Abstract
Nonstandard programming languages are the languages that deal with hyperreal numbers—real numbers extended with infinitesimal (i.e., smaller than any positive real) numbers—whose theory is given by nonstandard analysis (NSA). Nonstandard imperative language \(\mathbf {While}^\mathbf {dt}\) and nonstandard stream-processing language \(\mathbf {SProc}^\mathbf {dt}\) have been proposed so far and applied to hybrid system modeling and verification. We introduce a nonstandard functional language \(\textsc {NSF}\). We extend a simply typed functional language with a constant \(\mathbf {dt}\) that denotes an infinitesimal number and define its denotational semantics. The major challenge consists in giving the semantics of functions of an arbitrarily high order. To this end, we introduce *-cpos, a nonstandard counterpart of the standard domains.
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Notes
- 1.
The set
is also called the superstructure over 
in the literature [7]. - 2.
This lemma requires the condition
, which is stronger than 
. This stronger condition is indeed necessary. For example, if \(M_f = \mathbf {if}\ 0 \le t\ \mathbf {then}\ \mathbf {dt}\ \mathbf {else}\ 0\) and 
for all r, then they satisfy the weaker condition, but \([\![{{D}\ M_f} ]\!]_{\partial }(0) = 1 \ne 0 = f'(0)\). - 3.
\(\langle {M([D_i]_\mathcal {F}),M([\preceq _i]_\mathcal {F})}\rangle \) is an internal domain but not a *-cpo if \(D_i := \{{0,1,\dots ,i,\bot }\}\) and \({\preceq _i} \subseteq D_i \times D_i\) is defined by \(x \preceq _i y \iff x = \bot \vee x = y\).
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Acknowledgments
We appreciate the anonymous reviewers for their comments. This work was supported in part by JST CREST (Kojima), JST PRESTO No. JPMJPR15E5 (Suenaga), and MEXT KAKENHI Grant Number 15H05706 (Igarashi).
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Nakamura, H., Kojima, K., Suenaga, K., Igarashi, A. (2017). A Nonstandard Functional Programming Language. In: Chang, BY. (eds) Programming Languages and Systems. APLAS 2017. Lecture Notes in Computer Science(), vol 10695. Springer, Cham. https://doi.org/10.1007/978-3-319-71237-6_25
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DOI: https://doi.org/10.1007/978-3-319-71237-6_25
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