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This may require cleanup to meet Wikipedia's quality standards. Please improve this if you can. The talk page may contain suggestions. (September 2010) ATS Paradigm multi-paradigm: imperative, functional Designed by Hongwei Xi at the Boston University Influenced by ML, Objective Caml Website ATS (Applied Type System) is a programming language whose stated purpose is to support theorem proving in combination with practical programming through the use of advanced type systems.[1] The performance of ATS has been demonstrated to be comparable to that of the C and C++ programming languages.[2] Contents 1 History 2 Theorem proving 3 Data representation 4 Features 4.1 Basic types 4.2 Tuples and records 4.3 Common 4.4 Dictionary 4.5 propositions 4.6 Example 4.7 pattern matching exhaustivity 4.8 modules 4.9 dataview 4.10 datatype / dataviewtype 4.10.1 dataviewtype 4.11 variables 5 References 6 External links // History ATS is derived mostly from the ML and Objective Caml programming languages. An earlier language, Dependent ML, by the same author has been incorporated by the language. Theorem proving The primary focus of ATS is to support theorem proving in combination with practical programming.[3] Data representation According to the author (Hongwei Xi), ATS's efficiency[4] 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 is stored in a flat or unboxed representation rather than a boxed representation. Features Basic types bool (true, false) int ( 255, 0377, 0xFF), unary minus as ~ 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) @(...) flat tuple or variadic function parameters tuple (see example's printf) @[byte][BUFLEN] type of an array of BUFLEN values of type byte[5] @[byte][BUFLEN]() array instance @[byte][BUFLEN](0) array initialized to 0 Dictionary sort domain sortdef nat = {a: int | a >= 0 } // from prelude 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 polymorphic functions // {..}: ∀ 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. It supersets type 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[6] viewdef array_v (a:viewt@ype, n:int, l: addr) = @[a][n] @ l T?  possibly uninitialized type propositions dataprop expresses predicates as algebraic types predicates: FACT( n, r)    if and only if    fact(n) = r MUL( n, m, prod)    iff    n * m = prod given r = fact(n) ; r1 = fact(n-1) FACT(n, r) = FACT(0, 1) | FACT(n, r) iff FACT (n-1, r1) and MUL (n, r1, r) // for n > 0 in ATS code: dataprop FACT (int, int) = | FACTbas (0, 1) // basic case: FACT(0, 1) | {n:int | n > 0} {r,r1:int} FACTind (n, r) of (FACT (n-1, r1), MUL (n, r1, r)) // inductive case 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. // file fact1.dats (* [FACT (n, r)] implies [fact (n) = r] [MUL( n, m, prod)] implies [n * m = prod] Given: fact(0) = 1 ; r1 = fact(n-1) ; r = fact(n) = n * r1 ; for n > 0 FACT (0, 1) FACT (n, r) iff FACT (n-1, r1) and MUL (n, r1, r) *) (* to remember: {...} universal quantification [...] existential quantification (... | ...) (proof | value) @(...) flat tuple or variadic function parameters tuple *) dataprop FACT (int, int) = | FACTbas (0, 1) // basic case | {n:int | n > 0} {r,r1:int} FACTind (n, r) of (FACT (n-1, r1), MUL (n, r1, r)) // inductive case // [fact1] implements factorial without tail-recursion fun fact1 {n:nat} .< n >. (num: int(n)) (* {n:nat} domain .< n >. metrics (num: int(n)) parameter and type *) : [r:int] (FACT (n, r) | int(r)) // type of result as (proof type | value type) = if num > 0 then let val (pf_fact_n_minus_1 | r1) = fact1 (num-1) // pf_fact_n_minus_1 = FACT( num-1, r1) val (pf_mul | r) = num imul2 r1 // pf_mul = MUL( num, r1, r) in (FACTind (pf_fact_n_minus_1, pf_mul) | r) end else (FACTbas () | 1) // ''fn'' introduces a non recursive function ; ''fun'' introduces a recursive one fn abs {n:int} (num: int(n)): [r: nat] int(r) = if num >= 0 then num else ~num implement main (argc, argv) : void = if (argc <> 2) then printf ("usage: %s 9 (expected one argument only)\n", @(argv.[0])) else let val num = int1_of argv.[1] val nat_num = abs num val (pf_fact_n | res) = fact1 nat_num in printf ("factorial of %i = %i\n", @(nat_num, res)) end Compilation (compiles through gcc, without setting up garbage collection unless explicitly stated with -D_ATS_GCATS )[7] atscc fact1.dats -o fact1 ./fact1 4 compiles and gives the expected result 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 $ dynload "foo.dats" // loaded dynamically at run-time dataview Dataviews are often declared to encode recursively defined relations on linear resources.[8] 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[9] 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.[10] variables local variables var res: int with pf_res = 1 // introduces pf_res as an alias of view @ (res) on stack array allocation: var !p_buf with pf_buf = @[byte][BUFLEN](0) // pf_buf = @[byte][BUFLEN](0) @ p_buf [11] See val and var declarations[12] References ^ Combining Programming with Theorem Proving ^ ATS benchmarks | Computer Language Benchmarks Game ^ Combining Programming with Theorem Proving ^ Discussion about the language's efficiency (Language Shootout: ATS is the new top gunslinger. Beats C++.) ^ type @[T[I] of an array] ^ Manual, section 7.1. Safe Memory Access through Pointers ^ Compilation - Garbage collection ^ Dataview construct ^ Datatype construct ^ Dataviewtype construct ^ Manual - 7.3 Memory allocation on stack ^ Val and Var declarations External links ATS home page Manual Draft. Some examples refer to features or routines not present in the release (Anairiats-0.1.6) (e.g.: print overload for strbuf, and using its array examples gives errmsgs like "use of array subscription is not supported".) ATS for ML programmers and other articles