A Special Issue on Formal Proof
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Russell's 1903 – 1905 Anticipation of the Lambda Calculus
HISTORY AND PHILOSOPHY OF LOGIC, 24 (2003), 15–37 Russell’s 1903 – 1905 Anticipation of the Lambda Calculus KEVIN C. KLEMENT Philosophy Dept, Univ. of Massachusetts, 352 Bartlett Hall, 130 Hicks Way, Amherst, MA 01003, USA Received 22 July 2002 It is well known that the circumflex notation used by Russell and Whitehead to form complex function names in Principia Mathematica played a role in inspiring Alonzo Church’s ‘Lambda Calculus’ for functional logic developed in the 1920s and 1930s. Interestingly, earlier unpublished manuscripts written by Russell between 1903 and 1905—surely unknown to Church—contain a more extensive anticipation of the essential details of the Lambda Calculus. Russell also anticipated Scho¨ nfinkel’s Combinatory Logic approach of treating multi- argument functions as functions having other functions as value. Russell’s work in this regard seems to have been largely inspired by Frege’s theory of functions and ‘value-ranges’. This system was discarded by Russell due to his abandonment of propositional functions as genuine entities as part of a new tack for solving Rus- sell’s paradox. In this article, I explore the genesis and demise of Russell’s early anticipation of the Lambda Calculus. 1. Introduction The Lambda Calculus, as we know it today, was initially developed by Alonzo Church in the late 1920s and 1930s (see, e.g. Church 1932, 1940, 1941, Church and Rosser 1936), although it built heavily on work done by others, perhaps most notably, the early works on Combinatory Logic by Moses Scho¨ nfinkel (see, e.g. his 1924) and Haskell Curry (see, e.g. -
[Math.CV] 6 May 2005
HOLOMORPHIC FLEXIBILITY PROPERTIES OF COMPLEX MANIFOLDS FRANC FORSTNERICˇ Abstract. We obtain results on approximation of holomorphic maps by al- gebraic maps, the jet transversality theorem for holomorphic and algebraic maps between certain classes of manifolds, and the homotopy principle for holomorphic submersions of Stein manifolds to certain algebraic manifolds. 1. Introduction In the present paper we use the term holomorphic flexibility property for any of several analytic properties of complex manifolds which are opposite to Koba- yashi-Eisenman-Brody hyperbolicity, the latter expressing holomorphic rigidity. A connected complex manifold Y is n-hyperbolic for some n ∈{1,..., dim Y } if every entire holomorphic map Cn → Y has rank less than n; for n = 1 this means that every holomorphic map C → Y is constant ([4], [16], [46], [47]). On the other hand, all flexibility properties of Y will require the existence of many such maps. We shall center our discussion around the following classical property which was studied by many authors (see the surveys [53] and [25]): Oka property: Every continuous map f0 : X → Y from a Stein manifold X is homotopic to a holomorphic map f1 : X → Y ; if f0 is holomorphic on (a neighbor- hood of) a compact H(X)-convex subset K of X then a homotopy {ft}t∈[0,1] from f0 to a holomorphic map f1 can be chosen such that ft is holomorphic and uniformly close to f0 on K for every t ∈ [0, 1]. Here, H(X)-convexity means convexity with respect to the algebra H(X) of all holomorphic functions on X (§2). -
Moduli Spaces
spaces. Moduli spaces such as the moduli of elliptic curves (which we discuss below) play a central role Moduli Spaces in a variety of areas that have no immediate link to the geometry being classified, in particular in alge- David D. Ben-Zvi braic number theory and algebraic topology. Moreover, the study of moduli spaces has benefited tremendously in recent years from interactions with Many of the most important problems in mathemat- physics (in particular with string theory). These inter- ics concern classification. One has a class of math- actions have led to a variety of new questions and new ematical objects and a notion of when two objects techniques. should count as equivalent. It may well be that two equivalent objects look superficially very different, so one wishes to describe them in such a way that equiv- 1 Warmup: The Moduli Space of Lines in alent objects have the same description and inequiva- the Plane lent objects have different descriptions. Moduli spaces can be thought of as geometric solu- Let us begin with a problem that looks rather simple, tions to geometric classification problems. In this arti- but that nevertheless illustrates many of the impor- cle we shall illustrate some of the key features of mod- tant ideas of moduli spaces. uli spaces, with an emphasis on the moduli spaces Problem. Describe the collection of all lines in the of Riemann surfaces. (Readers unfamiliar with Rie- real plane R2 that pass through the origin. mann surfaces may find it helpful to begin by reading about them in Part III.) In broad terms, a moduli To save writing, we are using the word “line” to mean problem consists of three ingredients. -
