Summary Last Lecture. Automated Reasoning. Outline of the Lecture. Definition sequent calculus. Theorem (Normalisation and Strong Normalisation)
|
|
- Ruby Mason
- 8 years ago
- Views:
Transcription
1 Summary Summary Last Lecture sequent calculus Automated Reasoning Georg Moser Institute of Computer UIBK Winter 013 (Normalisation and Strong Normalisation) let Π be a proof in minimal logic 1 a reduction sequence Π = Π 1,, Π n computable upper bound n on the maximal length of any reduction sequence Corollary if T 0 is complete and T 1, T are satisfiable extensions of T 0, then T 1 T is satisfiable GM (Institute of Computer UIBK) Automated Reasoning 17/1 Summary let G be a set of universal sentences (of L) without =, then the following is equivalent 1 G is satisfiable G has a Herbrand model 3 finite G 0 Gr(G), G 0 has a Herbrand model Corollary x 1 x n G(x 1,, x n ) is valid iff there are ground terms t k 1,, tk n, k N and the following is valid: G(t 1 1,, t1 n) G(t k 1,, tk n ) formula, formula G not containing individual, nor function constants, nor = such that G GM (Institute of Computer UIBK) Automated Reasoning 18/1 Summary Outline of the Lecture Propositional Logic short reminder of propositional logic, soundness and completeness theorem, natural deduction, propositional resolution irst Order Logic introduction, syntax, semantics, undecidability of first-order, Löwenheim- Skolem, compactness, model existence theorem, natural deduction, completeness, sequent calculus, normalisation Properties of irst Order Logic Craig s Interpolation, Robinson s Joint Consistency, Herbrand s Limits and xtensions of irst Order Logic, Curry-Howard Isomorphism, Limits, Second-Order Logic GM (Institute of Computer UIBK) Automated Reasoning 19/1
2 Background: Russel s paradox according to naive set theory, any definable collection is a set; this is not a good idea Proof 1 let R := {x x x} as x x is a definition (= a predicate) this should be set 3 so either R R, or R R, but R R R R R R R R 4 hence R R R R, which is a contradiction 5 thus naive set theory is inconsistent Oops, what to do? Brouwer s Way Out (174) Change Mathematics! intuitionistic logic is a restriction of classical logic, where certain formulas are no longer derivable for example A A is no longer valid its interpretation in intuitionistic logic is: there is an argument for A or there is a argument for A (= from the assumption A we can prove a contradiction) GM (Institute of Computer UIBK) Automated Reasoning 130/1 GM (Institute of Computer UIBK) Automated Reasoning 131/1 A Problem with the xcluded Middle solutions of the equation x y = z with x and y irrational and z rational Proof 1 is an irrational number one of the following two cases has to occur: is rational, then is irrational, then x = y = z = x = y = z = ( ) = = prototypical example of a non-constructive proof : i introduction : i : i elimination : e G G : i G : e : e : e GM (Institute of Computer UIBK) Automated Reasoning 13/1 GM (Institute of Computer UIBK) Automated Reasoning 133/1
3 Remark note the absence of introduction elimination : i : e : e : e (Alternative) an equivalent formalisation of intutitionistic logic is given by sequent calculus with the following restriction: sequents Γ : 1 (Brower, Heyting, Kolmogorow (Kreisel) Interpretation) an argument for is an argument for and an argument for is an argument of or an argument for is a transformation of an argument for into an argument for is interpreted as no argument for can exist the formal definition needs Kripke models GM (Institute of Computer UIBK) Automated Reasoning 134/1 GM (Institute of Computer UIBK) Automated Reasoning 135/1 Kripke Models a frame is a pair (W, ), where W denotes a nonempty set of worlds and denotes a preorder on W a Kripke model on a frame = (W, ) is a triple K = (W,, (A p ) p W ) such that for all p: 1 A p = (A p, a p ) A p is a non-empty set (the domain in world p) 3 A p is a mapping that associates predicate constants to domains predicate symbols P, p, q W, (a 1,, a n ) A n p: p q, A p = P(a 1,, a n ) implies A q = P(a 1,, a n ) we set A = p W A p Convention suppose (x 1,, x n ) is formula with free variables x 1,, x n ; we write (a 1,, a n ) for the interpretation of x i by a i A in for a given Kripke model K = (W,, (A p ) p W ) the satisfaction relation is defined as follows: K, p K, p K, p P(a 1,, a n ) if A p = P(a 1,, a n ) K, p A B iff K, p A and K, p B K, p A B iff K, p A or K, p B K, p A B iff for all q p: K, q A implies K, q B a formula is valid in K if K, p for all p W GM (Institute of Computer UIBK) Automated Reasoning 136/1 GM (Institute of Computer UIBK) Automated Reasoning 137/1
