1218 lines
66 KiB
Plaintext
1218 lines
66 KiB
Plaintext
(*<*)
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theory "paper"
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imports "Isabelle_DOF.scholarly_paper"
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begin
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open_monitor*[this::article]
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declare[[ strict_monitor_checking = false]]
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declare[[ Definition_default_class = "definition"]]
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declare[[ Lemma_default_class = "lemma"]]
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declare[[ Theorem_default_class = "theorem"]]
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define_shortcut* hol \<rightleftharpoons> \<open>HOL\<close>
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isabelle \<rightleftharpoons> \<open>Isabelle/HOL\<close>
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dof \<rightleftharpoons> \<open>Isabelle/DOF\<close>
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LaTeX \<rightleftharpoons> \<open>LaTeX\<close>
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html \<rightleftharpoons> \<open>HTML\<close>
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csp \<rightleftharpoons> \<open>CSP\<close> \<comment>\<open>obsolete\<close>
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holcsp \<rightleftharpoons> \<open>HOL-CSP\<close> \<comment>\<open>obsolete\<close>
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ML\<open>
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fun boxed_text_antiquotation name (* redefined in these more abstract terms *) =
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DOF_lib.gen_text_antiquotation name DOF_lib.report_text
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(fn ctxt => DOF_lib.string_2_text_antiquotation ctxt
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#> DOF_lib.enclose_env false ctxt "isarbox")
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val neant = K(Latex.text("",\<^here>))
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fun boxed_theory_text_antiquotation name (* redefined in these more abstract terms *) =
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DOF_lib.gen_text_antiquotation name DOF_lib.report_theory_text
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(fn ctxt => DOF_lib.string_2_theory_text_antiquotation ctxt
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#> DOF_lib.enclose_env false ctxt "isarbox"
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(* #> neant *)) (*debugging *)
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fun boxed_sml_text_antiquotation name =
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DOF_lib.gen_text_antiquotation name (K(K()))
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(fn ctxt => Input.source_content
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#> Latex.text
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#> DOF_lib.enclose_env true ctxt "sml")
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(* the simplest conversion possible *)
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fun boxed_pdf_antiquotation name =
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DOF_lib.gen_text_antiquotation name (K(K()))
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(fn ctxt => Input.source_content
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#> Latex.text
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#> DOF_lib.enclose_env true ctxt "out")
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(* the simplest conversion possible *)
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fun boxed_latex_antiquotation name =
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DOF_lib.gen_text_antiquotation name (K(K()))
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(fn ctxt => Input.source_content
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#> Latex.text
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#> DOF_lib.enclose_env true ctxt "ltx")
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(* the simplest conversion possible *)
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fun boxed_bash_antiquotation name =
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DOF_lib.gen_text_antiquotation name (K(K()))
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(fn ctxt => Input.source_content
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#> Latex.text
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#> DOF_lib.enclose_env true ctxt "bash")
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(* the simplest conversion possible *)
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\<close>
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setup\<open>(* std_text_antiquotation \<^binding>\<open>my_text\<close> #> *)
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boxed_text_antiquotation \<^binding>\<open>boxed_text\<close> #>
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(* std_text_antiquotation \<^binding>\<open>my_cartouche\<close> #> *)
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boxed_text_antiquotation \<^binding>\<open>boxed_cartouche\<close> #>
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(* std_theory_text_antiquotation \<^binding>\<open>my_theory_text\<close>#> *)
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boxed_theory_text_antiquotation \<^binding>\<open>boxed_theory_text\<close> #>
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boxed_sml_text_antiquotation \<^binding>\<open>boxed_sml\<close> #>
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boxed_pdf_antiquotation \<^binding>\<open>boxed_pdf\<close> #>
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boxed_latex_antiquotation \<^binding>\<open>boxed_latex\<close>#>
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boxed_bash_antiquotation \<^binding>\<open>boxed_bash\<close>
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\<close>
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(*>*)
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title*[tit::title]\<open>A Framework for Proving Ontology-Relations and Runtime Testing Ontology Instances\<close>
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author*[idir,email="\<open>idir.aitsadoune@lri.fr\<close>",affiliation="\<open>LMF, CentraleSupelec\<close>"]\<open>Idir Ait-Sadoune\<close>
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author*[nic,email="\<open>nicolas.meric@lri.fr\<close>",affiliation="\<open>LRI, Université Paris-Saclay\<close>"]\<open>Nicolas Méric\<close>
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author*[bu,email="\<open>wolff@lri.fr\<close>",affiliation = "\<open>LRI, Université Paris-Saclay\<close>"]\<open>Burkhart Wolff\<close>
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abstract*[abs, keywordlist="[\<open>Ontologies\<close>,\<open>Formal Documents\<close>,\<open>Formal Development\<close>,\<open>\<^isabelle>\<close>,\<open>Ontology Mapping\<close>]"]
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\<open> \<^dof> is a novel ontology framework on top of Isabelle
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@{cite "brucker.ea:isabelledof:2019" and "brucker.ea:isabelle-ontologies:2018"}.
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\<^dof> allows for the formal development of ontologies as well as continuous checking that
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a formal document under development conforms to an underlying ontology.
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Such a document may contain text and code elements as well as formal Isabelle definitions and proofs.
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Thus, \<^dof> is designed to annotate and interact with typed meta-data
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within formal developments in Isabelle.
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% While prior versions of \<^dof> provided already a mechanism to check ontological \<^emph>\<open>rules\<close>
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% (in OWL terminology) or \<^emph>\<open>class invariants\<close> (in UML/OCL terminology) via hand-written SML test-code,
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% we provide in this paper a novel mechanism to specify \<^emph>\<open>invariants\<close> in \<^hol> via a reflection
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% mechanism.
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In this paper we extend \<^dof> with \<^emph>\<open>invariants\<close> via a reflection mechanism,
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which serve as equivalent to the concept of ontological \<^emph>\<open>rules\<close> (in OWL terminology) or
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\<^emph>\<open>class invariants\<close> (in UML/OCL terminology).
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This allows for both efficient run-time checking of abstract properties of formal
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content \<^bold>\<open>as well as\<close> formal proofs that establish mappings between different ontologies in
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general and specific ontology instances in concrete cases.
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With this feature widely called \<^emph>\<open>ontology mapping\<close> in the literature, our framework paves the
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way for a deeper integration of ontological information in, for example,
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the articles of the Archive of Formal Proofs.
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\<close>
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section*[introheader::introduction,main_author="Some(@{author ''bu''})"]
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\<open> Introduction \<close>
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text*[introtext::introduction]\<open>
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The linking of \<^emph>\<open>formal\<close> and \<^emph>\<open>informal\<close> information is perhaps the most pervasive challenge
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in the digitization of knowledge and its propagation. Unsurprisingly, this problem reappears
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in the libraries with formalized mathematics and engineering such as the Isabelle Archive of
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Formal Proofs @{cite "AFP-ref22"}, which passed the impressive numbers of 650 articles,
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written by 420 authors at the beginning of 2022. Still, while the problem of logical consistency
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even under system-changes and pervasive theory evolution is technically solved via continuous
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proof-checking, the problem of knowledge retrieval and of linking semi-formal explanations to
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definitions and proofs remains largely open.
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The \<^emph>\<open>knowledge\<close> problem of the increasingly massive \<^emph>\<open>digital information\<close> available
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incites numerous research efforts summarized under the labels ``semantic web'',
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``integrated document management'', or any form of advanced ``semantic'' text processing.
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These technologies are increasingly important in jurisprudence, medical research and
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life-sciences in order to tame their respective publication tsunamies. The central role
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in these technologies is played by \<^emph>\<open>document ontologies\<close>, \<^ie>, a machine-readable form
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of meta-data attached to document-elements as well as their document discourse. In order
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to make these techniques applicable to the area of \<^emph>\<open>formal theory development\<close>,
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the following is needed: \<^vs>\<open>0.2cm\<close>
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\<^item> a general mechanism to define and develop \<^emph>\<open>domain-specific\<close> ontologies,
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\<^item> ... that should be adapted to entities occurring in formal theories,
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\<^ie>, provide built-in support for types, terms, theorems, proofs, etc.,
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\<^item> ways to annotate meta-data generated by ontologies to the document elements,
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as ``deep'' as possible, together with strong validation checks,
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\<^item> a smooth integration into the theory document development process, and
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\<^item> ways to relate ontologies and ontology-conform documents along different
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ontologies by \<^emph>\<open>ontological mappings\<close> and \<^emph>\<open>data translations\<close>
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@{footnote \<open>We follow throughout this text the terminology established in
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@{cite "books/daglib/0032976"}, pp. 39 ff.\<close>}.
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\<close>
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text\<open> \<^vs>\<open>-0.2cm\<close>
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Recently, \<^dof> @{cite "brucker.ea:isabelledof:2019" and "brucker.ea:isabelle-ontologies:2018"}
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\<^footnote>\<open>The official releases are available at \<^url>\<open>https://zenodo.org/record/3370483#.YfWg6S-B1qt\<close>, the
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developer version at \<^url>\<open>https://github.com/logicalhacking/Isabelle_DOF\<close>.\<close>
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has been designed as an Isabelle component that attempts to answer these needs.
