A Lisp Subset Based on MathML

1
A Lisp Subset Based on MathML

• Yuzhen Xie Stephen M. Watt Luca Padovani
• June 2002

The University of Western Ontario
2
Introduction

• Interest XML representation for programming
  languages
• Exploration content MathML as an expression for
  a functional programming language, a Lisp subset
• Development XML-Scheme interpreter in Java
• Demonstration content MathML is quite sufficient
  for this purpose
• Suggestion potential of a web scripting language
  encoded in XML (possibly based on MathML)

3
Content MathML as the Nucleus of a Functional
Programming Language

• Focus of XML structure
• Potential to create a programming language
  framework in XML, i.e. control, data flow,
  functions, format of message passing, etc.
• Advantages of programs in XML format
  standardization of the encoding format,
  independence of the delivery medium, etc.

4
Content MathML as the Nucleus of a Functional
Programming Language (contd)

• Selection of the programming language
• Lisp S-Expressions
• Content markup of MathML as an expression
• - lambda constructs and definitions
• apply, lambda, piecewise, declare, and
  csymbol
• - recursive use of container elements
• - nested building of constructs
• - flexibility provided by csymbol, e.g. for
  cons, and ()
• - types

5
Content MathML as the Nucleus of a Functional
Programming Language (contd)

• Content MathML as an expression example
• MathML content markup of
• (define reverse-subtract (lambda (x y) (- y
  x))) is
• ltdeclaregt
• ltcigtreverse-subtractlt/cigt
• ltlambdagt
• ltbvargtltcigtxlt/cigtlt/bvargt
• ltbvargtltcigtylt/cigtlt/bvargt
• ltapplygt
• ltminus/gt
• ltcigtylt/cigt
• ltcigtxlt/cigt
• lt/applygt
• lt/lambdagt
• lt/declaregt

6
XML-Scheme Syntax and Grammar

• Most of the first class operators, constants and
  symbols, expressions, and function definitions
  are represented in the core content elements of
  MathML
• csymbol element is used to create symbols having
  a particular meaning in Scheme but that are not
  part of the core content elements in MathML
  cons, car, cdr, (), null?
• Additional top element xml-scheme
• Defined syntax rules for expression reduction and
  parameter passing e.g. apply and ci

7
A complete example of XML-Scheme program that
uses , -, , cons, if, lambda, and define

• lt!-- (define my_abs (lambda (x)
• (if (lt x 0) (- x) ( x)))) --gt
• ltdeclaregt
• ltcigtmy_abslt/cigt
• ltlambdagt
• ltbvargtltcigtxlt/cigtlt/bvargt
• ltbvargtltcigtxlt/cigtlt/bvargt
• ltpiecewisegt
• ltpiecegt
• ltapplygtltminus/gtltcigtxlt/cigtlt/applygt
• ltapplygtltlt/gtltcigtxlt/cigtltcngt0lt/cngt
• lt/applygt
• lt/piecegt
• ltotherwisegt
• ltapplygtltplus/gtltcigtxlt/cigtlt/applygt
  lt/otherwisegt
• lt/piecewisegt
• lt/lambdagt
• lt/declaregt

• lt?xml version1.0
• encodingUTF-8?gt
•  
• ltxml-schemegt
• lt!-- (define my_pair (cons 1 -1)) --gt
• ltdeclaregt
• ltcigtmy_pairlt/cigt
• ltapplygt
• ltcsymbolgtconslt/csymbolgt
• ltcngt1lt/cngt
• ltcngt-1lt/cngt
• lt/applygt
• lt/declaregt

8
A complete example of XML-Scheme program that
uses , -, , cons, if, lambda, and define

• lt!-- (define my_double
• (lambda (x) ( 2 x))) --gt
• ltdeclaregt
• ltcigtmy_doublelt/cigt
• ltlambdagt
• ltbvargtltcigtxlt/cigtlt/bvargt
• ltapplygt
• lttimes/gt
• ltcngt2lt/cngt
• ltcigtxlt/cigt
• lt/applygt
• lt/lambdagt
• lt/declaregt

