LOGIC DESIGN

In electronics, logic synthesis is a process by which an abstract form of desired circuit behavior, typically at register transfer level (RTL), is turned into a design implementation in terms of logic gates, typically by a computer program called a synthesis tool. Common examples of this process include synthesis of HDLs, including VHDL and Verilog.^[1]Some synthesis tools generate bitstreams for programmable logic devices such as PALs or FPGAs, while others target the creation of ASICs. Logic synthesis is one aspect of electronic design automation.

History of logic synthesis

The roots of logic synthesis can be traced to the treatment of logic by George Boole (1815 to 1864), in what is now termed Boolean algebra. In 1938, Claude Shannon showed that the two-valued Boolean algebra can describe the operation of switching circuits. In the early days, logic design involved manipulating the truth table representations as Karnaugh maps. The Karnaugh map-based minimization of logic is guided by a set of rules on how entries in the maps can be combined. A human designer can typically only work with Karnaugh maps containing up to four to six variables.

The first step toward automation of logic minimization was the introduction of the Quine–McCluskey algorithm that could be implemented on a computer. This exact minimization technique presented the notion of prime implicants and minimum cost covers that would become the cornerstone of two-level minimization. Nowadays, the much more efficient Espresso heuristic logic minimizer has become the standard tool for this operation.^{[needs update]} Another area of early research was in state minimization and encoding of finite state machines (FSMs), a task that was the bane of designers. The applications for logic synthesis lay primarily in digital computer design. Hence, IBM and Bell Labs played a pivotal role in the early automation of logic synthesis. The evolution from discrete logic components to programmable logic arrays (PLAs) hastened the need for efficient two-level minimization, since minimizing terms in a two-level representation reduces the area in a PLA.

However, two-level logic circuits are of limited importance in a very-large-scale integration (VLSI) design; most designs use multiple levels of logic. As a matter of fact, almost any circuit representation in RTL or Behavioural Description is a multi-level representation. An early system that was used to design multilevel circuits was LSS from IBM. It used local transformations to simplify logic. Work on LSS and the Yorktown Silicon Compiler spurred rapid research progress in logic synthesis in the 1980s. Several universities contributed by making their research available to the public, most notably SIS from University of California, Berkeley, RASP from University of California, Los Angeles and BOLD from University of Colorado, Boulder. Within a decade, the technology migrated to commercial logic synthesis products offered by electronic design automation companies.

Logic elements

Logic design is a step in the standard design cycle in which the functional design of an electronic circuit is converted into the representation which captures logic operations, arithmetic operations, control flow, etc. A common output of this step is RTL description. Logic design is commonly followed by the circuit design step. In modern electronic design automation parts of the logical design may be automated using high-level synthesis tools based on the behavioral description of the circuit.^[2]

Various representations of Boolean operations

Logic operations usually consist of boolean AND, OR, XOR and NAND operations, and are the most basic forms of operations in an electronic circuit. Arithmetic operations are usually implemented with the use of logic operators. Circuits such as a binary multiplier or a binary adder are examples of more complex binary operations that can be implemented using basic logic operators.

High-level synthesis or behavioral synthesis

With a goal of increasing designer productivity, research efforts on the synthesis of circuits specified at the behavioral level have led to the emergence of commercial solutions in 2004,^[3] which are used for complex ASIC and FPGA design. These tools automatically synthesize circuits specified using high-level languages, like ANSI C/C++ or SystemC, to a register transfer level (RTL) specification, which can be used as input to a gate-level logic synthesis flow.^[3] Using high-level synthesis, also known as ESL synthesis, the allocation of work to clock cycles and across structural components, such as floating-point ALUs, is done by the compiler using an optimisation procedure, whereas with RTL logic synthesis (even from behavioural Verilog or VHDL, where a thread of execution can make multiple reads and writes to a variable within a clock cycle) those allocation decisions have already been made.

Multi-level logic minimization

Typical practical implementations of a logic function utilize a multi-level network of logic elements. Starting from an RTL description of a design, the synthesis tool constructs a corresponding multilevel Boolean network.

Next, this network is optimized using several technology-independent techniques before technology-dependent optimizations are performed. The typical cost function during technology-independent optimizations is total literal count of the factored representation of the logic function (which correlates quite well with circuit area).

