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Computational Formula that can be measured in terms of True or False
language consisting of the true quantified Boolean formulas. A (fully) quantified Boolean formula is a formula in quantified propositional logic (also
True quantified Boolean formula
True_quantified_Boolean_formula
Problem of determining if a Boolean formula could be made true
Boolean formula. In other words, it asks whether the formula's variables can be consistently replaced by the values TRUE or FALSE to make the formula
Boolean satisfiability problem
Boolean_satisfiability_problem
Type of propositional logic
impredicative quantification, System F. Parigot (1997) showed how this calculus can be extended to admit classical logic. True quantified Boolean formula Second-order
Second-order propositional logic
Second-order_propositional_logic
Complexity class
NP-complete nor Undecidable. For instance, the language of true quantified Boolean formulas is decidable in polynomial space, but not in non-deterministic
NP-hardness
A formula game is an artificial game represented by a fully quantified Boolean formula such as ∃ x 1 ∀ x 2 ∃ x 3 … ψ {\displaystyle \exists x_{1}\forall
Formula_game
Mathematical use of "for all" and "there exists"
satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable
Quantifier_(logic)
Logical connective AND
And-inverter graph AND gate Bitwise AND Boolean algebra Boolean conjunctive query Boolean domain Boolean function Boolean-valued function Conjunction/disjunction
Logical_conjunction
Boolean satisfiability is NP-complete and therefore that NP-complete problems exist
(the recognition of true quantified Boolean formulas) that is PSPACE-complete. Analogously, dependency quantified boolean formulas encode computation with
Cook–Levin_theorem
Algebraic manipulation of "true" and "false"
Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values true and
Boolean_algebra
In logic, a statement which is always true
logic, where a tautology is defined as a propositional formula that is true under any possible Boolean valuation of its propositional variables. A key property
Tautology_(logic)
Function returning one of only two values
In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually {true, false}, {0,1} or {−1
Boolean_function
Mathematical use of "for all"
universal quantifier into an existential quantifier and negating the quantified formula. That is, ¬ ∀ x P ( x ) is equivalent to ∃ x ¬ P ( x ) {\displaystyle
Universal_quantification
Mathematical use of "there exists"
intersection and union of sets. A quantified propositional function is a statement; thus, like statements, quantified functions can be negated. The ¬
Existential_quantification
Type of logical system
truth value. Quantifiers can be applied to variables in a formula. The variable x in the previous formula can be universally quantified, for instance
First-order_logic
Overview of and topical guide to logic
Logical connective Logical matrix Product term True quantified Boolean formula Truth table Atomic formula Atomic sentence Domain of discourse Empty domain
Outline_of_logic
List of concepts in artificial intelligence
quantified Boolean formulas. A (fully) quantified Boolean formula is a formula in quantified propositional logic where every variable is quantified (or bound)
Glossary of artificial intelligence
Glossary_of_artificial_intelligence
Pair of logical equivalences
In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid
De_Morgan's_laws
Assignment of meaning to the symbols of a formal language
a formula. This is how we define logical connectives in propositional logic: ¬Φ is True iff Φ is False. (Φ ∧ Ψ) is True iff Φ is True and Ψ is True. (Φ
Interpretation_(logic)
Mathematical program specifications
verification. QBFEVAL is a biennial competition of solvers for true quantified Boolean formulas, which have applications to model checking. SV-COMP is an annual
Formal_methods
Set theory concept
c)\|} The completeness of the Boolean algebra is required to define truth values for quantified formulas. If φ(x) is a formula with free variable x (and possibly
Boolean-valued_model
Branch of logic
{L}}} , an interpretation, valuation, Boolean valuation, or case, is an assignment of semantic values to each formula of L {\displaystyle {\mathcal {L}}}
Propositional_logic
Computer program for the Boolean satisfiability problem
computer program which aims to solve the Boolean satisfiability problem (SAT). On input a formula over Boolean variables, such as "(x or y) and (x or not