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\<^dof> generates from ontology definitions directly integrated into Isabelle theories
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typed meta-data, that may be annotated to a number of document elements and that were
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validated ``on-the-fly'' during the general continuous type and proof-checking process
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in an IDE (Isabelle/PIDE). Thus, we extend the document-centric view on code, definitions,
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proofs, text-elements, etc., prevailing in the Isabelle system framework.
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In more detail, \<^dof> introduces a number of ``ontology aware'' text-elements with analogous
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syntax to standard ones. The difference is a bracket with meta-data of the form: \<^vs>\<open>-0.3cm\<close>
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@{theory_text [display,indent=10, margin=70]
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\<open>
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text*[label::classid, attr\<^sub>1=E\<^sub>1, ... attr\<^sub>n=E\<^sub>n]\<open> some semi-formal text \<close>
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ML*[label::classid, attr\<^sub>1=E\<^sub>1, ... attr\<^sub>n=E\<^sub>n]\<open> some SML code \<close>
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...
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\<close>}
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\<^vs>\<open>-0.3cm\<close> In these \<^dof> elements, a meta-data object is created and associated to it. This
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meta-data can be referenced via its label and used in further computations in text or code.
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%; the details will be explained in the subsequent section.
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Admittedly, Isabelle is not the first system that comes into one's mind when writing a scientific
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paper, a book, or a larger technical documentation. However, it has a typesetting system inside
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which is in the tradition of document generation systems such as mkd, Document! X, Doxygen,
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Javadoc, etc., and which embed formal content such as formula pretty-prints into semi-formal text
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or code. The analogous mechanism the Isabelle system provides is a machine-checked macro
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called \<^emph>\<open>antiquotation\<close> that depends on the logical context of the document element.
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With standard Isabelle antiquotations, for example, the following text element
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of the integrated source will appear in Isabelle/PIDE as follows: \<^vs>\<open>-0.3cm\<close>
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@{theory_text [display,indent=10, margin=70]
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\<open>
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text\<open> According to the reflexivity axiom @{thm refl}, we obtain in \<Gamma>
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for @{term "fac 5"} the result @{value "fac 5"}.\<close>
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\<close>}
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\<^vs>\<open>-0.5cm\<close> In the corresponding generated \<^LaTeX> or HTML output, this looks like this:
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\<^vs>\<open>-0.1cm\<close>
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@{theory_text [display,indent=10, margin=70]
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\<open>According to the reflexivity axiom \<open>x = x\<close>, we obtain in \<Gamma> for \<open>fac 5\<close> the result \<open>120\<close>.\<close>
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}
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\<^vs>\<open>-0.1cm\<close>where the meta-texts \<open>@{thm refl}\<close> (``give the presentation of theorem `refl'\,\!''),
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\<open>@{term "fac 5"}\<close> (``parse and type-check `fac 5' in the previous logical context'')
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and \<open>@{value "fac 5"}\<close> (``compile and execute `fac 5' according to its
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definitions'') are built-in antiquotations in \<^hol>.
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One distinguishing feature of \<^dof> is that specific antiquotations \<^emph>\<open>were generated from
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an ontology\<close> rather than being hard-coded into the Isabelle system infrastructure.
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\<close>
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(*
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text
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\<open>
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%too long !
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This leads to an evolution strategy we call "integrate the document, strengthen the
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links" (IDSL): integrate all sources into the Isabelle document model, and
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replace \<^emph>\<open>text\<close> by appropriate \<^emph>\<open>meta-text\<close> wherever you can.
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Developers are rewarded for applying IDSL by specific IDE-support,
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by additional semantic checks and thus by a more robust document consistency
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which is easier to maintain under collaborative changes.
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%For example, if someone changes the theorem name, an error would result when processing
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%the text. On the other hand, \<open>@{value "fac 5"}\<close> would not guard against a change of definition
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%of \<open>fac\<close>. If this is desirable, an antiquotation like \<open>@{assert "fac 5 = 120"}\<close> would be more appropriate.
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%too long !
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Antiquotations do not only occur in text-elements in Isabelle; they are also heavily used
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in the code-elements of Isabelle's SML implementation, or were specifically supported in
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C-program contexts in Isabelle/C @{cite "Tuong-IsabelleC:2019"}.
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However, programming antiquotations on the intern Isabelle API's is nothing for the
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faint-hearted. Recently, \<^dof> @{cite "brucker.ea:isabelledof:2019" and "brucker.ea:isabelle-ontologies:2018"}
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has been designed as an Isabelle component that \<^emph>\<open>generates\<close> antiquotation languages
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from a more abstract description, namely an \<^emph>\<open>ontology\<close> that provides typed meta-data
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and typed reference mechanisms inside text- and ML-contexts.
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*)
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text\<open>As novel contribution, this work extends prior versions of \<^dof> by
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\<^enum> support of antiquotions in a new class of contexts, namely \<^emph>\<open>term contexts\<close>
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(rather than SML code or semi-formal text). Thus, annotations generated
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from \<^dof> may also occur in \<open>\<lambda>\<close>-terms used to denote meta-data.
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\<^enum> formal, machine-checked invariants on meta-data, which correspond to the concept of
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``rules'' in OWL or ``constraints'' in UML, and which can be specified in
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common \<^hol> \<open>\<lambda>\<close>-term syntax.
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\<close>
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text\<open> For example, the \<^dof> evaluation command taking a \<^hol>-expression:
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@{theory_text [display,indent=6, margin=70]
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\<open> value*[ass::Assertion, relvce=2::int] \<open>filter (\<lambda> \<sigma>. relvce \<sigma> > 2) @{Assertion-instances}\<close>\<close>
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}
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where \<^dof> command \<open>value*\<close> type-checks, expands in an own validation phase
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the \<open>Assertion-instances\<close>-term antiquotation, and evaluates the resulting \<^hol> expression
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above. Assuming an ontology providing the class \<open>Assertion\<close> having at least the
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integer attribute \<open>relvce\<close>, the command finally creates an instance of \<open>Assertion\<close> and
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binds this to the label \<open>ass\<close> for further use.
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Beyond the gain of expressivity in \<^dof> ontologies, term-antiquotations pave the way
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for advanced queries of elements inside an integrated source, and invariants
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allow for formal proofs over the relations/translations of ontologies and ontology-instances.
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The latter question raised scientific interest under the label ``ontology mapping'' for
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which we therefore present a formal solution. To sum up, we completed \<^dof> to
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a fairly rich, ITP-oriented ontology language, which is a concrete proposal for the
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ITP community allowing a deeper structuring of mathematical libraries
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such as the Archive of Formal Proofs (AFP).
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\<close>
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(*<*)
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declare_reference*[casestudy::text_section]
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(*>*)
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section*[bgrnd::background,main_author="Some(@{docitem ''bu''}::author)"] \<open> Background\<close>
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(*
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subsection\<open>Isabelle/DOF Design and Implementation\<close>
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text\<open>
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In this section, we provide a guided tour through the underlying technologies of this paper:
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\begin{inparaenum}
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\item Isabelle and Isabelle/HOL,
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\item \<^dof> and its Ontology Definition Language (ODL).
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\end{inparaenum}
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\<close>
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subsection*[bgrnd_isabelle::text_section]\<open>Isabelle and HOL\<close>
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text\<open>
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While still widely perceived as an interactive theorem proving environment, Isabelle
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@{cite "nipkow.ea:isabelle:2002"} has become a generic system framework providing
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an infrastructure for plug-ins. This comprises extensible
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state components, extensible syntax, code-generation, and advanced documentation support.
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The plugin Isabelle/HOL offers a modeling language similar to functional programming languages
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extended by a logic and automated proof and animation techniques.
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\<close>
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*)
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subsection*[bgrnd_isadof::background]\<open>The \<^dof> Framework\<close>
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text\<open>
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\<^dof>~@{cite "brucker.ea:isabelle-ontologies:2018" and
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"brucker.ea:isabelledof:2019"}
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is a document ontology framework that extends \<^isabelle>.
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\<^dof> offers basically two things: a language called Ontology Definition Language (ODL)
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to \<^emph>\<open>specify\<close> a formal ontology,
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and ways to \<^emph>\<open>annotate\<close> an integrated document written in \<^isabelle> with the specified
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meta-data. Additionally, \<^dof> generates from an ontology a family of
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\<^emph>\<open>antiquotations\<close> allowing to establish machine-checked links between classified entities.
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% Unlike UML, however, \<^dof> allows for integrated documents with informal and formal elements
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% including the necessary management of logical contexts.
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The perhaps most attractive aspect of \<^dof> is its deep integration into the IDE of Isabelle
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(Isabelle/PIDE), which allows a hypertext-like navigation as well as fast user-feedback
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during development and evolution of the integrated source. This includes rich editing support,
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including on-the-fly semantics checks, hinting, or auto-completion.
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\<^dof> supports \<^LaTeX> - based document generation as well as ontology-aware ``views'' on
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the integrated document, \<^ie>, specific versions of generated PDF addressing,
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for example, different stake-holders.
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\<close>
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(*<*)
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figure*[isadof_screenshot::figure, relative_width="100", src="''figures/cicm2018-combined''"]\<open>
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The \<^dof> IDE (left) and the corresponding PDF(right).
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\<close>
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text*[description_scrrenshot::background]\<open>
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@{docitem \<open>isadof_screenshot\<close>} shows \<^dof> in action: the left-hand side shows the IDE of
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\<^dof> in the context of a user session maintaining our case study
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(see @{docitem (unchecked) "casestudy"})
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where a user is editing a semi-formal requirement. The right-hand side show the
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generated PDF document that can be used within a certification process.
