∵ Bitwise r←[x](F∵)y

F is a function that either accepts boolean vectors and returns a scalar, or both accepts boolean vectors and returns a number vector and accepts boolean scalars and returns a boolean scalar; applies to simple scalars of x and y which are integers.

If x is not provided, F is called on the bits of y, except that if y is negative the ¯1 "bit" returned by ⊤ Encode is dropped. If the result is a scalar, it is returned. If the result is a vector, F is called again with y<0. The result is then the base 2 decode of the first result of y, catenating ¯1 if the second result was 1.

If x is provided, F is called on the bits of x and y, without ¯1s, padded to be of the same length, padding with 1s if the number is negative and 0s otherwise. If the result is a scalar, it is returned. If the result is a vector, F is called again with x<0 and y<0. The result is then the base 2 decode of the first result of y, catenating ¯1 if the second result was 1.

Common usages

• ~∵ is bitwise NOT
• ∧∵ is bitwise AND
• ∨∵ is bitwise OR
• ≠∵ is bitwise XOR
• ⍲∵ is bitwise NAND
• ⍱∵ is bitwise NOR
• =∵ is bitwise XNOR
• ≤∵ is bitwise implication
• ≥∵ is bitwise converse implication
• >∵ is bitwise non-implication
• <∵ is bitwise converse non-implication
• +/∵ is popcnt (amount of bits set)
• ≢∵ is bit length