* Abdel Alim Kamal and Amr M. Youssef, Fault analysis of the NTRUSign digital signature scheme, Journal of Cryptography and Communications, Discrete Structures, Boolean Functions and Sequences, 4 ( 2 ): 131 – 144, 2012.

Marshall Harvey Stone ( April 8, 1903, New York City – January 9, 1989, Madras, India ) was an American mathematician who contributed to real analysis, functional analysis, and the study of Boolean algebras.

Solving the set of constraints can be done by Boolean solvers ( e. g. SAT-solvers based on the Boolean satisfiability problem ) or by numerical analysis, like the Gaussian elimination.

Interests: Boolean algebras, mathematical logic, functional analysis, theories of distribution, measure theory, general topology ( descriptive set theory ).

The Cook – Levin theorem states that the Boolean satisfiability problem is NP-complete, and in fact, this was the first decision problem proved to be NP-complete.

Victor Shestakov at Moscow State University ( 1907 – 1987 ) proposed a theory of electric switches based on Boolean logic even earlier than Claude Shannon in 1935 on the testimony of Soviet logicians and mathematicians Yanovskaya, Gaaze-Rapoport, Dobrushin, Lupanov, Medvedev and Uspensky, though they presented their academic theses in the same year, 1938.

* Mathematical logic – Boolean logic and other ways of modeling logical queries ; the uses and limitations of formal proof methods

The mathematical structure of modal logic, namely Boolean algebras augmented with unary operations ( often called modal algebras ), began to emerge with J. C. C. McKinsey's 1941 proof that S2 and S4 are decidable, and reached full flower in the work of Alfred Tarski and his student Bjarni Jonsson ( Jonsson and Tarski 1951 – 52 ).

* Jónsson, B. and Tarski, A., 1951 – 52, " Boolean Algebra with Operators I and II ", American Journal of Mathematics 73: 891-939 and 74: 129 – 62.

If H < sub > 1 </ sub > is a separable space ( in particular, if it is a Euclidean space ) the result is true in Zermelo – Fraenkel set theory ; for the fully general case, it appears to need some form of the axiom of choice ; the Boolean prime ideal theorem is known to be sufficient.

In computational complexity theory, the Cook – Levin theorem, also known as Cook's theorem, states that the Boolean satisfiability problem is NP-complete.

If the theory is propositional and its set of logical connectives is functionally complete, the Lindenbaum – Tarski algebra is the free Boolean algebra generated by the set of propositional variables.

Bryant, Graph-based algorithms for Boolean function manipulation, IEEE Transactions on Computers., C-35, pp. 677 – 691, 1986.

Bitmap indexes have traditionally been considered to work well for data such as Boolean values, which have a modest number of distinct values – in this case, boolean True and False-but many occurrences of those values.

This is in sharp contrast to Boolean algebras, whose only SI is the two-element one, which on its own therefore suffices for all counterexamples to non-laws of Boolean algebra, the basis for the simple truth table decision method.

In logic, the term decidable refers to the decision problem, the question of the existence of an effective method for determining membership in a set of formulas, or, more precisely, an algorithm that can and will return a Boolean true or false value ( instead of looping indefinitely ).

In Boolean logic, the algebraic normal form ( ANF ) is a method of standardizing and normalizing logical formulas.

In mathematics, Shannon's expansion or the Shannon decomposition is a method by which a Boolean function can be represented by the sum of two sub-functions of the original.

Since the ATPG problem is NP-complete ( by reduction from the Boolean satisfiability problem ) there will be cases where patterns exist, but ATPG gives up since it will take an incredibly long time to find them ( assuming P ≠ NP, of course ).

For example, P < sup > SAT </ sup > is the class of problems solvable in polynomial time by a deterministic Turing machine with an oracle for the Boolean satisfiability problem.

It is functionally identical to Karnaugh mapping, but the tabular form makes it more efficient for use in computer algorithms, and it also gives a deterministic way to check that the minimal form of a Boolean function has been reached.

That is, any problem in NP can be reduced in polynomial time by a deterministic Turing machine to the problem of determining whether a Boolean formula is satisfiable.

An important consequence of the theorem is that if there exists a deterministic polynomial time algorithm for solving Boolean satisfiability, then there exists a deterministic polynomial time algorithm for solving all problems in NP.

SAT is in NP because any assignment of Boolean values to Boolean variables that is claimed to satisfy the given expression can be verified in polynomial time by a deterministic Turing machine.

A homomorphism between two Boolean algebras A and B is a function f: A → B such that for all a, b in A:

In 1937, Claude Shannon showed there is a one-to-one correspondence between the concepts of Boolean logic and certain electrical circuits, now called logic gates, which are now ubiquitous in digital computers.

