Logic and Computation
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Mathematics Calculation Policy ÷
Mathematics Calculation Policy ÷ Commissioned by The PiXL Club Ltd. June 2016 This resource is strictly for the use of member schools for as long as they remain members of The PiXL Club. It may not be copied, sold nor transferred to a third party or used by the school after membership ceases. Until such time it may be freely used within the member school. All opinions and contributions are those of the authors. The contents of this resource are not connected with nor endorsed by any other company, organisation or institution. © Copyright The PiXL Club Ltd, 2015 About PiXL’s Calculation Policy • The following calculation policy has been devised to meet requirements of the National Curriculum 2014 for the teaching and learning of mathematics, and is also designed to give pupils a consistent and smooth progression of learning in calculations across the school. • Age stage expectations: The calculation policy is organised according to age stage expectations as set out in the National Curriculum 2014 and the method(s) shown for each year group should be modelled to the vast majority of pupils. However, it is vital that pupils are taught according to the pathway that they are currently working at and are showing to have ‘mastered’ a pathway before moving on to the next one. Of course, pupils who are showing to be secure in a skill can be challenged to the next pathway as necessary. • Choosing a calculation method: Before pupils opt for a written method they should first consider these steps: Should I use a formal Can I do it in my Could I use -
COMPSCI 501: Formal Language Theory Insights on Computability Turing Machines Are a Model of Computation Two (No Longer) Surpris
Insights on Computability Turing machines are a model of computation COMPSCI 501: Formal Language Theory Lecture 11: Turing Machines Two (no longer) surprising facts: Marius Minea Although simple, can describe everything [email protected] a (real) computer can do. University of Massachusetts Amherst Although computers are powerful, not everything is computable! Plus: “play” / program with Turing machines! 13 February 2019 Why should we formally define computation? Must indeed an algorithm exist? Back to 1900: David Hilbert’s 23 open problems Increasingly a realization that sometimes this may not be the case. Tenth problem: “Occasionally it happens that we seek the solution under insufficient Given a Diophantine equation with any number of un- hypotheses or in an incorrect sense, and for this reason do not succeed. known quantities and with rational integral numerical The problem then arises: to show the impossibility of the solution under coefficients: To devise a process according to which the given hypotheses or in the sense contemplated.” it can be determined in a finite number of operations Hilbert, 1900 whether the equation is solvable in rational integers. This asks, in effect, for an algorithm. Hilbert’s Entscheidungsproblem (1928): Is there an algorithm that And “to devise” suggests there should be one. decides whether a statement in first-order logic is valid? Church and Turing A Turing machine, informally Church and Turing both showed in 1936 that a solution to the Entscheidungsproblem is impossible for the theory of arithmetic. control To make and prove such a statement, one needs to define computability. In a recent paper Alonzo Church has introduced an idea of “effective calculability”, read/write head which is equivalent to my “computability”, but is very differently defined. -
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. -
Use Formal and Informal Language in Persuasive Text
Author’S Craft Use Formal and Informal Language in Persuasive Text 1. Focus Objectives Explain Using Formal and Informal Language In this mini-lesson, students will: Say: When I write a persuasive letter, I want people to see things my way. I use • Learn to use both formal and different kinds of language to gain support. To connect with my readers, I use informal language in persuasive informal language. Informal language is conversational; it sounds a lot like the text. way we speak to one another. Then, to make sure that people believe me, I also use formal language. When I use formal language, I don’t use slang words, and • Practice using formal and informal I am more objective—I focus on facts over opinions. Today I’m going to show language in persuasive text. you ways to use both types of language in persuasive letters. • Discuss how they can apply this strategy to their independent writing. Model How Writers Use Formal and Informal Language Preparation Display the modeling text on chart paper or using the interactive whiteboard resources. Materials Needed • Chart paper and markers 1. As the parent of a seventh grader, let me assure you this town needs a new • Interactive whiteboard resources middle school more than it needs Old Oak. If selling the land gets us a new school, I’m all for it. Advanced Preparation 2. Last month, my daughter opened her locker and found a mouse. She was traumatized by this experience. She has not used her locker since. If you will not be using the interactive whiteboard resources, copy the Modeling Text modeling text and practice text onto chart paper prior to the mini-lesson. -
An Arbitrary Precision Computation Package David H