4 Some Transfer Results Natural Deduction vs Sequent Calculus Natural Deduction for Minimal Logic natural deduction (and the sequent calculus) for intuitionistic logic is sound and complete introduction A A elimination natural deduction is strongly normalising sequent calculus admits cut-eliminiation Γ Γ Γ Γ, G Γ, G Γ G Craig s interpolation theoremm holds for intutitionistic logic Γ, Γ GM (Institute of Computer UIBK) Automated Reasoning 138/1 GM (Institute of Computer UIBK) Automated Reasoning 139/1 Natural Deduction vs Sequent Calculus A Sequent Calculus for Minimal Logic left right, Γ C, Γ C : l Γ 1 Γ Γ 1, Γ, Γ C, Γ C : l, Γ 1 C, Γ C, Γ 1, Γ C : l : r Γ : r Γ 1, Γ C, Γ 1, Γ C : l Γ, : l : r Natural Deduction vs Sequent Calculus Lemma let S = (Γ C) be a sequent; proof Π of S in natural deduction iff proof Ψ of S in the sequent calculus Proof direction from left to right is shown by induction on the length of Π, ie, on the number of sequents in Π 1 the base case is immediate as Π A A iff Ψ A A for the step case, consider the case that Π has the following form: Π 0 by induction hypothesis a sequent calculus proof Ψ 0 of GM (Institute of Computer UIBK) Automated Reasoning 140/1 GM (Institute of Computer UIBK) Automated Reasoning 141/1
5 Natural Deduction vs Sequent Calculus Proof (cont d) 3 the following is a correct proof: Natural Deduction vs Sequent Calculus Natural Deduction Structural Rules Ψ 0 : l cut 4 all other cases are similar the other direction follows by induction on the length of Ψ contraction weakening A, A, Γ C A, Γ C Γ C A, Γ C Question is this really correct? Answer no, we forgot about the structural rules in the direction from right to left Observations note the restriction to one formula in the succedent contraction and weakening can also be represented by changed axioms and representation of sequents GM (Institute of Computer UIBK) Automated Reasoning 14/1 GM (Institute of Computer UIBK) Automated Reasoning 143/1 Natural Deduction vs Sequent Calculus Outline of the Lecture Propositional Logic short reminder of propositional logic, soundness and completeness theorem, natural deduction, propositional resolution irst Order Logic introduction, syntax, semantics, undecidability of first-order, Löwenheim- Skolem, compactness, model existence theorem, natural deduction, completeness, sequent calculus, normalisation Properties of irst Order Logic Craig s Interpolation, Robinson s Joint Consistency, Herbrand s Limits and xtensions of irst Order Logic, Curry-Howard Isomorphism, Second-Order Logic Typed λ-calculus Typed λ-calculus (types and terms) we define the set of types T and typed λ-terms as follows: a variable type: α,, γ, if σ, τ are types, then (σ τ) is a (product) type if σ, τ are types, then (σ τ) is a (function) type any (typed) variable x : σ is a (typed) term if M : σ, N : τ are terms, then M, N : σ τ is a term if M : σ τ is a term, then fst(m) : σ and snd(m) : τ are terms if M : τ is a term, x : σ a variable, then the abstraction (λx σ M) : σ τ is a term if M : σ τ, N : σ are terms, then the application (MN) : τ is a term GM (Institute of Computer UIBK) Automated Reasoning 144/1 GM (Institute of Computer UIBK) Automated Reasoning 145/1
