LAMBDABEAM: neural program search with higher-order functions and lambdas

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 ...