(** {1 Range Sum Queries} We are interested in specifying and proving correct data structures that support efficient computation of the sum of the values over an arbitrary range of an array. Concretely, given an array of integers [a], and given a range delimited by indices [i] (inclusive) and [j] (exclusive), we wish to compute the value: [\sum_{k=i}^{j-1} a[k]]. In the first part, we consider a simple loop for computing the sum in linear time. In the second part, we introduce a cumulative sum array that allows answering arbitrary range queries in constant time. In the third part, we explore a tree data structure that supports modification of values from the underlying array [a], with logarithmic time operations. *) (** {2 Specification of Range Sum} *) module ArraySum use export int.Int use export array.Array (** [sum a i j] denotes the sum [\sum_{i <= k < j} a[k]]. It is axiomatizated by the two following axioms expressing the recursive definition if [i <= j] then [sum a i j = 0] if [i < j] then [sum a i j = a[i] + sum a (i+1) j] *) function sum (array int) int int : int axiom sum_def_empty : forall a : array int, i j : int. j <= i -> sum a i j = 0 axiom sum_def_non_empty : forall a: array int, i j : int. i < j /\ 0 <= i < a.length -> sum a i j = a[i] + sum a (i+1) j (** lemma for summation from the right: if [i < j] then [sum a i j = sum a i (j-1) + a[j-1]] *) lemma sum_right : forall a : array int, i j : int. 0 <= i < j <= a.length -> sum a i j = sum a i (j-1) + a[j-1] end (** {2 First algorithm, a linear one} *) module Simple use import ArraySum use import ref.Ref (** [query a i j] returns the sum of elements in [a] between index [i] inclusive and index [j] exclusive *) let query (a:array int) (i j:int) : int requires { 0 <= i <= j <= a.length } ensures { result = sum a i j } = let s = ref 0 in for k=i to j-1 do invariant { !s = sum a i k } s := !s + a[k] done; !s end (** {2 Additional lemmas on [sum]} needed in the remaining code *) module ExtraLemmas use array.Array use import ArraySum (** summation in adjacent intervals *) lemma sum_concat : forall a:array int, i j k:int. 0 <= i <= j <= k <= a.length -> sum a i k = sum a i j + sum a j k (** Frame lemma for [sum], that is [sum a i j] depends only of values of [a[i..j-1]] *) lemma sum_frame : forall a1 a2 : array int, i j : int. 0 <= i <= j -> j <= a1.length -> j <= a2.length -> (forall k : int. i <= k < j -> a1[k] = a2[k]) -> sum a1 i j = sum a2 i j (** Updated lemma for [sum]: how does [sum a i j] changes when [a[k]] is changed for some [k] in [[i..j-1]] *) lemma sum_update : forall a:array int, i v l h:int. 0 <= l <= i < h <= a.length -> sum (a[i<-v]) l h = sum a l h + v - a[i] end (** {2 Algorithm 2: using a cumulative array} creation of cumulative array is linear query is in constant time array update is linear *) module CumulativeArray use array.Array use import ArraySum use ExtraLemmas predicate is_cumulative_array_for (c:array int) (a:array int) = c.length = a.length + 1 /\ forall i. 0 <= i < c.length -> c[i] = sum a 0 i (** [create a] builds the cumulative array associated with [a]. *) let create (a:array int) : array int ensures { is_cumulative_array_for result a } = let l = a.length in let s = Array.make (l+1) 0 in for i=1 to l do invariant { forall k. 0 <= k < i -> s[k] = sum a 0 k } s[i] <- s[i-1] + a[i-1] done; s (** [query c i j a] returns the sum of elements in [a] between index [i] inclusive and index [j] exclusive, in constant time *) let query (c:array int) (i j:int) (ghost a:array int): int requires { is_cumulative_array_for c a } requires { 0 <= i <= j < c.length } ensures { result = sum a i j } = c[j] - c[i] (** [update