UNITY (programming language)

The UNITY programming language 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 causes change if run. A correct program converges into a "fix-point".

Description

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" &gt;, "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 $\{0,1\}$. This can be fixed 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

Rank-sort

You can sort in $Theta(log\; n)$ time with rank-sort. You need $Theta(n^2)$ processors, and do $Theta(n^2)$ work.

Program ranksort declare n: integer, A,R: array [0..n-1] of integer initially n = 15 # <|| i : 0 <= i < n :: A [i] , R [i] = rand() % 100, i > assign <|| i : 0 <= i < n :: R [i] := <+ j : 0 <= j < n and (A [j] < A [i] or (A [j] = A [i] and j < i)) :: 1 > > # <|| i : 0 <= i < n :: A [R [i] := A [i] > 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

We can do this even faster. The following programs computes all pairs shortest path in $Theta(log^2\; n)$ time, using $Theta(n^3)$ processors and $Theta(n^3\; log\; n)$ work.

Program shortestpath2 declare n: integer, D: array [0..n-1, 0..n-1] of integer initially n = 10 # <|| i,j : 0 <= i < n and 0 <= j < n :: D [i,j] = rand() % 10 > assign <|| i,j : 0 <= i < n and 0 <= j < n :: D [i,j] := min(D [i,j] , ) > end

After round $r$, D [i,j] contains the length of the shortest path from $i$ to $j$ of length $0\; dots\; r$. In the next round, of length $0\; dots\; 2r$, and so on.

References

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