6 Typed λ-calculus Typed λ-calculus the following are (well-formed, typed) terms λfxfx : (σ τ) σ τ λxx, λyy : (σ σ) (τ τ) but λxxx cannot be typed! the set of free variables of a term is defined as follows V(x) = {x} V(λxM) = V(M) {x} V(MN) = V( M, N ) = V(M) V(N) V(fst(M)) = V(snd(M)) = V(M) (informal) occurrences of x in the scope of λ are called bound GM (Institute of Computer UIBK) Automated Reasoning 146/1 (substitution) M[x := N] denotes the result of substituting N for x in M x[x := N] = N and if x y, then y[x := N] = y (λxm)[x := N] = λxm (λym)[x := N] = λy(m[x := N]), if x y and y V(N) (M 1 M )[x := N] = (M 1 [x := N])(M [x := N]) M 1, M [x := N] = M 1 [x := N], M [x := N] fst(m)[x := N] = fst(m[x := N]) snd(m)[x := N] = snd(m[x := N]) (-reduction) (λxm)n M[x := N] fst( M, N ) M snd( M, N ) N GM (Institute of Computer UIBK) Automated Reasoning 147/1 Typed λ-calculus Lemma -reduction is closed under context: LM LN ML NL λxm λxn M N = M, L N, L L, M L, N fst(m) fst(n) snd(m) snd(n) (λf λxfx)(λxx + 1)0 (λx(λxx + 1)x)0 (λxx + 1)0 1 GM (Institute of Computer UIBK) Automated Reasoning 148/1 Curry-Howard Isomorphism Type Checking Γ M : σ Γ N : τ Γ M, N : σ τ Remarks x : σ, Γ x : σ ref pair Γ M : σ τ Γ fst(m) : σ fst Γ M : σ τ Γ snd(m) : τ snd Γ, x : σ M : τ Γ λxm : σ τ abs Γ M : σ τ Γ N : σ Γ MN : τ 1 different to type checking system in functional programming we have type assignment for product types weakening is incorporated into the axiom, sequents arepresented as sets GM (Institute of Computer UIBK) Automated Reasoning 149/1 app
7 Types as ormulas Types as ormulas (Types as ormulas) (ref) (Ax) + structural rules (pair) ( : i) (abs) ( : i) (fst) ( : e) (app) ( : e) (snd) ( : e) Question what is the correspondence to? Answer sum types! a (binary) sum type describes a set of values drawn from exactly two given types GM (Institute of Computer UIBK) Automated Reasoning 150/1 Types as ormulas Type System for Sum Types Γ M : σ Γ inl(m) : σ + τ Γ N : τ Γ inr(n) : σ + τ Γ M : σ + τ Γ, x : σ N 1 : γ Γ, y : τ N : γ Γ case M of inl(x) N 1 inr(y) N : γ (-reduction, cont d) (λxm)n M[x := N] fst( M, N ) snd( M, N ) M N case inl(m) of inl(x) N 1 inr(y) N N1 [x := M] case inr(n) of inl(x) N 1 inr(y) N N [y := N] GM (Institute of Computer UIBK) Automated Reasoning 151/1 Types as ormulas Curry-Howard Correspondence Proofs as Programs Proofs as Programs (Types as ormulas (cont d)) (ref) (Ax) + structural rules (pair) ( : i) (abs) ( : i) (fst) ( : e) (app) ( : e) (snd) ( : e) (inl) ( : i) (inr) ( : i) (case) ( : e) (Curry-Howard) (normalisation) Π 1 Π Γ M : σ Γ N : τ Γ M, N : σ τ Γ fst( M, N ) : σ = Π 1 Γ M : σ the Curry-Howard correspondence (aka Curry-Howard isomorophism) consists of the following parts: 1 formulas = types proof = programs 3 normalisation = computation (-reduction) fst( M, N ) M GM (Institute of Computer UIBK) Automated Reasoning 15/1 GM (Institute of Computer UIBK) Automated Reasoning 153/1
8 Proofs as Programs Proofs as Programs (normalisation) Π 1 Γ, x : σ M : τ Γ λxm : σ τ Γ (λxm)n : τ Π Γ N : σ = Π 1 [x\π ] Γ M[x := N] : τ Discussion act the Curry-Howard correspondence extends to many systems, for example intuitionistic logic and λ-calculus Hilbert axioms and combinatory logic linear logic and interaction nets the proof Π 1 [x\π ] represents the proof obtained from Π 1 by substituting Π into Π 1 instead of the use of ref wrt x (-reduction) (λxm)n M Observations the Curry-Howard correspondence 1 links logic with programming, ie, provides an explanation for the sucess of logic in computer science allows to mutual enrich both areas 3 provides a formally verified form of programming 4 GM (Institute of Computer UIBK) Automated Reasoning 154/1 GM (Institute of Computer UIBK) Automated Reasoning 155/1 Proofs as Programs strong normalisation of simply typed λ-calculus is typically proved via strong normalisation of minimal logic similarily, undecidablilty of type inhabitation of dependent types follows from undeciabilty of intuitionistic predicate logic correspondencence between interaction nets and linear logic provides type system to interaction nets formalisation of the theory of forbidden patterns for rewrite strategies in Isabelle provides a machine-checked theory code export from Isabelle provides OCaml code that has been integrated into T T T GM (Institute of Computer UIBK) Automated Reasoning 156/1
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