c i v a] updates cell [a[i]] to value [v] and updates the cumulative array [c] accordingly *) let update (c:array int) (i:int) (v:int) (ghost a:array int) : unit requires { is_cumulative_array_for c a } requires { 0 <= i < a.length } writes { c, a } ensures { is_cumulative_array_for c a } ensures { a[i] = v } ensures { forall k. 0 <= k < a.length /\ k <> i -> a[k] = (old a)[k] } = 'Init: let incr = v - c[i+1] + c[i] in a[i] <- v; for j=i+1 to c.length-1 do invariant { forall k. j <= k < c.length -> c[k] = sum a 0 k - incr } invariant { forall k. 0 <= k < j -> c[k] = sum a 0 k } c[j] <- c[j] + incr done end (** {2 Algorithm 3: using a cumulative tree} creation is linear query is logarithmic update is logarithmic *) module CumulativeTree use array.Array use import ArraySum use ExtraLemmas use import int.ComputerDivision type indexes = { low : int; high : int; isum : int; } type tree = Leaf indexes | Node indexes tree tree function indexes (t:tree) : indexes = match t with | Leaf ind -> ind | Node ind _ _ -> ind end predicate is_indexes_for (ind:indexes) (a:array int) (i j:int) = ind.low = i /\ ind.high = j /\ 0 <= i < j <= a.length /\ ind.isum = sum a i j predicate is_tree_for (t:tree) (a:array int) (i j:int) = match t with | Leaf ind -> is_indexes_for ind a i j /\ j = i+1 | Node ind l r -> is_indexes_for ind a i j /\ i = l.indexes.low /\ j = r.indexes.high /\ let m = l.indexes.high in m = r.indexes.low /\ i < m < j /\ m = div (i+j) 2 /\ is_tree_for l a i m /\ is_tree_for r a m j end (** {3 creation of cumulative tree} *) let rec tree_of_array (a:array int) (i j:int) : tree requires { 0 <= i < j <= a.length } variant { j - i } ensures { is_tree_for result a i j } = if i+1=j then begin Leaf { low = i; high = j; isum = a[i] } end else begin let m = div (i+j) 2 in assert { i < m < j }; let l = tree_of_array a i m in let r = tree_of_array a m j in let s = l.indexes.isum + r.indexes.isum in assert { s = sum a i j }; Node { low = i; high = j; isum = s} l r end let create (a:array int) : tree requires { a.length >= 1 } ensures { is_tree_for result a 0 a.length } = tree_of_array a 0 a.length (** {3 query using cumulative tree} *) let rec query_aux (t:tree) (ghost a: array int) (i j:int) : int requires { is_tree_for t a t.indexes.low t.indexes.high } requires { 0 <= t.indexes.low <= i < j <= t.indexes.high <= a.length } variant { t } ensures { result = sum a i j } = match t with | Leaf ind -> ind.isum | Node ind l r -> let k1 = ind.low in let k3 = ind.high in if i=k1 && j=k3 then ind.isum else let m = l.indexes.high in if j <= m then query_aux l a i j else if i >= m then query_aux r a i j else query_aux l a i m + query_aux r a m j end let query (t:tree) (ghost a: array int) (i j:int) : int requires { 0 <= i <= j <= a.length } requires { is_tree_for t a 0 a.length } ensures { result = sum a i j } = if i=j then 0 else query_aux t a i j (** frame lemma for predicate [is_tree_for] *) lemma is_tree_for_frame : forall t:tree, a:array int, k v i j:int. 0 <= k < a.length -> k < i \/ k >= j -> is_tree_for t a i j -> is_tree_for t a[k<-v] i j (** {3 update cumulative tree} *) let rec update_aux (t:tree) (i:int) (ghost a :array int) (v:int) : (tree,int) requires { is_tree_for t a t.indexes.low t.indexes.high } requires { t.indexes.low <= i < t.indexes.high } variant { t } returns { (t',delta) -> delta = v - a[i] /\ t'.indexes.low = t.indexes.low /\ t'.indexes.high = t.indexes.high /\ is_tree_for t' a[i<-v] t'.indexes.low t'.indexes.high } = match t with | Leaf ind -> assert { i = ind.low }; (Leaf { ind with isum = v }, v - ind.isum) | Node ind l r -> let m = l.indexes.high in if i < m then let l',delta = update_aux