Finally, technology-dependent optimization transforms the technology-independent circuit into a network of gates in a given technology. The simple cost estimates are replaced by more concrete, implementation-driven estimates during and after technology mapping. Mapping is constrained by factors such as the available gates (logic functions) in the technology library, the drive sizes for each gate, and the delay, power, and area characteristics of each gate.

Boolean algebra (structure)

     In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution).

Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle.

HISTORY

      The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English mathematician. He introduced the algebraic system initially in a small pamphlet, The Mathematical Analysis of Logic, published in 1847 in response to an ongoing public controversy between Augustus De Morgan and William Hamilton, and later as a more substantial book, The Laws of Thought, published in 1854. Boole's formulation differs from that described above in some important respects. For example, conjunction and disjunction in Boole were not a dual pair of operations. Boolean algebra emerged in the 1860s, in papers written by William Jevons and Charles Sanders Peirce. The first systematic presentation of Boolean algebra and distributive lattices is owed to the 1890 Vorlesungen of Ernst Schröder. The first extensive treatment of Boolean algebra in English is A. N. Whitehead's 1898 Universal Algebra. Boolean algebra as an axiomatic algebraic structure in the modern axiomatic sense begins with a 1904 paper by Edward V. Huntington. Boolean algebra came of age as serious mathematics with the work of Marshall Stone in the 1930s, and with Garrett Birkhoff's 1940 Lattice Theory. In the 1960s, Paul Cohen, Dana Scott, and others found deep new results in mathematical logic and axiomatic set theory using offshoots of Boolean algebra, namely forcing and Boolean-valued models.

DEFINATION:

                     A Boolean algebra is a six-tuple consisting of a set A, equipped with two binary operations ∧ (called "meet" or "and"), ∨ (called "join" or "or"), a unary operation ¬ (called "complement" or "not") and two elements 0 and 1 (called "bottom" and "top", or "least" and "greatest" element, also denoted by the symbols ⊥ and ⊤, respectively), such that for all elements a, b and c of A, the following axioms hold:^[2]

a ∨ (b ∨ c) = (a ∨ b) ∨ c	a ∧ (b ∧ c) = (a ∧ b) ∧ c	associativity
a ∨ b = b ∨ a	a ∧ b = b ∧ a	commutativity
a ∨ (a ∧ b) = a	a ∧ (a ∨ b) = a	absorption
a ∨ 0 = a	a ∧ 1 = a	identity
a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)	a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)	distributivity
a ∨ ¬a = 1	a ∧ ¬a = 0	complements

Note, however, that the absorption law can be excluded from the set of axioms as it can be derived by the other axioms.

A Boolean algebra with only one element is called a trivial Boolean algebra or a degenerate Boolean algebra. (Some authors require 0 and 1 to be distinct elements in order to exclude this case.)

It follows from the last three pairs of axioms above (identity, distributivity and complements), or from the absorption axiom, that

a = b ∧ a     if and only if     a ∨ b = b.

The relation ≤ defined by a ≤ b if these equivalent conditions hold, is a partial order with least element 0 and greatest element 1. The meet a ∧ b and the join a ∨ b of two elements coincide with their infimum and supremum, respectively, with respect to ≤.

The first four pairs of axioms constitute a definition of a bounded lattice.

It follows from the first five pairs of axioms that any complement is unique.

The set of axioms is self-dual in the sense that if one exchanges ∨ with ∧ and 0 with 1 in an axiom, the result is again an axiom. Therefore, by applying this operation to a Boolean algebra (or Boolean lattice), one obtains another Boolean algebra with the same elements; it is called its dual.

EXAMPLE:

• The simplest non-trivial Boolean algebra, the two-element Boolean algebra, has only two elements, 0 and 1, and is defined by the rules:

∧ 0 1 0 0 0 1 0 1

∨ 0 1 0 0 1 1 1 1

a 0 1 ¬a 1 0

• It has applications in logic, interpreting 0 as false, 1 as true, ∧ as and, ∨ as or, and ¬ as not. Expressions involving variables and the Boolean operations represent statement forms, and two such expressions can be shown to be equal using the above axioms if and only if the corresponding statement forms are logically equivalent.