ISSP Data Report – Report Data ISSP Current This Research
Das International Social Survey Programme (ISSP) erhebt jährlich Umfragedaten zu sozialwissenschaftlich relevanten Themen. Der vorliegende ISSP Data Report – Religious Attitudes and Religious Change beruht auf ISSP-Daten, die zu drei verschiedenen Zeitpunkten innerhalb von 17 Jahren in bis zu 42 Mitgliedsländern zu Einstellungen gegenüber Kirche und Religion im weitesten Sinne gesammelt wurden. Jedes Kapitel wurde von unterschiedlichen Autoren der ISSP-Gemeinschaft geschrieben und beleuchtet mit Hilfe der ISSP-Daten spezielle Aspekte religiöser Einstellungen und religiösen Wandels im internationalen Vergleich. In der Gesamtschau ergeben sich sowohl Einblicke in das religiöse Leben verschiedener Länder, als auch insbesondere Erkenntnisse zu den Einflussfaktoren religiösen Wandels innerhalb von fast zwei Dekaden. The annual survey of the International Social Survey Programme (ISSP) provides data on topics relevant in social research. This current ISSP Data Report – Religious Attitudes and Religious Change examines data collected at three different points over 17 years, from up to 42 ISSP member countries, covering various facets of respondents’ attitudes towards Church and Religion. Individual chapters were written by different members of the ISSP community thereby offering a cross-national, comparative perspective on particular aspects of religious attitudes and religious change via ISSP data. Overall, this report offers insights into the religious landscapes of various countries and in particular information about the factors influencing -
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A Thesis Submitted for the Degree of PhD at the University of Warwick Permanent WRAP URL: http://wrap.warwick.ac.uk/145151 Copyright and reuse: This thesis is made available online and is protected by original copyright. Please scroll down to view the document itself. Please refer to the repository record for this item for information to help you to cite it. Our policy information is available from the repository home page. For more information, please contact the WRAP Team at: [email protected] warwick.ac.uk/lib-publications APPENDIX A Containing Violence to What End? The Political Economy of Amnesty in Nigeria’s Oil-Rich Niger Delta (2009-2016) by Elvis Nana Kwasi Amoateng A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Politics and International Studies University of Warwick, Department of Politics and International Studies February 2020 Contents Acronyms ...................................................................................................................... 4 List of Figures ............................................................................................................... 5 Acknowledgement ........................................................................................................ 7 Abstract ......................................................................................................................... 8 Introduction: ............................................................................................................... -
Variations on the Sensitivity Conjecture
THEORY OF COMPUTING LIBRARY GRADUATE SURVEYS 4 (2011), pp. 1–27 www.theoryofcomputing.org Variations on the Sensitivity Conjecture Pooya Hatami Raghav Kulkarni Denis Pankratov Received: November 2, 2010; published: June 22, 2011. Abstract: The sensitivity of a Boolean function f of n Boolean variables is the maximum over all inputs x of the number of positions i such that flipping the i-th bit of x changes the value of f (x). Permitting to flip disjoint blocks of bits leads to the notion of block sensitivity, known to be polynomially related to a number of other complexity measures of f , including the decision-tree complexity, the polynomial degree, and the certificate complexity. A long-standing open question is whether sensitivity also belongs to this equivalence class. A positive answer to this question is known as the Sensitivity Conjecture. We present a selection of known as well as new variants of the Sensitivity Conjecture and point out some weaker versions that are also open. Among other things, we relate the problem to Communication Complexity via recent results by Sherstov (QIC 2010). We also indicate new connections to Fourier analysis. ACM Classification: F.0, F.1, F.2 AMS Classification: 68-02, 68Q05, 68Q25 Key words and phrases: sensitivity, block sensitivity, complexity measures of Boolean functions 1 The Sensitivity Conjecture For Boolean strings x;y 2 f0;1gn let x ⊕ y 2 f0;1gn denote the coordinate-wise exclusive or of x and y. n th Let ei 2 f0;1g denote an n-bit Boolean string whose i bit is 1 and the rest of the bits are 0: Let f : f0;1gn ! f0;1g be a Boolean function. -
Haskell Before Haskell. an Alternative Lesson in Practical Logics of the ENIAC.∗
