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Compiling Hilbert's epsilon operator

unpublished

*Hilbert's*

*epsilon*

*operator*is a binder that picks an arbitrary element froma nonempty set. ... More specifically, the paper presents the design choicesmade around supporting this

*operator*in the verification-aware language Dafny. ... In this sense, it isn't really

*Hilbert's*ε

*operator*, because

*Hilbert's*ε

*operator*is axiomatized to be deterministic. We'll look at that next. ...

##
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Using PVS to prove a Z refinement: A case study
[chapter]

1997
*
Lecture Notes in Computer Science
*

a small, but non trivial

doi:10.1007/3-540-63533-5_30
fatcat:o7fwem25tbhxld7w5ch7bdrg7e
*compiler*by hand in the Z specication language. ... This is especially true of*compilers*, which are a weak link between the source code produced and the ob ject code which is executed. ... The second option can be achieved by the use of*Hilbert's**epsilon**operator*[13] , which, given a predicate, returns a value which satises that predicate (if possible) and if it is not satisable returns ...##
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The Mechanical Verification of a DPLL-Based Satisfiability Solver

2011
*
Electronical Notes in Theoretical Computer Science
*

It uses

doi:10.1016/j.entcs.2011.03.002
fatcat:fpgxzw3blvf5lp4y2nrntyszom
*Hilbert's**epsilon*, axiomatized in the PVS prelude library so that for any predicate p over a nonempty type T, if there is any a such that p(a) holds, then p(*epsilon*(p)) holds The main DPLL search ... Examples of these include the McCarthy-Painter proof of correctness of a*compiler*for arithmetic expressions, the correctness of a satisfiability solver for conditional expressions, and the Turingcompleteness ...##
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Subtypes for specifications: predicate subtyping in PVS

1998
*
IEEE Transactions on Software Engineering
*

The first,

doi:10.1109/32.713327
fatcat:mut73sijtzbw3ghwa4yi33jreq
*epsilon*, is simply*Hilbert's*ε*operator*. Given a set p over a nonempty type T,*epsilon*(p) is some member of p, if any such exist, otherwise it is just some value of type T. ... The full definition is actually choose(p): (p) =*epsilon*(p); this additionally specifies that choose(p) returns the same value as*epsilon*(p), which is useful in specifications that use both*epsilon*and ...##
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Verifying hybrid systems modeled as timed automata: A case study
[chapter]

1997
*
Lecture Notes in Computer Science
*

Our ultimate solution, which t o o k a d v antage of this fact, uses PVS's implementation

doi:10.1007/bfb0014724
fatcat:os3s6l7sbnclhho3x5iloccy6e
*epsilon*" of*Hilbert's*choice*operator*in expressing the transition relation as a function. ... Instead, we aim to*compile*a list of facts that can eventually be used automatically in an improved real-arithmetic" PVS strategy. ...##
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Verifying the Correctness and Amortized Complexity of a Union-Find Implementation in Separation Logic with Time Credits

2017
*
Journal of automated reasoning
*

We define p(x) using

doi:10.1007/s10817-017-9431-7
fatcat:opju5dkdmfcypel4zst5iuwcaq
*Hilbert's*ε*operator*: Definition p F x :=*epsilon*(fun y ⇒ F x y). ...*Hilbert's*ε*operator*applies to a predicate P of type A → Prop, where A is an inhabited type. ...##
###
A synthetic axiomatization of Map Theory

2016
*
Theoretical Computer Science
*

Map Theory axiomatizes lambda calculus plus

doi:10.1016/j.tcs.2015.11.028
fatcat:px36k53cyjh4nokyjjvjlcfudq
*Hilbert's**epsilon**operator*. ... All theorems of ZFC set theory including the axiom of foundation are provable in Map Theory, and if one omits*Hilbert's**epsilon**operator*from Map Theory then one is left with a computer programming language ... Syntactically, Map Theory comprises a computer programming language plus*Hilbert's**epsilon**operator*. ...##
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A Most Interesting, but Revoked Draft for Hilbert and Bernays' "Grundlagen der Mathematik" that never found its way into any publication, and 2 CV of Gisbert Hasenjaeger
[article]