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\<close>
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(*>*)
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subsection*[bgrnd_ODL::background]\<open>A Guided Tour through ODL\<close>
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text\<open>
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\<^dof> provides a strongly typed OLD that provides the usual
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concepts of ontologies such as
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\<^item> \<^emph>\<open>document class\<close> (using the \<^theory_text>\<open>doc_class\<close> keyword) that describes a concept,
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\<^item> \<^emph>\<open>attributes\<close> specific to document classes (attributes might be initialized with default
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values), and
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\<^item> a special link, the reference to a super-class,
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establishes an \<^emph>\<open>is-a\<close> relation between classes.
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% classes may refer to other classes via a regular expression in an optional \<^emph>\<open>where\<close> clause
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% (a class with a where clause is called \<^emph>\<open>monitor\<close>).
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\fixIsarList The types of attributes are \<^hol>-types. Thus, ODL can refer to any predefined type
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from the \<^hol> library, \<^eg>, \<^type>\<open>string\<close>, \<^type>\<open>int\<close> as well as parameterized types, \<^eg>,
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\<^type>\<open>option\<close>, \<^type>\<open>list\<close>. As a consequence of the Isabelle document model, ODL definitions
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may be arbitrarily mixed with standard \<^hol> type definitions. Document class definitions are
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\<^hol>-types, allowing for formal \<^emph>\<open>links\<close> to and between ontological concepts. For example, the
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basic concept of requirements from CENELEC 50128~@{cite "bsi:50128:2014"} is captured in ODL as
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follows:
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@{theory_text [display,indent=10, margin=70]
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\<open>
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doc_class requirement = text_element + (* derived from text_element *)
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long_name ::"string option" (* an optional string attribute *)
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is_concerned::"role set" (* roles working with this req. *)
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\<close>}
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This ODL class definition maybe part of one or more Isabelle theory--files capturing the entire
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ontology definition. Isabelle's session management allows for pre-compiling them before being
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imported in the actual target documentation required to be compliant to this ontology.
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%\vspace{-0.7cm}\<close>
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side_by_side_figure*["text-elements"::side_by_side_figure,anchor="''fig-Req-Def-ex''",
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caption="''A Text-Element as Requirement.''",relative_width="48",
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src="''figures/Req-Def-ex''",anchor2="''fig-Req-Appl-ex''",
|
|
caption2="''Referencing a Requirement.''",relative_width2="48",
|
|
src2="''figures/Req-Appl-ex''"]\<open>Referencing a Requirement. \<close>
|
|
|
|
text\<open>\autoref{text-elements} shows an ontological annotation of a requirement and the referencing
|
|
via an antiquotation \<^theory_text>\<open>requirement \<open>req1\<close>\<close> generated by \<^dof> from the above
|
|
class definition. Undefined or ill-typed references were rejected, the high-lighting displays
|
|
the hyperlinking which is activated on a click. Revising the actual definition of \<open>requirement\<close>,
|
|
it suffices to click on its keyword: the IDE will display the class-definition and its surrounding
|
|
documentation in the ontology.\<close>
|
|
|
|
(*
|
|
|
|
text\<open>\<^dof>'s generated antiquotations are part of a general mechanism of
|
|
Isabelle's standard antiquotations heavily used in various papers and technical reports.
|
|
For example, in the following informal text, the antiquotation \<^verbatim>\<open>thm refl\<close> refers
|
|
to the reflexivity axiom from HOL:
|
|
@{theory_text [display,indent=10, margin=70]
|
|
\<open>
|
|
text<Open>According to the reflexivity axiom <@>{thm refl}, we obtain in \<Gamma>
|
|
for <@>{term <Open>fac 5<Close>} the result <@>{value <Open>fac 5<Close>}.<Close>\<close>}
|
|
In the PDF output, this is represented as follows:
|
|
\begin{out}
|
|
According to the reflexivity axiom $x = x$, we obtain in \<open>\<Gamma>\<close> for $\operatorname{fac} 5$ the result $120$.
|
|
\end{out}
|
|
The antiquotation \inlineisar|<@>{value <Open>fac 5<Close>}| refers to a function that is defined in the
|
|
preceding logical context (and parsed as inner syntax) to compute the value of $5!$, \ie, $120$.
|
|
Summing up, antiquotations can refer to formal content, can be type-checked before being displayed
|
|
and can be used for calculations before actually being typeset. All these features can be
|
|
used for the calculation of attribute values (as in @{docitem \<open>text-elements\<close>}, observe the value
|
|
\<open>UNIV\<close> used to set the attribute \<open>is_concerned\<close> is a HOL-constant denoting the universal set).
|
|
|
|
Finally, for each ontological concept, a custom representation, using \<^LaTeX>-notation, for the
|
|
generated PDF document can be defined. The latter includes, \<^eg>, the possibility to automatically
|
|
generated glossaries or lists of concepts.
|
|
\<close>
|
|
|
|
*)
|
|
(*<*)
|
|
type_synonym A = int
|
|
type_synonym B = int
|
|
record T =
|
|
x :: A
|
|
y :: B
|
|
|
|
term "\<lparr>x = a,y = b\<rparr>"
|
|
(*>*)
|
|
|
|
subsection\<open>Meta-Objects as Extensible Records\<close>
|
|
(* too fat ? what do we need of this ? *)
|
|
text\<open>
|
|
\<^isabelle> supports both fixed and schematic records at the level of terms and
|
|
types. The notation for terms and types is as follows:
|
|
|
|
\<^item> fixed record terms \<^term>\<open>\<lparr>x = a,y = b\<rparr>\<close>; fixed record types \<open>\<lparr>x::A, y::B\<rparr>\<close>,
|
|
\<^item> schematic record terms \<^term>\<open>\<lparr>x = a,y = b, \<dots> = m::'a\<rparr>\<close>;
|
|
schematic record types: \<open>\<lparr>x::A, y::B, \<dots> = 'a\<rparr>\<close> which were usually abbreviated
|
|
to \<^typ>\<open>'a T_scheme\<close>,
|
|
\<^item> selectors are written \<^term>\<open>x(R::'a T_scheme)\<close>, \<^term>\<open>y(R::'a T_scheme)\<close>, and
|
|
\<^item> updates were denoted \<^theory_text>\<open>r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<close> or just \<^term>\<open>r\<lparr>x:=a, y:=b\<rparr>\<close>.
|
|
\<close>
|
|
|
|
text\<open> ... where the so-called more-field \<open>\<dots>\<close> is used to 'fill-in' record-extensions.
|
|
Schematic record types @{cite "naraschewski1998object"} allow for simulating object-oriented features such as
|
|
(single-)inheritance while maintaining a compositional style of verification:
|
|
it is possible to prove a property \<^term>\<open>P (a::'a T_scheme)\<close>
|
|
which will remain true for all extensions (which are just instances of the
|
|
\<^typ>\<open>'a T_scheme\<close>-type).
|
|
|
|
\<close>
|
|
|
|
text\<open>In \<^dof>, \<^verbatim>\<open>onto_class\<close>es and the logically equivalent \<^verbatim>\<open>doc_class\<close>es were
|
|
represented by schematic record types and instances thereof by schematic terms.
|
|
Invariants of an \<^verbatim>\<open>onto_class\<close> are thus predicates over schematic record
|
|
types and can therefore be inherited in a subclass. \<^dof> handles the parametric
|
|
polymorphism implicitly.
|
|
\<close>
|
|
|
|
subsection\<open>Advanced Evaluation in Isabelle\<close>
|
|
text\<open>Besides the powerful, but relatively slow rewriting-based proof method
|
|
\<^theory_text>\<open>simp\<close>, there are basically two other techniques for the evaluation of terms:
|
|
\<^item> evaluation via reflection into SML (\<^theory_text>\<open>eval\<close>), and
|
|
\<^item> normalization by evaluation @{cite "AehligHN12"} (\<^theory_text>\<open>nbe\<close>).\<close>
|
|
|
|
text\<open>
|
|
The former is based on a nearly one-to-one compilation of datatype specification
|
|
constructs and recursive function definitions into SML datatypes and functions.
|
|
The generated code is directly compiled by the underlying SML compiler of the
|
|
Isabelle platform. This way, pattern-matching becomes natively compiled rather
|
|
than interpreted as in the matching process of \<^theory_text>\<open>simp\<close>. @{cite "AehligHN12"}
|
|
describes scenarios where \<^theory_text>\<open>eval\<close> is five orders of magnitude faster than \<^theory_text>\<open>simp\<close>.
|
|
However, it is restricted to ground terms.
|
|
|
|
The latter is based on a compilation of datatype specifications into a uniform
|
|
data-universe enriched by closures and explicit variables. Recursive function
|
|
definitions in \<^hol> were compiled to SML functions where the argument terms
|
|
were represented in the data-universe. Pattern-matching is still compiled to
|
|
native functions executed if closures are completed. \<^theory_text>\<open>nbe\<close> is not restricted
|
|
to ground terms, but lies in its efficiency between \<^theory_text>\<open>simp\<close> and \<^theory_text>\<open>eval\<close>.