ARPREC: An Arbitrary Precision Computation Package David H. Bailey, Yozo Hida, Xiaoye S. Li and Brandon Thompson1 Lawrence Berkeley National Laboratory, Berkeley, CA 94720 Draft Date: 2002-10-14 Abstract This paper describes a new software package for performing arithmetic with an arbitrarily high level of numeric precision. It is based on the earlier MPFUN package [2], enhanced with special IEEE floating-point numerical techniques and several new functions. This package is written in C++ code for high performance and broad portability and includes both C++ and Fortran-90 translation modules, so that conventional C++ and Fortran-90 programs can utilize the package with only very minor changes. This paper includes a survey of some of the interesting applications of this package and its predecessors. 1. Introduction For a small but growing sector of the scientific computing world, the 64-bit and 80-bit IEEE floating-point arithmetic formats currently provided in most computer systems are not sufficient. As a single example, the field of climate modeling has been plagued with non-reproducibility, in that climate simulation programs give different results on different computer systems (or even on different numbers of processors), making it very difficult to diagnose bugs. Researchers in the climate modeling community have assumed that this non-reproducibility was an inevitable conse- quence of the chaotic nature of climate simulations. Recently, however, it has been demonstrated that for at least one of the widely used climate modeling programs, much of this variability is due to numerical inaccuracy in a certain inner loop. When some double-double arithmetic rou- tines, developed by the present authors, were incorporated into this program, almost all of this variability disappeared [12]. -
Neural Correlates of Arithmetic Calculation Strategies
Cognitive, Affective, & Behavioral Neuroscience 2009, 9 (3), 270-285 doi:10.3758/CABN.9.3.270 Neural correlates of arithmetic calculation strategies MIRIA M ROSENBERG -LEE Stanford University, Palo Alto, California AND MARSHA C. LOVETT AND JOHN R. ANDERSON Carnegie Mellon University, Pittsburgh, Pennsylvania Recent research into math cognition has identified areas of the brain that are involved in number processing (Dehaene, Piazza, Pinel, & Cohen, 2003) and complex problem solving (Anderson, 2007). Much of this research assumes that participants use a single strategy; yet, behavioral research finds that people use a variety of strate- gies (LeFevre et al., 1996; Siegler, 1987; Siegler & Lemaire, 1997). In the present study, we examined cortical activation as a function of two different calculation strategies for mentally solving multidigit multiplication problems. The school strategy, equivalent to long multiplication, involves working from right to left. The expert strategy, used by “lightning” mental calculators (Staszewski, 1988), proceeds from left to right. The two strate- gies require essentially the same calculations, but have different working memory demands (the school strategy incurs greater demands). The school strategy produced significantly greater early activity in areas involved in attentional aspects of number processing (posterior superior parietal lobule, PSPL) and mental representation (posterior parietal cortex, PPC), but not in a numerical magnitude area (horizontal intraparietal sulcus, HIPS) or a semantic memory retrieval area (lateral inferior prefrontal cortex, LIPFC). An ACT–R model of the task successfully predicted BOLD responses in PPC and LIPFC, as well as in PSPL and HIPS. A central feature of human cognition is the ability to strategy use) and theoretical (parceling out the effects of perform complex tasks that go beyond direct stimulus– strategy-specific activity). -
Scheme Representation for First-Logic
Scheme representation for first-order logic Spencer Breiner Carnegie Mellon University Advisor: Steve Awodey Committee: Jeremy Avigad Hans Halvorson Pieter Hofstra February 12, 2014 arXiv:1402.2600v1 [math.LO] 11 Feb 2014 Contents 1 Logical spectra 11 1.1 The spectrum M0 .......................... 11 1.2 Sheaves on M0 ............................ 20 1.3 Thegenericmodel .......................... 24 1.4 The spectral groupoid M = Spec(T)................ 27 1.5 Stability, compactness and definability . 31 1.6 Classicalfirst-orderlogic. 37 2 Pretopos Logic 43 2.1 Coherentlogicandpretoposes. 43 2.2 Factorization in Ptop ........................ 51 2.3 Semantics, slices and localization . 57 2.4 Themethodofdiagrams. .. .. .. .. .. .. .. .. .. 61 2.5 ClassicaltheoriesandBooleanpretoposes . 70 3 Logical Schemes 74 3.1 StacksandSheaves.......................... 74 3.2 Affineschemes ............................ 80 3.3 Thecategoryoflogicalschemes . 87 3.4 Opensubschemesandgluing . 93 3.5 Limitsofschemes........................... 98 1 4 Applications 110 4.1 OT asasite..............................111 4.2 Structuresheafasuniverse . 119 4.3 Isotropy ................................125 4.4 ConceptualCompleteness . 134 2 Introduction Although contemporary model theory has been called “algebraic geometry mi- nus fields” [15], the formal methods of the two fields are radically different. This dissertation aims to shrink that gap by presenting a theory of “logical schemes,” geometric entities which relate to first-order logical theories in much the same way that -
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. -
Chapter 1 Logic and Set Theory