A map between two Boolean rings is a ring homomorphism if and only if it is a homomorphism of the corresponding Boolean algebras.

Indeed, it is not hard to see that it is equivalent to the Boolean prime ideal theorem ( BPIT ), a well-known intermediate point between the axioms of Zermelo-Fraenkel set theory ( ZF ) and the ZF theory augmented by the axiom of choice ( ZFC ).

Firebird index buckets aren ’ t subject to two-phase locking, and Boolean “ and ” and “ or ” operations can be performed on intermediate bitmaps at a negligible cost, eliminating the need for the optimizer to choose between alternative indexes.

It is due to this intermediate status between ZF and ZF + AC ( ZFC ) that the Boolean prime ideal theorem is often taken as an axiom of set theory.

One obtains a duality between the category of Boolean algebras ( with their homomorphisms ) and Stone spaces ( with continuous mappings ).

Restating the theorem using the language of category theory ; the theorem states that there is a duality between the category of Boolean algebras and the category of Stone spaces.

This duality means that in addition to the isomorphisms between Boolean algebras and their Stone spaces, each homomorphism from a Boolean algebra A to a Boolean algebra B corresponds in a natural way to a continuous function from S ( B ) to S ( A ).

It was first introduced by Tarski in 1935 as a device to establish correspondence between classical propositional calculus and Boolean algebras.

Given two interior algebras A and B, a map f: A → B is an interior algebra homomorphism if and only if f is a homomorphism between the underlying Boolean algebras of A and B, that also preserves interiors and closures.

A map f: A → B is a topomorphism if and only if f is a homomorphism between the Boolean algebras underlying A and B, that also preserves the open and closed elements of A.

The area of mathematics known as Stone duality is founded on the fact that the Stone representation of a Boolean algebra can be recovered purely from the corresponding Stone space whence a duality exists between Boolean algebras and Boolean spaces.

Viewing a variety V and its homomorphisms as a category, a subclass U of V that is itself a variety is a subvariety of V implies that U is a full subcategory of V, meaning that for any objects a, b in U, the homomorphisms from a to b in U are exactly those from a to b in V. On the other hand there is a sense in which Boolean algebras and Boolean rings can be viewed as subvarieties of each other even though they have different signatures, because of the translation between them allowing every Boolean algebra to be understood as a Boolean ring and conversely ; in this sort of situation the homomorphisms between corresponding structures are the same.

In computer science, satisfiability ( often written in all capitals or abbreviated SAT ) is the problem of determining if the variables of a given Boolean formula can be assigned in such a way as to make the formula evaluate to TRUE.

In complexity theory, the satisfiability problem ( SAT ) is a decision problem, whose instance is a Boolean expression written using only AND, OR, NOT, variables, and parentheses.

The satisfiability problem becomes more difficult ( PSPACE-complete ) if we allow both " for all " and " there exists " quantifiers to bind the Boolean variables.

* Boolean satisfiability problem, determining if the variables of a Boolean formula can be assigned to make the formula evaluate to True

One simple example of a co-NP-complete problem is tautology, the problem of determining whether a given Boolean formula is a tautology ; that is, whether every possible assignment of true / false values to variables yields a true statement.

The satisfiability problem is the problem of whether there are assignments of truth values to variables that make a Boolean expression true.

The quantified Boolean formula problem differs in allowing both universal and existential quantification over the values of the variables:

* Boolean network, a certain network consisting of a set of Boolean variables whose state is determined by other variables in the network

Synonyms for the term dummy variables include design variables, Boolean indicators, proxies, indicator variables, categorical variables and qualitative variables.

Boolean flags are the only kind of variables, making arithmetic quite difficult ; the programmer is forced to be creative with algorithms, often relying on physical movement, possibly in invisible ( but still physically present ) objects, rather than arithmetic calculations.

In computer science, 2-satisfiability ( abbreviated as 2-SAT or just 2SAT ) is the problem of determining whether a collection of two-valued ( Boolean or binary ) variables with constraints on pairs of variables can be assigned values satisfying all the constraints.

It is a special case of the general Boolean satisfiability problem, which can involve constraints on more than two variables, and of constraint satisfaction problems, which can allow more than two choices for the value of each variable.

A 2-SAT problem may be described using a Boolean expression with a special restricted form: a conjunction of disjunctions ( and of ors ), where each disjunction ( or operation ) has two arguments that may either be variables or the negations of variables.

An instance of the Boolean satisfiability problem is a Boolean expression that combines Boolean variables using Boolean operators.