• lt!-- (define compose-fn (lambda (f1 f2)
• lambda (n) (f1 (f2 n))))) --gt
• ltdeclaregt
• ltcigtcompose-fnlt/cigt
• ltlambdagt
• ltbvargtltcigtf1lt/cigtlt/bvargt
• ltbvargtltcigtf2lt/cigtlt/bvargt
• ltlambdagt
• ltbvargtltcigtnlt/cigtlt/bvargt
• ltapplygt
• ltcigtf1lt/cigt
• ltapplygt
• ltcigtf2lt/cigt
• ltcigtnlt/cigt
• lt/applygt
• lt/applygt
• lt/lambdagt
• lt/lambdagt
• lt/declaregt
  

9
A complete example of XML-Scheme program that
uses , -, , cons, if, lambda, and define
(contd)

• lt!-- (define compose-fn (lambda (f1 f2)
• (lambda (n) (f1 (f2 n))))) --gt
• ltdeclaregt
• ltcigtcompose-fnlt/cigt
• ltlambdagt
• ltbvargtltcigtf1lt/cigtlt/bvargt
• ltbvargtltcigtf2lt/cigtlt/bvargt
• ltlambdagt
• ltbvargtltcigtnlt/cigtlt/bvargt
• ltapplygt
• ltcigtf1lt/cigt
• ltapplygt
• ltcigtf2lt/cigt
• ltcigtnlt/cigt
• lt/applygt
• lt/applygt
• lt/lambdagt
• lt/lambdagt
• lt/declaregt
  

• lt!-- (define myfun
• (compose-fn my_abs my_double))--gt
• ltdeclaregt
• ltcigtmyfunlt/cigt
• ltapplygt
• ltcigtcompose-fnlt/cigt
• ltcigtmy_abslt/cigt
• ltcigtmy_doublelt/cigt
• lt/applygt
• lt/declaregt
•  lt!-- (myfun (car my_pair)) --gt
• ltapplygt
• ltcigtmyfunlt/cigt
• ltapplygt
• ltcsymbolgtcarlt/csymbolgt
• ltcigtmy_pairlt/cigt
• lt/applygt
• lt/applygt
•  

10
XML-Scheme Interpreter
11
Quotation

• A PHP fragment
• lt?
• print("ltulgt")
• for (ix 0 ix lt count(stuff) ix)
  
• print("ltligt ix. stuff(ix) lt/ligt\n")
  
• ?gt
• Structure difficult to maintain -- note missing
  lt/ulgt.
• Potential using Lisp-style quasi-quoting
  mechanism for XML documents in a natural,
  structured manner
• Lisp example
• (list ,( 1 2) 4) gt (list 3 4)

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Quotation (contd)

• Proposition
• - A scripting model for XHTML documents as
  quasi-quoted data
• - Scripting components are in XML syntax
• - Evaluation is indicated by the unquoting
  mechanism as comma in Scheme

ltligt ltxsunquotegt ltmapplygt
ltmcigtmy_funlt/mcigt
ltmapplygt
ltmcsymbolgtcdrlt/mcsymbolgt
ltmcigtmy_pairlt/mcigt lt/mapplygt
lt/mapplygt lt/xsunquotegt lt/ligt
lt/ulgt
lt...xmlnsxs "http//www.orcca.on.ca/MathML/XML-S
cheme" xmlnsm"http//www.w3.org/TR/MathML2"gt
ltulgt ltligt ltpgt A fragment of
quasi-quoted data in XML format lt/pgt lt/ligt
ltligt ltxsunquotegt ltmapplygt
ltmcigtmy_funlt/mcigt
ltmapplygt ltmcsymbolgtcarlt/mcsymbo
lgt ltmcigtmy_pairlt/mcigt
lt/mapplygt lt/mapplygt
lt/xsunquotegt lt/ligt
13
Concluding Remarks

• Prototype of XML-Scheme interpreter
• Implementation of a basic set of Scheme
  operators, lambda calculus and definitions
• Feasibility of content MathML to represent a
  simple functional programming language, Scheme
• Potential uses of programming languages
  represented in XML, in particular, a new family
  of scripting languages for Web pages