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Erschienen in:
Verified Over-Approximation of the Diameter of Propositionally Factored Transition Systems Kapitel lesen Erstes Kapitel lesen

2015 | OriginalPaper | Buchkapitel

Proof-Producing Reflection for HOL

With an Application to Model Polymorphism

verfasst von : Benja Fallenstein, Ramana Kumar

Erschienen in: Interactive Theorem Proving

Verlag: Springer International Publishing

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Abstract

We present a reflection principle of the form “If \(\ulcorner \varphi \urcorner \) is provable, then \(\varphi \)” implemented in the HOL4 theorem prover, assuming the existence of a large cardinal. We use the large-cardinal assumption to construct a model of HOL within HOL, and show how to ensure \(\varphi \) has the same meaning both inside and outside of this model. Soundness of HOL implies that if \(\ulcorner \varphi \urcorner \) is provable, then it is true in this model, and hence \(\varphi \) holds. We additionally show how this reflection principle can be extended, assuming an infinite hierarchy of large cardinals, to implement model polymorphism, a technique designed for verifying systems with self-replacement functionality.

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Vorheriges Kapitel Formalizing Size-Optimal Sorting Networks: Extracting a Certified Proof Checker
Nächstes Kapitel Improved Tool Support for Machine-Code Decompilation in HOL4
Fußnoten
1
\(\ulcorner \varphi \urcorner \) refers to \(\varphi \) as a syntactic object, represented e.g. by its Gödel number or by an abstract syntax tree.
 
2
Including, as in Gordon [9], polymorphism (type variables and polymorphic constants) and defined constants and type operators.
 
3
This is akin to working within an Isabelle [23] locale which fixes \(\mu \) and \({ {mem}}\) and assumes \(\mathsf{is\_set\_theory }\;{ {mem}}\). (We assume infinity only when necessary).
 
4
The abstract syntax here is the inner HOL rendition of our example term. \(\mathsf{Bool }\) and \(\mathsf{Fun }\;{ {a}}\;{ {b}}\) are abbreviations for \(\mathsf{Tyapp }\;{\text {``bool''}}\;\mathsf{[\,] }\) and \(\mathsf{Tyapp }\;{\text {``fun''}}\;[{ {a}};\;{ {b}}]\).
 
5
To understand the three turnstiles: \(\vdash \) denotes provability in outer HOL (and applies to the whole formula), infix |- denotes provability in inner HOL, and infix \(\models \) states that a sequent is valid according to the semantics of inner HOL.
 
6
To be pedantic, \(\mathsf{to\_inner }\) also depends on the pervasive \({ {mem}}\) relation of the set theory.
 
7
We follow the convention of treating predicates in outer HOL as sets. To be clear \(\mathcal {U}({:}\alpha )\) is a term of type https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-22102-1_11/340828_1_En_11_IEq84_HTML.gif , and \(\left\{ \right. {}{ {a}}\;|{}\;{ {a}}\;\mathsf{\lessdot }\;{ {x}}\left. \right\} {}\) is a term of type https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-22102-1_11/340828_1_En_11_IEq86_HTML.gif . These sets-as-predicates are distinct from the Zermelo sets, i.e., the terms of type \(\mu \).
 
8
Also known as indefinite choice, the Hilbert choice principle in HOL provides a constant \((\mathsf{\varepsilon {} })\), usually written as a binder, together with the axiom \((\mathsf{\exists \,{} }{ {x}}.\;{ {P}}\;{ {x}})\;\mathsf{\Rightarrow {} }\;{ {P}}\;(\mathsf{\varepsilon {} }{ {x}}.\;{ {P}}\;{ {x}})\) which holds for any predicate \({ {P}}\).
 
9
\(\mathsf{Num }\) is an abbreviation for \(\mathsf{Tyapp }\;{\text {``num''}}\;\mathsf{[\,] }\).
 
10
When a new constant or type operator is defined, theorems produced by the definition are considered axioms of the resulting theory. The other source of axioms is new-axiom updates, which we do not support since they are not sound in general.
 
