ICMS’2006 slides - - a system for web publication of mathematics Klaus Grue September 1, 2006
Logiweb from a users perspective, 1 Case: A mathematician M comes up with a marvelous new lemma: Lemma 1 0 · x = x M wants to web publish a formal, machine checked proof of that.
Logiweb from a users perspective, 2 M decides to prove 0 · x = 0 in Peano arithmetic. Furthermore, M decides to publish using Logiweb. So M Googles “Peano arithmetic logiweb”. M gets 162 hits, and after some browsing, M finds http://logiweb.eu:8080/logiweb/page/check/fixed/index.html .
Logiweb main menu of check General: Up Help Starting point: Reference After fetching: Vector After unpacking: Body Bibliography Dictionary Lisp After codifying: Codex Expansion Header After verifying: The page is correct Where it all began: Source Click ’Body’ to see the page
Logiweb body of check Up Pdf Toc Pyk TeX Help Click ’Pdf’ to see the page
A Logiweb proof checker Klaus Grue GRD-2006-08-30.UTC:17:55:57.700479 Contents 1 Introduction 2 1.1 Electronic appendices . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Referenced Logiweb pages . . . . . . . . . . . . . . . . . . . . . . 2 · · ·
Logiweb from a users perspective, 3 M has found a Logiweb page which defines a proof checker but, accidentally, that page also defines Peano arithmetic and proves x + y = y + x . To author a new Logiweb page, M must download and install the Logiweb software. As an alternative, M may use Logiwikiweb, but that is not so convenient in the long run. So M downloads and installs the latest version of Logiweb from logiweb.eu . The software uses CLISP so M downloads and installs that as well.
Logiweb from a users perspective, 4 To author a new Logiweb page, M must prepare a pyk source file and run it through the pyk compiler from the Logiweb software suite. To do so M opens a text editor and writes: PAGE proof of my lemma Now the page has a name.
Logiweb from a users perspective, 5 The page at http://logiweb.eu:8080/logiweb/page/check/fixed/index.html is named check and references another Logiweb page named base . M browses check and finds that it defines a construct named System S which denotes Peano arithmetic expressed as an axiomatic system. M sees that 0, x · y and x = y is defined on base rather than check .
Logiweb from a users perspective, 6 So M must references both check and base : BIBLIOGRAPHY ”check” ”http:/ /logiweb.eu:8080/logiweb/page/base/fixed/vector/page.lgw”, ”base” ”http:/ /logiweb.eu:8080/logiweb/page/check/fixed/vector/page.lgw”
Logiweb from a users perspective, 7 M must now import the constructs needed from check and base . But M is lazy and just imports all constructs. M also defines a new construct named my lemma . To do so, M looks up the header of check and base and merge them as shown on the next slide.
Logiweb from a users perspective, 8 PREASSOCIATIVE PREASSOCIATIVE "base" base "base" " Times " "check" check ... "" my lemma PREASSOCIATIVE PREASSOCIATIVE "base" " = " "base" " factorial "check" " member " "check" " plist ( " ) ... ...
Logiweb from a users perspective, 9 Now M writes the body of the page: BODY \end{document} "File page.tex ---- ---- latex page \documentclass{article} latex page ... dvipdfm page" \begin{document} ...
Logiweb from a users perspective, 10 M now defines how my lemma should be rendered somewhere between \begin{document} and \end{document} : "[ math tex define my lemma as " \mathbf{Multiplication\ by\ Zero}" end define end math ]"
Logiweb from at users perspective, 11 M now formulates the lemma somewhere between \begin{document} and \end{document} : Our goal is to prove: "[ System S lemma my lemma : All meta x : 0 * meta x = 0 end lemma ]" We do so by induction in "[ math meta x end math ]".
Logiweb from a users perspective, 12 The pyk source on the previous slide is rendered thus: Our goal is to prove: System S lemma Multiplication by Zero : Π x : 0 · x = 0 ✷ We do so by induction in x .
Logiweb from a users perspective, 13 Then M proves Multiplication by Zero as shown on the next slide.
"[ System S proof of my lemma : line L01 : Axiom S7 >> 0 * 0 = 0 ; line L02 : Block >> Begin ; line L03 : Arbitrary >> meta x ; line L04 : Premise >> 0 * meta x = 0 ; line L05 : Axiom S8 >> 0 * meta x suc = 0 * meta x + 0 ; line L06 : Axiom S5 >> 0 * meta x + 0 = 0 * meta x ; line L07 : 3.2c ponens L05 ponens L06 >> 0 * meta x suc = 0 * meta x line L08 : 3.2c ponens L07 ponens L04 >> 0 * meta x suc = 0 ; line L09 : Block >> End ; line L10 : Deduction ponens L09 >> 0 * var x = 0 imply 0 * var x suc line L11 : Induction at var x ponens L01 ponens L10 >> 0 * var x = 0 line L12 : Arbitrary >> meta x ; line L13 : Deduction ponens L11 >> 0 * meta x = 0 qed ]"
Logiweb from a users perspective, 14 The next slide shows how the pyk source on the previous slide is rendered.
System S proof of Multiplication by Zero : L01: Axiom S7 � 0 · 0 = 0 ; L02: Block � Begin ; L03: Arbitrary � ; x L04: Premise � 0 · x = 0 ; 0 · x ′ = 0 · x + 0 L05: Axiom S8 � ; L06: Axiom S5 � 0 · x + 0 = 0 · x ; 0 · x ′ = 0 · x L07: 3 . 2c ✄ L05 ✄ L06 � ; 0 · x ′ = 0 L08: 3 . 2c ✄ L07 ✄ L04 � ; L09: Block � End ; 0 · x = 0 ⇒ 0 · x ′ = 0 L10: Deduction ✄ L09 � ; L11: Induction @ x ✄ L01 ✄ L10 � 0 · x = 0 ; L12: Arbitrary � ; x L13: Deduction ✄ L11 � 0 · x = 0 ✷
Logiweb from a users perspective, 15 M saves the pyk source as MyLemma.pyk. Then M runs pyk MyLemma M corrects errors until pyk accepts the paper. Finally M runs pyk MyLemma -level submit That publishes the page on Logiweb so that anyone in the world can use Multiplication by Zero just by referencing M ’s page.
Logiweb from a designers perspective That sounds easy. It is—for the user. A designer must consider many things, among other: What if somebody changes or deletes the check page on which M relies? That is solved by a complex referencing/mirroring system based on world-wide cooperation between Logiweb servers which ensures that the check page M references cannot change. M may ensure the continued existence of check by mirroring it locally.
Status of the Logiweb software Logiweb has been used on several courses: For first year students in 2004 and for graduate students in 2005 and 2006. Predecessors of Logiweb have been used internally for teaching at Diku since 1986. The releases are still alpha releases. Missing before beta-release: A Logiweb tutorial and Unicode support (is essentially implemented).
Status of theorems proved Graduate students and the author have written proofs in Peano arithmetic. A consistency proof for ZFC in map theory (300 pages) awaits the first beta release of Logiweb. The present slides are produced and checked by Logiweb. A 500 page math text book used at Diku since 1992 was written for Logiweb and also awaits the first beta release.
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