SAT_solver
Logical connective OR
false. Because this semantics allows a disjunctive formula to be true when both of its disjuncts are true, it is an inclusive interpretation of disjunction
Logical_disjunction
Representation of data types in lambda calculus
a\end{aligned}}} Church Booleans encode the Boolean values true and false. Some programming languages use these as an implementation model for Boolean arithmetic;
Church_encoding
Calculations of the game complexity of Go
Without ko, Go is PSPACE-hard. This is proved by reducing True Quantified Boolean Formula, which is known to be PSPACE-complete, to generalized geography
Go_and_mathematics
Form of logic that allows quantification over predicates
be universally and/or existentially quantified over, to build up formulas. Thus there are many kinds of quantifiers, two for each sort of variables. A
Second-order_logic
SAT solving algorithm
the Boolean satisfiability problem (SAT). Given a Boolean formula, the SAT problem asks for an assignment of variables so that the entire formula evaluates
Conflict-driven clause learning
Conflict-driven_clause_learning
Logic problem, AND of pairwise ORs
may also be used to evaluate fully quantified Boolean formulae in which the formula being quantified is a 2-CNF formula. A number of exact and approximate
2-satisfiability
Logical connective
interpreted as material implication, a formula P → Q {\displaystyle P\to Q} is true unless P {\displaystyle P} is true and Q {\displaystyle Q} is false. Material
Material_conditional
Topics referred to by the same term
from Cornell University Kochi University (disambiguation) True quantified Boolean formula, also known as QSAT Kusat, a village in Oman This disambiguation
CUSAT_(disambiguation)
Properties linking logical conjunction and disjunction
In propositional logic and Boolean algebra, there is a duality between conjunction and disjunction, also called the duality principle. It is the most
Conjunction/disjunction duality
Conjunction/disjunction_duality
Symbol connecting formulas in logic
statements, so one can speak about n-ary logical connectives. The boolean constants True and False can be thought of as nullary operators. Negation is a
Logical_connective
Value indicating the relation of a proposition to truth
languages, any expression can be evaluated in a context that expects a Boolean data type. Typically (though this varies by programming language) expressions
Truth_value
Mathematical theory of data types
include the natural number 0 {\displaystyle 0} , the Boolean value true {\displaystyle {\texttt {true}}} , and functions such as the successor function
Type_theory
Boolean algebra
two-element Boolean algebra is the Boolean algebra whose underlying set (or universe or carrier) B is the Boolean domain. The elements of the Boolean domain
Two-element_Boolean_algebra
Logical operation
to a canonical Boolean, ie. an integer with a value of either 0 or 1 and no other. Although any integer other than 0 is logically true in C and 1 is not
Negation
Logical quantifier
Uniqueness quantification can be expressed in terms of the existential and universal quantifiers of predicate logic, by defining the formula ∃ ! x ( P
Uniqueness_quantification
Type of formal logic
modal logics treat the formula ◻ P → P {\displaystyle \Box P\rightarrow P} as a tautology, representing the principle that only true statements can count
Modal_logic
System including an indeterminate value
indicating true, false, and some third value. This is contrasted with the more commonly known bivalent logics (such as classical sentential or Boolean logic)
Three-valued_logic
Relationship where one statement follows from another
penguin}. Abstract algebraic logic Ampheck Boolean algebra (logic) Boolean domain Boolean function Boolean logic Causality Deductive reasoning Logic gate
Logical_consequence
Method of deriving conclusions
or true statements. It uses metavariables—placeholders that can be replaced by specific terms or formulas to generate an infinite number of true statements
Rule_of_inference
Type of decision problem in computer science
expressions and context-sensitive grammars, determining the truth of quantified Boolean formulas, step-by-step changes between solutions of combinatorial optimization
PSPACE-complete
Standard form of Boolean function
In Boolean algebra, a formula is in conjunctive normal form (CNF) or clausal normal form if it is a conjunction of one or more clauses, where a clause