11
Any valuation would do for this input since it has no free type or term variables.
 
12
Our proof constructs a potentially slightly different model of inner HOL for each value of t; this is, roughly, the origin of the term model polymorphism.
 
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Allen, S.F., Constable, R.L., Howe, D.J., Aitken, W.E.: The semantics of reflected proof. In: Proceedings of the LICS, pp. 95–105, IEEE Computer Society (1990)
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Dybjer, P., Setzer, A.: A finite axiomatization of inductive-recursive definitions. In: Girard, J.-Y. (ed.) TLCA 1999. LNCS, vol. 1581, pp. 129–146. Springer, Heidelberg (1999) CrossRef
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Fallenstein, B., Soares, N.: Vingean reflection. Technical report, Machine Intelligence Research Institute, Berkeley, CA (2015)
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Feferman, S.: Transfinite recursive progressions of axiomatic theories. J. Symb. Log. 27(3), 259–316 (1962)MathSciNetCrossRef
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Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte fr Mathematik und Physik 38(1), 173–198 (1931)CrossRef
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Gonthier, G.: The four colour theorem: engineering of a formal proof. In: Kapur, D. (ed.) ASCM 2007. LNCS (LNAI), vol. 5081, pp. 333–333. Springer, Heidelberg (2008) CrossRef
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Gonthier, G., Mahboubi, A.: An introduction to small scale reflection in Coq. J. Form. Reasoning 3(2), 95–152 (2010)MATHMathSciNet
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Gordon, M.: From LCF to HOL: a short history. In: Plotkin, G.D., Stirling, C., Tofte, M. (eds.) Proof, Language, and Interaction, Essays in Honour of Robin Milner, pp. 169–186. The MIT Press, Cambridge (2000)
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Harrison, J.: Towards self-verification of HOL Light. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 177–191. Springer, Heidelberg (2006) CrossRef
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Harrison, J.: HOL Light: an overview. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 60–66. Springer, Heidelberg (2009) CrossRef
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Kumar, R., Arthan, R., Myreen, M.O., Owens, S.: HOL with definitions: semantics, soundness, and a verified implementation. In: Klein, G., Gamboa, R. (eds.) ITP 2014. LNCS, vol. 8558, pp. 308–324. Springer, Heidelberg (2014)
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Kumar, R., Arthan, R., Myreen, M.O., Owens, S.: Self-formalisation of higher-order logic. J. Autom. Reasoning (2015), submitted. Preprint at https://​cakeml.​org
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Mohamed, O.A., Muñoz, C.A., Tahar, S. (eds.): Theorem Proving in Higher Order Logics. Springer, Heidelberg (2008) MATH
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Myreen, M.O., Davis, J.: The reflective Milawa theorem prover is sound - (down to the machine code that runs it). In: Klein and Gamboa [14], pp. 421–436
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Myreen, M.O., Owens, S.: Proof-producing translation of higher-order logic into pure and stateful ML. J. Funct. Program. 24(2–3), 284–315 (2014)MATHMathSciNetCrossRef
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Norrish, M., Huffman, B.: Ordinals in HOL: transfinite arithmetic up to (and beyond) \(\omega _{1}\). In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 133–146. Springer, Heidelberg (2013) CrossRef
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Slind, K., Norrish, M.: A brief overview of HOL4. In: Mohamed et al. [17], pp. 28–32
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Turing, A.M.: Systems of logic based on ordinals. Proc. LMS 2(1), 161–228 (1939)MathSciNet
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Wenzel, M., Paulson, L.C., Nipkow, T.: The Isabelle framework. In: Mohamed et al. [17], pp. 33–38
Metadaten
Titel
Proof-Producing Reflection for HOL
verfasst von
Benja Fallenstein
Ramana Kumar
Copyright-Jahr
2015
Buch
Interactive Theorem Proving

Print ISBN: 978-3-319-22101-4

Electronic ISBN: 978-3-319-22102-1

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-3-319-22102-1_11

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