|
|
|
|
\<^dof> uses a combination of these three techniques in order to check invariants
|
|
for ontological classes on the fly during editing, paving the way for both
|
|
a uniform specification language of ontological data --- namely \<^hol> --- as
|
|
well as the possibility to \<^emph>\<open>prove\<close> properties over and relations between
|
|
classes.\<close>
|
|
|
|
|
|
section*[invariants::technical,main_author="Some(@{docitem ''nic''}::author)"]
|
|
\<open>Term-Context Support, Invariants and Queries in DOF\<close>
|
|
|
|
(*<*)
|
|
(* Ontology example for mathematical papers *)
|
|
|
|
doc_class myauthor =
|
|
email :: "string" <= "''''"
|
|
doc_class mytext_section =
|
|
authored_by :: "myauthor set" <= "{}"
|
|
level :: "int option" <= "None"
|
|
doc_class myintroduction = mytext_section +
|
|
authored_by :: "myauthor set" <= "UNIV"
|
|
uses :: "string set"
|
|
invariant author_finite :: "finite (authored_by \<sigma>)"
|
|
and force_level :: "the (level \<sigma>) > 1"
|
|
doc_class myclaim = myintroduction +
|
|
based_on :: "string list"
|
|
doc_class mytechnical = text_section +
|
|
formal_results :: "thm list"
|
|
|
|
datatype kind = expert_opinion | argument | "proof"
|
|
|
|
doc_class myresult = mytechnical +
|
|
evidence :: kind
|
|
property :: "thm list" <= "[]"
|
|
invariant has_property :: "evidence \<sigma> = proof \<longleftrightarrow> property \<sigma> \<noteq> []"
|
|
doc_class myconclusion = text_section +
|
|
establish :: "(myclaim \<times> myresult) set"
|
|
invariant establish_defined :: "\<forall> x. x \<in> Domain (establish \<sigma>)
|
|
\<longrightarrow> (\<exists> y \<in> Range (establish \<sigma>). (x, y) \<in> establish \<sigma>)"
|
|
|
|
declare[[invariants_checking = true]]
|
|
declare[[invariants_checking_with_tactics = true]]
|
|
|
|
text*[church::myauthor, email="\<open>church@lambda.org\<close>"]\<open>\<close>
|
|
|
|
text*[resultProof::myresult, evidence = "proof", property="[@{thm \<open>HOL.refl\<close>}]"]\<open>\<close>
|
|
|
|
text*[resultProof2::myresult, evidence = "proof", property="[@{thm \<open>HOL.sym\<close>}]"]\<open>\<close>
|
|
|
|
(*text*[introduction1::myintroduction, authored_by = "{@{myauthor \<open>church\<close>}}", level = "Some 0"]\<open>\<close>*)
|
|
|
|
(*text*[claimNotion::myclaim, authored_by = "{@{myauthor \<open>church\<close>}}", based_on= "[\<open>Notion1\<close>, \<open>Notion2\<close>]", level = "Some 0"]\<open>\<close>*)
|
|
|
|
text*[introduction2::myintroduction, authored_by = "{@{myauthor \<open>church\<close>}}", level = "Some 2"]\<open>\<close>
|
|
|
|
declare[[invariants_checking = false]]
|
|
declare[[invariants_checking_with_tactics = false]]
|
|
(*>*)
|
|
|
|
text\<open>
|
|
To offer a smooth integration into the \<^emph>\<open>formal\<close> theory development process,
|
|
\<^isabelle> should be able to dynamically interpret the source document.
|
|
But the specific antiquotations introduced by \<^dof> are not directly recognized
|
|
by \<^isabelle>, and the process of term checking and evaluation must be enriched.
|
|
Previous works~@{cite "brucker.ea:isabelle-ontologies:2018" and "brucker.ea:isabelledof:2019"}
|
|
added a validation step for the SML and semi-formal text contexts.
|
|
To support \<^dof> antiquotations in the term contexts, the validation step should
|
|
be improved and a new step, which we call \<^emph>\<open>elaboration\<close> must be added to allow
|
|
these antiquotations in \<open>\<lambda>\<close>-terms.
|
|
The resulting process encompasses the following steps:
|
|
\<^item> Parse the term which represents the object in \<^hol>,
|
|
\<^item> Infer the type of the term,
|
|
\<^item> Certify the term,
|
|
\<^item> Validate the term: the \<^dof> antiquotations terms are parsed and type-checked,
|
|
\<^item> Elaborate: the \<^dof> antiquotations terms are expanded.
|
|
The antiquotations are replaced by the object in \<^hol> they reference
|
|
i.e. a \<open>\<lambda>\<close>-calculus term,
|
|
\<^item> Pass on the information to Isabelle/PIDE, and
|
|
\<^item> Code generation:
|
|
\<^vs>\<open>-0.3cm\<close>
|
|
\<^item> Evaluation of \<^hol> expressions with ontological annotations,
|
|
\<^item> Generation of ontological invariants processed simultaneously
|
|
with the generation of the document (a theory in \<^hol>).
|
|
|
|
\<^isabelle> provides commands to type-check and print terms (the command \<^theory_text>\<open>term\<close>)
|
|
and evaluate and print a term (the command \<^theory_text>\<open>value\<close>).
|
|
We provide the equivalent commands, respectively \<^theory_text>\<open>term*\<close> and \<^theory_text>\<open>value*\<close>, which support
|
|
the validation and elaboration phase.
|
|
This allow a smooth integration into the \<^emph>\<open>formal\<close> theory development process.
|
|
\<close>
|
|
|
|
(*<*)
|
|
declare_reference*["type-checking-example"::side_by_side_figure]
|
|
(*>*)
|
|
|
|
text\<open>
|
|
If we take back the ontology example for mathematical papers
|
|
of~@{cite "brucker.ea:isabelledof:2019"} shown in \autoref{fig-ontology-example}
|
|
\begin{figure}
|
|
@{boxed_theory_text [display] \<open>
|
|
doc_class myauthor =
|
|
email :: "string" <= "''''"
|
|
doc_class mytext_section =
|
|
authored_by :: "myauthor set" <= "{}"
|
|
level :: "int option" <= "None"
|
|
doc_class myintroduction = mytext_section +
|
|
authored_by :: "myauthor set" <= "UNIV"
|
|
uses :: "string set"
|
|
invariant author_finite :: "finite (authored_by \<sigma>)"
|
|
and force_level :: "the (level \<sigma>) > 1"
|
|
doc_class myclaim = myintroduction +
|
|
based_on :: "string list"
|
|
doc_class mytechnical = text_section +
|
|
formal_results :: "thm list"
|
|
|
|
datatype kind = expert_opinion | argument | "proof"
|
|
|
|
doc_class myresult = mytechnical +
|
|
evidence :: kind
|
|
property :: "thm list" <= "[]"
|
|
invariant has_property :: "evidence \<sigma> = proof \<longleftrightarrow> property \<sigma> \<noteq> []"
|
|
doc_class myconclusion = text_section +
|
|
establish :: "(myclaim \<times> myresult) set"
|
|
invariant establish_defined :: "\<forall> x. x \<in> Domain (establish \<sigma>)
|
|
\<longrightarrow> (\<exists> y \<in> Range (establish \<sigma>). (x, y) \<in> establish \<sigma>)"
|
|
\<close>}
|
|
\caption{Excerpt of an ontology example for mathematical papers.}
|
|
\label{fig-ontology-example}
|
|
\end{figure}
|
|
we can define some class instances for this ontology with the \<^theory_text>\<open>text*\<close> command,
|
|
as in \autoref{fig-instances-example}.
|
|
\begin{figure}
|
|
@{boxed_theory_text [display] \<open>
|
|
text*[church::myauthor, email="\<open>church@lambda.org\<close>"]\<open>\<close>
|
|
|
|
text*[resultProof::myresult, evidence = "proof", property="[@{thm \<open>HOL.refl\<close>}]"]\<open>\<close>
|
|
|
|
text*[resultProof2::myresult, evidence = "proof", property="[@{thm \<open>HOL.sym\<close>}]"]\<open>\<close>
|
|
|
|
text*[introduction1::myintroduction, authored_by = "{@{myauthor \<open>church\<close>}}", level = "Some 0"]\<open>\<close>
|
|
text*[introduction2::myintroduction, authored_by = "{@{myauthor \<open>church\<close>}}", level = "Some 2"]\<open>\<close>
|
|
|
|
text*[claimNotion::myclaim, authored_by = "{@{myauthor \<open>church\<close>}}", based_on= "[\<open>Notion1\<close>, \<open>Notion2\<close>]", level = "Some 0"]\<open>\<close>
|
|
\<close>}
|
|
\caption{Some instances of the classes of the ontology in the \autoref{fig-ontology-example}.}
|
|
\label{fig-instances-example}
|
|
\end{figure}
|
|
In the instance \<^theory_text>\<open>introduction1\<close>, \<^term>\<open>@{myauthor \<open>church\<close>}\<close> denotes
|
|
the instance \<^theory_text>\<open>church\<close> of the class \<^typ>\<open>myauthor\<close>.
|
|
One can now reference a class instance in a \<^theory_text>\<open>term*\<close> command.
|
|
In the command \<^theory_text>\<open>term*\<open>@{myauthor \<open>church\<close>}\<close>\<close>
|
|
the term \<^term>\<open>@{myauthor \<open>church\<close>}\<close> is type-checked, \<^ie>, the command \<^theory_text>\<open>term*\<close> checks that
|
|
\<^theory_text>\<open>church\<close> references a term of type \<^typ>\<open>myauthor\<close>
|
|
(see \<^side_by_side_figure>\<open>type-checking-example\<close>).