• The two-element Boolean algebra is also used for circuit design in electrical engineering; here 0 and 1 represent the two different states of one bit in a digital circuit, typically high and low voltage. Circuits are described by expressions containing variables, and two such expressions are equal for all values of the variables if and only if the corresponding circuits have the same input-output behavior. Furthermore, every possible input-output behavior can be modeled by a suitable Boolean expression.

• The two-element Boolean algebra is also important in the general theory of Boolean algebras, because an equation involving several variables is generally true in all Boolean algebras if and only if it is true in the two-element Boolean algebra (which can be checked by a trivial brute force algorithm for small numbers of variables). This can for example be used to show that the following laws (Consensus theorems) are generally valid in all Boolean algebras:
  • (a ∨ b) ∧ (¬a ∨ c) ∧ (b ∨ c) ≡ (a ∨ b) ∧ (¬a ∨ c)
  • (a ∧ b) ∨ (¬a ∧ c) ∨ (b ∧ c) ≡ (a ∧ b) ∨ (¬a ∧ c)

• The power set (set of all subsets) of any given nonempty set S forms a Boolean algebra, an algebra of sets, with the two operations ∨ := ∪ (union) and ∧ := ∩ (intersection). The smallest element 0 is the empty set and the largest element 1 is the set S itself.

• After the two-element Boolean algebra, the simplest Boolean algebra is that defined by the power set of two atoms:

∧ 0 a b 1 0 0 0 0 0 a 0 a 0 a b 0 0 b b 1 0 a b 1

∨ 0 a b 1 0 0 a b 1 a a a 1 1 b b 1 b 1 1 1 1 1 1

x 0 a b 1 ¬x 1 b a 0

• The set of all subsets of S that are either finite or cofinite is a Boolean algebra, an algebra of sets.
• Starting with the propositional calculus with κ sentence symbols, form the Lindenbaum algebra (that is, the set of sentences in the propositional calculus modulo tautology). This construction yields a Boolean algebra. It is in fact the free Boolean algebra on κ generators. A truth assignment in propositional calculus is then a Boolean algebra homomorphism from this algebra to the two-element Boolean algebra.
• Given any linearly ordered set L with a least element, the interval algebra is the smallest algebra of subsets of L containing all of the half-open intervals [a, b) such that a is in L and b is either in L or equal to ∞. Interval algebras are useful in the study of Lindenbaum-Tarski algebras; every countable Boolean algebra is isomorphic to an interval algebra.

Hasse diagram of the Boolean algebra of divisors of 30.

• For any natural number n, the set of all positive divisors of n, defining a≤b if a divides b, forms a distributive lattice. This lattice is a Boolean algebra if and only if n is square-free. The bottom and the top element of this Boolean algebra is the natural number 1 and n, respectively. The complement of a is given by n/a. The meet and the join of a and b is given by the greatest common divisor (gcd) and the least common multiple (lcm) of a and b, respectively. The ring addition a+b is given by lcm(a,b)/gcd(a,b). The picture shows an example for n = 30. As a counter-example, considering the non-square-free n=60, the greatest common divisor of 30 and its complement 2 would be 2, while it should be the bottom element 1.
• Other examples of Boolean algebras arise from topological spaces: if X is a topological space, then the collection of all subsets of X which are both open and closed forms a Boolean algebra with the operations ∨ := ∪ (union) and ∧ := ∩ (intersection).
• If R is an arbitrary ring and we define the set of central idempotents by
  A = { e ∈ R : e² = e, ex = xe, ∀x ∈ R }
  then the set A becomes a Boolean algebra with the operations e ∨ f := e + f - ef and e ∧ f := ef.

Boolean algebra

         In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively. Instead of elementary algebra where the values of the variables are numbers, and the prime operations are addition and multiplication, the main operations of Boolean algebra are the conjunction and denoted as the disjunction or denoted as ∨, and the negation not denoted as. It is thus a formalism for describing logical relations in the same way that ordinary algebra describes numeric relations.

Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic (1847), and set forth more fully in his An Investigation of the Laws of Thought (1854).According to Huntington, the term "Boolean algebra" was first suggested by Sheffer in 1913.

