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1 Dependent Types for Termination Verification Hongwei Xi University of Cincinnati

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2 Overview Motivation Program error detection at compile-time Compilation certification Termination Verification for Dependent ML Theoretical development Practical applications Conclusion

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3 Program Error Detection Unfortunately one often pays a price for [languages which impose no disciplines of types] in the time taken to find rather inscrutable bugs — anyone who mistakenly applies CDR to an atom in LISP and finds himself absurdly adding a property list to an integer, will know the symptoms. -- Robin Milner A Theory of Type Polymorphism in Programming Therefore, a stronger type discipline allows for capturing more program errors at compile-time.

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4 Some Advantages of Types Detecting program errors at compile-time Enabling compiler optimizations Facilitating program verification Using types to encode program properties Verifying the encoded properties through type- checking Serving as program documentation Unlike informal comments, types are formally verified and can thus be fully trusted

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5 Compiler Correctness How can we prove the correctness of a (realistic) compiler? Verifying that the semantics of e is the same as the semantics of |e| for every program e But this simply seems too challenging (and is unlikely to be feasible) Source program e Target code | e | compilation |. ||. |

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6 Compilation Certification Assume that e holds, i.e., e has the property Then e should hold, too A compiler can be designed to produce a certificate to assert that e does have the property Target code e : e holds Source program e: e holds compilation |. ||. |

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7 Narrowing the Gap NuPrl Coq Program Extraction Proof Synthesis ML Dependent ML

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8 Termination Verification Termination is a liveness property can not be verified at run-time is often proven with a well-founded metric that decreases whenever a recursive function call is made

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9 Primitive Recursion datatype Nat with nat = Z(0) | {n:nat} S(n+1) of Nat(n) (* Z: Nat(0) S: {n:nat} Nat(n) -> Nat(n+1) *) fun (‘a) R Z u v = u | R (S n) u v = v n (R n u v) withtype {n:nat} Nat(n) -> ‘a -> (Nat -> ‘a -> ‘a) -> ‘a (* Nat is for [n:nat] Nat(n) *) =>

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10 Ackermann Function in DML fun ack (m, n) = if m = 0 then n+1 else if n = 0 then ack (m-1, 1) else ack (m-1, ack (m, n-1)) withtype {m:nat,n:nat} int(m) * int(n) -> nat (* Note: nat = [a:int | a >=0] int(a) *) =>

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11 What’s really new here? We present a formalism that allows the programmer to supply crucial information for facilitating program termination verification can smoothly integrate into a realistic programming language

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12 ML 0: start point base types ::= int | bool | (user defined datatypes) types ::= | | patterns p ::= x | c(p) | <> | match clauses ms ::= (p e) | (p e | ms) expressions e ::= x | f | c | if (e, e 1, e 2 ) | <> | | lam x: . e | fix f: . e | e 1 (e 2 ) | let x=e 1 in e 2 end | case e of ms values v ::= x | c | | lam x: . e context ::=. | , x:

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13 Integer Constraint Domain We use a for index variables index expressions i, j ::= a | c | i + j | i – j | i * j | i / j | … index propositions P, Q ::= i j | i >= j | i = j | i <> j | P Q | P Q index sorts ::= int | {a : | P } index variable contexts ::=. | , a: | , P index constraints ::= P | P | a:

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14 Dependent Types dependent types ::=... | (i) | a: . | a: . For instance, int(0), bool array(16); nat = [a:int | a >= 0] int(a); {a:int | a >= 0} int list(a) -> int list(a)

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15 DML 0 ML 0 + dependent types expressions e ::=... | a: .v | e[i] | | open e 1 as in e 2 end values v ::=... | a: .v | typing judgment e

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16 A Typing Rule e bool(i) i e 1: i e 2: type-if if (e, e 1, e 2 ):

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17 Metric Typing Judgments Definition (Metric) Let = be a tuple of index expressions. We write : metric if we have i j :nat for 1 j n. We use a for a decorated type We use the judgemnt e f to mean that for each occurrence of f[i] in e, [a->i] holds, where f is declared in to have type a

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18 Some Metric Typing Rules The rule ( app) is: e 1 f e 2 f e 1( e 2): f The rule ( lab) is: i a i f a f i a i f

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19 DML 0, The following typing rule is for forming functions: a f a |- e: f type-fun) fun f a is e: a

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20 Reducibility Definition Suppose that e is a closed expression of type and e * v holds for some value v. is a base type. Then e is reducible Then e is reducible if e(v 1 ) is reducible for every reducible value v 1 of type . Then e is reducible if v= and v 1, v 2 are reducible. a: . Then e is reducible if e[i] is reducible for every i: . a: . Then e is reducible if v= and v 1 is reducible.

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21 -reducibility Definition Let e be a well-typed closed function fun f[a: ]: is v and be a closed metric. e is -reducible if e[i] is reducible for each i satisfying [a->i] . Theorem Every closed expression e is reducible if it is well-typed in DML 0,

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22 Quicksort in DML fun qs [] = [] | qs (x :: xs) = par (x, xs, [], []) withtype {n:nat} int list(n) -> int list(n) and par (x, [], l, g) = qs (l) @ (x :: qs (g)) | par (x, y :: ys, l, g) = if y <= x then par (x, ys, y :: l, g) else par (x, ys, l, y :: g) withtype {p:nat,q:nat,r:nat} int * int list(p) * int list(q) * int list(r) -> int list(p+q+r+1) =>

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23 Contributions Novel type system design to facilitate termination verification Unobtrusive programming Theoretical foundation Prototype implementation and evaluation

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24 Some Closely Related Work Termination and Reduction Checking in Logic Framework, Pientka and Pfenning Termination Analysis for Mercury, Speirs, Somogyi and S Ø ndergarrd Proving correctness of reactive systems using sized types, Hughes, Pareto and Sabry. Cost recurrences for DML programs, Bernd Grobauer.

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25 End of the Talk Thank You! Questions?

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