|
|
\<close>
|
|
|
|
|
|
side_by_side_figure*[
|
|
"type-checking-example"::side_by_side_figure
|
|
, anchor="''fig-term-type-checking-ex''"
|
|
, caption="''church is an existing instance.''"
|
|
, relative_width="48"
|
|
, src="''figures/term-context-checking-example''"
|
|
, anchor2="''fig-term-type-checking-failed-ex''"
|
|
, caption2="''The churche instance is not defined.''"
|
|
, relative_width2="48"
|
|
, src2="''figures/term-context-failed-checking-example''"
|
|
]\<open>Type-checking of antiquotations in term context.\<close>
|
|
|
|
(*<*)
|
|
declare_reference*["evaluation-example"::side_by_side_figure]
|
|
(*>*)
|
|
|
|
text\<open>
|
|
The command \<^theory_text>\<open>value*\<open>email @{author \<open>church\<close>}\<close>\<close>
|
|
type-checks the antiquotation \<^term>\<open>@{myauthor \<open>church\<close>}\<close>,
|
|
and then returns the value of the attribute \<^const>\<open>email\<close> for the \<^theory_text>\<open>church\<close> instance,
|
|
\<^ie> the \<^hol> string \<^term>\<open>''church@lambda.org''\<close>
|
|
(see \<^side_by_side_figure>\<open>evaluation-example\<close>).
|
|
\<close>
|
|
|
|
side_by_side_figure*[
|
|
"evaluation-example"::side_by_side_figure
|
|
, anchor="''fig-term-evaluation-ex''"
|
|
, caption="''The evaluation succeeds and returns the value.''"
|
|
, relative_width="48"
|
|
, src="''figures/term-context-evaluation-example''"
|
|
, anchor2="''fig-term-failed-evaluation-ex''"
|
|
, caption2="''The evaluation fails, due to the undefined instance.''"
|
|
, relative_width2="48"
|
|
, src2="''figures/term-context-failed-evaluation-example''"
|
|
]\<open>Evaluation of antiquotations in term context.\<close>
|
|
|
|
(*
|
|
figure*[
|
|
"term-context-checking-example-figure"::figure
|
|
, relative_width="99"
|
|
, src="''figures/term-context-checking-example''"
|
|
]\<open>The invariant \<^theory_text>\<open>force_level\<close> of the class claim is inherited
|
|
from the class \<^theory_text>\<open>introduction\<close> and is violated by the instance \<^theory_text>\<open>claimNotion\<close>.
|
|
\<close>
|
|
|
|
figure*[
|
|
"term-context-evaluation-figure"::figure
|
|
, relative_width="99"
|
|
, src="''figures/term-context-evaluation-example''"
|
|
]\<open>The invariant \<^theory_text>\<open>force_level\<close> of the class claim is inherited
|
|
from the class \<^theory_text>\<open>introduction\<close> and is violated by the instance \<^theory_text>\<open>claimNotion\<close>.
|
|
\<close>
|
|
*)
|
|
|
|
(*<*)
|
|
declare_reference*["term-context-equality-evaluation"::figure]
|
|
(*>*)
|
|
|
|
text\<open>
|
|
We can even compare class instances. \<^figure>\<open>term-context-equality-evaluation\<close>
|
|
shows that the two class instances \<^theory_text>\<open>resultProof\<close> and \<^theory_text>\<open>resultProof2\<close> are not equivalent
|
|
because their attribute \<^term>\<open>property\<close> differ.
|
|
\<close>
|
|
|
|
figure*[
|
|
"term-context-equality-evaluation"::figure
|
|
, relative_width="80"
|
|
, src="''figures/term-context-equality-evaluation-example''"
|
|
]\<open>Evaluation of the equivalence of two class instances.
|
|
\<close>
|
|
|
|
text\<open>
|
|
A novel mechanism to specify constraints as invariants is implemented
|
|
and can now be specified in common \<^hol> syntax, using the keyword \<^theory_text>\<open>invariant\<close>
|
|
in the class definition (recall \autoref{fig-ontology-example}).
|
|
Following the constraints proposed in @{cite "brucker.ea:isabelle-ontologies:2018"},
|
|
one can specify that any instance of a class \<^typ>\<open>myresult\<close>
|
|
finally has a non-empty property list, if its \<^typ>\<open>kind\<close> is \<^const>\<open>proof\<close>
|
|
(see the \<^theory_text>\<open>invariant has_property\<close>), or that
|
|
the relation between \<^typ>\<open>myclaim\<close> and \<^typ>\<open>myresult\<close> expressed in the attribute \<^const>\<open>establish\<close>
|
|
must be defined when an instance
|
|
of the class \<^typ>\<open>myconclusion\<close> is defined (see the \<^theory_text>\<open>invariant establish_defined\<close>).
|
|
|
|
In our example, the \<^theory_text>\<open>invariant author_finite\<close> of the class \<^typ>\<open>myintroduction\<close> enforces
|
|
that the user sets the \<^const>\<open>authored_by\<close> set.
|
|
The \<open>\<sigma>\<close> symbol is reserved and references the future class instance.
|
|
By relying on the implementation of the Records
|
|
in \<^isabelle>~@{cite "wenzel:isabelle-isar:2020"},
|
|
one can reference an attribute of an instance using its selector function.
|
|
For example, \<^term>\<open>establish \<sigma>\<close> denotes the value
|
|
of the attribute \<^const>\<open>establish\<close>
|
|
of the future instance of the class \<^typ>\<open>myconclusion\<close>.
|
|
\<close>
|
|
|
|
(*<*)
|
|
declare_reference*["invariant-checking-figure"::figure]
|
|
(*>*)
|
|
|
|
text\<open>
|
|
The value of each attribute defined for the instances is checked at run-time
|
|
against their class invariants.
|
|
The instance \<^theory_text>\<open>resultProof\<close> respects the \<^theory_text>\<open>invariant has_property\<close>,
|
|
because we specify its attribute \<^const>\<open>evidence\<close> to the \<^typ>\<open>kind\<close> \<^const>\<open>proof\<close>,
|
|
we also specify its attribute \<^const>\<open>property\<close> with a suited value
|
|
as a \<^type>\<open>list\<close> of \<^type>\<open>thm\<close>.
|
|
In \<^figure>\<open>invariant-checking-figure\<close>,
|
|
we try to specify a new instance \<^theory_text>\<open>introduction1\<close> of the class \<^typ>\<open>myintroduction\<close>.
|
|
But an invariant checking error is triggered because we do not respect the
|
|
constraint specified in the \<^theory_text>\<open>invariant force_level\<close>,
|
|
when we specify the attribute \<^const>\<open>level\<close> of \<^theory_text>\<open>introduction1\<close> to \<^term>\<open>Some (0::int)\<close>.
|
|
The \<^theory_text>\<open>invariant force_level\<close> forces the value of the argument
|
|
of the attribute \<^const>\<open>level\<close> to be greater than 1.
|
|
\<close>
|
|
|
|
figure*[
|
|
"invariant-checking-figure"::figure
|
|
, relative_width="99"
|
|
, src="''figures/invariant-checking-violated-example''"
|
|
]\<open>The \<^theory_text>\<open>invariant force_level\<close> of the class \<^typ>\<open>myintroduction\<close> is violated by
|
|
the instance \<^theory_text>\<open>introduction1\<close>.\<close>
|
|
|
|
(*<*)
|
|
declare_reference*["inherited-invariant-checking-figure"::figure]
|
|
(*>*)
|
|
|
|
text\<open>
|
|
Classes inherit the invariants from their super-class.
|
|
As the class \<^typ>\<open>myclaim\<close> is a subclass
|
|
of the class \<^typ>\<open>myintroduction\<close>, it inherits the \<^typ>\<open>myintroduction\<close> invariants.
|
|
Hence the \<^theory_text>\<open>invariant force_level\<close> is checked
|
|
when the instance \<^theory_text>\<open>claimNotion\<close> is defined,
|
|
like in \<^figure>\<open>inherited-invariant-checking-figure\<close>.
|
|
\<close>
|
|
|
|
figure*[
|
|
"inherited-invariant-checking-figure"::figure
|
|
, relative_width="99"
|
|
, src="''figures/inherited-invariant-checking-violated-example''"
|
|
]\<open>The \<^theory_text>\<open>invariant force_level\<close> of the class \<^typ>\<open>myclaim\<close> is inherited
|
|
from the class \<^typ>\<open>myintroduction\<close> and is violated by the instance \<^theory_text>\<open>claimNotion\<close>.
|
|
\<close>
|
|
|
|
(*<*)
|
|
value*\<open>map (myresult.property) @{myresult-instances}\<close>
|
|
value*\<open>map (mytext_section.authored_by) @{myintroduction-instances}\<close>
|
|
|
|
value*\<open>filter (\<lambda>\<sigma>. myresult.evidence \<sigma> = proof) @{myresult-instances}\<close>
|
|
value*\<open>filter (\<lambda>\<sigma>. the (mytext_section.level \<sigma>) > 1) @{myintroduction-instances}\<close>
|
|
value*\<open>filter (\<lambda>\<sigma>. myresult.evidence \<sigma> = argument) @{myresult-instances}\<close>
|
|
(*>*)
|
|
|
|
text\<open>
|
|
A new mechanism to make query on instances is available and uses
|
|
the \<^hol> implementation of \<^type>\<open>list\<close>s.