Conjunctive_normal_form
Theories in mathematical logic
first-order properties of Boolean algebras: Atomic: ∀x x = 0 ∨ ∃y y ≤ x ∧ atom(y) Atomless: ∀x ¬atom(x) The theory of atomless Boolean algebras is ω-categorical
List_of_first-order_theories
Graphical set representation involving overlapping shapes
read as "true", 0 as "false" ~ for NOT and abbreviated to ′ when illustrating the minterms e.g. x′ =defined NOT x, + for Boolean OR (from Boolean algebra:
Euler_diagram
Problem in computational complexity theory
of a given Boolean formula in conjunctive normal form, that can be made true by an assignment of truth values to the variables of the formula. It is a generalization
Maximum satisfiability problem
Maximum_satisfiability_problem
Algebraization of first-order logic with equality
equality. This is comparable to the role Boolean algebras play for propositional logic. Cylindric algebras are Boolean algebras equipped with additional cylindrification
Cylindric_algebra
Impossible task in computing
interest for program verification and circuit verification. Pure Boolean logical formulas are usually decided using SAT-solving techniques based on the DPLL
Entscheidungsproblem
Tool for proving a logical formula
universally quantified, so universal quantifiers over these variables can be added, resulting in a formula with no free variables. A first-order formula ∀ x
Method_of_analytic_tableaux
Area of mathematical logic
= (×,+,−,0,1) has quantifier elimination. This means that in an algebraically closed field, every formula is equivalent to a Boolean combination of equations
Model_theory
In mathematical logic, a well-formed formula with no free variables
mathematical logic, a sentence (or closed formula) of a predicate logic is a Boolean-valued well-formed formula with no free variables. A sentence can be
Sentence_(mathematical_logic)
Syntactically correct logical formula
first-order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any free variables in φ have
Well-formed_formula
Limitative results in mathematical logic
a formula in the language of arithmetic consisting of a number of leading universal quantifiers followed by a quantifier-free body (these formulas are
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Collection of sets in mathematics that can be defined based on a property of its members
classes, so each formula with classes must be reduced syntactically to a formula without classes. For example, one can reduce the formula A = { x ∣ x = x
Class_(set_theory)
Statement that is taken to be true
mathematicians of the 19th century and the developers of systems such as Boolean algebra made elaborate efforts to derive them from traditional arithmetic
Axiom
Form of mathematical proof
n=k\geq 0} , the single case P ( k ) {\displaystyle P(k)} is true. Using the angle addition formula and the triangle inequality, we deduce: | sin ( k + 1
Mathematical_induction
Symbolic description of a mathematical object
{\displaystyle 8x-5\geq 3} is a formula. However, formulas are often considered as expressions that can be evaluated to the Boolean values true or false. To evaluate
Expression_(mathematics)
Branch of mathematical logic
sets of natural numbers can be defined by simple formulas in the language of set theory (which can quantify over arbitrary sets). In the context of second-order
Reverse_mathematics
Various systems of symbolic logic
A corresponding theorem is true for intuitionistic logic, but instead of assigning each formula a value from a Boolean algebra, one uses values from
Intuitionistic_logic
3-volume treatise on mathematics, 1910–1913
English-language nonfiction books of the 20th century. Axiomatic set theory Boolean algebra Information Processing Language – first computational demonstration
Principia_Mathematica
Complexity class used to classify decision problems
The Boolean satisfiability problem (SAT), where we want to know whether or not a certain formula in propositional logic with Boolean variables is true for
NP_(complexity)
Attempt to persuade or to determine the truth of a conclusion
Philosophy portal Abductive reasoning Argument map Bayes' theorem Belief bias Boolean logic Cosmological argument Evidence-based policy Logical reasoning Practical
Argument
Method in mathematical logic
Fraïssé limit of the class of nontrivial finite Boolean algebras is the unique countable atomless Boolean algebra. The class K {\displaystyle \mathbf {K}
Fraïssé_limit
Subfield of automated reasoning and mathematical logic
