The Gate of Twenty-Six Dials

The north gate of bitwick is held shut by twenty-six brass dials, lettered a through z. Every dial stands at 0 when the gate is locked. A traveller hands the gatekeeper an inscription; the gate opens to the single number the inscription reads out.

readGate(expr) takes the inscription as a string and returns its value as an integer.

Tokens. expr holds only digits, lowercase letters a-z, the characters + - * ^ = ,, round brackets, and spaces. Spaces may sit anywhere between tokens and mean nothing. A run of digits is a non-negative literal. A single letter names a dial and reads its current value.

Forms, listed from loosest-binding to tightest:

• x , y — left-associative. Evaluate x, then y; the value is the value of y.
• d = x — right-associative. Store the value of x in dial d; the value is that stored number.
• x + y and x - y — left-associative sum and difference.
• x * y — left-associative product.
• - x — prefix negation, and it may be stacked.
• x ^ y — right-associative. x raised to the power y.

So prefix negation binds looser than ^, and ^ binds tighter than every other form. Brackets ( ... ) may enclose any inscription, including one that contains , or =, and a bracketed inscription is an operand like any other.

Order of evaluation. Operands are evaluated left to right, and a write to a dial takes effect at once, so any operand evaluated later reads what an earlier one stored.

Guarantees. expr is non-empty and well-formed. The complete left operand of every = is a bare dial letter. Every exponent (the right operand of ^) evaluates to an integer from 0 to 12, and every value that arises stays within ±1000000000.

Return the integer value of the whole inscription.