|
|
Complex queries can then be defined using functions over the instances list.
|
|
To get the list of the properties of the instances of the class \<^typ>\<open>myresult\<close>,
|
|
and the list of the authors of the instances of the class \<^typ>\<open>myintroduction\<close>,
|
|
one can use the commands:
|
|
@{theory_text [display,indent=10, margin=70] \<open>
|
|
value*\<open>map (myresult.property) @{myresult-instances}\<close>
|
|
value*\<open>map (mytext_section.authored_by) @{myintroduction-instances}\<close>
|
|
\<close>}
|
|
To get the list of the instances of the class \<^typ>\<open>myresult\<close>
|
|
whose \<^const>\<open>evidence\<close> is a \<^const>\<open>proof\<close>, on can use the command:
|
|
@{theory_text [display,indent=10, margin=70] \<open>
|
|
value*\<open>filter (\<lambda>\<sigma>. myresult.evidence \<sigma> = proof) @{myresult-instances}\<close>
|
|
\<close>}
|
|
To get the list of the instances of the class \<^typ>\<open>myintroduction\<close> whose \<^const>\<open>level\<close> > 1,
|
|
on can use the command:
|
|
@{theory_text [display,indent=10, margin=70] \<open>
|
|
value*\<open>filter (\<lambda>\<sigma>. the (mytext_section.level \<sigma>) > 1) @{myintroduction-instances}\<close>
|
|
\<close>}
|
|
\<close>
|
|
|
|
section*["morphisms"::technical,main_author="Some(@{docitem ''idir''}::author)"] \<open>Proving Morphisms on Ontologies\<close>
|
|
|
|
text\<open>
|
|
\<^dof> framework does not assume that all documents reference the same ontology.
|
|
Each document may build its local ontology without any external reference.
|
|
It may also build it based upon one or several reference ontologies (i.e., standard ones).
|
|
|
|
The relationship between the local ontology and the reference one is formalised using a morphism function.
|
|
More precisely, a class of a local ontology may be described as a consequence of a transformation applied
|
|
to one or several other class(es) defined in other ontologies. This means that each instance of the former can be
|
|
computed from one or more instances of the latter.
|
|
|
|
Thanks to the morphism relationship, the obtained class may either import properties (definitions are preserved)
|
|
or map properties (the properties are different but are semantically equivalent) that are defined in the referenced class(es).
|
|
It may also define additional properties.
|
|
\<close>
|
|
|
|
text\<open>
|
|
\begin{figure}
|
|
@{boxed_theory_text [display] \<open>
|
|
definition sum
|
|
where "sum S = (fold (+) S 0)"
|
|
|
|
datatype Hardware_Type =
|
|
Motherboard |
|
|
Expansion_Card |
|
|
Storage_Device |
|
|
Fixed_Media |
|
|
Removable_Media |
|
|
Input_Device |
|
|
Output_Device
|
|
|
|
onto_class Resource =
|
|
name :: string
|
|
|
|
onto_class Electronic = Resource +
|
|
provider :: string
|
|
manufacturer :: string
|
|
|
|
onto_class Component = Electronic +
|
|
mass :: int
|
|
dimensions :: "int list"
|
|
|
|
onto_class Simulation_Model = Electronic +
|
|
type :: string
|
|
|
|
onto_class Informatic = Resource +
|
|
description :: string
|
|
|
|
onto_class Hardware = Informatic +
|
|
type :: Hardware_Type
|
|
mass :: int
|
|
composed_of :: "Component list"
|
|
invariant c1 :: "mass \<sigma> = sum(map Component.mass (composed_of \<sigma>))"
|
|
|
|
onto_class R_Software = Informatic +
|
|
version :: int
|
|
\<close>}
|
|
\caption{An extract of a reference ontology.}
|
|
\label{fig-Reference-Ontology-example}
|
|
\end{figure}
|
|
|
|
To illustrate this process, we use the reference ontology (considered as a standard) described
|
|
in the listing \autoref{fig-Reference-Ontology-example}, defining the \<open>Resource\<close>, \<open>Electronic\<close>,
|
|
\<open>Component\<close>, \<open>Informatic\<close>, \<open>Hardware\<close> and \<open>Software\<close> concepts (or classes).
|
|
Each class contains a set of attributes or properties and some local invariants.
|
|
|
|
In our example, we focus on the \<open>Hardware\<close> class containing a \<open>mass\<close> attribute and composed of a list
|
|
of components with a \<open>mass\<close> attribute formalising the mass value of each component. The \<open>Hardware\<close>
|
|
class also contains a local invariant \<open>c1\<close> to define a constraint linking the global mass of
|
|
a \<open>Hardware\<close> object with the masses of its own components.
|
|
\<close>
|
|
|
|
text\<open>
|
|
\begin{figure}
|
|
@{boxed_theory_text [display] \<open>
|
|
onto_class Item =
|
|
name :: string
|
|
|
|
onto_class Product = Item +
|
|
serial_number :: int
|
|
provider :: string
|
|
mass :: int
|
|
|
|
onto_class Computer_Software = Item +
|
|
version :: int
|
|
|
|
onto_class Electronic_Component = Product +
|
|
dimensions :: "int set"
|
|
|
|
onto_class Computer_Hardware = Product +
|
|
type :: Hardware_Type
|
|
composed_of :: "Product list"
|
|
invariant c2 :: "Product.mass \<sigma> = sum(map Product.mass (composed_of \<sigma>))"
|
|
\<close>}
|
|
\caption{An extract of a local (user) ontology.}
|
|
\label{fig-Local-Ontology-example}
|
|
\end{figure}
|
|
|
|
|
|
For the needs of our document, we have defined a simple ontology to classify Software and Hardware
|
|
objects. This ontology is described in the figure \autoref{fig-Local-Ontology-example} and
|
|
defines the \<open>Item\<close>, \<open>Product\<close>, \<open>Computer_Software\<close>
|
|
and \<open>Computer_Hardware\<close> concepts. As for the reference ontology, we focus on the \<open>Computer_Hardware\<close>
|
|
class defined as a list of products characterised by their mass value. This class contains a local
|
|
invariant \<open>c2\<close> to guarantee that its own mass value equals the sum of all the masses of the products
|
|
composing the object.
|
|
\<close>
|
|
|
|
|
|
text\<open>
|
|
\begin{figure}
|
|
@{boxed_theory_text [display] \<open>
|
|
definition Item_to_Resource_morphism :: "'a Item_scheme \<Rightarrow> Resource"
|
|
("_ \<langle>Resource\<rangle>\<^sub>I\<^sub>t\<^sub>e\<^sub>m" [1000]999)
|
|
where "\<sigma> \<langle>Resource\<rangle>\<^sub>I\<^sub>t\<^sub>e\<^sub>m = \<lparr> Resource.tag_attribute = 0::int ,
|
|
Resource.name = name \<sigma> \<rparr>"
|
|
|
|
definition Product_to_Component_morphism :: "'a Product_scheme \<Rightarrow> Component"
|
|
("_ \<langle>Component\<rangle>\<^sub>P\<^sub>r\<^sub>o\<^sub>d\<^sub>u\<^sub>c\<^sub>t" [1000]999)
|
|
where " \<sigma> \<langle>Component\<rangle>\<^sub>P\<^sub>r\<^sub>o\<^sub>d\<^sub>u\<^sub>c\<^sub>t = \<lparr> Resource.tag_attribute = 1::int ,
|
|
Resource.name = name \<sigma> ,
|
|
Electronic.provider = Product.provider \<sigma> ,
|
|
Electronic.manufacturer = '''' ,
|
|
Component.mass = Product.mass \<sigma> ,
|
|
Component.dimensions = [] \<rparr>"
|
|
|
|
definition Computer_Hardware_to_Hardware_morphism :: "'a Computer_Hardware_scheme \<Rightarrow> Hardware"
|
|
("_ \<langle>Hardware\<rangle>\<^sub>C\<^sub>o\<^sub>m\<^sub>p\<^sub>u\<^sub>t\<^sub>e\<^sub>r\<^sub>H\<^sub>a\<^sub>r\<^sub>d\<^sub>w\<^sub>a\<^sub>r\<^sub>e" [1000]999)
|
|
where "\<sigma> \<langle>Hardware\<rangle>\<^sub>C\<^sub>o\<^sub>m\<^sub>p\<^sub>u\<^sub>t\<^sub>e\<^sub>r\<^sub>H\<^sub>a\<^sub>r\<^sub>d\<^sub>w\<^sub>a\<^sub>r\<^sub>e =
|
|
\<lparr> Resource.tag_attribute = 2::int ,
|
|
Resource.name = name \<sigma> ,
|
|
Informatic.description = ''no description'',
|
|
Hardware.type = Computer_Hardware.type \<sigma> ,
|
|
Hardware.mass = mass \<sigma> ,
|
|
Hardware.composed_of = map Product_to_Component_morphism
|
|
(Computer_Hardware.composed_of \<sigma>)
|
|
\<rparr>"
|
|
\<close>}
|
|
\caption{An extract of a mapping definitions.}
|
|
\label{fig-mapping-example}
|
|
\end{figure}
|
|
|
|
|
|
To check the coherence of our local ontology, we define a relationship between the local ontology
|
|
and the reference ontology using morphism functions (or mapping rules). These rules are applied to
|
|
define the relationship between one class of the local ontology to one or several other class(es)
|
|
described in the reference ontology.
