Admin/page/main-content/logics.content
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%title%
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Isabelle's Logics
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%body%
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<h3>What is Isabelle?</h3>
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Isabelle can be viewed from two main perspectives.  On the one hand it
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may serve as a generic framework for rapid prototyping of deductive
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systems.  On the other hand, major existing logics like
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<strong>Isabelle/HOL</strong> provide a theorem proving environment
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ready to use for sizable applications.
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<h2>Isabelle's Logics</h2>
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The Isabelle distribution includes a large body of object logics and
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other examples (see the <a href="library/">Isabelle theory
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library</a>).
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<dl>
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<dt><a href="library/HOL/"><strong>Isabelle/HOL</strong></a><dd> is a
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version of classical higher-order logic resembling that of the <A
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HREF="http://www.cl.cam.ac.uk/Research/HVG/HOL/HOL.html">HOL
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System</A>.
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<dt><a href="library/HOLCF/"><strong>Isabelle/HOLCF</strong></a><dd>
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adds Scott's Logic for Computable Functions (domain theory) to HOL.
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<dt><a href="library/FOL/"><strong>Isabelle/FOL</strong></a><dd>
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provides basic classical and intuitionistic first-order logic.  It is
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polymorphic.
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<dt><a href="library/ZF/"><strong>Isabelle/ZF</strong></a><dd> offers
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a formulation of Zermelo-Fraenkel set theory on top of FOL.
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</dl>
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<p>
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Isabelle/HOL is currently the best developed object logic, including
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an extensive library of (concrete) mathematics, and various packages
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for advanced definitional concepts (like (co-)inductive sets and
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types, well-founded recursion etc.).  The distribution also includes
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some large applications, for example correctness proofs of
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cryptographic protocols (<a href="library/HOL/Auth/">HOL/Auth</a>) or
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communication protocols (<a href="library/HOLCF/IOA/">HOLCF/IOA</a>).
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<p>
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Isabelle/ZF provides another starting point for applications, with a
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slightly less developed library.  Its definitional packages are
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similar to those of Isabelle/HOL.  Untyped ZF provides more advanced
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constructions for sets than simply-typed HOL.
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<p>
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There are a few minor object logics that may serve as further
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examples: <a href="library/CTT/">CTT</a> is an extensional version of
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Martin-L&ouml;f's Type Theory, <a href="library/Cube/">Cube</a> is
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Barendregt's Lambda Cube.  There are also some sequent calculus
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examples under <a href="library/Sequents/">Sequents</a>, including
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modal and linear logics.  Again see the <a href="library/">Isabelle
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theory library</a> for other examples.
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<h3>Defining Logics</h3>
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Logics are not hard-wired into Isabelle, but formulated within
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Isabelle's meta logic: <strong>Isabelle/Pure</strong>.  There are
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quite a lot of syntactic and deductive tools available in generic
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Isabelle.  Thus defining new logics or extending existing ones
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basically works as follows:
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<ol>
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<li> declare concrete syntax (via mixfix grammar and syntax macros),
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<li> declare abstract syntax (as higher-order constants),
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<li> declare inference rules (as meta-logical propositions),
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<li> instantiate generic automatic proof tools (simplifier, classical
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tableau prover etc.),
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<li> manually code special proof procedures (via tacticals or
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hand-written ML).
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</ol>
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The first three steps above are fully declarative and involve no ML
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programming at all.  Thus one already gets a decent deductive
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environment based on primitive inferences (by employing the built-in
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mechanisms of Isabelle/Pure, in particular higher-order unification
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and resolution).
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For sizable applications some degree of automated reasoning is
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essential.  Instantiating existing tools like the classical tableau
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prover involves only minimal ML-based setup.  One may also write
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arbitrary proof procedures or even theory extension packages in ML,
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without breaching system soundness (Isabelle follows the well-known
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<em>LCF system approach</em> to achieve a secure system).