lo48576 / xslt10-lambda-calculus

Lambda calculus by XSLT 1.0 (and `node-set()` of EXSLT)

xslt10-lambda-calculus

About

This project provides XSLT 1.0 stylesheets to evaluate terms of Lambda Calculus.

Files and directories

xsl/ directory

Contains XSLT files.

For users (for use by CLI XSLT processor):

• conv-to-de-bruijn-term.xsl
  • Front-end of /lib/conv-to-de-bruijn-term.xsl.
• desugar.xsl
  • Front-end of /lib/desugar.xsl.
• eta-reduction.xsl
  • Front-end of /lib/eta-reduction.xsl.
• full-reduction.xsl
  • Front-end of /lib/full-reduction.xsl.
• onestep-reduction.xsl
  • Front-end of /lib/onestep-reduction.xsl.
• pretty-print.xsl
  • Front-end of /lib/pretty-print.xsl.

For developers (for use by <xsl:import> or <xsl:include>):

• lib/
  • conv-to-de-bruijn-term.xsl
    • Converts the given term into a de Bruijn term.
  • desugar-apply-at-once.xsl
    • Desugars (a b c ...) into (...((a b) c) ...).
  • desugar-let-expr.xsl
    • Desugars (let a=x, b=y in foobar) into ((λa. λb. foobar) x y).
    • let* is also supported.
  • desugar-multiple-params.xsl
    • Desugars (λx y z. foobar) into (λx. λy. λz. foobar).
  • desugar.xsl
    • Applies all supported desugaring in correct order.
    • Users usually want to use this for desugaring (not desugar-*.xsl ones).
  • eta-reduction.xsl
    • Applies eta (η) reduction as possible.
  • full-reduction.xsl
    • Evaluates the given de Bruijn term as possible.
  • has-beta-redex.xsl
    • Returns whether the given de Bruijn term has beta redex, by string value 'true' or 'false'.
  • onestep-reduction.xsl
    • Does a reduction of single beta redex of the given de Bruijn term.
    • This is leftmost outermost reduction.
  • pretty-print.xsl
    • Prints the given term into the text.
    • Both normal terms and de Bruijn terms are supported.
    • This does not add trailing newline character (
).
  • reduction-steps.xsl
    • Returns results of each reduction steps.
    • For examples, see /tests/reduction-steps/*.txt.
    • By default, it applies eta reduction as the last step. This can be disabled by setting parameter eta-reduction to 'no'.
  • shift.xsl
    • Does shift operation for the given de Bruijn term.
    • For detail, see chapter 6.2 of TaPL.
  • substitute.xsl
    • Does substitution for the given de Bruijn term.
    • For detail, see chapter 6.2 of TaPL.

tests/ directory

Test scripts and helpers:

• reduction-steps.xsl
  • Front-end of reduction-steps feature with pretty-printing enabled.
• test.sh
  • Script to run tests.
  • Nothing other than running test-case names should be printed on success.

Test cases:

• de-bruijn-term/*.txt
  • Expected results for conv-to-de-bruijn-term feature.
• desugar/*.txt
  • Expected results for desugar feature.
• expr/*.xml
  • Original (starting) term to be used as test cases.
• plain/*.txt
  • Expected results for pretty-print feature.
• plain/*.eta.txt
  • Expected results for eta-reduction feature.
• reduction-steps/*.txt
  • Expected results for reduction-steps and full-reduction feature.
• reduction-steps/*.final.txt
  • Expected results for full-reduction feature.
  • These files are used when the result files for reduction-steps feature was not found.
  • Maybe useful when reduction steps are too long and people cannot (or don't want to) prepare the expected results.

Namespaces

• http://www.cardina1.red/_ns/xslt-lambda-calculus
  • For lambda calculus terms.
• http://www.cardina1.red/_ns/xslt-lambda-calculus/stylesheet
  • For stylesheet-specific stuff.
  • Template modes, template names, etc.
• http://www.cardina1.red/_ns/xslt-lambda-calculus/_internal
  • For internal stuff.
  • This should not be used by end users, or stylesheet may use inconsistent data and result in broken calculation.

Lambda terms representation

Consider xmlns:l="http://www.cardina1.red/_ns/xslt-lambda-calculus".

Named variable

<l:var>{{varname}}</l:var>

{{varname}} is variable name. It should be string, and should not contain space characters.

l:var can be used in any terms.

Variable with de Bruijn index

<l:de-bruijn-var index="{{index}} />

{{index}} is de Bruijn index. It should be integer >= 1.

The index is 1-based, because De Bruijn index - Wikipedia uses 1-based index (at 2018-09-16 edition).

l:de-bruijn-var can only be used in de Bruijn terms.

Lambda abstraciton

<l:lambda>
  <l:param>{{varname}}</l:param>
  <l:body>{{term}}</l:body>
</l:lambda>

{{varname}} is parameter name. It should be string, and should not contain space characters.

{{term}} is XML representation of body term. It can be any (non-de-Bruijn) term, it should be just one XML element.

l:lambda can only be used in non-de-Bruijn terms.

Syntax sugar

With desugar-multiple-params feature:

• Zero or more <l:param>s can be used for single <l:lambda>.
• In any case, <l:body> should be the last child of <l:lambda>.

Lambda abstraction of de Bruijn term

<l:de-bruijn-lambda>
  {{term}}
</l:de-bruijn-lambda>

{{term}} is XML representation of body term. It can be any de Bruijn term, but it should be just one XML element.

l:lambda can only be used in de Bruijn terms.

Application

<l:apply>
  {{term1}}
  {{term2}}
</l:apply>

{{term1}} and {{term2}} are XML representations of applied term and argument. They can be any terms (which can appear in l:apply's position), but each of them should be just one XML element.

l:apply can only be used in any terms.

Syntax sugar

With desugar-apply-at-once feature:

• One or more terms can be inside <l:apply>.

Let expressions (syntax sugar)

With desugar-let-expr feature, <l:let> can be available.

<!-- binding-mode is optional. -->
<l:let mode="binding-mode">
  <l:bind>
    <l:var>var1</l:var>
    {{term1}}
  </l:bind>
  <l:bind>
    <l:var>var2</l:var>
    {{term2}}
  </l:bind>
  ...
  {{term-body}}
</l:let>

• Zero or more <l:bind> can be used.
  • Put a <l:var> with the variable name, then put a term.
• Just one body expression ({{term-body}} in example) should be placed at the last of <l:let>.

You can choose binding mode:

• independent (default)
  • In this mode, <l:let> behaves as let of Scheme programming language.
• one-by-one
  • In this mode, <l:let> behaves as let* of Scheme programming language.

Use of EXSLT (node-set())

Some of stylesheets uses node-set() function of EXSLT.

This is required for multi-step tree processing, because plain XSLT 1.0 does not allow result node-set of template to be bound to variables as node-set. This means, you cannot do "pass result tree of some template A to other template B" without some extension.

EXSLT is widely known and node-set() is supported by many XSLT processors.

License

Licensed under either of

• Apache License, Version 2.0, (LICENSE-APACHE.txt or https://www.apache.org/licenses/LICENSE-2.0)
• MIT license (LICENSE-MIT.txt or https://opensource.org/licenses/MIT)

at your option.

Contribution

Unless you explicitly state otherwise, any contribution intentionally submitted for inclusion in the work by you, as defined in the Apache-2.0 license, shall be dual licensed as above, without any additional terms or conditions.