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Imperative Programming with Dependent Types Hongwei Xi University of Cincinnati

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A Wish List We would like to have a programming language that should – be simple and general – support extensive error checking – facilitate proofs of program properties – possess correct and efficient implementation –......

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Reality Invariably, there are many conflicts among this wish list These conflicts must be resolved with careful attention to the needs of the user

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Advantages of Types Capturing errors at compile-time Enabling compiler optimizations Facilitating program verification Serving as program documentation

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Limitations of (Simple) Types Not general enough – Many correct programs cannot be typed Not specific enough – Many interesting properties cannot be captured

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Dependent Types Dependent types are types that are – more refined – dependent on the values of expressions Examples – int(i): singleton type containing only integer i – array(n): type for integer arrays of size n

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Type System Design A practically useful type system should be – Scalable – Applicable – Comprehensible – Unobtrusive – Flexible

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Xanadu Xanadu is a dependently typed imperative programming language with C-like syntax The type of a variable in Xanadu can change during execution The programmer may need to provide dependent type annotations for type- checking purpose

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Early Design Decisions Practical type-checking Realistic programming features Conservative extension Pay-only-if-you-use policy

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Examples of Dependent Types in Xanadu int(a): singleton types containing the only integer equal to a, where a ranges over all integers array(a): types for arrays of size a in which all elements are of type ‘a, where a ranges over all natural numbers

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Examples of Dependent Types in Xanadu int(i,j) is defined as [a:int | i < a < j] int(a), that is, the sum of all types int(a) for i < a < j int[i,j), int(i,j], int[i,j] are defined similarly

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A Xanadu Program {n:nat} unit init (int vec[n]) { var int ind, size;; size = arraysize(vec); invariant: [i:nat] (ind: int(i)) for (ind=0; ind

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A Slight Variation {n:nat} unit init (int vec[n]) { var nat ind, size;; size = arraysize(vec); for (ind=0; ind

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Dependent Record Types A polymorphic type for arrays {n:nat} array(n) { size: int(n); data[n]: ‘a }

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Binary Search in Xanadu {n:nat} int bs(key: int, vec: array(n)) { var: l: int [0, n], h: int [-1, n); m: int; x: float;; l = 0; h = vec.size - 1; while (l <= h) { m = (l + h) / 2; x = vec.data[m]; if (x < key) { l = m - 1; } else if (x > key) { h = m + 1; } else { return m; } } return –1; }

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Dependent Record Types A polymorphic type for 2-dimensional arrays: {n:nat} array2(m,n) { row: int(m); col: int(n); data[m][n]: ‘a }

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Dependent Record Types A polymorphic type for sparse arrays: sparseArray(m,n) { row: int(m); col: int(n); data[m]: list }

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Dependent Union Types A polymorphic type for lists: union list with nat = { Nil(0); {n:nat} Cons(n+1) of ‘a list(n) } Nil: list(0) Cons: {n:nat} ‘a * list(n) -> ‘a list(n+1)

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Dependent Union Types A polymorphic type for binary trees: union tree with (nat,nat) = { E(0,0); {sl:nat,sr:nat,hl:nat,hr:nat} B(sl+sr+1,1+max(hl,hr)) of tree(sl,hl) ‘a tree(sr,hr) }

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Typing Judgment in Xanadu e

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Typing Assignment e x e x unit

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Typing Loop e bool i i e unit while e e i unit

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Reverse Append in Xanadu (‘a) {m:nat,n:nat} list(m+n) revApp (xs: list(m),ys: list(n)) { var: ‘a x;; invariant: [m1:nat,n1:nat | m1+n1=m+n] (xs: list(m1), ys: list(n1)) while (true) { switch (xs) { case Nil: return ys; case Cons (x, xs): ys = Cons(x, ys); } } exit; /* can never be reached */ }

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Constraint Generation in Type- checking The following integer constraint is generated when the revApp example is type-checked: m:nat,n:nat, m1:nat,n1:nat, m1+n1=m+n, a:nat, m1=a+1 |= a+(n1+1)=m+n

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Conclusion It is still largely an elusive goal in practice to verify the correctness of a program It is therefore important to identify those program properties that can be effectively verified for realistic programs

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Conclusion We have designed a type-theoretic approach to capturing simple arithmetic reasoning The preliminary studies indicate that this approach allows the programmer to capture many more properties in realistic programs

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Future Work Adding more program features into Xanadu Constructing a compiler for Xanadu that can compile dependent types from source level into bytecode level Incorporating dependent types into Java and …

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