(2019-01-01). "The SMT Competition 2015–2018". Journal on Satisfiability, Boolean Modeling and Computation. 11 (1): 221–259. doi:10.3233/SAT190123. In recent
Automated_theorem_proving
Set whose elements all belong to another set
included (or contained) in B. A k-subset is a subset with k elements. When quantified, A ⊆ B {\displaystyle A\subseteq B} is represented as ∀ x ( x ∈ A ⇒ x
Subset
Formal system of logic
higher-order logics in the sense that for every formula of a higher-order logic, one can find an equisatisfiable formula for it in second-order logic. The term
Higher-order_logic
Process in digital electronics and integrated circuit design
structures on an integrated circuit. In terms of Boolean algebra, the optimization of a complex Boolean expression is a process of finding a simpler one
Logic_optimization
Mathematical-logic system based on functions
following two definitions (known as Church Booleans) are used for the Boolean values TRUE and FALSE: TRUE := λx.λy.x FALSE := λx.λy.y Then, with these
Lambda_calculus
Study of correct reasoning
logic assigns the formula P ∧ Q {\displaystyle P\land Q} the denotation "true" whenever P {\displaystyle P} and Q {\displaystyle Q} are true. From the semantic
Logic
Subfield of mathematics
such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize
Mathematical_logic
Basic framework of mathematics
algebra, now called Boolean algebra, that allows expressing Aristotle's logic in terms of formulas and algebraic operations. Boolean algebra is the starting
Foundations_of_mathematics
Basic notion of sameness in mathematics
a boolean-valued expression, or relational operator, which returns 1 and 0 for true and false respectively. An identity is an equality that is true for
Equality_(mathematics)
Mathematical table used in logic
mathematical table used in logic—specifically in connection with Boolean algebra, Boolean functions, and propositional calculus—which sets out the functional
Truth_table
Computer science field
for the FSM; instead, they represent the graph implicitly using a formula in quantified propositional logic. The use of binary decision diagrams (BDDs)
Model_checking
Axiomatic system
well-formed formulas, and provided that no variable which is free in one is quantified in the other, then the following are all well-formed formulas <x∧y>,
Typographical_Number_Theory
English mathematician and philosopher (1815–1864)
known as the author of The Laws of Thought (1854), which contains Boolean algebra. Boolean logic, essential to computer programming, is credited with helping
George_Boole
Variable that can either be true or false
(that can either be true or false) of a truth function. Propositional variables are the basic building-blocks of propositional formulas, used in propositional
Propositional_variable
Form of second-order logic
whether a Boolean MSO formula is satisfied by an input finite tree, this problem can be solved in linear time in the tree, by translating the Boolean MSO formula
Monadic_second-order_logic
Mathematics, EMS Press Ratschan, Stefan (2006). "Efficient Solving of Quantified Inequality Constraints over the Real Numbers". ACM Transactions on Computational
Decidability of first-order theories of the real numbers
Decidability_of_first-order_theories_of_the_real_numbers
Mathematical concept
<0} , P ( β ) {\displaystyle P(\beta )} is true. A class function is a rule (specifically, a logical formula) assigning each element in the lefthand class
Transfinite_induction
Branch of computational complexity theory
since a Boolean formula can be efficiently converted to a Boolean circuit. Note that the opposite is not true in general, since the equivalent Boolean formula
Parameterized_complexity
Type of logical formula
atomic formula A is logically implied by D if and only if A is true in M. It follows that a problem P represented by an existentially quantified conjunction
Horn_clause
Concept in mathematical logic
connectives or Boolean operators is one that can be used to express all possible truth tables by combining members of the set into a Boolean expression.
Functional_completeness
formula or a mathematical expression. More formally, a mathematical symbol is any grapheme used in mathematical formulas and expressions. As formulas
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Sequence of words formed by specific rules
formula can be derived from the interpretation of its terms; a model for a formula is an interpretation of terms such that the formula becomes true.