Boolean algebra has been fundamental in the development of digital electronics, and is provided for in all modern programming languages. It is also used in set theory and statistics

HISTORY:

               Boole's algebra predated the modern developments in abstract algebra and mathematical logic; it is however seen as connected to the origins of both fields.^[4] In an abstract setting, Boolean algebra was perfected in the late 19th century by Jevons, Schröder, Huntington, and others until it reached the modern conception of an (abstract) mathematical structure.^[4] For example, the empirical observation that one can manipulate expressions in the algebra of sets by translating them into expressions in Boole's algebra is explained in modern terms by saying that the algebra of sets is a Boolean algebra (note the indefinite article). In fact, M. H. Stone proved in 1936 that every Boolean algebra is isomorphic to a field of sets.

In the 1930s, while studying switching circuits, Claude Shannon observed that one could also apply the rules of Boole's algebra in this setting, and he introduced switching algebra as a way to analyze and design circuits by algebraic means in terms of logic gates. Shannon already had at his disposal the abstract mathematical apparatus, thus he cast his switching algebra as the two-element Boolean algebra. In circuit engineering settings today, there is little need to consider other Boolean algebras, thus "switching algebra" and "Boolean algebra" are often used interchangeably.^[5][6][7] Efficient implementation of Boolean functions is a fundamental problem in the design of combinational logic circuits. Modern electronic design automation tools for VLSI circuits often rely on an efficient representation of Boolean functions known as (reduced ordered) binary decision diagrams (BDD) for logic synthesis and formal verification.

VALUES:

          Whereas in elementary algebra expressions denote mainly numbers, in Boolean algebra they denote the truth values false and true. These values are represented with the bits (or binary digits), namely 0 and 1. They do not behave like the integers 0 and 1, for which 1 + 1 = 2, but may be identified with the elements of the two-element field GF(2), that is, integer arithmetic modulo 2, for which 1 + 1 = 0. Addition and multiplication then play the Boolean roles of XOR (exclusive-or) and AND (conjunction) respectively, with disjunction x∨y (inclusive-or) definable as x + y + xy.

Boolean algebra also deals with functions which have their values in the set {0, 1}. A sequence of bits is a commonly used such function. Another common example is the subsets of a set E: to a subset F of E is associated the indicator function that takes the value 1 on F and 0 outside F. The most general example is the elements of a Boolean algebra, with all of the foregoing being instances thereof.

As with elementary algebra, the purely equational part of the theory may be developed without considering explicit values for the variables.

Basic operations

The basic operations of Boolean calculus are as follows.

• AND (conjunction), denoted x∧y (sometimes x AND y or Kxy), satisfies x∧y = 1 if x = y = 1 and x∧y = 0 otherwise.
• OR (disjunction), denoted x∨y (sometimes x OR y or Axy), satisfies x∨y = 0 if x = y = 0 and x∨y = 1 otherwise.
• NOT (negation), denoted ¬x (sometimes NOT x, Nx or !x), satisfies ¬x = 0 if x = 1 and ¬x = 1 if x = 0.

Alternatively the values of x∧y, x∨y, and ¬x can be expressed by tabulating their values with truth tables as follows.

${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x\wedge y}$ ${\displaystyle x\vee y}$ 0 0 0 0 1 0 0 1 0 1 0 1 1 1 1 1

${\displaystyle x}$ ${\displaystyle \neg x}$ 0 1 1 0

If the truth values 0 and 1 are interpreted as integers, these operations may be expressed with the ordinary operations of arithmetic, or by the minimum/maximum functions:

${\displaystyle {\begin{aligned}x\wedge y&=x\times y=\min(x,y)\\x\vee y&=x+y-(x\times y)=\max(x,y)\\\neg x&=1-x\end{aligned}}}$

One may consider that only the negation and one of the two other operations are basic, because of the following identities that allow to define the conjunction in terms of the negation and the disjunction, and vice versa:

${\displaystyle {\begin{aligned}x\wedge y&=\neg (\neg x\vee \neg y)\\x\vee y&=\neg (\neg x\wedge \neg y)\end{aligned}}}$

Secondary operations[edit]

The three Boolean operations described above are referred to as basic, meaning that they can be taken as a basis for other Boolean operations that can be built up from them by composition, the manner in which operations are combined or compounded. Operations composed from the basic operations include the following examples:

${\displaystyle x\rightarrow y=\neg {x}\vee y}$

${\displaystyle x\oplus y=(x\vee y)\wedge \neg {(x\wedge y)}}$

${\displaystyle x\equiv y=\neg {(x\oplus y)}}$

These definitions give rise to the following truth tables giving the values of these operations for all four possible inputs.