|
|
|
|
For example,\<open>Product_to_Component_morphism\<close> and \<open>Computer_Hardware_to_Hardware_morphism\<close> definitions,
|
|
detailed in the figure \autoref{fig-mapping-example}, specify how \<open>Product\<close> or \<open>Computer_Hardware\<close>
|
|
objects are mapped to \<open>Component\<close>
|
|
or \<open>Hardware\<close> objects defined in the reference ontology. This mapping shows that the structure
|
|
of a (user) ontology may be quite different from the one of a standard ontology she references.
|
|
\<close>
|
|
|
|
|
|
(* Bu, can you take care of commenting on these last lemmas? *)
|
|
|
|
text\<open>
|
|
\begin{figure}
|
|
@{boxed_theory_text [display] \<open>
|
|
lemma inv_c2_preserved :
|
|
"c2_inv \<sigma> \<Longrightarrow> c1_inv (\<sigma> \<langle>Hardware\<rangle>\<^sub>C\<^sub>o\<^sub>m\<^sub>p\<^sub>u\<^sub>t\<^sub>e\<^sub>r\<^sub>H\<^sub>a\<^sub>r\<^sub>d\<^sub>w\<^sub>a\<^sub>r\<^sub>e)"
|
|
unfolding c1_inv_def c2_inv_def
|
|
Computer_Hardware_to_Hardware_morphism_def Product_to_Component_morphism_def
|
|
by (auto simp: comp_def)
|
|
|
|
lemma Computer_Hardware_to_Hardware_morphism_total :
|
|
"Computer_Hardware_to_Hardware_morphism ` ({X::Computer_Hardware. c2_inv X}) \<subseteq> ({X::Hardware. c1_inv X})"
|
|
using inv_c2_preserved
|
|
by auto
|
|
\<close>}
|
|
\caption{xxxx.}
|
|
\label{fig-xxx}
|
|
\end{figure}
|
|
\<close>
|
|
|
|
|
|
section*[ontoexample::text_section,main_author="Some(@{docitem ''idir''}::author)"] \<open>Applications\<close>
|
|
|
|
(* subsection\<open>Engineering Example : An Extract from PLib\<close>
|
|
Already done in the previous section
|
|
*)
|
|
|
|
subsection\<open>Mathematics Example : An Extract from OntoMath\textsuperscript{PRO}\<close>
|
|
|
|
(*<*)
|
|
(* OntoMathPro_Ontology example *)
|
|
|
|
datatype dc = creator string | title string
|
|
|
|
datatype owl =
|
|
backwardCompatibleWith string
|
|
| deprecated string
|
|
| incompatibleWith string
|
|
| priorVersion string
|
|
| versionInfo string
|
|
|
|
datatype rdfs =
|
|
comment string
|
|
| isDefinedBy string
|
|
| label string
|
|
| seeAlso string
|
|
|
|
datatype annotation = DC dc | OWL owl | RDFS rdfs
|
|
|
|
onto_class Top =
|
|
Annotations :: "annotation list"
|
|
|
|
onto_class Field_of_mathematics =
|
|
Annotations :: "annotation list"
|
|
invariant restrict_annotation_F ::"set (Annotations \<sigma>) \<subseteq>
|
|
range (RDFS o label) \<union> range (RDFS o comment)"
|
|
|
|
onto_class Mathematical_knowledge_object =
|
|
Annotations :: "annotation list"
|
|
invariant restrict_annotation_M ::"set (Annotations \<sigma>) \<subseteq>
|
|
range (RDFS o label) \<union> range (RDFS o comment)"
|
|
|
|
onto_class assoc_F_M =
|
|
contains:: "(Field_of_mathematics \<times> Mathematical_knowledge_object) set"
|
|
invariant contains_defined :: "\<forall> x. x \<in> Domain (contains \<sigma>)
|
|
\<longrightarrow> (\<exists> y \<in> Range (contains \<sigma>). (x, y) \<in> contains \<sigma>)"
|
|
|
|
onto_class assoc_M_F =
|
|
belongs_to :: "(Mathematical_knowledge_object \<times> Field_of_mathematics) set"
|
|
invariant belong_to_defined :: "\<forall> x. x \<in> Domain (belongs_to \<sigma>)
|
|
\<longrightarrow> (\<exists> y \<in> Range (belongs_to \<sigma>). (x, y) \<in> belongs_to \<sigma>)"
|
|
|
|
onto_class assoc_M_M =
|
|
is_defined_by :: "(Mathematical_knowledge_object \<times> Mathematical_knowledge_object) set"
|
|
invariant is_defined_by_defined :: "\<forall> x. x \<in> Domain (is_defined_by \<sigma>)
|
|
\<longrightarrow> (\<exists> y \<in> Range (is_defined_by \<sigma>). (x, y) \<in> is_defined_by \<sigma>)"
|
|
(*typedef 'a A'_scheme =
|
|
"{x :: 'a A_scheme. "
|
|
*)
|
|
|
|
onto_class assoc_M_M' =
|
|
"defines" :: "(Mathematical_knowledge_object \<times> Mathematical_knowledge_object) set"
|
|
invariant defines_defined :: "\<forall> x. x \<in> Domain (defines \<sigma>)
|
|
\<longrightarrow> (\<exists> y \<in> Range (defines \<sigma>). (x, y) \<in> defines \<sigma>)"
|
|
|
|
onto_class assoc_M_M_see_also =
|
|
see_also :: "(Mathematical_knowledge_object rel) set"
|
|
invariant see_also_trans :: "\<forall> r \<in> (see_also \<sigma>). trans r"
|
|
invariant see_also_sym :: "\<forall> r \<in> (see_also \<sigma>). sym r"
|
|
|
|
onto_class Problem = Mathematical_knowledge_object +
|
|
Annotations :: "annotation list"
|
|
|
|
onto_class Method = Mathematical_knowledge_object +
|
|
Annotations :: "annotation list"
|
|
|
|
onto_class assoc_Problem_Method =
|
|
is_solved_by :: "(Problem \<times> Method) set"
|
|
invariant is_solved_by_defined :: "\<forall> x. x \<in> Domain (is_solved_by \<sigma>)
|
|
\<longrightarrow> (\<exists> y \<in> Range (is_solved_by \<sigma>). (x, y) \<in> is_solved_by \<sigma>)"
|
|
|
|
onto_class assoc_Method_Problem =
|
|
solves :: "(Method \<times> Problem) set"
|
|
invariant solves_defined :: "\<forall> x. x \<in> Domain (solves \<sigma>)
|
|
\<longrightarrow> (\<exists> y \<in> Range (solves \<sigma>). (x, y) \<in> solves \<sigma>)"
|
|
|
|
(*>*)
|
|
|
|
text\<open>
|
|
The ontology \<^emph>\<open>OntoMath\textsuperscript{PRO}\<close> @{cite "DBLP:journals/corr/NevzorovaZKL14"}
|
|
is an OWL ontology of mathematical knowledge concepts.
|
|
It posits the IS-A semantics for hierarchies of mathematical knowledge elements,
|
|
and defines these elements as two hierarchies of classes:
|
|
a taxonomy of the fields of mathematics and a taxonomy of mathematical knowledge objects.
|
|
It defines two main type of relations for these two taxonomies:
|
|
directed relations between elements of the two hierarchies
|
|
like \<^const>\<open>belongs_to\<close>, \<^const>\<open>contains\<close>, \<^const>\<open>defines\<close>, \<^const>\<open>is_defined_by\<close>, \<^const>\<open>solves\<close>,
|
|
\<^const>\<open>is_solved_by\<close>, and a symmetric transitive associative relation \<^const>\<open>see_also\<close>
|
|
between two mathematical knowledge objects.
|
|
It also represents links with external resources such as DBpedia
|
|
with annotations properties @{cite "Parsia:12:OWO"}.
|
|
\<^dof> covers a wide range of the OWL concepts used by the ontology OntoMath\textsuperscript{PRO}.
|
|
We can represent the annotations properties as datatypes and
|
|
then attach them as an attributes list to a class.
|
|
For example the declaration:
|
|
@{theory_text [display,indent=10, margin=70] \<open>
|
|
datatype dc = creator string | title string
|
|
datatype owl = backwardCompatibleWith string
|
|
| deprecated string
|
|
| incompatibleWith string
|
|
| priorVersion string
|
|
| versionInfo string
|
|
datatype rdfs = comment string
|
|
| isDefinedBy string
|
|
| label string
|
|
| seeAlso string
|
|
datatype annotation = DC dc | OWL owl | RDFS rdfs
|
|
|
|
onto_class Field_of_mathematics =
|
|
Annotations :: "annotation list"
|
|
invariant restrict_annotation_F ::"set (Annotations \<sigma>) \<subseteq>
|
|
range (RDFS o label) \<union> range (RDFS o comment)"
|
|
|
|
\<close>}
|
|
defines the class \<^typ>\<open>Field_of_mathematics\<close> with an attribute \<^const>\<open>Annotations\<close>
|
|
which is a \<^type>\<open>list\<close> of \<^typ>\<open>annotation\<close>s.
|
|
We can even constraint the type of allowed \<^typ>\<open>annotation\<close>s with an invariant.