More informationSchedule. Logic (master program) Literature & Online Material. gic. Time and Place. Literature. Exercises & Exam. Online Material
OLC mputational gic Schedule Time and Place Thursday, 8:15 9:45, HS E Logic (master program) Georg Moser Institute of Computer Science @ UIBK week 1 October 2 week 8 November 20 week 2 October 9 week 9
More informationSummary of Last Lecture. Automated Reasoning. Outline of the Lecture. Definition. Example. Lemma non-redundant superposition inferences are liftable
Summary Automated Reasoning Georg Moser Institute of Computer Science @ UIBK Summary of Last Lecture C A D B (C D)σ C s = t D A[s ] (C D A[t])σ ORe OPm(L) C s = t D u[s ] v SpL (C D u[t] v)σ C s t Cσ ERR
More informationFoundational Proof Certificates
An application of proof theory to computer science INRIA-Saclay & LIX, École Polytechnique CUSO Winter School, Proof and Computation 30 January 2013 Can we standardize, communicate, and trust formal proofs?
More informationMATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.
MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P
More information(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 (,,,,
More informationA Propositional Dynamic Logic for CCS Programs
A Propositional Dynamic Logic for CCS Programs Mario R. F. Benevides and L. Menasché Schechter {mario,luis}@cos.ufrj.br Abstract This work presents a Propositional Dynamic Logic in which the programs are
More informationFirst-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
More informationNon-deterministic Semantics and the Undecidability of Boolean BI
1 Non-deterministic Semantics and the Undecidability of Boolean BI DOMINIQUE LARCHEY-WENDLING, LORIA CNRS, Nancy, France DIDIER GALMICHE, LORIA Université Henri Poincaré, Nancy, France We solve the open
More informationLecture 13 of 41. More Propositional and Predicate Logic
Lecture 13 of 41 More Propositional and Predicate Logic Monday, 20 September 2004 William H. Hsu, KSU http://www.kddresearch.org http://www.cis.ksu.edu/~bhsu Reading: Sections 8.1-8.3, Russell and Norvig
More informationCS510 Software Engineering
CS510 Software Engineering Propositional Logic Asst. Prof. Mathias Payer Department of Computer Science Purdue University TA: Scott A. Carr Slides inspired by Xiangyu Zhang http://nebelwelt.net/teaching/15-cs510-se
More informationAutomated Theorem Proving - summary of lecture 1
Automated Theorem Proving - summary of lecture 1 1 Introduction Automated Theorem Proving (ATP) deals with the development of computer programs that show that some statement is a logical consequence of
More informationHow To Write Intuitionistic Logic
AN INTUITIONISTIC EPISTEMIC LOGIC FOR ASYNCHRONOUS COMMUNICATION by Yoichi Hirai A Master Thesis Submitted to the Graduate School of the University of Tokyo on February 10, 2010 in Partial Fulfillment
More informationHandout #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
More informationON 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
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationMathematical 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
More informationPROOFS AND TYPES JEAN-YVES GIRARD PAUL TAYLOR YVES LAFONT. CAMBRIDGE UNIVERSITY PRESS Cambridge. Translated and with appendices by
PROOFS AND TYPES JEAN-YVES GIRARD Translated and with appendices by PAUL TAYLOR YVES LAFONT CAMBRIDGE UNIVERSITY PRESS Cambridge New York Melbourne New Rochelle Sydney ii Published by the Press Syndicate
More information3. 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)
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationFIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.
FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is
More informationTuring Degrees and Definability of the Jump. Theodore A. Slaman. University of California, Berkeley. CJuly, 2005
Turing Degrees and Definability of the Jump Theodore A. Slaman University of California, Berkeley CJuly, 2005 Outline Lecture 1 Forcing in arithmetic Coding and decoding theorems Automorphisms of countable
More informationIntroduction to Finite Fields (cont.)