${\displaystyle x}$	${\displaystyle y}$	${\displaystyle x\rightarrow y}$	${\displaystyle x\oplus y}$	${\displaystyle x\equiv y}$
0	0	1	0	1
1	0	0	1	0
0	1	1	1	0
1	1	1	0	1

The first operation, x → y, or Cxy, is called material implication. If x is true then the value of x → y is taken to be that of y. But if x is false then the value of y can be ignored; however the operation must return some truth value and there are only two choices, so the return value is the one that entails less, namely true. (Relevance logic addresses this by viewing an implication with a false premise as something other than either true or false.)

The second operation, x ⊕ y, or Jxy, is called exclusive or (often abbreviated as XOR) to distinguish it from disjunction as the inclusive kind. It excludes the possibility of both x and y. Defined in terms of arithmetic it is addition mod 2 where 1 + 1 = 0.

The third operation, the complement of exclusive or, is equivalence or Boolean equality: x ≡ y, or Exy, is true just when x and y have the same value. Hence x ⊕ y as its complement can be understood as x ≠ y, being true just when x and y are different. Equivalence counterpart in arithmetic mod 2 is x + y + 1.

Given two operands, each with two possible values, there are 2² = 4 possible combinations of inputs. Because each output can have two possible values, there are a total of 2⁴ = 16 possible binary Boolean operations.

Laws

A law of Boolean algebra is an identity such as x∨(y∨z) = (x∨y)∨z between two Boolean terms, where a Boolean term is defined as an expression built up from variables and the constants 0 and 1 using the operations ∧, ∨, and ¬. The concept can be extended to terms involving other Boolean operations such as ⊕, →, and ≡, but such extensions are unnecessary for the purposes to which the laws are put. Such purposes include the definition of a Boolean algebra as any model of the Boolean laws, and as a means for deriving new laws from old as in the derivation of x∨(y∧z) = x∨(z∧y) from y∧z = z∧y as treated in the section on axiomatization.

Monotone laws

Boolean algebra satisfies many of the same laws as ordinary algebra when one matches up ∨ with addition and ∧ with multiplication. In particular the following laws are common to both kinds of algebra:^[13]

Associativity of ${\displaystyle \vee }$:		${\displaystyle x\vee (y\vee z)}$	${\displaystyle =(x\vee y)\vee z}$
Associativity of ${\displaystyle \wedge }$:		${\displaystyle x\wedge (y\wedge z)}$	${\displaystyle =(x\wedge y)\wedge z}$
Commutativity of ${\displaystyle \vee }$:		${\displaystyle x\vee y}$	${\displaystyle =y\vee x}$
Commutativity of ${\displaystyle \wedge }$:		${\displaystyle x\wedge y}$	${\displaystyle =y\wedge x}$
Distributivity of ${\displaystyle \wedge }$ over ${\displaystyle \vee }$:		${\displaystyle x\wedge (y\vee z)}$	${\displaystyle =(x\wedge y)\vee (x\wedge z)}$
Identity for ${\displaystyle \vee }$:		${\displaystyle x\vee 0}$	${\displaystyle =x}$
Identity for ${\displaystyle \wedge }$:		${\displaystyle x\wedge 1}$	${\displaystyle =x}$
Annihilator for ${\displaystyle \wedge }$:		${\displaystyle x\wedge 0}$	${\displaystyle =0}$

The following laws hold in Boolean Algebra, but not in ordinary algebra:

Annihilator for ${\displaystyle \vee }$:		${\displaystyle x\vee 1}$	${\displaystyle =1}$
Idempotence of ${\displaystyle \vee }$:		${\displaystyle x\vee x}$	${\displaystyle =x}$
Idempotence of ${\displaystyle \wedge }$:		${\displaystyle x\wedge x}$	${\displaystyle =x}$
Absorption 1:		${\displaystyle x\wedge (x\vee y)}$	${\displaystyle =x}$
Absorption 2:		${\displaystyle x\vee (x\wedge y)}$	${\displaystyle =x}$
Distributivity of ${\displaystyle \vee }$ over ${\displaystyle \wedge }$:		${\displaystyle x\vee (y\wedge z)}$	${\displaystyle =(x\vee y)\wedge (x\vee z)}$

Taking x = 2 in the third law above shows that it is not an ordinary algebra law, since 2×2 = 4. The remaining five laws can be falsified in ordinary algebra by taking all variables to be 1, for example in Absorption Law 1 the left hand side would be 1(1+1) = 2 while the right hand side would be 1, and so on.