Haskell before Haskell. An alternative lesson in practical logics of the ENIAC.∗ Liesbeth De Mol† Martin Carl´e‡ Maarten Bullynck§ January 21, 2011 Abstract This article expands on Curry’s work on how to implement the prob- lem of inverse interpolation on the ENIAC (1946) and his subsequent work on developing a theory of program composition (1948-1950). It is shown that Curry’s hands-on experience with the ENIAC on the one side and his acquaintance with systems of formal logic on the other, were conduc- tive to conceive a compact “notation for program construction” which in turn would be instrumental to a mechanical synthesis of programs. Since Curry’s systematic programming technique pronounces a critique of the Goldstine-von Neumann style of coding, his “calculus of program com- position”not only anticipates automatic programming but also proposes explicit hardware optimisations largely unperceived by computer history until Backus famous ACM Turing Award lecture (1977). This article frames Haskell B. Curry’s development of a general approach to programming. Curry’s work grew out of his experience with the ENIAC in 1946, took shape by his background as a logician and finally materialised into two lengthy reports (1948 and 1949) that describe what Curry called the ‘composition of programs’. Following up to the concise exposition of Curry’s approach that we gave in [28], we now elaborate on technical details, such as diagrams, tables and compiler-like procedures described in Curry’s reports. We also give a more in-depth discussion of the transition from Curry’s concrete, ‘hands-on’ experience with the ENIAC to a general theory of combining pro- grams in contrast to the Goldstine-von Neumann method of tackling the “coding and planning of problems” [18] for computers with the help of ‘flow-charts’. -
Tractatus Logico-Philosophicus</Em>
University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School 8-6-2008 Three Wittgensteins: Interpreting the Tractatus Logico-Philosophicus Thomas J. Brommage Jr. University of South Florida Follow this and additional works at: https://scholarcommons.usf.edu/etd Part of the American Studies Commons Scholar Commons Citation Brommage, Thomas J. Jr., "Three Wittgensteins: Interpreting the Tractatus Logico-Philosophicus" (2008). Graduate Theses and Dissertations. https://scholarcommons.usf.edu/etd/149 This Dissertation is brought to you for free and open access by the Graduate School at Scholar Commons. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. Three Wittgensteins: Interpreting the Tractatus Logico-Philosophicus by Thomas J. Brommage, Jr. A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Philosophy College of Arts and Sciences University of South Florida Co-Major Professor: Kwasi Wiredu, B.Phil. Co-Major Professor: Stephen P. Turner, Ph.D. Charles B. Guignon, Ph.D. Richard N. Manning, J. D., Ph.D. Joanne B. Waugh, Ph.D. Date of Approval: August 6, 2008 Keywords: Wittgenstein, Tractatus Logico-Philosophicus, logical empiricism, resolute reading, metaphysics © Copyright 2008 , Thomas J. Brommage, Jr. Acknowledgments There are many people whom have helped me along the way. My most prominent debts include Ray Langely, Billy Joe Lucas, and Mary T. Clark, who trained me in philosophy at Manhattanville College; and also to Joanne Waugh, Stephen Turner, Kwasi Wiredu and Cahrles Guignon, all of whom have nurtured my love for the philosophy of language. -
First Order Logic
Artificial Intelligence CS 165A Mar 10, 2020 Instructor: Prof. Yu-Xiang Wang ® First Order Logic 1 Recap: KB Agents True sentences TELL Knowledge Base Domain specific content; facts ASK Inference engine Domain independent algorithms; can deduce new facts from the KB • Need a formal logic system to work • Need a data structure to represent known facts • Need an algorithm to answer ASK questions 2 Recap: syntax and semantics • Two components of a logic system • Syntax --- How to construct sentences – The symbols – The operators that connect symbols together – A precedence ordering • Semantics --- Rules the assignment of sentences to truth – For every possible worlds (or “models” in logic jargon) – The truth table is a semantics 3 Recap: Entailment Sentences Sentence ENTAILS Representation Semantics Semantics World Facts Fact FOLLOWS A is entailed by B, if A is true in all possiBle worlds consistent with B under the semantics. 4 Recap: Inference procedure • Inference procedure – Rules (algorithms) that we apply (often recursively) to derive truth from other truth. – Could be specific to a particular set of semantics, a particular realization of the world. • Soundness and completeness of an inference procedure – Soundness: All truth discovered are valid. – Completeness: All truth that are entailed can be discovered. 5 Recap: Propositional Logic • Syntax:Syntax – True, false, propositional symbols – ( ) , ¬ (not), Ù (and), Ú (or), Þ (implies), Û (equivalent) • Semantics: – Assigning values to the variables. Each row is a “model”. • Inference rules: – Modus Pronens etc. Most important: Resolution 6 Recap: Propositional logic agent • Representation: Conjunctive Normal Forms – Represent them in a data structure: a list, a heap, a tree? – Efficient TELL operation • Inference: Solve ASK question – Use “Resolution” only on CNFs is Sound and Complete. -
