# UNITY (programming language)

UNITY is a programming language constructed by K. Mani Chandy and Jayadev Misra for their book Parallel Program Design: A Foundation. It is a theoretical language which focuses on what, instead of where, when or how. The language contains no method of flow control, and program statements run in a nondeterministic way until statements cease to cause changes during execution. This allows for programs to run indefinitely, such as auto-pilot or power plant safety systems, as well as programs that would normally terminate (which here converge to a fixed point).

## Description

All statements are assignments, and are separated by #. A statement can consist of multiple assignments, of 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 ${\displaystyle \Theta (n)}$ expected time, ${\displaystyle \Theta (n)}$ processors and ${\displaystyle \Theta (n^{2})}$ expected work. The reason you only have ${\displaystyle \Theta (n)}$ expected time, is that k is always chosen randomly from ${\displaystyle \{0,1\}}$. 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 ${\displaystyle \Theta (\log n)}$ time with rank-sort. You need ${\displaystyle \Theta (n^{2})}$ processors, and do ${\displaystyle \Theta (n^{2})}$ 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 algorithm

Using the Floyd–Warshall algorithm all pairs shortest path algorithm, we include intermediate nodes iteratively, and get ${\displaystyle \Theta (n)}$ time, using ${\displaystyle \Theta (n^{2})}$ processors and ${\displaystyle \Theta (n^{3})}$ 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 ${\displaystyle \Theta (\log ^{2}n)}$ time, using ${\displaystyle \Theta (n^{3})}$ processors and ${\displaystyle \Theta (n^{3}\log n)}$ 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], <min k : 0 <= k < n :: D[i,k] + D[k,j] >) >
end

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

## References

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