All of the laws treated so far have been for conjunction and disjunction. These operations have the property that changing either argument either leaves the output unchanged or the output changes in the same way as the input. Equivalently, changing any variable from 0 to 1 never results in the output changing from 1 to 0. Operations with this property are said to be monotone. Thus the axioms so far have all been for monotonic Boolean logic. Nonmonotonicity enters via complement as follows.

Applications[edit]

Boolean algebra as the calculus of two values is fundamental to computer circuits, computer programming, and mathematical logic, and is also used in other areas of mathematics such as set theory and statistics.^[3]

Computers[edit]

In the early 20th century, several electrical engineers intuitively recognized that Boolean algebra was analogous to the behavior of certain types of electrical circuits. Claude Shannon formally proved such behavior was logically equivalent to Boolean algebra in his 1937 master's thesis, A Symbolic Analysis of Relay and Switching Circuits.

Today, all modern general purpose computers perform their functions using two-value Boolean logic; that is, their electrical circuits are a physical manifestation of two-value Boolean logic. They achieve this in various ways: as voltages on wires in high-speed circuits and capacitive storage devices, as orientations of a magnetic domain in ferromagnetic storage devices, as holes in punched cards or paper tape, and so on. (Some early computers used decimal circuits or mechanisms instead of two-valued logic circuits.)

Of course, it is possible to code more than two symbols in any given medium. For example, one might use respectively 0, 1, 2, and 3 volts to code a four-symbol alphabet on a wire, or holes of different sizes in a punched card. In practice, the tight constraints of high speed, small size, and low power combine to make noise a major factor. This makes it hard to distinguish between symbols when there are several possible symbols that could occur at a single site. Rather than attempting to distinguish between four voltages on one wire, digital designers have settled on two voltages per wire, high and low.

Computers use two-value Boolean circuits for the above reasons. The most common computer architectures use ordered sequences of Boolean values, called bits, of 32 or 64 values, e.g. 01101000110101100101010101001011. When programming in machine code, assembly language, and certain other programming languages, programmers work with the low-level digital structure of the data registers. These registers operate on voltages, where zero volts represents Boolean 0, and a reference voltage (often +5V, +3.3V, +1.8V) represents Boolean 1. Such languages support both numeric operations and logical operations. In this context, "numeric" means that the computer treats sequences of bits as binary numbers (base two numbers) and executes arithmetic operations like add, subtract, multiply, or divide. "Logical" refers to the Boolean logical operations of disjunction, conjunction, and negation between two sequences of bits, in which each bit in one sequence is simply compared to its counterpart in the other sequence. Programmers therefore have the option of working in and applying the rules of either numeric algebra or Boolean algebra as needed. A core differentiating feature between these families of operations is the existence of the carry operation in the first but not the second.

Two-valued logic

Other areas where two values is a good choice are the law and mathematics. In everyday relaxed conversation, nuanced or complex answers such as "maybe" or "only on the weekend" are acceptable. In more focused situations such as a court of law or theorem-based mathematics however it is deemed advantageous to frame questions so as to admit a simple yes-or-no answer—is the defendant guilty or not guilty, is the proposition true or false—and to disallow any other answer. However much of a straitjacket this might prove in practice for the respondent, the principle of the simple yes-no question has become a central feature of both judicial and mathematical logic, making two-valued logic deserving of organization and study in its own right.

A central concept of set theory is membership. Now an organization may permit multiple degrees of membership, such as novice, associate, and full. With sets however an element is either in or out. The candidates for membership in a set work just like the wires in a digital computer: each candidate is either a member or a nonmember, just as each wire is either high or low.

Algebra being a fundamental tool in any area amenable to mathematical treatment, these considerations combine to make the algebra of two values of fundamental importance to computer hardware, mathematical logic, and set theory.