Chapter 6 Introduction to Finite Fields (cont.) 6.1 Recall Theorem. Z m is a field m is a prime number. Theorem (Subfield Isomorphic to Z p ). Every finite field has the order of a power of a prime number
More informationON THE DEGREE OF MAXIMALITY OF DEFINITIONALLY COMPLETE LOGICS
Bulletin of the Section of Logic Volume 15/2 (1986), pp. 72 79 reedition 2005 [original edition, pp. 72 84] Ryszard Ladniak ON THE DEGREE OF MAXIMALITY OF DEFINITIONALLY COMPLETE LOGICS The following definition
More informationParametric Domain-theoretic models of Linear Abadi & Plotkin Logic
Parametric Domain-theoretic models of Linear Abadi & Plotkin Logic Lars Birkedal Rasmus Ejlers Møgelberg Rasmus Lerchedahl Petersen IT University Technical Report Series TR-00-7 ISSN 600 600 February 00
More informationResearch Note. Bi-intuitionistic Boolean Bunched Logic
UCL DEPARTMENT OF COMPUTER SCIENCE Research Note RN/14/06 Bi-intuitionistic Boolean Bunched Logic June, 2014 James Brotherston Dept. of Computer Science University College London Jules Villard Dept. of
More informationHow To Understand The Theory Of Hyperreals
Ultraproducts and Applications I Brent Cody Virginia Commonwealth University September 2, 2013 Outline Background of the Hyperreals Filters and Ultrafilters Construction of the Hyperreals The Transfer
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More informationRigorous Software Development CSCI-GA 3033-009
Rigorous Software Development CSCI-GA 3033-009 Instructor: Thomas Wies Spring 2013 Lecture 11 Semantics of Programming Languages Denotational Semantics Meaning of a program is defined as the mathematical
More information4. CLASSES OF RINGS 4.1. Classes of Rings class operator A-closed Example 1: product Example 2:
4. CLASSES OF RINGS 4.1. Classes of Rings Normally we associate, with any property, a set of objects that satisfy that property. But problems can arise when we allow sets to be elements of larger sets
More informationBindings, mobility of bindings, and the -quantifier
ICMS, 26 May 2007 1/17 Bindings, mobility of bindings, and the -quantifier Dale Miller, INRIA-Saclay and LIX, École Polytechnique This talk is based on papers with Tiu in LICS2003 & ACM ToCL, and experience
More informationDiscrete 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
More informationGREATEST COMMON DIVISOR
DEFINITION: GREATEST COMMON DIVISOR The greatest common divisor (gcd) of a and b, denoted by (a, b), is the largest common divisor of integers a and b. THEOREM: If a and b are nonzero integers, then their
More informationHow To Know If A Domain Is Unique In An Octempo (Euclidean) Or Not (Ecl)
Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness
More informationINTRODUCTORY 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,
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More information! " # The Logic of Descriptions. Logics for Data and Knowledge Representation. Terminology. Overview. Three Basic Features. Some History on DLs
,!0((,.+#$),%$(-&.& *,2(-$)%&2.'3&%!&, Logics for Data and Knowledge Representation Alessandro Agostini agostini@dit.unitn.it University of Trento Fausto Giunchiglia fausto@dit.unitn.it The Logic of Descriptions!$%&'()*$#)
More informationCONTRIBUTIONS TO ZERO SUM PROBLEMS
CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg
More informationMAP-I Programa Doutoral em Informática. Rigorous Software Development
MAP-I Programa Doutoral em Informática Rigorous Software Development Unidade Curricular em Teoria e Fundamentos Theory and Foundations (UCTF) DI-UM, DCC-FCUP May, 2012 Abstract This text presents a UCTF
More informationDEFINABLE TYPES IN PRESBURGER ARITHMETIC
DEFINABLE TYPES IN PRESBURGER ARITHMETIC GABRIEL CONANT Abstract. We consider the first order theory of (Z, +,
More informationo-minimality and Uniformity in n 1 Graphs
o-minimality and Uniformity in n 1 Graphs Reid Dale July 10, 2013 Contents 1 Introduction 2 2 Languages and Structures 2 3 Definability and Tame Geometry 4 4 Applications to n 1 Graphs 6 5 Further Directions
More informationit is easy to see that α = a
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore
More informationCorrespondence analysis for strong three-valued logic
Correspondence analysis for strong three-valued logic A. Tamminga abstract. I apply Kooi and Tamminga s (2012) idea of correspondence analysis for many-valued logics to strong three-valued logic (K 3 ).