Mathematics Clinic Handbook
Mathematics Clinic Mathematics Clinic Handbook 2019 Copyright © 2019 Department of Mathematics, Harvey Mudd College. This material is provided for the sole use of participants in Harvey Mudd College’s Mathematics Clinic program. Any other usage requires prior permission from the Department. 2019 Edition; Revision 1.0. (April 30, 2019) Calendar Note that dates and deadlines are subject to change. Date Description Page September 4 (Tue) Student Orientation Day (4:15 p.m.–5:30 p.m.; 5 Sprague 3) September 4 (Tue) Project Managers Meeting (5:30 p.m.–6:30 p.m. in Sprague 3) September 6–18 Marathon Push (Including Preparation of Your 8 Statement of Work) September 6 (Thu) Clinic/Sponsor Orientation Day (11:00 a.m.– 6 1:00 p.m.; Sprague 3 workrooms) Be prepared to ask your liaisons questions. September 7 (Fri) All Forms Due by 4:00 p.m. to Molly Reeves 5 (SCTL 2404) September Site Visits with Sponsor (Arranged by Teams and 9 Sponsors) September 11 (Tue) Professional Development: Teleconference and 10 Site Visit Etiquette (11:00–12:15 p.m.; Galileo McAlister; Combined with CS) September 18 (Tue) Statement of Work Peer Review (11:00–12:15 p.m.; 8 Sprague 3) Bring Advisor-Previewed Draft. September 18 (Tue) Project Managers Meeting (11:00–12:15 p.m.; Sprague 3) September 20 (Thu) Engineering Career Fair September 21 (Fri) Peer Reviews of Statement of Work Due to Team 8 September 24 (Mon) Advisor-Previewed Project Budget Due to Clinic 53 Director September 25 (Tue) Review Statement of Work According to Slides 10 on Project Management & Planning Sent -
Prizes and Awards Session
PRIZES AND AWARDS SESSION Wednesday, July 12, 2021 9:00 AM EDT 2021 SIAM Annual Meeting July 19 – 23, 2021 Held in Virtual Format 1 Table of Contents AWM-SIAM Sonia Kovalevsky Lecture ................................................................................................... 3 George B. Dantzig Prize ............................................................................................................................. 5 George Pólya Prize for Mathematical Exposition .................................................................................... 7 George Pólya Prize in Applied Combinatorics ......................................................................................... 8 I.E. Block Community Lecture .................................................................................................................. 9 John von Neumann Prize ......................................................................................................................... 11 Lagrange Prize in Continuous Optimization .......................................................................................... 13 Ralph E. Kleinman Prize .......................................................................................................................... 15 SIAM Prize for Distinguished Service to the Profession ....................................................................... 17 SIAM Student Paper Prizes .................................................................................................................... -
System Profile
Steve Sample’s Power Mac G5 6/16/08 9:13 AM Hardware: Hardware Overview: Model Name: Power Mac G5 Model Identifier: PowerMac11,2 Processor Name: PowerPC G5 (1.1) Processor Speed: 2.3 GHz Number Of CPUs: 2 L2 Cache (per CPU): 1 MB Memory: 12 GB Bus Speed: 1.15 GHz Boot ROM Version: 5.2.7f1 Serial Number: G86032WBUUZ Network: Built-in Ethernet 1: Type: Ethernet Hardware: Ethernet BSD Device Name: en0 IPv4 Addresses: 192.168.1.3 IPv4: Addresses: 192.168.1.3 Configuration Method: DHCP Interface Name: en0 NetworkSignature: IPv4.Router=192.168.1.1;IPv4.RouterHardwareAddress=00:0f:b5:5b:8d:a4 Router: 192.168.1.1 Subnet Masks: 255.255.255.0 IPv6: Configuration Method: Automatic DNS: Server Addresses: 192.168.1.1 DHCP Server Responses: Domain Name Servers: 192.168.1.1 Lease Duration (seconds): 0 DHCP Message Type: 0x05 Routers: 192.168.1.1 Server Identifier: 192.168.1.1 Subnet Mask: 255.255.255.0 Proxies: Proxy Configuration Method: Manual Exclude Simple Hostnames: 0 FTP Passive Mode: Yes Auto Discovery Enabled: No Ethernet: MAC Address: 00:14:51:67:fa:04 Media Options: Full Duplex, flow-control Media Subtype: 100baseTX Built-in Ethernet 2: Type: Ethernet Hardware: Ethernet BSD Device Name: en1 IPv4 Addresses: 169.254.39.164 IPv4: Addresses: 169.254.39.164 Configuration Method: DHCP Interface Name: en1 Subnet Masks: 255.255.0.0 IPv6: Configuration Method: Automatic AppleTalk: Configuration Method: Node Default Zone: * Interface Name: en1 Network ID: 65460 Node ID: 139 Proxies: Proxy Configuration Method: Manual Exclude Simple Hostnames: 0 FTP Passive Mode: