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Admin/page/main-content/logics.content

author | berghofe |

Mon, 05 Aug 2002 14:35:33 +0200 | |

changeset 13455 | f88a91ff8ac6 |

parent 11109 | ce1cefc6c14c |

child 14659 | a68de9a2770a |

permissions | -rw-r--r-- |

Removed theory NatDef.

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