The rules for determining required parentheses as opposed to parentheses
that can be omitted are more complicated than just showing parentheses
whenever a child expression has lower precedence than its parent. This
necessitated a rewrite.
This new implementation determines whether an expression requires to be
parenthesized with individual rules for each type of expression, which
may or may not depend on the type of the parent expression and the
position of a child within its parent expression. For example,
implication is defined to be right-associative, which means that the
parentheses in the formula
(F -> G) -> H
cannot be ommitted. When determining whether the subformula (F -> G)
needs to be parenthesized, the new algorithm notices that the subformula
is contained as the antecedent of another implication and concludes that
parentheses are required.
Furthermore, this adds extensive unit tests for both formula and term
formatting. The general idea is to test all types of expressions
individually and, in addition to that, all combinations of parent and
child expression types.
Unit testing made it clear that the formatting of empty and 1-ary
conjunctions, disjunctions, and biconditionals needs to be well-defined
even though these types of formulas may be unconventional. The same
applies to existentially and universally quantified formulas where the
list of parameters is empty. Now, empty conjunctions and biconditionals
are rendered like true Booleans, empty disjunctions like false Booleans,
and 1-ary conjunctions, disjunctions, biconditionals, as well as
quantified expressions with empty parameter lists as their singleton
argument.
The latter formulas can be considered neutral intermediates. That is,
they should not affect whether their singleton arguments are
parenthesized or not. To account for that, all unit tests covering
combinations of formulas are tested with any of those five neutral
intermediates additionally.
The type aliases for vectors of formulas and terms were defined as
vectors of boxed formulas and terms, respectively. This is an
unnecessary indirection, so store the formulas and terms directly.
These reference-counted arguments were taken by reference, which made it
necessary to clone them. If a reference-counted object is created for the
sole purpose of being passed to one of these methods, it would be cloned
unnecessarily. This changes the signatures to take these arguments by
value, shifting the responsibility of cloning the reference-counted
objects to the users of these methods.
Existential and universal quantification used redundant data
representations, while they actually share the same structure. This
unifies both into a single QuantifiedFormula type.
For convenience, support biconditionals with more than one argument.
An n-ary “if and only if” statement
F_1 <-> F_2 <-> ... <-> F_n
is to be interpreted as
F_1 <-> F_2 and F2 <-> F3 and ... and F_(n - 1) <-> F_n
This provides an abstract syntax tree for first-order logic with integer
arithmetics. Initially, the following types of formulas are supported:
- Booleans values (true and false)
- predicates
- negated formulas
- comparisons of terms (<, ≤, >, ≥, =, ≠)
- implications and biconditionals
- conjunctions and disjunctions of formulas
- existentially and universally quantified formulas
In addition, these types of terms are provided:
- Boolean values (true and false)
- integers
- strings
- special integers (infimum and supremum)
- symbolic functions
- variables
- binary operations (addition, subtraction, multiplication, division,
modulo, exponentiation)
- unary operations (absolute value, numeric negation)
