From f7a0d4b54a7f1907fa2598cb65135cdbb3387514 Mon Sep 17 00:00:00 2001 From: Alan Jeffrey Date: Tue, 7 Nov 2017 15:39:59 -0600 Subject: [PATCH] v1.1 --- .gitignore | 2 + finite-dtypes.bib | 15 ++++++ finite-dtypes.lagda | 109 ++++++++++++++++++++++---------------------- finite-dtypes.pdf | Bin 499140 -> 500455 bytes 4 files changed, 71 insertions(+), 55 deletions(-) diff --git a/.gitignore b/.gitignore index 65ee745..1f07b18 100644 --- a/.gitignore +++ b/.gitignore @@ -3,6 +3,8 @@ *.bbl *.blg *.dvi +*.fdb* +*.fls *.log *.out *~ diff --git a/finite-dtypes.bib b/finite-dtypes.bib index 19d4d2f..57bb784 100644 --- a/finite-dtypes.bib +++ b/finite-dtypes.bib @@ -136,3 +136,18 @@ @Misc{self howpublished = {\url{https://github.com/asajeffrey/finite-dtypes}}, } +@Article{AP16, + author = {A. Abel and B. Pientka}, + title = {Well-founded recursion with copatterns and sized types}, + journal = {J. Functional Programming}, + year = {2016}, + volume = {26}, +} + +@InProceedings{FGGvW18, + author = {D. Frumin and H. Geuvers and L. Gondelman and N. van der Weide}, + title = {Finite Sets in Homotopy Type Theory}, + year = {2018}, + note = {Submitted for publication}, +} + diff --git a/finite-dtypes.lagda b/finite-dtypes.lagda index 1e06697..5b50737 100644 --- a/finite-dtypes.lagda +++ b/finite-dtypes.lagda @@ -315,67 +315,66 @@ In particular, systems programs are usually parameterized by $\kw{WORDSIZE}$, and assume that data fits into memory (for example that arrays are indexed by a word, not by a natural number). -A simple language of finite types is given in Figure~\ref{simple-types}. -In this language, all types are finite, in particular -$\kw{FSet}(n) \in \kw{FSet}(\kw{one} + n)$ -(the size is slightly arbitrary, we could have chosen any increasing -size, for instance $\kw{FSet}(n) \in \kw{FSet}(\kw{one} \ll n)$). - -The system is based on a theory of binary arithmetic, but even -that is definable within the language, for example the type -of bitstrings is definable by induction: +An overview of approaches to finite sets in dependent type theory is +given in~\cite{FGGvW18}, which defines +a type constructor $\mathcal{K}(A)$ (the +Kuratowski-finite subsets of $A$), and defines the finite +types to be a type $A$ together with an $X \in \mathcal{K}(A)$ +such that every $x:A$ has $x \in X$. +Type constructors such as dependent functions and pairs, preserve +finiteness, but the type of finite types is not itself finite. + +In Figure~\ref{simple-types}, we posit a type system with +a type-of-types $\kw{FSet}(n)$, +containing types with at most $2^n$ elements. +In this system, all types are finite, in particular $\kw{FSet}(n) +\in \kw{FSet}(\kw{one} + n)$. We expect that this explicit bound on +the size of $\kw{FSet}(n)$ cannot be expressed internally inside an +off-the-shelf dependent type theory, but we conjecture that an extension +like this is consistent. + +This system is based on a theory of binary arithmetic, but even that is +definable within the language, in the presence of an appropriate +induction principal for binary arithmetic. For example the type of +bitstrings is definable by induction: \begin{code} -/binary/ = /indn/ +/binary/ = & /indn/ (/FSet/) (/unitp/) - (/fn/ n /cdot/ /fn/ A /cdot/ (/bool/ /times/ A)) -\end{code} -For example: -\begin{code} + (/fn/ n /cdot/ /fn/ A /cdot/ (/bool/ /times/ A)) \\ %WORDSIZE = /WORDSIZE/ &/in/ /binary/(/WORDSIZE/) \end{code} -Some issues finite types raise include: -\begin{description} - -\item[Induction:] - As the type for $\kw{WORDSIZE}$ suggests, - this is all spookily cyclic. In particular, binary numbers - are parameterized by their bitlength, which is itself - a binary number. An induction principal is needed, - even if it is extra-logical, similar to - Agda's universe polymorphism~\cite{agda}. We hypothesize - that a theory of irrelevant natural numbers might suffice. - -\item[Path types:] Dependent type systems usually come with an identity type $a - \equiv_A b$ where $a : A$ and $b : A$. - If identity types are interpreted as paths as in Homotopy Type Theory~\cite{hott}, - then the size of $A \equiv_{\kw{FSet}(n)} B$ is at most the size of $A \rightarrow B$, - which would suggest considering $(a \equiv_A b) \in \kw{FSet}(n \ll n)$ - when $A \in \kw{FSet}(n)$. - This makes the type of identities over $A$ much larger than the type of $A$, - which may give problems with, for example, codings of indexed types. - -\item[Theory of binary arithmetic:] - The theory of finite types is very dependent on the theory of bitstrings, - and it is very easy to end up type checking modulo - properties such as associativity and commutativity of $+$. - Such theorems could be handled by an SMT solver, but this has its own - issues, such as type inference and complexity. - -\item[Pointers:] - In systems programming languages such as Rust, cyclic data structures - are mediated by pointers. In a finite type system, we could allow a - type of pointers $\&(A)$ where $\&(A) \in \kw{FSet}(\kw{WORDSIZE})$ - when $A \in \kw{FSet}(n)$. - Pointer creation can fail in low-memory situations, so should be - encapsulated in a monad, similar to Haskell's $\kw{ST}$ monad. - -\item[Error messages:] - Type systems should not just reject incorrect programs, - they should provide useful hits about fixing them. - -\end{description} +As