Chi: a scalable and programmable control plane for distributed stream processing systems

All the results given in the paper hold true. In the proof of Theorem 1, change steps IV, V, and VI to

IV. for every $a,b,c \in T$ , add $(\langle a\rangle,\langle b \rangle,\langle c \rangle,\$) \rightarrow (\langle 0a \rangle,\langle 0b \rangle,\langle 0c \rangle,\S)$ to P ;

V. for every $a,b,c,d \in T$ , add $(Y, \langle 0a \rangle, Y, \langle 0b \rangle, Y, \langle 0c \rangle, \S) \rightarrow \# , \langle 0a \rangle, X, \langle 0b \rangle, Y, \langle 0c \rangle, \S)$ , $(\langle 0a\rangle, \langle 0b \rangle, \langle 0c \rangle, \S) \rightarrow (\langle 4a \rangle, \langle 1b \rangle, \langle 2c \rangle, \S)$ , $(\langle 4a \rangle, X, \langle 1b \rangle, Y, \langle 2c \rangle, \S) \rightarrow (\langle 4a \rangle, \# , \langle 1b \rangle, X, \langle 2c \rangle, \S)$ , $(\langle 4a \rangle, \langle 1b \rangle, \langle 2c \rangle, \langle d \rangle, \S \rightarrow (a, \langle 4b \rangle, \langle 1c \rangle, \langle 2d \rangle, \S)$ , $(\langle 4a \rangle, \langle 1b \rangle, \langle 2c \rangle, \S) \rightarrow (a, \langle 1b \rangle, \langle 3c \rangle, \S)$ , $(\langle 1a \rangle, X, \langle 3b \rangle, Y, \S) \rightarrow (\langle 1a \rangle, \# , \langle 3b \rangle, \# , \S)$ to P ;

VI. for every $a, b \in T$ , add $(\langle 1a \rangle, X, \langle 3b \rangle, \S) \rightarrow (a, \# , b, \# )$ to P .

The paper is concerned with sums of the type \[S_{n,j} = \sum {x_1^{a_1 } x_2^{a_2 } \cdots x_n^{a_n } } \quad (n > 1),\] where the summation is over either \[( * )\qquad ja_i \leqq a_1 + a_2 + \cdots + a_n \quad (1 \leqq j \leqq n;1 \leqq i \leqq ...