|
|
Here the \<^theory_text>\<open>invariant restrict_annotation_F\<close> forces the \<^typ>\<open>annotation\<close>s to be
|
|
a \<^const>\<open>label\<close> or a \<^const>\<open>comment\<close>.
|
|
Subsumption relation is straightforward.
|
|
The ontology OntoMath\textsuperscript{PRO} defines
|
|
a \<^typ>\<open>Problem\<close> and a \<^typ>\<open>Method\<close>
|
|
as subclasses of the class \<^typ>\<open>Mathematical_knowledge_object\<close>.
|
|
It gives in \<^dof>:
|
|
@{theory_text [display,indent=10, margin=70] \<open>
|
|
onto_class Problem = Mathematical_knowledge_object +
|
|
Annotations :: "annotation list"
|
|
|
|
onto_class Method = Mathematical_knowledge_object +
|
|
Annotations :: "annotation list"
|
|
\<close>}
|
|
We can express the relations between the two taxonomies
|
|
with association \<^theory_text>\<open>onto_class\<close>es which specify
|
|
the relation as an attribute and enforces the relation with an \<^theory_text>\<open>invariant\<close>.
|
|
The two directed relations \<^const>\<open>is_solved_by\<close> and \<^const>\<open>solves\<close>
|
|
between \<^typ>\<open>Problem\<close> and a \<^typ>\<open>Method\<close> can be represented in \<^dof> like this:
|
|
@{theory_text [display,indent=10, margin=70] \<open>
|
|
onto_class assoc_Problem_Method =
|
|
is_solved_by :: "(Problem \<times> Method) set"
|
|
invariant is_solved_by_defined :: "\<forall> x. x \<in> Domain (is_solved_by \<sigma>)
|
|
\<longrightarrow> (\<exists> y \<in> Range (is_solved_by \<sigma>). (x, y) \<in> is_solved_by \<sigma>)"
|
|
|
|
onto_class assoc_Method_Problem =
|
|
solves :: "(Method \<times> Problem) set"
|
|
invariant solves_defined :: "\<forall> x. x \<in> Domain (solves \<sigma>)
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\<longrightarrow> (\<exists> y \<in> Range (solves \<sigma>). (x, y) \<in> solves \<sigma>)"
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\<close>}
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|
where the attributes \<^const>\<open>is_solved_by\<close> and \<^const>\<open>solves\<close> define the relations
|
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of the classes and the invariants \<^theory_text>\<open>is_solved_by_defined\<close> and \<^theory_text>\<open>solves_defined\<close> enforce
|
|
the existence of the relations when one define instances of the classes.
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|
|
|
\<^dof> allows to define ontologies and specify constraints
|
|
on their concepts.
|
|
Additionally it dynamically checks at run-time the concepts when defining instances.
|
|
It offers an environment to define and test ontologies in an integrated document,
|
|
where all the knowledge and the proof-checking can be specified,
|
|
and thus can be a solution to go over
|
|
the trade-off between plain vocabularies and highly formalized models.
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|
\<close>
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|
|
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section*[concl::conclusion]\<open>Conclusion\<close>
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|
text\<open>We presented \<^dof>, an ontology framework
|
|
deeply integrating continuous-check/continuous-build functionality into
|
|
the formal development process in \<^hol>. The novel feature of term-contexts in \<^dof>,
|
|
which permits term-antiquotations elaborated in the parsing process, paves the
|
|
way for the abstract specification of meta-data constraints as well the possibility
|
|
of advanced search in the meta-data of document elements. Thus it profits and
|
|
extends Isabelle's document-centric view on formal development.
|
|
|
|
Many ontological languages such OWL @{cite "owl2012"} as well as the meta-modeling technology
|
|
available for UML/OCL provide concepts for semantic rules and constraints, but
|
|
leave the validation checking usually to external tools (if implementing them at all).
|
|
This limits their practical usefulness drastically. Our approach treats invariants as
|
|
first-class citizens, and turns them into an object of formal study in, for example,
|
|
ontological mappings. Such a technology exists, to our knowledge, for the first time.
|
|
|
|
Our experiments with adaptions of existing ontologies from engineering and mathematics
|
|
show that \<^dof>'s ODL has sufficient expressive power to cover all aspects
|
|
of languages such as OWL (with perhaps the exception of multiple inheritance on classes).
|
|
However, these ontologies have been developed specifically \<^emph>\<open>in\<close> OWL and target
|
|
its specific support, the Protege editor @{cite "protege"}. We argue that \<^dof> might ask
|
|
for a re-engineering of these ontologies: less deep hierarchies, rather deeper structure
|
|
in meta-data and stronger invariants.
|
|
\<close>
|
|
|
|
|
|
subsection*[rw::related_work]\<open>Related Work\<close>
|
|
|
|
text\<open>There are a number of approaches to use ontologies for structuring the link between
|
|
information and knowledge, and to make it amenable to
|
|
"semantic" search in or consistency checking of documents:
|
|
|
|
\<^item> The search engine \<^url>\<open>http://shinh.org/wfs\<close> uses a quite brute-force,
|
|
but practical approach. It is getting its raw-data from Wikipedia and PlanetMath
|
|
and similar sites, and uses clever text-based search methods in
|
|
a large number of formulas, agnostic of their logical context, and of formal proof.
|
|
|
|
\<^item> The OAF project @{cite "KohlhaseR21"} comes closest to our ideal of wide-spread
|
|
use of ontologies for search in mathematical libraries.
|
|
The project developed a common ontological format, called OMDoc/MMT, and
|
|
developed six \<^emph>\<open>export\<close> functions from major ITP systems into it. The more difficult task to
|
|
develop \<^emph>\<open>import\<close> functions has not been addressed, not to mention the construction
|
|
of imported proofs in a native proof format. Rather, the emphasis was put on building
|
|
a server infrastructure based on conventional, rather heavy-weight
|
|
database- and OWL technology.
|
|
Our approach targets so far only one ITP system and its libraries, and emphasizes
|
|
type-safeness, expressive power and "depth" of meta-data, which is adapted
|
|
to the specific needs of ITP systems and theory developments.
|
|
|
|
\<^item> There are meanwhile a number of proposals of ontologies targeting mathematics:
|
|
\<^item> \<open>OntoMath\<^sup>P\<^sup>R\<^sup>O\<close> @{cite "Nevzorova2014OntoMathPO"} proposes a
|
|
``taxonomy of the fields of mathematics'' (pp 110). In total,
|
|
\<open>OntoMath\<^sup>P\<^sup>R\<^sup>O\<close> contains the daunting number of 3,449 classes, which is in part due to
|
|
the need to compensate the lack of general datatype definition methods for meta-data.
|
|
It is inspired from a translation of the Russian Federal Standard for Higher Education
|
|
on mathematics for master students \<^url>\<open>http://www.edu.ru/db-mon/mo/Data/d_10/prm40-1.pdf\<close>,
|
|
which makes it nevertheless an interesting starting point for a future development
|
|
of a mathematics ontology in the \<^dof> framework.
|
|
\<^item> Other ontologies worth mentioning are DBpedia @{cite "10.1007/978-3-540-76298-0_52"},
|
|
which provides with the \<^emph>\<open>SPARQL endpoint\<close> \<^url>\<open>http://dbpedia.org/sparql\<close> a search engine,
|
|
and the more general ScienceWISE (see \<^url>\<open>http://sciencewise.info/ontology/\<close> linked to the
|
|
ArXiv.org project) that allows users to introduce their own category concepts.
|
|
Both suffer from the lack of deeper meta-data modeling, and the latter is still at the beginning
|
|
(ScienceWISE marks the Mathematics part as "under construction").
|
|
|
|
% \<^url>\<open>https://github.com/CLLKazan/OntoMathPro\<close>
|
|
%
|
|
% ITEM The "Ontology for Engineering Mathematics"
|
|
% \<^url>\<open>https://tomgruber.org/writing/an-ontology-for-engineering-mathematics\<close> is
|
|
% is unfortunately only a half-baked approach to model physical quantities
|
|
% and SI-measurements. Instead of using ontologies for this purpose, there
|
|
% exist approaches based on strong type systems
|
|
\<close>
|
|
|
|
subsection*[fw::related_work]\<open>Future Work\<close>
|
|
text\<open> We plan to complement \<^dof> with incremental LaTeX generation and a previewing facility
|
|
that will further increase the usability of our framework for the ontology-conform editing
|
|
of formal content, be it in the engineering or the mathematics domain.
|
|
|
|
Another line of future application is to increase the "depth" of build-in term antiquotations such
|
|
\<open>@{typ \<open>'\<tau>\<close>}\<close>, \<open>@{term \<open>a + b\<close>}\<close> and \<open>@{thm \<open>HOL.refl\<close>}\<close>, which are currently implemented just as
|
|
validations in the current logical context. In the future, they could optionally be expanded to
|
|
the types, terms and theorems (with proof objects attached) in a meta-model of the Isabelle Kernel
|
|
such as the one presented in @{cite "10.1007/978-3-030-79876-5_6"} (also available in the AFP).
|
|
This will allow for the HOL-specification of query-functions in, \<^eg>, proof-objects, and
|
|
pave the way to annotate them with typed meta-data. Such a technology could be relevant
|
|
for the interoperability of proofs across different ITP platforms.
|
|
\<close>
|
|
|
|
(*<*)
|
|
close_monitor*[this]
|
|
|
|
end
|
|
(*>*)
|