the type for $\kw{WORDSIZE}$ suggests, this is all spookily +cyclic. In particular, binary numbers are parameterized by their +bitlength, which is itself a binary number. A bootstrapping induction principal is +needed, similar to Agda universe +polymorphism~\cite{agda}. We hypothesize that irrelevant +natural numbers might suffice. + +Potential advantages of explicitly finite dependent types +include: +\begin{itemize} + +\item A better fit with systems programming languages such as Rust, + where sized types are the default. + +\item Domain specific techniques for arithmetic can better support + type inference. + +\item Simpler induction schemes, such as over the irrelevant naturals, + similar to the use of sized types in taming coinduction~\cite{AP16}, + but without the use of ordinals. + +\end{itemize} +Dependent type systems usually come with an identity type $a \equiv_A +b$ where $a : A$ and $b : A$. If identity types are interpreted as +paths as in Homotopy Type Theory~\cite{hott}, then the size of $A +\equiv_{\kw{FSet}(n)} B$ is at most the size of $A \rightarrow B$, +which would suggest considering $(a \equiv_A b) \in \kw{FSet}(n \ll +n)$ when $A \in \kw{FSet}(n)$. This makes the type of identities over +$A$ much larger than the type of $A$, which may give problems with, +for example, codings of indexed types. \section{Conclusions} diff --git a/finite-dtypes.pdf b/finite-dtypes.pdf index 55a7874a644a13afdbfaaa9ea41af6f3cfe506e9..69b7bd70e61b541efc951b9a19a6467c9d4bfe73 100644 GIT binary patch delta 61427 zcmV((K;XZ`yB_Do98Ug{xlO_+-krPx`{UgR zwYjI~sr_(3b}v` z{soNk-1|A%%B2%A-_?u4-P5WJW3OD+f3S)m$~JZCp%Q~ki@ic3uG-fn*~Jcwk`wFefX%jx!FLhvo5>~f^;gRKeJ(B(4|EAlz>u+ zq+%YySUOCD`PEJ#5(VnhVjM^qw^`mE3FB*1B!ipp_sUIu+CqF)Xt|uY+S1K3uB`XV@ZKk7 z#j*dW0{OdK-UdijOz;LJI0DI(B-&%=Gg+y|Z;I)=vhnRSb2OG#mVPFOunA^}u)(Mc z4{nD}=v&6R4rSwitM7^$e=tYjol?O6Y(L*4$!)EU8UzV_`eBl|B9Dp-h4w6!G)TsR zK$c}jj~xq^J|i_Y9P!<=+p1VCroMAJ_cOn%k9|iCYEk-uo(0f1U%NQOxYe2LLrG{9 zBr7Z+w+lF-T8x`ep7&3lNZaOKte_7h#5cC}wpcn*fw$$61nF{gf7ErGmBXmAT5qrC zevngA*VlTgj9L<|kcyf5`tlb~cF@e0={~VeHxr5``oY!BVj0IQR-xWISPco&Nf0|G zk_|6=ZX;|aIO7Wd3ICU;JodQ@ve>}{XJU|oF#_f_MG1j-| zPkwzeR0X0Dg;;C4dh+%Hk7n@B&T01-98XM3D_`g#U%p6 zzzG#fOK025)uQQmZF7|vDc)=^@o z=Rk-Z&}+I^f7-7gBeCn`#fprct$~;yTbPpobkc+BNCk`EQ7U1%G-+pcl+K&xrvB>W zWV$JVr1vElht;~O(Ry=z0%Vp|UDhX;wn5o7Hh!_n;ipdz)c$pE-&#pJ` z>_ae?&>1FZH43aIDEY=W3;%QK9dK96gzry1l77I`f2P?iF1KN$2RZ;%-*^Gqx~Uwe zY}9DGUWOS7i54$KUURT|De{)(g*P{wn3IZ&G*kGnvC>kdslv-e6S91_IUCe;SP<=8 zO3#V)1+Oc>_B0)y-c!U8MFM4Wk<11umpvoNAe_LU2ZCZG;`XSsjGoEN^afLiRINz* zP2n&?f6(3tFtg0W5kF&0b6_Cg#Uc*+CX7h15-<_zzz7oO;>jQ(TWtQ617j!ZgNZ~5 zkq~778wrVGs2BX0!x_R8t5>~Yzvbyy-3sj?o?>-i4GrzDA<2OOmNWkFH5^T}M3(-E z4a?!0=HhCHJOuG=YFY3$@oW?NMWPz-Sw<$L#M# ze~)$6hV7W;7MtJlVC>ng{t8bNNrWmj;<#h$r}d`tQ)SP+*!qegVOpwy!{B5D+Vr?bgTR~vPd|7O)b$`2p5X*=Plt@+zi z-if$#Yl1BpxaM+@fQ9DJWq6c8ZbJK$IG$f(8je>QA% z31Ph0jRQ16eAOfX`*Hp-)t zTVZfl=J8a&-OB6_Vx%1utAEYB9M0;)cx1ZcAm%3t`lqb{vSa!q$x| zs=T~oIbw*$8xvIyYlxf)A5D70f6ArX&#m`err6nl<)Kar|3kAABmW$~oE1&nhkaJ2Th=XB_7VvWo9crrIwYNv^e$Z3 zecq+7GtX}Q4DG(V3daRW*k$!6qU!!0ao8FlW4~`PS+)lam;Ep8rsgD`f8yw?KzD=Z zt+_bv9Zzwao3{R(x$N_nUL4$w4!0`${0i<3%-HZfBMq_9-5}w$SK+uS^LDOG910xLhp5br#__O$7&7O}X(q ztc`z@?v3(xZ?yKW!(APCe^G><+cRLs8Z1WqA0XcuMHn8nb7wU!64@w`R6?wVr`Z{_C5U)v=qh*B&f!TNd5w_4ghu0^7Qkq=6+%j_>t+e~LQ}Zt?pnXVgJM zl|@S;7UVxtpQ#F13&`lO{0k9d-?(8Q4?K&K(RNdjn-~n11~b-3Op_7hWECl&tw|Jkk{MTRF6ckJ-)S)15LOA#} zva?&!u3eHrbHpgmf7JvPD1dr`7U($W6_OCfQA=IWKS+WqUnJP|B&R~ zB_Rkcz|mOa>V!$}3X|DU!(9yaU2C)PiPUmLHA7hN_AqC-mKKiBZ<@k|pj`TxbiJs; zHMb&c;cT1qihxQya*UpPH|3gecTSEMgFAE!f9PSEjgKK~@CTz^)V?01FWqgkepoq; z7rl~*p?_>)e}=-L{j1(hTi=|OE=4S}kpT5)E^H{kMTk&N&w56gr-Laahc<#ABw{}u zF+P6%=Gobc|NeH5|I~zJDtKa0!PA|A%)m^Qj)PS;hm}s@`%NtGm^cjwiH)I!Uv3!> zKR5GuYNemVwgF#zU@tXTwrI!l7Au((ESBkFpO`MUe-avQ{|)Akae7u{CWg4B@iLQj zf`el&kkyMnqV_Z~crQGN(nve$SsotFSJ?)&EN&jdf%la#0R_Q~kq2whyf@z+C1}~{ z)>*!O&V~Mr)%;Jex0pD~{RLtBCoc}{9CY6DBLdKU*?H>*uz;X6xU19oa&c3*KhnYV z;B2S8f8S+KSSBBc;n_|m03AL6dlCPtMcrO%u*H!o-nEW*zXwG_`!ZX|uou6J@bz^# z3^>FikY`W5+QPi|a=}qe>f)-jl?Ktfghmwi)f?Neo@!G!6Tg&Y=~kxINgbK-)mwM7Rv_Yyc$uZIt%*is_w_hZp`T zdkw*k4=axMGFX73nQ&{s^Uv7R*fj)NwxnX8PLJ&64SQAJK5c8qKxoy#tE#b;HT%>} zf9b1Ih72@-D-97IO{XMgh#A+)o;puPT_8Wl*`68e76=>C$L*DjP+V=7%T~KK_NGsr zGd&3$0Wq&Vyx)iOHfxO9FYKTHNk51ivMMc9n9lHgohxN9$lY4AeY*z+-vMC~5WL&* z{QV%c8l1d6}pBWUQ37qx%kcU9~g$1dAd7pwBFRpIvX@}5Q6e_P*{ z7)dr*`>3II()Rw?uiHicE}6q@?Z>oKW59@{U*Wg*;CeC%!Z`!a@qZ< zrnXKk`gc&gcvk**AihPYlXnKmKffi%_`)~aaVY0ZN|y`hBktpc`nA99eRb zQk4KU`gr~N^}D+#J-aze&%XZZ>E&0CUXoOuWl1h2KfAm>6D(1AcBZIE1kcYd*JnR3 z-tMoq)n8I3%JZdKtk0KJq>F#+W@ldR_E(!~RdiKTUzn0N&AD29IL{X4_Ue2|(nV>j z(v+x+^LLlOKjV@llo5bJkMLwu)TYYsOBjglurYLUkCnelEw~74=E0gK0BH*Uhyz>Z zS;7Q4yWB#j-_-ywrD{=a?>6POtUC=PmSu~!>{{#``V@J*Pr^k)hzNf>y|j(y03m`T znHWLRH3ksG27!r1#wD)3A_*s9^L<@)$A-;h!kG%&UD#3gotblYYY|c-Zhz@c#sn-( zl8t*mh6yTE@+%NQl}ZSsVS7VP4U%EAIW~UIl9c9Q2agX8sj?&`0CC0=DuroK*anh% z4S2!`bF}+&tV@=VoVtG+#MV$M=p$9RtMOm)J`|I#Mmp0+c6t{iJ{2N+Fy6U7xX%$1 zK!~gHKeh&!UX3gfxpFoBl8Q723Qh7TUv@}3udNs7yUQ-Y&klHlZn2<|f=B*uF}$`gM{_K*&*i%nZbJCaFG zz8viZ43hwl9LZ3rk{lL9v%jbN?Oj#hoG%%bMt(3V7Tc;u>5pYv%I1)cRu# zmK@%)v!z$9t^VFiYZ{U&<7~X)gU(dKRX?$hKrRhxck5Vh6x9qekZej~a{yiltM%|X zT*ih39fgq7;eYwI>|%Qj@E{&00zHg|Ju#@IVj{x_8(4pl1O*zOuvzot2xFgsu)oJf z20in1bX~@K$SMDPi>yLIjgLqvIVhC^sQ)9jE;0d|=s_zyLvM`roZ{c2+5iE)chX4! 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