Two-valued logic can be extended to multi-valued logic, notably by replacing the Boolean domain {0, 1} with the unit interval [0,1], in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically, negation (NOT) is replaced with 1 − x, conjunction (AND) is replaced with multiplication (${\displaystyle xy}$), and disjunction (OR) is defined via De Morgan's law. Interpreting these values as logical truth values yields a multi-valued logic, which forms the basis for fuzzy logic and probabilistic logic. In these interpretations, a value is interpreted as the "degree" of truth – to what extent a proposition is true, or the probability that the proposition is true.

Boolean operations

The original application for Boolean operations was mathematical logic, where it combines the truth values, true or false, of individual formulas.

Natural languages such as English have words for several Boolean operations, in particular conjunction (and), disjunction (or), negation (not), and implication (implies). But not is synonymous with and not. When used to combine situational assertions such as "the block is on the table" and "cats drink milk," which naively are either true or false, the meanings of these logical connectives often have the meaning of their logical counterparts. However, with descriptions of behavior such as "Jim walked through the door", one starts to notice differences such as failure of commutativity, for example the conjunction of "Jim opened the door" with "Jim walked through the door" in that order is not equivalent to their conjunction in the other order, since and usually means and then in such cases. Questions can be similar: the order "Is the sky blue, and why is the sky blue?" makes more sense than the reverse order. Conjunctive commands about behavior are like behavioral assertions, as in get dressed and go to school. Disjunctive commands such love me or leave me or fish or cut bait tend to be asymmetric via the implication that one alternative is less preferable. Conjoined nouns such as tea and milk generally describe aggregation as with set union while tea or milk is a choice. However context can reverse these senses, as in your choices are coffee and tea which usually means the same as your choices are coffee or tea (alternatives). Double negation as in "I don't not like milk" rarely means literally "I do like milk" but rather conveys some sort of hedging, as though to imply that there is a third possibility. "Not not P" can be loosely interpreted as "surely P", and although P necessarily implies "not not P" the converse is suspect in English, much as with intuitionistic logic. In view of the highly idiosyncratic usage of conjunctions in natural languages, Boolean algebra cannot be considered a reliable framework for interpreting them.

Boolean operations are used in digital logic to combine the bits carried on individual wires, thereby interpreting them over {0,1}. When a vector of n identical binary gates are used to combine two bit vectors each of n bits, the individual bit operations can be understood collectively as a single operation on values from a Boolean algebra with 2ⁿ elements.

Naive set theory interprets Boolean operations as acting on subsets of a given set X. As we saw earlier this behavior exactly parallels the coordinate-wise combinations of bit vectors, with the union of two sets corresponding to the disjunction of two bit vectors and so on.

The 256-element free Boolean algebra on three generators is deployed in computer displays based on raster graphics, which use bit blit to manipulate whole regions consisting of pixels, relying on Boolean operations to specify how the source region should be combined with the destination, typically with the help of a third region called the mask. Modern video cards offer all 2²³ = 256 ternary operations for this purpose, with the choice of operation being a one-byte (8-bit) parameter. The constants SRC = 0xaa or 10101010, DST = 0xcc or 11001100, and MSK = 0xf0 or 11110000 allow Boolean operations such as (SRC^DST)&MSK (meaning XOR the source and destination and then AND the result with the mask) to be written directly as a constant denoting a byte calculated at compile time, 0x60 in the (SRC^DST)&MSK example, 0x66 if just SRC^DST, etc. At run time the video card interprets the byte as the raster operation indicated by the original expression in a uniform way that requires remarkably little hardware and which takes time completely independent of the complexity of the expression.

Solid modeling systems for computer aided design offer a variety of methods for building objects from other objects, combination by Boolean operations being one of them. In this method the space in which objects exist is understood as a set S of voxels (the three-dimensional analogue of pixels in two-dimensional graphics) and shapes are defined as subsets of S, allowing objects to be combined as sets via union, intersection, etc. One obvious use is in building a complex shape from simple shapes simply as the union of the latter. Another use is in sculpting understood as removal of material: any grinding, milling, routing, or drilling operation that can be performed with physical machinery on physical materials can be simulated on the computer with the Boolean operation x ∧ ¬y or x − y, which in set theory is set difference, remove the elements of y from those of x. Thus given two shapes one to be machined and the other the material to be removed, the result of machining the former to remove the latter is described simply as their set difference.