2021
*
arXiv
*
pre-print

A simplified and improved free-variable framework for

arXiv:1803.01386v4
fatcat:ai5itqzjnzddpnum2shh5johua
*Hilbert's**epsilon*as an*operator*of indefinite committed choice. IFCoLog J. of Logics and Their Applications, 4:435-526. Received Oct. 23, 2015. ...*Hilbert's*program. ...##
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Abstracts

1974
*
Historia Mathematica
*

KEMENY, John G 1973 Mathematical
models and the
computer
Pi Mu

doi:10.1016/0315-0860(74)90196-7
fatcat:vz3xwkbqevg3xni7jbwhyyolby
*Epsilon*Journal 5, 373-386 COMPUTERS. CURRENT HISTORY. MATHEMATICAL MODELS. PREDICTIONS. ... MITCHAM, Carl & MACKEY, Robert,*Compilers*1973 Bibliography of the Philosophy of Technology Chicago (Univ of Chicago Press) 205 p BIBLIOGRAPHIES. TECHNOLOGY. PHILOSOPHY OF TECHNOLOGY. ...##
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Formalization and Execution of Linear Algebra: From Theorems to Algorithms
[chapter]

2014
*
Lecture Notes in Computer Science
*

(traversing its indexing type), algorithms traversing the universe of the rows or columns indexing types, and also

doi:10.1007/978-3-319-14125-1_1
fatcat:f47wcfrze5hzjn5l4ehx6krpqm
*operations*that involve logical*operators*(∀, ∃) or the*Hilbert's**operator*, such as every ... A priori, nite (enumerable) types have nice computational features, since mathematical and logical*operations*(traversing,*epsilon**operator*, universal or existential quantiers) over them can be executed ...##
###
Passages of Proof
[article]

2004
*
arXiv
*
pre-print

However, from a practical point of view, the chance of a computer error that appears consistently in exactly the same way on all runs of our programs on all the

arXiv:math/0305213v2
fatcat:ja3dy6x7ojfnpgpt4h7ndmhpxq
*compilers*under all the*operating*systems ... In particular, we have not proved the correctness of the*compiler*we*compiled*our programs on, nor have we proved the infallibility of the hardware we ran our programs on. ...##
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A Proposal for a Formal OCL Semantics in Isabelle/HOL
[chapter]

2002
*
Lecture Notes in Computer Science
*

specification into HOL-OCL 2 leads to the following specification triangle spec of the

doi:10.1007/3-540-45685-6_8
fatcat:nx6vit5dwzaofgpb3qvd6ifyhe
*operation*triangle(): For the actual test-case generation, we define triangle, which selects via*Hilbert's**epsilon*... The task of implementing a*compiler*that converts representations of UML-diagrams (for example, formats produced by ArgoUML) is desirable but not in the focus of this research. ...##
###
ABSTRACTS

2002
*
Historia Mathematica
*

Galois Theory in the Work of Serret [in Spanish],

doi:10.1006/hmat.2002.2376
fatcat:vpqzokff2vhvxi6hhi3iq7atxa
*Epsilon*16 (3) (2000), 377-408. ... The texts are: 46 problems*compiled*by Francisco di Cortisi (1420); 35 problems*compiled*by Altovino da Firenzo (1446-1447); and an anonymous collection of 63 problems written circa 1420. ...##
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Formal Verification of an Executable LTL Model Checker with Partial Order Reduction
[chapter]

2016
*
Lecture Notes in Computer Science
*

Performing a step like this in a formal proof requires precise reasoning and in our case the use of

doi:10.1007/978-3-319-40648-0_23
fatcat:e4caxriogfdjpme3gwsbnrzwlq
*Hilbert's**epsilon**operator*. ... Among other steps, this includes*compilation*of the model to a more efficient representation. ...##
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Verified Bytecode Subroutines
[chapter]

2003
*
Lecture Notes in Computer Science
*

Since succs returns lists and not sets, we use

doi:10.1007/10930755_4
fatcat:jdc274pbrzgdxdtncpqgstgayu
*Hilbert's**epsilon**operator*SOME to pick any list that converts to this set. ... List*operations*may be unfamiliar: the "cons"*operator*is the infix #, concatenation the infix @; head and tail are hd and tl. ...
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