Chapter 1 Logic and Set Theory To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. – Ian Stewart Does God play dice? The mathematics of chaos In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. That is, a proof is a logical argument, not an empir- ical one. One must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture. Mathematical logic is the framework upon which rigorous proofs are built. It is the study of the principles and criteria of valid inference and demonstrations. Logicians have analyzed set theory in great details, formulating a collection of axioms that affords a broad enough and strong enough foundation to mathematical reasoning. The standard form of axiomatic set theory is denoted ZFC and it consists of the Zermelo-Fraenkel (ZF) axioms combined with the axiom of choice (C). Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can lead to contradictions. Avoiding such contradictions was one of the original motivations for the axiomatization of set theory. 1 2 CHAPTER 1. LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. -
Continuous First-Order Logic
Continuous First-Order Logic Wesley Calvert Calcutta Logic Circle 4 September 2011 Wesley Calvert (SIU / IMSc) Continuous First-Order Logic 4 September 2011 1 / 45 Definition A first-order theory T is said to be stable iff there are less than the maximum possible number of types over T , up to equivalence. Theorem (Shelah's Main Gap Theorem) If T is a first-order theory and is stable and . , then the class of models looks like . Otherwise, there's no hope. Model-Theoretic Beginnings Problem Given a theory T , describe the structure of models of T . Wesley Calvert (SIU / IMSc) Continuous First-Order Logic 4 September 2011 2 / 45 Theorem (Shelah's Main Gap Theorem) If T is a first-order theory and is stable and . , then the class of models looks like . Otherwise, there's no hope. Model-Theoretic Beginnings Problem Given a theory T , describe the structure of models of T . Definition A first-order theory T is said to be stable iff there are less than the maximum possible number of types over T , up to equivalence. Wesley Calvert (SIU / IMSc) Continuous First-Order Logic 4 September 2011 2 / 45 Model-Theoretic Beginnings Problem Given a theory T , describe the structure of models of T . Definition A first-order theory T is said to be stable iff there are less than the maximum possible number of types over T , up to equivalence. Theorem (Shelah's Main Gap Theorem) If T is a first-order theory and is stable and . , then the class of models looks like . Otherwise, there's no hope. -
Chapter 6 Formal Language Theory
Chapter 6 Formal Language Theory In this chapter, we introduce formal language theory, the computational theories of languages and grammars. The models are actually inspired by formal logic, enriched with insights from the theory of computation. We begin with the definition of a language and then proceed to a rough characterization of the basic Chomsky hierarchy. We then turn to a more de- tailed consideration of the types of languages in the hierarchy and automata theory. 6.1 Languages What is a language? Formally, a language L is defined as as set (possibly infinite) of strings over some finite alphabet. Definition 7 (Language) A language L is a possibly infinite set of strings over a finite alphabet Σ. We define Σ∗ as the set of all possible strings over some alphabet Σ. Thus L ⊆ Σ∗. The set of all possible languages over some alphabet Σ is the set of ∗ all possible subsets of Σ∗, i.e. 2Σ or ℘(Σ∗). This may seem rather simple, but is actually perfectly adequate for our purposes. 6.2 Grammars A grammar is a way to characterize a language L, a way to list out which strings of Σ∗ are in L and which are not. If L is finite, we could simply list 94 CHAPTER 6. FORMAL LANGUAGE THEORY 95 the strings, but languages by definition need not be finite. In fact, all of the languages we are interested in are infinite. This is, as we showed in chapter 2, also true of human language. Relating the material of this chapter to that of the preceding two, we can view a grammar as a logical system by which we can prove things. -
John P. Burgess Department of Philosophy Princeton University Princeton, NJ 08544-1006, USA [email protected]
John P. Burgess Department of Philosophy Princeton University Princeton, NJ 08544-1006, USA [email protected] LOGIC & PHILOSOPHICAL METHODOLOGY Introduction For present purposes “logic” will be understood to mean the subject whose development is described in Kneale & Kneale [1961] and of which a concise history is given in Scholz [1961]. As the terminological discussion at the beginning of the latter reference makes clear, this subject has at different times been known by different names, “analytics” and “organon” and “dialectic”, while inversely the name “logic” has at different times been applied much more broadly and loosely than it will be here. At certain times and in certain places — perhaps especially in Germany from the days of Kant through the days of Hegel — the label has come to be used so very broadly and loosely as to threaten to take in nearly the whole of metaphysics and epistemology. Logic in our sense has often been distinguished from “logic” in other, sometimes unmanageably broad and loose, senses by adding the adjectives “formal” or “deductive”. The scope of the art and science of logic, once one gets beyond elementary logic of the kind covered in introductory textbooks, is indicated by two other standard references, the Handbooks of mathematical and philosophical logic, Barwise [1977] and Gabbay & Guenthner [1983-89], though the latter includes also parts that are identified as applications of logic rather than logic proper. The term “philosophical logic” as currently used, for instance, in the Journal of Philosophical Logic, is a near-synonym for “nonclassical logic”. There is an older use of the term as a near-synonym for “philosophy of language”.