# Changeset 8d76f2b for doc/theses/mike_brooks_MMath/array.tex

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Apr 11, 2022, 1:02:54 PM (8 months ago)
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13888c0, 437b8b5
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a08443b
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Adding runtime bound checking for array subscripts and showing the optimizer removing them.

Adding draft thesis content on dependent types and bound checks.

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 ra08443b enq( N, S, arpk(N', S', E_i', E_b), E_b ) & = & arpk( N', S', enq(N, S, E_i', E_b), E_b ) \end{eqnarray*} \section{Bound checks, added and removed} \CFA array subscripting is protected with runtime bound checks.  Having dependent typing causes the opimizer to remove more of these bound checks than it would without them.  This section provides a demonstration of the effect. The experiment compares the \CFA array system with the padded-room system [todo:xref] most typically exemplified by Java arrays, but also reflected in the C++ pattern where restricted vector usage models a checked array.  The essential feature of this padded-room system is the one-to-one correspondence between array instances and the symbolic bounds on which dynamic checks are based.  The experiment compares with the C++ version to keep access to generated assembly code simple. As a control case, a simple loop (with no reused dimension sizes) is seen to get the same optimization treatment in both the \CFA and C++ versions.  When the programmer treats the array's bound correctly (making the subscript obviously fine''), no dynamic bound check is observed in the program's optimized assembly code.  But when the bounds are adjusted, such that the subscript is possibly invalid, the bound check appears in the optimized assemly, ready to catch an occurrence the mistake. TODO: paste source and assemby codes Incorporating reuse among dimension sizes is seen to give \CFA an advantage at being optimized.  The case is naive matrix multiplication over a row-major encoding. TODO: paste source codes \section{Comparison with other arrays} \CFA's array is the first lightweight application of dependently-typed bound tracking to an extension of C.  Other extensions of C that apply dependently-typed bound tracking are heavyweight, in that the bound tracking is part of a linearly typed ownership system that further helps guarantee statically the validity of every pointer deference.  These systems, therefore, ask the programmer to convince the typechecker that every pointer dereference is valid.  \CFA imposes the lighter-weight obligation, with the more limited guarantee, that initially-declared bounds are respected thereafter. \CFA's array is also the first extension of C to use its tracked bounds to generate the pointer arithmetic implied by advanced allocation patterns.  Other bound-tracked extensions of C either forbid certain C patterns entirely, or address the problem of \emph{verifying} that the user's provided pointer arithmetic is self-consistent.  The \CFA array, applied to accordion structures [TOD: cross-reference] \emph{implies} the necessary pointer arithmetic, generated automatically, and not appearing at all in a user's program. \subsction{Safety in a padded room} Java's array [todo:cite] is a straightforward example of assuring safety against undefined behaviour, at a cost of expressiveness for more applied properties.  Consider the array parameter declarations in: \begin{tabular}{rl} C      &  @void f( size_t n, size_t m, float a[n][m] );@ \\ Java   &  @void f( float[][] a );@ \end{tabular} Java's safety against undefined behaviour assures the callee that, if @a@ is non-null, then @a.length@ is a valid access (say, evaluating to the number $\ell$) and if @i@ is in $[0, \ell)$ then @a[i]@ is a valid access.  If a value of @i@ outside this range is used, a runtime error is guaranteed.  In these respects, C offers no guarantess at all.  Notably, the suggestion that @n@ is the intended size of the first dimension of @a@ is documentation only.  Indeed, many might prefer the technically equivalent declarations @float a[][m]@ or @float (*a)[m]@ as emphasizing the no guarantees'' nature of an infrequently used language feature, over using the opportunity to explain a programmer intention.  Moreover, even if @a[0][0]@ is valid for the purpose intended, C's basic infamous feature is the possibility of an @i@, such that @a[i][0]@ is not valid for the same purpose, and yet, its evaluation does not produce an error. Java's lack of expressiveness for more applied properties means these outcomes are possible: \begin{itemize} \item @a[0][17]@ and @a[2][17]@ are valid accesses, yet @a[1][17]@ is a runtime error, because @a[1]@ is a null pointer \item the same observation, now because @a[1]@ refers to an array of length 5 \item execution times vary, because the @float@ values within @a@ are sometimes stored nearly contiguously, and other times, not at all \end{itemize} C's array has none of these limitations, nor do any of the array language'' comparators discussed in this section. This Java level of safety and expressiveness is also exemplified in the C family, with the commonly given advice [todo:cite example], for C++ programmers to use @std::vector@ in place of the C++ language's array, which is essentially the C array.  The advice is that, while a vector is also more powerful (and quirky) than an arry, its capabilities include options to preallocate with an upfront size, to use an available bound-checked accessor (@a.at(i)@ in place of @a[i]@), to avoid using @push_back@, and to use a vector of vectors.  Used with these restrictions, out-of-bound accesses are stopped, and in-bound accesses never exercise the vector's ability to grow, which is to say, they never make the program slow to reallocate and copy, and they never invalidate the program's other references to the contained values.  Allowing this scheme the same referential integrity assumption that \CFA enjoys [todo:xref], this scheme matches Java's safety and expressiveness exactly.  [TODO: decide about going deeper; some of the Java expressiveness concerns have mitigations, up to even more tradeoffs.] \subsection{Levels of dependently typed arrays} The \CFA array and the field of array language'' comparators all leverage dependent types to improve on the expressiveness over C and Java, accommodating examples such as: \begin{itemize} \item a \emph{zip}-style operation that consumes two arrays of equal length \item a \emph{map}-style operation whose produced length matches the consumed length \item a formulation of matrix multiplication, where the two operands must agree on a middle dimension, and where the result dimensions match the operands' outer dimensions \end{itemize} Across this field, this expressiveness is not just an avaiable place to document such assumption, but these requirements are strongly guaranteed by default, with varying levels of statically/dynamically checked and ability to opt out.  Along the way, the \CFA array also closes the safety gap (with respect to bounds) that Java has over C. Dependent type systems, considered for the purpose of bound-tracking, can be full-strength or restricted.  In a full-strength dependent type system, a type can encode an arbitrarily complex predicate, with bound-tracking being an easy example.  The tradeoff of this expressiveness is complexity in the checker, even typically, a potential for its nontermination.  In a restricted dependent type system (purposed for bound tracking), the goal is to check helpful properties, while keeping the checker well-behaved; the other restricted checkers surveyed here, including \CFA's, always terminate.  [TODO: clarify how even Idris type checking terminates] Idris is a current, general-purpose dependently typed programming language.  Length checking is a common benchmark for full dependent type stystems.  Here, the capability being considered is to track lengths that adjust during the execution of a program, such as when an \emph{add} operation produces a collection one element longer than the one on which it started.  [todo: finish explaining what Data.Vect is and then the essence of the comparison] POINTS: here is how our basic checks look (on a system that deosn't have to compromise); it can also do these other cool checks, but watch how I can mess with its conservativeness and termination Two current, state-of-the-art array languages, Dex\cite{arr:dex:long} and Futhark\cite{arr:futhark:tytheory}, offer offer novel contributions concerning similar, restricted dependent types for tracking array length.  Unlike \CFA, both are garbage-collected functional languages.  Because they are garbage-collected, referential integrity is built-in, meaning that the heavyweight analysis, that \CFA aims to avoid, is unnecessary.  So, like \CFA, the checking in question is a leightweight bounds-only analysis.  Like \CFA, their checks that are conservatively limited by forbidding arithmetic in the depended-upon expression. The Futhark work discusses the working language's connection to a lambda calculus, with typing rules and a safety theorem proven in reference to an operational semantics.  There is a particular emphasis on an existential type, enabling callee-determined return shapes. Dex uses a novel conception of size, embedding its quantitative information completely into an ordinary type. Futhark and full-strength dependently typed lanaguages treat array sizes are ordinary values.  Futhark restricts these expressions syntactically to variables and constants, while a full-strength dependent system does not. CFA's hybrid presentation, @forall( [N] )@, has @N@ belonging to the type system, yet has no instances.  Belonging to the type system means it is inferred at a call site and communicated implicitly, like in Dex and unlike in Futhark.  Having no instances means there is no type for a variable @i@ that constrains @i@ to be in the range for @N@, unlike Dex, [TODO: verify], but like Futhark. \subsection{Static safety in C extensions} \section{Future Work} \subsection{Declaration syntax} \subsection{Range slicing} \subsection{With a module system} \subsection{With described enumerations} A project in \CFA's current portfolio will improve enumerations.  In the incumbent state, \CFA has C's enumerations, unmodified.  I will not discuss the core of this project, which has a tall mission already, to improve type safety, maintain appropriate C compatibility and offer more flexibility about storage use.  It also has a candidate stretch goal, to adapt \CFA's @forall@ generic system to communicate generalized enumerations: \begin{lstlisting} forall( T | is_enum(T) ) void show_in_context( T val ) { for( T i ) { string decorator = ""; if ( i == val-1 ) decorator = "< ready"; if ( i == val   ) decorator = "< go"   ; sout | i | decorator; } } enum weekday { mon, tue, wed = 500, thu, fri }; show_in_context( wed ); \end{lstlisting} with output \begin{lstlisting} mon tue < ready wed < go thu fri \end{lstlisting} The details in this presentation aren't meant to be taken too precisely as suggestions for how it should look in \CFA.  But the example shows these abilities: \begin{itemize} \item a built-in way (the @is_enum@ trait) for a generic routine to require enumeration-like information about its instantiating type \item an implicit implementation of the trait whenever a user-written enum occurs (@weekday@'s declaration implies @is_enum@) \item a total order over the enumeration constants, with predecessor/successor (@val-1@) available, and valid across gaps in values (@tue == 1 && wed == 500 && tue == wed - 1@) \item a provision for looping (the @for@ form used) over the values of the type. \end{itemize} If \CFA gets such a system for describing the list of values in a type, then \CFA arrays are poised to move from the Futhark level of expressiveness, up to the Dex level. [TODO: indroduce Ada in the comparators] In Ada and Dex, an array is conceived as a function whose domain must satisfy only certain structural assumptions, while in C, C++, Java, Futhark and \CFA today, the domain is a prefix of the natural numbers.  The generality has obvious aesthetic benefits for programmers working on scheduling resources to weekdays, and for programmers who prefer to count from an initial number of their own choosing. This change of perspective also lets us remove ubiquitous dynamic bound checks.  [TODO: xref] discusses how automatically inserted bound checks can often be otimized away.  But this approach is unsatisfying to a programmer who believes she has written code in which dynamic checks are unnecessary, but now seeks confirmation.  To remove the ubiquitious dynamic checking is to say that an ordinary subscript operation is only valid when it can be statically verified to be in-bound (and so the ordinary subscript is not dynamically checked), and an explicit dynamic check is available when the static criterion is impractical to meet. [TODO, fix confusion:  Idris has this arrangement of checks, but still the natural numbers as the domain.] The structural assumptions required for the domain of an array in Dex are given by the trait (there, interface'') @Ix@, which says that the parameter @n@ is a type (which could take an argument like @weekday@) that provides two-way conversion with the integers and a report on the number of values.  Dex's @Ix@ is analogous the @is_enum@ proposed for \CFA above. \begin{lstlisting} interface Ix n get_size n : Unit -> Int ordinal : n -> Int unsafe_from_ordinal n : Int -> n \end{lstlisting} Dex uses this foundation of a trait (as an array type's domain) to achieve polymorphism over shapes.  This flavour of polymorphism lets a function be generic over how many (and the order of) dimensions a caller uses when interacting with arrays communicated with this funciton.  Dex's example is a routine that calculates pointwise differences between two samples.  Done with shape polymorphism, one function body is equally applicable to a pair of single-dimensional audio clips (giving a single-dimensional result) and a pair of two-dimensional photographs (giving a two-dimensional result).  In both cases, but with respectively dimensoned interpretations of size,'' this function requries the argument sizes to match, and it produces a result of the that size. The polymorphism plays out with the pointwise-difference routine advertizing a single-dimensional interface whose domain type is generic.  In the audio instantiation, the duration-of-clip type argument is used for the domain.  In the photograph instantiation, it's the tuple-type of $\langle \mathrm{img\_wd}, \mathrm{img\_ht} \rangle$.  This use of a tuple-as-index is made possible by the built-in rule for implementing @Ix@ on a pair, given @Ix@ implementations for its elements \begin{lstlisting} instance {a b} [Ix a, Ix b] Ix (a & b) get_size = \(). size a * size b ordinal = \(i, j). (ordinal i * size b) + ordinal j unsafe_from_ordinal = \o. bs = size b (unsafe_from_ordinal a (idiv o bs), unsafe_from_ordinal b (rem o bs)) \end{lstlisting} and by a user-provided adapter expression at the call site that shows how to indexing with a tuple is backed by indexing each dimension at a time \begin{lstlisting} img_trans :: (img_wd,img_ht)=>Real img_trans.(i,j) = img.i.j result = pairwise img_trans \end{lstlisting} [TODO: cite as simplification of example from https://openreview.net/pdf?id=rJxd7vsWPS section 4] In the case of adapting this pattern to \CFA, my current work provides an adapter from successively subscripted'' to subscripted by tuple,'' so it is likely that generalizing my adapter beyond subscripted by @ptrdiff_t@'' is sufficient to make a user-provided adapter unnecessary. \subsection{Retire pointer arithmetic}