l i a v in assert { is_tree_for l' a[i<-v] t.indexes.low m }; assert { is_tree_for r a[i<-v] m t.indexes.high }; (Node {ind with isum = ind.isum + delta } l' r, delta) else let r',delta = update_aux r i a v in assert { is_tree_for l a[i<-v] t.indexes.low m }; assert { is_tree_for r' a[i<-v] m t.indexes.high }; (Node {ind with isum = ind.isum + delta} l r',delta) end let update (t:tree) (ghost a:array int) (i v:int) : tree requires { 0 <= i < a.length } requires { is_tree_for t a 0 a.length } writes { a } ensures { a[i] = v } ensures { forall k. 0 <= k < a.length /\ k <> i -> a[k] = (old a)[k] } ensures { is_tree_for result a 0 a.length } = let t,_ = update_aux t i a v in a[i] <- v; t (** {2 complexity analysis} We would like to prove that [query] is really logarithmic. This is non-trivial because there are two recursive calls in some cases. So far, we are only able to prove that [update] is logarithmic We express the complexity by passing a credit'' as a ghost parameter. We pose the precondition that the credit is at least equal to the depth of the tree. *) (** preliminaries: definition of the depth of a tree, and showing that it is indeed logarithmic in function of the number of its elements *) use import int.MinMax function depth (t:tree) : int = match t with | Leaf _ -> 1 | Node _ l r -> 1 + max (depth l) (depth r) end lemma depth_min : forall t. depth t >= 1 use import bv.Pow2int let rec lemma depth_is_log (t:tree) (a :array int) (k:int) requires { k >= 0 } requires { is_tree_for t a t.indexes.low t.indexes.high } requires { t.indexes.high - t.indexes.low <= pow2 k } variant { t } ensures { depth t <= k+1 } = match t with | Leaf _ -> () | Node _ l r -> depth_is_log l a (k-1); depth_is_log r a (k-1) end (** [update_aux] function instrumented with a credit *) use import ref.Ref let rec update_aux_complexity (t:tree) (i:int) (ghost a :array int) (v:int) (ghost c:ref int): (tree,int) requires { is_tree_for t a t.indexes.low t.indexes.high } requires { t.indexes.low <= i < t.indexes.high } variant { t } ensures { !c - old !c <= depth t } returns { (t',delta) -> delta = v - a[i] /\ t'.indexes.low = t.indexes.low /\ t'.indexes.high = t.indexes.high /\ is_tree_for t' a[i<-v] t'.indexes.low t'.indexes.high } = c := !c + 1; match t with | Leaf ind -> assert { i = ind.low }; (Leaf { ind with isum = v }, v - ind.isum) | Node ind l r -> let m = l.indexes.high in if i < m then let l',delta = update_aux_complexity l i a v c in assert { is_tree_for l' a[i<-v] t.indexes.low m }; assert { is_tree_for r a[i<-v] m t.indexes.high }; (Node {ind with isum = ind.isum + delta } l' r, delta) else let r',delta = update_aux_complexity r i a v c in assert { is_tree_for l a[i<-v] t.indexes.low m }; assert { is_tree_for r' a[i<-v] m t.indexes.high }; (Node {ind with isum = ind.isum + delta} l r',delta) (*>*) end (** [query_aux] function instrumented with a credit *) let rec query_aux_complexity (t:tree) (ghost a: array int) (i j:int) (ghost c:ref int) : int requires { is_tree_for t a t.indexes.low t.indexes.high } requires { 0 <= t.indexes.low <= i < j <= t.indexes.high <= a.length } variant { t } ensures { !c - old !c <= if i = t.indexes.low /\ j = t.indexes.high then 1 else if i = t.indexes.low \/ j = t.indexes.high then 2 * depth t else 4 * depth t } ensures { result = sum a i j } = c := !c + 1; match t with | Leaf ind -> ind.isum | Node ind l r -> let k1 = ind.low in let k3 = ind.high in if i=k1 && j=k3 then ind.isum else let m = l.indexes.high in if j <= m then query_aux_complexity l a i j c else if i >= m then query_aux_complexity r a i j c else query_aux_complexity l a i m c + query_aux_complexity r a m j c end end