀k (P(k) → P(k + 1))) → ∀n P(n), where the domain is the set of positive integers. -
First Order Logic and Nonstandard Analysis
First Order Logic and Nonstandard Analysis Julian Hartman September 4, 2010 Abstract This paper is intended as an exploration of nonstandard analysis, and the rigorous use of infinitesimals and infinite elements to explore properties of the real numbers. I first define and explore first order logic, and model theory. Then, I prove the compact- ness theorem, and use this to form a nonstandard structure of the real numbers. Using this nonstandard structure, it it easy to to various proofs without the use of limits that would otherwise require their use. Contents 1 Introduction 2 2 An Introduction to First Order Logic 2 2.1 Propositional Logic . 2 2.2 Logical Symbols . 2 2.3 Predicates, Constants and Functions . 2 2.4 Well-Formed Formulas . 3 3 Models 3 3.1 Structure . 3 3.2 Truth . 4 3.2.1 Satisfaction . 5 4 The Compactness Theorem 6 4.1 Soundness and Completeness . 6 5 Nonstandard Analysis 7 5.1 Making a Nonstandard Structure . 7 5.2 Applications of a Nonstandard Structure . 9 6 Sources 10 1 1 Introduction The founders of modern calculus had a less than perfect understanding of the nuts and bolts of what made it work. Both Newton and Leibniz used the notion of infinitesimal, without a rigorous understanding of what they were. Infinitely small real numbers that were still not zero was a hard thing for mathematicians to accept, and with the rigorous development of limits by the likes of Cauchy and Weierstrass, the discussion of infinitesimals subsided. Now, using first order logic for nonstandard analysis, it is possible to create a model of the real numbers that has the same properties as the traditional conception of the real numbers, but also has rigorously defined infinite and infinitesimal elements. -
5 Deduction in First-Order Logic
5 Deduction in First-Order Logic The system FOLC. Let C be a set of constant symbols. FOLC is a system of deduction for # the language LC . Axioms: The following are axioms of FOLC. (1) All tautologies. (2) Identity Axioms: (a) t = t for all terms t; (b) t1 = t2 ! (A(x; t1) ! A(x; t2)) for all terms t1 and t2, all variables x, and all formulas A such that there is no variable y occurring in t1 or t2 such that there is free occurrence of x in A in a subformula of A of the form 8yB. (3) Quantifier Axioms: 8xA ! A(x; t) for all formulas A, variables x, and terms t such that there is no variable y occurring in t such that there is a free occurrence of x in A in a subformula of A of the form 8yB. Rules of Inference: A; (A ! B) Modus Ponens (MP) B (A ! B) Quantifier Rule (QR) (A ! 8xB) provided the variable x does not occur free in A. Discussion of the axioms and rules. (1) We would have gotten an equivalent system of deduction if instead of taking all tautologies as axioms we had taken as axioms all instances (in # LC ) of the three schemas on page 16. All instances of these schemas are tautologies, so the change would have not have increased what we could 52 deduce. In the other direction, we can apply the proof of the Completeness Theorem for SL by thinking of all sententially atomic formulas as sentence # letters. The proof so construed shows that every tautology in LC is deducible using MP and schemas (1){(3). -
Tarski's Threat to the T-Schema
Tarski’s Threat to the T-Schema MSc Thesis (Afstudeerscriptie) written by Cian Chartier (born November 10th, 1986 in Montr´eal, Canada) under the supervision of Prof. dr. Frank Veltman, and submitted to the Board of Examiners in partial fulfillment of the requirements for the degree of MSc in Logic at the Universiteit van Amsterdam. Date of the public defense: Members of the Thesis Committee: August 30, 2011 Prof. Dr. Frank Veltman Prof. Dr. Michiel van Lambalgen Prof. Dr. Benedikt L¨owe Dr. Robert van Rooij Abstract This thesis examines truth theories. First, four relevant programs in philos- ophy are considered. Second, four truth theories are compared according to a range of criteria. The truth theories are categorised according to Leitgeb’s eight criteria for a truth theory. The four truth theories are then compared with each-other based on three new criteria. In this way their relative use- fulness in pursuing some of the aims of the four programs is evaluated. This presents a springboard for future work on truth: proposing ideas for differ- ent truth theories that advance unambiguously different programs on the basis of the four truth theories proposed. 1 Acknowledgements I first thank my supervisor Prof. Dr. Frank Veltman for regularly read- ing and providing constructive feedback, recommending some papers along the way. This thesis has been much improved by his corrections and input, including extensive discussions and suggestions of the ideas proposed here. His patience in working with me on the thesis has been much appreciated. Among the other professors in the ILLC, Dr. Robert van Rooij also provided helpful discussion.