More informationThis chapter is all about cardinality of sets. At first this looks like a
CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },
More informationDegrees of Truth: the formal logic of classical and quantum probabilities as well as fuzzy sets.
Degrees of Truth: the formal logic of classical and quantum probabilities as well as fuzzy sets. Logic is the study of reasoning. A language of propositions is fundamental to this study as well as true
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More informationGROUPS ACTING ON A SET
GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for
More informationH/wk 13, Solutions to selected problems
H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.
More informationTheory of Automated Reasoning An Introduction. Antti-Juhani Kaijanaho
Theory of Automated Reasoning An Introduction Antti-Juhani Kaijanaho Intended as compulsory reading for the Spring 2004 course on Automated Reasononing at Department of Mathematical Information Technology,
More informationFixed-Point Logics and Computation
1 Fixed-Point Logics and Computation Symposium on the Unusual Effectiveness of Logic in Computer Science University of Cambridge 2 Mathematical Logic Mathematical logic seeks to formalise the process of
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More informationType 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
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationCartesian Products and Relations
Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special
More informationBasic Proof Techniques
Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document
More informationSolving Linear Systems, Continued and The Inverse of a Matrix
, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights
More informationComputational Methods for Database Repair by Signed Formulae
Computational Methods for Database Repair by Signed Formulae Ofer Arieli (oarieli@mta.ac.il) Department of Computer Science, The Academic College of Tel-Aviv, 4 Antokolski street, Tel-Aviv 61161, Israel.
More informationSoftware Modeling and Verification
Software Modeling and Verification Alessandro Aldini DiSBeF - Sezione STI University of Urbino Carlo Bo Italy 3-4 February 2015 Algorithmic verification Correctness problem Is the software/hardware system
More informationHow To Prove The Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationDEGREES OF CATEGORICITY AND THE HYPERARITHMETIC HIERARCHY
DEGREES OF CATEGORICITY AND THE HYPERARITHMETIC HIERARCHY BARBARA F. CSIMA, JOHANNA N. Y. FRANKLIN, AND RICHARD A. SHORE Abstract. We study arithmetic and hyperarithmetic degrees of categoricity. We extend
More informationAlgorithmic Software Verification
Algorithmic Software Verification (LTL Model Checking) Azadeh Farzan What is Verification Anyway? Proving (in a formal way) that program satisfies a specification written in a logical language. Formal
More information= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that
Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without
More information1 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
More informationOn the generation of elliptic curves with 16 rational torsion points by Pythagorean triples
On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples Brian Hilley Boston College MT695 Honors Seminar March 3, 2006 1 Introduction 1.1 Mazur s Theorem Let C be a
More informationScalable Automated Symbolic Analysis of Administrative Role-Based Access Control Policies by SMT solving
Scalable Automated Symbolic Analysis of Administrative Role-Based Access Control Policies by SMT solving Alessandro Armando 1,2 and Silvio Ranise 2, 1 DIST, Università degli Studi di Genova, Italia 2 Security
More informationNotes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand
Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such
More informationChapter ML:IV. IV. Statistical Learning. Probability Basics Bayes Classification Maximum a-posteriori Hypotheses
Chapter ML:IV IV. Statistical Learning Probability Basics Bayes Classification Maximum a-posteriori Hypotheses ML:IV-1 Statistical Learning STEIN 2005-2015 Area Overview Mathematics Statistics...... Stochastics
More information1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain
Notes on real-closed fields These notes develop the algebraic background needed to understand the model theory of real-closed fields. To understand these notes, a standard graduate course in algebra is
More informationDedekind s forgotten axiom and why we should teach it (and why we shouldn t teach mathematical induction in our calculus classes)
Dedekind s forgotten axiom and why we should teach it (and why we shouldn t teach mathematical induction in our calculus classes) by Jim Propp (UMass Lowell) March 14, 2010 1 / 29 Completeness Three common
More informationINCIDENCE-BETWEENNESS GEOMETRY
INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full
More informationLogic, Algebra and Truth Degrees 2008. Siena. A characterization of rst order rational Pavelka's logic
Logic, Algebra and Truth Degrees 2008 September 8-11, 2008 Siena A characterization of rst order rational Pavelka's logic Xavier Caicedo Universidad de los Andes, Bogota Under appropriate formulations,
More informationCatalan Numbers. Thomas A. Dowling, Department of Mathematics, Ohio State Uni- versity.
