ATS (programming language)

History

ATS is derived mostly from the ML and OCaml programming languages. An earlier language, Dependent ML, by the same author has been incorporated by the language.

The latest version of ATS1 (Anairiats) was released as v0.2.12 on 2015-01-20. The first version of ATS2 (Postiats) was released in September 2013.

Theorem proving

The primary focus of ATS is to support theorem proving in combination with practical programming. With theorem proving one can prove, for instance, that an implemented function does not produce memory leaks. It also prevents other bugs that might otherwise only be found during testing. It incorporates a system similar to those of proof assistants which usually only aim to verify mathematical proofs—except ATS uses this ability to prove that the implementations of its functions operate correctly, and produce the expected output.

As a simple example, in a function using division, the programmer may prove that the divisor will never equal zero, preventing a division by zero error. Let's say, the divisor 'X' was computed as 5 times the length of list 'A'. One can prove, that in the case of a non-empty list, 'X' is non-zero, since 'X' is the product of two non-zero numbers (5 and the length of 'A'). A more practical example would be proving through reference counting that the retain count on an allocated block of memory is being counted correctly for each pointer. Then one can know, and quite literally prove, that the object will not be deallocated prematurely, and that memory leaks will not occur.

The benefit of the ATS system is that since all theorem proving occurs strictly within the compiler, it has no effect on the speed of the executable program. ATS code is often harder to compile than standard C code, but once it compiles the programmer can be certain that it is running correctly to exactly the degree specified by their proofs.

In ATS proofs are separate from implementation, so it is possible to implement a function without proving it if the programmer so desires.

Data representation

According to the author (Hongwei Xi), ATS's efficiency is largely due to the way that data is represented in the language and tail-call optimizations (which are generally important for the efficiency of functional programming languages). Data can be stored in a flat or unboxed representation rather than a boxed representation.

Propositions

dataprop expresses predicates as algebraic types.

Predicates in pseudo‑code somewhat similar to ATS source (see below for valid ATS source):

FACT(n, r) iff fact(n) = r MUL(n, m, prod) iff n * m = prod FACT(n, r) = FACT(0, 1) | FACT(n, r) iff FACT(n-1, r1) and MUL(n, r1, r) // for n > 0 // expresses fact(n) = r iff r = n * r1 and r1 = fact(n-1)

In ATS code:

dataprop FACT (int, int) = | FACTbas (0, 1) // basic case: FACT(0, 1) // inductive case | {n:int | n > 0} {r,r1:int} FACTind (n, r) of (FACT (n-1, r1), MUL (n, r1, r))

where FACT (int, int) is a proof type

Example

Non tail-recursive factorial with proposition or "Theorem" proving through the construction dataprop.

The evaluation of fact1(n-1) returns a pair (proof_n_minus_1 | result_of_n_minus_1) which is used in the calculation of fact1(n). The proofs express the predicates of the proposition.

Part 1 (algorithm and propositions)

[FACT (n, r)] implies [fact (n) = r] [MUL (n, m, prod)] implies [n * m = prod] FACT (0, 1) FACT (n, r) iff FACT (n-1, r1) and MUL (n, r1, r) forall n > 0

To remember:

{...} universal quantification [...] existential quantification (... | ...) (proof | value) @(...) flat tuple or variadic function parameters tuple .<...>. termination metric

Part 2 (routines and test)

This can all be added to a single file and compiled as follows. Compilation should work with various back end C compilers, e.g. gcc. Garbage collection is not used unless explicitly stated with -D_ATS_GCATS )

patscc fact1.dats -o fact1 ./fact1 4

compiles and gives the expected result

Basic types

bool (true, false)

int (literals: 255, 0377, 0xFF), unary minus as ~ (as in ML)

double

char 'a'

string "abc"

Tuples and records

prefix @ or none means direct, flat or unboxed allocation

val x : @(int, char) = @(15, 'c') // x.0 = 15 ; x.1 = 'c' val @(a, b) = x // pattern matching binding, a= 15, b='c' val x = @{first=15, second='c'} // x.first = 15 val @{first=a, second=b} = x // a= 15, b='c' val @{second=b, ...} = x // with omission, b='c'

prefix ' means indirect or boxed allocation

val x : '(int, char) = '(15, 'c') // x.0 = 15 ; x.1 = 'c' val '(a, b) = x // a= 15, b='c' val x = '{first=15, second='c'} // x.first = 15 val '{first=a, second=b} = x // a= 15, b='c' val '{second=b, ...} = x // b='c'

special

With '|' as separator, some functions return wrapped the result value with an evaluation of predicates

val ( predicate_proofs | values) = myfunct params

Common

{...} universal quantification [...] existential quantification (...) parenthetical expression or tuple (... | ...) (proofs | values) .<...>. termination metric @(...) flat tuple or variadic function parameters tuple (see example's printf) @[byte][BUFLEN] type of an array of BUFLEN values of type byte @[byte][BUFLEN]() array instance @[byte][BUFLEN](0) array initialized to 0

Dictionary

sort

domain sortdef nat = {a: int | a >= 0 } // from prelude: ∀ a ∈ int ... typedef String = [a:nat] string(a) // [..]: ∃ a ∈ nat ...

type (as sort)

generic sort for elements with the length of a pointer word, to be used in type parameterized polymorphic functions. Also "boxed types" // {..}: ∀ a,b ∈ type ... fun {a,b:type} swap_type_type (xy: @(a, b)): @(b, a) = (xy.1, xy.0)

t@ype

linear version of previous type with abstracted length. Also unboxed types.

viewtype

a domain class like type with a view (memory association)

viewt@ype

linear version of viewtype with abstracted length. It supersets viewtype

view

relation of a Type and a memory location. The infix @ is its most common constructor T @ L asserts that there is a view of type T at location L fun {a:t@ype} ptr_get0 {l:addr} (pf: a @ l | p: ptr l): @(a @ l | a) fun {a:t@ype} ptr_set0 {l:addr} (pf: a? @ l | p: ptr l, x: a): @(a @ l | void) the type of ptr_get0 (T) is ∀ l : addr . ( T @ l | ptr( l ) ) -> ( T @ l | T) // see manual, section 7.1. Safe Memory Access through Pointers viewdef array_v (a:viewt@ype, n:int, l: addr) = @[a][n] @ l

T? 

possibly uninitialized type

pattern matching exhaustivity

as in case+, val+, type+, viewtype+, ...

with suffix '+' the compiler issues an error in case of non exhaustive alternatives

without suffix the compiler issues a warning

with '-' as suffix, avoids exhaustivity control

modules

staload "foo.sats" // foo.sats is loaded and then opened into the current namespace staload F = "foo.sats" // to use identifiers qualified as $F.bar dynload "foo.dats" // loaded dynamically at run-time

dataview

Dataviews are often declared to encode recursively defined relations on linear resources.

dataview array_v (a: viewt@ype+, int, addr) = | {l: addr} array_v_none (a, 0, l) | {n: nat} {l: addr} array_v_some (a, n+1, l) of (a @ l, array_v (a, n, l+sizeof a))

datatype / dataviewtype

Datatypes

datatype workday = Mon | Tue | Wed | Thu | Fri

lists

datatype list0 (a:t@ype) = list0_cons (a) of (a, list0 a) | list0_nil (a)

dataviewtype

A dataviewtype is similar to a datatype, but it is linear. With a dataviewtype, the programmer is allowed to explicitly free (or deallocate) in a safe manner the memory used for storing constructors associated with the dataviewtype.

variables

local variables

var res: int with pf_res = 1 // introduces pf_res as an alias of view @ (res)

on stack array allocation:

#define BUFLEN 10 var !p_buf with pf_buf = @[byte][BUFLEN](0) // pf_buf = @[byte][BUFLEN](0) @ p_buf

See val and var declarations