Formal_language
Argument whose conclusion must be true if its premises are
categorized as an invalid argument. A formula of a formal language is a valid formula if and only if it is true under every possible interpretation of
Validity_(logic)
Existence of values making formula true
mathematical logic, a formula is satisfiable if it is true under some assignment of values to its variables. For example, the formula x + 3 = y {\displaystyle
Satisfiability
Reasoning about equations with free variables
tacitly universally quantified over some universe of discourse. There are no existentially quantified variables or open formulas; Terms are built up from
Algebraic_logic
Thesis on the nature of computability
unsolvable: there is no algorithm that can determine whether a well formed formula has a beta normal form. Many years later in a letter to Davis (c. 1965)
Church–Turing_thesis
Mathematical logic concept
logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper propositional structure, that is, a formula that contains
Atomic_formula
In mathematics, a statement that has been proven
or not all of its theorems are also validities. A validity is a formula that is true under any possible interpretation (for example, in classical propositional
Theorem
Diagram that shows all possible logical relations between a collection of sets
to him "till much later", while attempting to adapt Euler diagrams to Boolean logic. In the opening sentence of his 1880 article Venn wrote that Euler
Venn_diagram
Infinite cardinal number
{\displaystyle \omega _{\alpha }} is strictly greater than α. For example, it is true for any successor ordinal: α + 1 ≤ ω α < ω α + 1 {\displaystyle \alpha +1\leq
Aleph_number
System of formal deduction in logic
those axioms. A generalization of a formula is obtained by prefixing zero or more universal quantifiers on the formula; for example ∀ y ( ∀ x P x y → P t
Hilbert_system
Concept in model theory
precisely, it is a set of first-order formulas in a language L with free variables x1, x2,..., xn that are true of a set of n-tuples of an L-structure
Type_(model_theory)
Logical incompatibility between two or more propositions
truth value "false", as symbolized, for instance, by "0" (as is common in Boolean algebra). It is not uncommon to see Q.E.D., or some of its variants, immediately
Contradiction
Mathematical logic concept
logic". Encyclopedia Britannica. Retrieved 2019-11-26. "Predicates and Quantified Statements II". www.csm.ornl.gov. Retrieved 2019-11-26. Brody, Bobuch
Contraposition
Class of formal logics
semantics. In Boolean-valued semantics (for classical propositional logic), the truth values are the elements of an arbitrary Boolean algebra; "true" corresponds
Classical_logic
Directed graph representing a Boolean expression
(1979). "A linear-time algorithm for testing the truth of certain quantified boolean formulas". Information Processing Letters. 8 (3): 121–123. doi:10
Implication_graph
Axiom of set theory
of countable choice.) Stone's representation theorem for Boolean algebras needs the Boolean prime ideal theorem. The Nielsen–Schreier theorem, that every
Axiom_of_choice
TRUE QUANTIFIED-BOOLEAN-FORMULA
TRUE QUANTIFIED-BOOLEAN-FORMULA
Boy/Male
Irish
Puppy.
Girl/Female
Norse German
Strong.
Girl/Female
Australian, Danish, Finnish, French, German, Norse, Scandinavian, Swedish
Strength; Spear Maiden; Strong Spear; Diminutive of Gertrude; Strength of a Spear; From Gertrude; Beloved Warrior
Surname or Lastname
French
French : topographic name for someone who lived on a track or pathway, Old French rue (Latin ruga ‘crease’, ‘fold’).English : variant of Rowe 1, from the Old English byform rǣw, or a habitational name from places in Devon and Isle of Wight called Rew from this word.Norwegian : habitational name from any of over fifteen farmsteads so named, notably in Telemark, from Old Norse ruð ‘clearing’.
Boy/Male
American, British, English
Manly; Abbreviation of Andrew
Boy/Male
American, Australian, British, English, Jamaican
Three
Boy/Male
Indian
The qualified person
Surname or Lastname
English
English : variant of Bowerman.
