Functional completeness


In logic, a functionally complete set of logical connectives or Boolean operators is one which can be used to express all possible truth tables by combining members of the set into a Boolean expression. A well-known complete set of connectives is, consisting of binary conjunction and negation. Each of the singleton sets and is functionally complete.
A gate or set of gates which is functionally complete can also be called a universal gate / gates.
A functionally complete set of gates may utilise or generate 'garbage bits' as part of its computation which are either not part of the input or not part of the output to the system.
In a context of propositional logic, functionally complete sets of connectives are also called adequate.
From the point of view of digital electronics, functional completeness means that every possible logic gate can be realized as a network of gates of the types prescribed by the set. In particular, all logic gates can be assembled from either only binary NAND gates, or only binary NOR gates.

Introduction

Modern texts on logic typically take as primitive some subset of the connectives: conjunction ; disjunction ; negation ; material conditional ; and possibly the biconditional. Further connectives can be defined, if so desired, by defining them in terms of these primitives. For example, NOR can be expressed as conjunction of two negations:
Similarly, the negation of the conjunction, NAND, can be defined in terms of disjunction and negation. It turns out that every binary connective can be defined in terms of, so this set is functionally complete.
However, it still contains some redundancy: this set is not a minimal functionally complete set, because the conditional and biconditional can be defined in terms of the other connectives as
It follows that the smaller set is also functionally complete. But this is still not minimal, as can be defined as
Alternatively, may be defined in terms of in a similar manner, or may be defined in terms of :
No further simplifications are possible. Hence, every two-element set of connectives containing and one of is a minimal functionally complete subset of.

Formal definition

Given the Boolean domain B = , a set F of Boolean functions ƒi: Bni → B is functionally complete if the clone on B generated by the basic functions ƒi contains all functions ƒ: Bn → B, for all strictly positive integers. In other words, the set is functionally complete if every Boolean function that takes at least one variable can be expressed in terms of the functions ƒi. Since every Boolean function of at least one variable can be expressed in terms of binary Boolean functions, F is functionally complete if and only if every binary Boolean function can be expressed in terms of the functions in F.
A more natural condition would be that the clone generated by F consist of all functions ƒ: Bn → B, for all integers. However, the examples given above are not functionally complete in this stronger sense because it is not possible to write a nullary function, i.e. a constant expression, in terms of F if F itself does not contain at least one nullary function. With this stronger definition, the smallest functionally complete sets would have 2 elements.
Another natural condition would be that the clone generated by F together with the two nullary constant functions be functionally complete or, equivalently, functionally complete in the strong sense of the previous paragraph. The example of the Boolean function given by S = z if x = y and S = x otherwise shows that this condition is strictly weaker than functional completeness.

Characterization of functional completeness

proved that a set of logical connectives is functionally complete if and only if it is not a subset of any of the following sets of connectives:
In fact, Post gave a complete description of the lattice of all clones on the two-element set, nowadays called Post's lattice, which implies the above result as a simple corollary: the five mentioned sets of connectives are exactly the maximal clones.

Minimal functionally complete operator sets

When a single logical connective or Boolean operator is functionally complete by itself, it is called a Sheffer function or sometimes a sole sufficient operator. There are no unary operators with this property. NAND and NOR, which are dual to each other, are the only two binary Sheffer functions. These were discovered, but not published, by Charles Sanders Peirce around 1880, and rediscovered independently and published by Henry M. Sheffer in 1913.
In digital electronics terminology, the binary NAND gate and the binary NOR gate are the only binary universal logic gates.
The following are the minimal functionally complete sets of logical connectives with arity ≤ 2:
;One element:,.
;Two elements:,,,,,,,,,,,,,,,,,
;Three elements:,,,,,
There are no minimal functionally complete sets of more than three at most binary logical connectives. In order to keep the lists above readable, operators that ignore one or more inputs have been omitted. For example, an operator that ignores the first input and outputs the negation of the second could be substituted for a unary negation.

Examples

Note that an electronic circuit or a software function can be optimized by reuse, to reduce the number of gates. For instance, the "A ∧ B" operation, when expressed by ↑ gates, is implemented with the reuse of "A ↑ B",

In other domains

Apart from logical connectives, functional completeness can be introduced in other domains. For example, a set of reversible gates is called functionally complete, if it can express every reversible operator.
The 3-input Fredkin gate is functionally complete reversible gate by itself – a sole sufficient operator. There are many other three-input universal logic gates, such as the Toffoli gate.
In quantum computing, the Hadamard gate and the T gate are universal, albeit with a slightly more restrictive definition than that of functional completeness.

Set theory

There is an isomorphism between the algebra of sets and the Boolean algebra, that is, they have the same structure. Then, if we map boolean operators into set operators, the "translated" above text are valid also for sets: there are many "minimal complete set of set-theory operators" that can generate any other set relations. The more popular "Minimal complete operator sets" are and. If the universal set is forbidden, set operators are restricted to being falsity- preserving, and cannot be equivalent to functionally complete Boolean algebra.