7 Catalan Numbers Thomas A. Dowling, Department of Mathematics, Ohio State Uni- Author: versity. Prerequisites: The prerequisites for this chapter are recursive definitions, basic counting principles,
More informationDegrees that are not degrees of categoricity
Degrees that are not degrees of categoricity Bernard A. Anderson Department of Mathematics and Physical Sciences Gordon State College banderson@gordonstate.edu www.gordonstate.edu/faculty/banderson Barbara
More informationIntroduction to Algebraic Geometry. Bézout s Theorem and Inflection Points
Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a
More information2. 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
More informationCODING TRUE ARITHMETIC IN THE MEDVEDEV AND MUCHNIK DEGREES
CODING TRUE ARITHMETIC IN THE MEDVEDEV AND MUCHNIK DEGREES PAUL SHAFER Abstract. We prove that the first-order theory of the Medvedev degrees, the first-order theory of the Muchnik degrees, and the third-order
More informationThis 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
More informationSemantics and Verification of Software
Semantics and Verification of Software Lecture 21: Nondeterminism and Parallelism IV (Equivalence of CCS Processes & Wrap-Up) Thomas Noll Lehrstuhl für Informatik 2 (Software Modeling and Verification)
More informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More informationBi-approximation Semantics for Substructural Logic at Work
Bi-approximation Semantics for Substructural Logic at Work Tomoyuki Suzuki Department of Computer Science, University of Leicester Leicester, LE1 7RH, United Kingdom tomoyuki.suzuki@mcs.le.ac.uk Abstract
More informationNOTES ON CATEGORIES AND FUNCTORS
NOTES ON CATEGORIES AND FUNCTORS These notes collect basic definitions and facts about categories and functors that have been mentioned in the Homological Algebra course. For further reading about category
More informationSet theory as a foundation for mathematics
V I I I : Set theory as a foundation for mathematics This material is basically supplementary, and it was not covered in the course. In the first section we discuss the basic axioms of set theory and the
More informationThe Banach-Tarski Paradox
University of Oslo MAT2 Project The Banach-Tarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the Banach-Tarski paradox states that for any ball in R, it is possible
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty
More informationA first step towards modeling semistructured data in hybrid multimodal logic
A first step towards modeling semistructured data in hybrid multimodal logic Nicole Bidoit * Serenella Cerrito ** Virginie Thion * * LRI UMR CNRS 8623, Université Paris 11, Centre d Orsay. ** LaMI UMR
More informationInterpersonal Comparisons of Utility: An Algebraic Characterization of Projective Preorders and Some Welfare Consequences
DISCUSSION PAPER SERIES IZA DP No. 2594 Interpersonal Comparisons of Utility: An Algebraic Characterization of Projective Preorders and Some Welfare Consequences Juan Carlos Candeal Esteban Induráin José
More informationOPERATIONAL TYPE THEORY by Adam Petcher Prepared under the direction of Professor Aaron Stump A thesis presented to the School of Engineering of
WASHINGTON NIVERSITY SCHOOL OF ENGINEERING AND APPLIED SCIENCE DEPARTMENT OF COMPTER SCIENCE AND ENGINEERING DECIDING JOINABILITY MODLO GROND EQATIONS IN OPERATIONAL TYPE THEORY by Adam Petcher Prepared
More informationIntroduction. Appendix D Mathematical Induction D1
Appendix D Mathematical Induction D D Mathematical Induction Use mathematical induction to prove a formula. Find a sum of powers of integers. Find a formula for a finite sum. Use finite differences to
More informationDegree Hypergroupoids Associated with Hypergraphs
Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationLecture 8: Resolution theorem-proving
Comp24412 Symbolic AI Lecture 8: Resolution theorem-proving Ian Pratt-Hartmann Room KB2.38: email: ipratt@cs.man.ac.uk 2014 15 In the previous Lecture, we met SATCHMO, a first-order theorem-prover implemented
More informationCOMPUTER SCIENCE TRIPOS
CST.98.5.1 COMPUTER SCIENCE TRIPOS Part IB Wednesday 3 June 1998 1.30 to 4.30 Paper 5 Answer five questions. No more than two questions from any one section are to be answered. Submit the answers in five
More informationSo 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
More informationSystem BV is NP-complete
System BV is NP-complete Ozan Kahramanoğulları 1,2 Computer Science Institute, University of Leipzig International Center for Computational Logic, TU Dresden Abstract System BV is an extension of multiplicative
More information