UNITY (programming language)

The programming language UNITY was constructed by K. Mani Chandy and Jayadev Misra for their book Parallel Program Design: A Foundation. It is a rather theoretical language, which tries to focus on what, instead of where, when or how. The peculiar thing about the language is that it has no flow control. The statements in the program run in a random order, until none of the statements cause change if run. A correct program converges into a fix-point.

All statements are assignments, and are separated by #. A statement can consist of multiple assignments, on the form a,b,c := x,y,z, or a := x || b := y || c := z. You can also have a quantified statement list, <# x,y : expression :: statement>, where x and y are chosen randomly among the values that satisfy expression. A quantified assignment is similar. In <|| x,y : expression :: statement >, statement is executed simultaneously for all pairs of x and y that satisfy expression.




Examples

Bubble sort

Bubble sort the array by comparing adjacent numbers, and swapping them if they are in the wrong order. Using $\Theta (n)$ expected time, $\Theta (n)$ processors and $\Theta (n^{2})$ expected work. The reason you only have $\Theta (n)$ expected time, is that k is always chosen randomly from $\{0,1\}$. You could fix this by flipping k manually.

Program bubblesort
declare
    n: integer,
    A: array [0..n-1] of integer
initially
    n = 20 #
    <# i : 0 <= i and i < n :: A[i] = rand() % 100 >
assign
    <# k : 0 <= k < 2 ::
        <|| i : i % 2 = k and 0 <= i < n - 1 ::
            A[i], A[i+1] := A[i+1], A[i] 
                if A[i] > A[i+1] > >
end

Merge sort

Merge sort the array using a constant time merge. Using $\Theta (\log n)$ time, $\Theta (n^{2})$ processors and $\Theta (n^{2})$ work. It should have CREW PRAM complexity.

Program mergesort
declare
    n,m: integer,
    A,S,N: array [0..n-1] of integer
initially
    n = 20 #
    m = n ||
    <|| i : 0 <= i < n :: A[i], N[i], S[i] = rand() % 100, 1, i >
assign
    <|| s : 0 <= s < m - 1 and s % 2 = 0 ::
        <|| a,b : (a = 0 and b = 1) or (a = 1 and b = 0) ::
            <|| sa : 0 <= sa < N[s+a] ::
                A[S[s]+sa] := A[S[s+a]+sa] 
                    if a = 0 and A[S[s+a]+sa] <= A[S[s+b]] 
                    or a = 1 and A[S[s+a]+sa] <  A[S[s+b]] || 
                <|| sb : 0 < sb < N[s+b] ::
                    A[S[s]+sa+sb] := A[S[s+a]+sa] 
                        if a = 0 and A[S[s+b]+sb-1] <  A[S[s+a]+sa] <= A[S[s+b]+sb]      
                        or a = 1 and A[S[s+b]+sb-1] <= A[S[s+a]+sa] <  A[S[s+b]+sb] > || 
                A[S[s]+sa+N[s+b]] := A[S[s+a]+sa] 
                    if a = 0 and A[S[s+b]+N[s+b]-1] <  A[S[s+a]+sa]
                    or a = 1 and A[S[s+b]+N[s+b]-1] <= A[S[s+a]+sa] > > ||
        S[s/2] := S[s] ||
        N[s/2] := N[s] + N[s+1] > ||
    S[(m-1)/2], N[(m-1)/2] := S[m-1], N[m-1] if m%2 = 1 ||
    m := m/2 + m%2
end

Floyd-Warshall

Using the Floyd-Warshall all pairs shortest path algorithm, we include intermediate nodes iteratively, and get $\Theta (n)$ time, using $\Theta (n^{2})$ processors and $\Theta (n^{3})$ work.

Program shortestpath
declare
    n,k: integer,
    D: array [0..n-1, 0..n-1] of integer
initially
    n = 10 #
    k = 0 #
    <|| i,j : 0 <= i < n and 0 <= j < n :: 
        D[i,j] = rand() % 100 >
assign
    <|| i,j : 0 <= i < n and 0 <= j < n ::
        D[i,j] := min(D[i,j], D[i,k] + D[k,j]) > ||
    k := k + 1 if k < n - 1
end

References

• K. Mani Chandy and Jayadev Misra (1988) Parallel Program Design: A Foundation.