Surname or Lastname
English
English : variant of Wool.Americanized form of Jewish Wollman or German Wollmann (see Wollman).
Surname or Lastname
English
English : variant of Trow, mainly of 1.
Girl/Female
English
Prudence. One of the many qualities and virtues that the Puritans adopted as names after the...
Boy/Male
British, English
Loyal
Male
Swedish
Danish and Swedish form of Scandinavian Tore, TURE means "thunder."
Boy/Male
English
Abbreviation of Andrew 'manly.
Girl/Female
Australian, Swedish
Behind
Boy/Male
Muslim
The qualified person
Boy/Male
American, British, English
Lives at the Buck Meadow
Boy/Male
American, Australian, Chinese
Three
Surname or Lastname
English (mainly southeastern)
English (mainly southeastern) : topographic name for someone who lived near a conspicuous tree, Middle English tre(w).
Surname or Lastname
English
English : nickname for a redoubtable warrior, from Middle English prou(s) ‘brave’, ‘valiant’ (Old French proux, preux).Americanized spelling of French Prou (see Proulx).
TRUE QUANTIFIED-BOOLEAN-FORMULA
TRUE QUANTIFIED-BOOLEAN-FORMULA
Boy/Male
Muslim
Partner
Boy/Male
Hindu
Lord Venkateshwara, Residence of Goddess of wealth, Abode of wealth
Girl/Female
Muslim
Beautiful, Graceful
Girl/Female
American, Australian, British, Christian, Danish, English, Finnish, French, German, Greek, Hebrew, Indian, Jamaican, Swedish, Swiss
From Doris; Dorian Woman; Woman of the Sea; Gift; Gift from God; Name of a Place
Boy/Male
Biblical
Sprinkling the chamber.
Boy/Male
Tamil
Vedic hymns
Boy/Male
English
warrior.
Boy/Male
Indian, Sanskrit
Approved by the Gods
Girl/Female
Hindu
To neglect
Girl/Female
Arabic, Muslim
Fruit
TRUE QUANTIFIED-BOOLEAN-FORMULA
TRUE QUANTIFIED-BOOLEAN-FORMULA
TRUE QUANTIFIED-BOOLEAN-FORMULA
TRUE QUANTIFIED-BOOLEAN-FORMULA
TRUE QUANTIFIED-BOOLEAN-FORMULA
n.
Something constructed in the form of, or considered as resembling, a tree, consisting of a stem, or stock, and branches; as, a genealogical tree.
a.
Modified; limited; as, a qualified statement.
a.
Of a genuine or right breed; as, a true-bred beast.
a.
Of inflexible honesty and fidelity; -- a term derived from the true, or Coventry, blue, formerly celebrated for its unchanging color. See True blue, under Blue.
n.
Right to precision; conformable to a rule or pattern; exact; accurate; as, a true copy; a true likeness of the original.
n.
Actual; not counterfeit, adulterated, or pretended; genuine; pure; real; as, true balsam; true love of country; a true Christian.
a.
Of or pertaining to wool or woolen cloths; as, woolen manufactures; a woolen mill; a woolen draper.
v. t.
To place upon a tree; to fit with a tree; to stretch upon a tree; as, to tree a boot. See Tree, n., 3.
n.
Conformable to fact; in accordance with the actual state of things; correct; not false, erroneous, inaccurate, or the like; as, a true relation or narration; a true history; a declaration is true when it states the facts.
n.
A cross or gallows; as Tyburn tree.
v. t.
To drive to a tree; to cause to ascend a tree; as, a dog trees a squirrel.
n.
Steady in adhering to friends, to promises, to a prince, or the like; unwavering; faithful; loyal; not false, fickle, or perfidious; as, a true friend; a wife true to her husband; an officer true to his charge.
a.
Made of wool; consisting of wool; as, woolen goods.
a.
Of genuine birth; having a right by birth to any title; as, a true-born Englishman.
a.
Being of real breeding or education; as, a true-bred gentleman.