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Calculemus Autumn School Approaches On Integration 10/28/02

Calculemus Autumn School Approaches On Integration 10/28/02. Sabina Petride. MathWeb: Web-Based Math (www.mathweb.org). Goals: Integrate different mechanized reasoning systems (theorem provers, computer algebra systems, model checkers)

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Calculemus Autumn School Approaches On Integration 10/28/02

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  1. Calculemus Autumn School Approaches On Integration10/28/02 Sabina Petride

  2. MathWeb: Web-Based Math(www.mathweb.org) • Goals: • Integrate different mechanized reasoning systems (theorem provers, computer algebra systems, model checkers) • Have math clients sending different requests in the network and being serviced by the appropiate systems, with correctness guarantee. • Multiple problems! Example: the logic used by a client C is “incompatible” with the logic used by the server S. • What can be used in the existing network: • PVS (automated theorem prover) • RDL (rewriting engine) • Omega, Lambda-Clam (semiautomated theorem provers) • Maple, GAP (computer algebra systems) • MBase (math knowledge base)

  3. The Logic Broker Architecture(Calculemus ’02, speakers: Alessandro Armando, Corrado Giromini) • The Logic Broker Architecture (LBA): MathWeb Clients MathWeb Servers Client1 Broker1 Maple Broker2 Client2 MBase Broker3 • Uses • KQML (Knowledge Query and Manipulation Language) • Language and protocol for exchanging information and knowledge ( agent “polite” communication language) • Can do theorem proofs in KQML • OpenMath

  4. LBA Project • More details: C S LB LS matcher DB Req: prove(,) subscribe register

  5. Open Math(Calculemus ’02, speaker: Olga Caprotti, Linz Research Institute for Symbolic Computation)(www.openmath.org) • Emerging standard for representing mathematical objects with their semantics • Strong relantionship with MathML, but… • MathML deals principally with the presentation of mathematical objects • OpenMath solely concerned with the content of mathematical objects • What can we do with OpenMath objects? • Display in a browser • Exchange between different systems • Verify if they are mathematically sound • Cut and paste and use them in different contexts

  6. Phrasebooks and CDs • CDs (Content Dictionaries): assign semantics to OMobjects • Public • Written in XML according to a DTD of less then 20 lines • examples: arith1, calculus1, complex1, fns1, linalg1(2), trans1, logic1… • Phrasebooks: interface programs that make the conversion of OMobjects to/from internal representation of math objects in a software application; the application has to declare the CDs it understands • There are phrasebooks for • Mathematica (developed by INRIA) • Maple • AXIOM 2.3 • GAP 4

  7. OpenMath Objects, Encodings and Libraries • Basic objects: integers, symbols, variables, floating-point numbers, strings, byte arrays • Application(A1, A2 ,… An) • Binding(B, v1, …, vn , C) • Attribution(A, S1 A1,…, Sn An) • Error(S, A1,…, An) where Ai, A, B, C are OMobjects, vi variable, and Si symbol. • There are two possible encodings of OMobjects: XML and binary. • Available libraries providing API for writing phrasebooks • C library (INRIA) • Java library (INRIA) (a more detailed version under development)

  8. Possible Scenarios • A multiple integrator • Problem: given a function f, compute a closed form integral of f • Suggestion: send the request to different systems (AXIOM, Maple) using OpenMath • Why better? • You don’t have to rewrite f before submitting; this is done by the specific phrasebooks • Get multiple answers and compare them • Tehnical conversations • Should be able to cut some math object from one context and paste it in another context. • Math databases

  9. MBase(Calculemus ’02, speaker: Michael Kohlhase, Carnegie Mellon University) • Web-based, distributed knowledge base of mathematical facts (definitions, theorems, proofs), implemented as a service provided by MathWeb • Examples of queries that MBase solves: • Search for a/all lemma(s)/theorem(s)/conjecture(s)/corollary(s) with the name N or matching the substring s. • Search for a/all theories with the name N or matching string s. • The answer consists of links to collections and theory matching the query, plus links to the XML encoding of the OM object represented by the theory; you can also generate a CD of the theory found. • Search for occurences of some symbol in some specified CDs, or search for all symbols in some CDs. • The demos available at http://mbase.mathweb.org:8080/mbase/ prove that response time is very good.

  10. More on MBase • Information is stored in the OMDoc format (a standard for Open Mathematical Documents) • Nov.1 2000 OMDoc 1.0 released • Queries supported: • SQL-like queries based on the structure of OMDoc • Queries that match formulae under a given equality theory • Ex: if we have int f(x) dx in the knowledge base, then a query for int f(y) dy should also be found • Queries with meta variables • What has been entered in the databse? • OpenMath CDs • Some CDs for logics • OMEGA • TPS • Working on: • Induction Challenge Problems

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