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The intersection non-emptiness problem, also known as finite automaton intersection problem or the non-emptiness of intersection problem, is a PSPACE-complete
Intersection non-emptiness problem
Intersection_non-emptiness_problem
of that question, such as the emptiness problem for non-erasing stack automata, are PSPACE-complete. The emptiness problem in machine learning and formal
Emptiness_problem
Set of elements common to all of some sets
In set theory, the intersection of two sets A {\displaystyle A} and B , {\displaystyle B,} denoted by A ∩ B , {\displaystyle A\cap B,} is the set containing
Intersection_(set_theory)
Mathematical set containing no elements
any set A: The empty set is a subset of A The union of A with the empty set is A The intersection of A with the empty set is the empty set The Cartesian
Empty_set
Theory of discrimination
Intersectionality is an analytical framework for understanding how groups' and individuals' social and political identities result in unique combinations
Intersectionality
Identities and relationships involving sets
properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion
Algebra_of_sets
nondeterministic finite automata Word problem and emptiness problem for non-erasing stack automata Emptiness of intersection of an unbounded number of deterministic
List of PSPACE-complete problems
List_of_PSPACE-complete_problems
Common elements of two or more sets
determination of an intersection leads to non-linear equations, which can be solved numerically, for example using Newton iteration. Intersection problems between
Intersection
Problem in computer science
In computability theory, the halting problem is the decision problem of determining, from a description of an arbitrary computer program and an input
Halting_problem
Property in general topology
to have the finite intersection property (FIP) if any finite subfamily of A {\displaystyle {\mathcal {A}}} has non-empty intersection. It has the strong
Finite_intersection_property
Proposition in mathematical logic
problems in set theory, and establishing its truth or falsehood was the first of Hilbert's 23 problems presented in 1900. The answer to this problem is
Continuum_hypothesis
Algorithm for finding shortest paths
the new ARMAC computer. His objective was to choose a problem and a computer solution that non-computing people could understand. He designed the shortest
Dijkstra's_algorithm
Shape formed from points common to other shapes
determination of an intersection leads to non-linear equations, which can be solved numerically, for example using Newton iteration. Intersection problems between
Intersection_(geometry)
Complexity class used to classify decision problems
NP. The "no"-answer version of this problem is stated as: "given a finite set of integers, does every non-empty subset have a nonzero sum?". The verifier-based
NP_(complexity)
Branch of algebraic geometry
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties
Intersection_theory
Mathematical use of "there exists"
{\displaystyle \cap } and ∪ {\displaystyle \cup } to respectively denote the intersection and union of sets. A quantified propositional function is a statement;
Existential_quantification
Complexity class
theory, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. Somewhat more precisely, a problem is NP-complete
NP-completeness
Theorem in mathematical logic
sets in a compact space has a non-empty intersection if every finite subcollection has a non-empty intersection. The compactness theorem is one of the
Compactness_theorem
Common point(s) shared by two lines in Euclidean geometry
In Euclidean geometry, the intersection of a line and a line can be the empty set, a single point, or a line (if they coincide). Distinguishing these cases
Line–line_intersection
Set whose elements all belong to another set
transfinite cardinal number. A set A is a subset of B if and only if their intersection is equal to A. Formally: A ⊆ B if and only if A ∩ B = A . {\displaystyle
Subset
Set of elements in any of some sets
by union, intersection, and complementation, is a Boolean algebra. In this Boolean algebra, union can be expressed in terms of intersection and complementation
Union_(set_theory)
Relationship where one statement follows from another
The Undecidable, Basic Papers on Undecidable Propositions, Unsolvable Problems And Computable Functions, New York: Raven Press, ISBN 9780486432281. Papers
Logical_consequence
Mathematical set formed from two given sets
{\displaystyle Q} . Their intersection will yield a C-system containing all non-empty intersections of each C-n-tuple from P {\displaystyle P} with each C-n-tuple
Cartesian_product
One-to-one correspondence
Chapman & Hall/ CRC Press. D'Angelo; West (2000). Mathematical Thinking: Problem Solving and Proofs. Prentice Hall. Cupillari (1989). The Nuts and Bolts
Bijection
Sequence of words formed by specific rules
letter/word metaphor and replaces it by a word/sentence metaphor. Given a non-empty set Σ {\displaystyle \Sigma } , a formal language L {\displaystyle L}
Formal_language
Formal language generated by context-free grammar
decidable (see "Emptiness" below). Containment: is L ( A ) ⊆ L ( B ) {\displaystyle L(A)\subseteq L(B)} ? Again, the variant of the problem where B is a
Context-free_language
Maximal proper filter
filter subbase is a non-empty family of sets that has the finite intersection property (i.e. all finite intersections are non-empty). Equivalently, a filter
Ultrafilter_on_a_set
Logic theorem
noncontradiction (LNC; also known as the law of contradiction, principle of non-contradiction (PNC), or the principle of contradiction) states that for any
Law_of_noncontradiction
Function that preserves distinctness
of choice, as the existence of a {\displaystyle a} is implied by the non-emptiness of the domain. However, this statement may fail in less conventional
Injective_function
Symbol representing a mathematical object
Papyrus (c. 1500 BC) which described problems with unknowns rhetorically, called the "Aha problems". The "Aha problems" involve finding unknown quantities
Variable_(mathematics)
Assignment of meaning to the symbols of a formal language
interpretations of the theories people study have non-empty domains. Empty relations do not cause any problem for first-order interpretations, because there
Interpretation_(logic)
Mathematical set of all subsets of a set
the empty set as the identity element and each set being its own inverse), and a commutative monoid when considered with the operation of intersection (with
Power_set
Informal set theories
the empty set {}. For any set A, the power set P ( A ) {\displaystyle P(A)} is a Boolean algebra under the operations of union and intersection. Intuitively
Naive_set_theory
Reasoning for mathematical statements
establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning that establish "reasonable expectation". Presenting
Mathematical_proof
Mathematical-logic system based on functions
(tail). The predicate NULL returns TRUE for the value NIL, and FALSE for a non-empty list: NIL := λf.TRUE NULL := λp.p (λx.λy.FALSE) Alternatively, with NIL :=
Lambda_calculus
Yes-or-no question that cannot ever be solved by a computer
theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm
Undecidable_problem
by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and
Mathematical_object
Fewest cliques covering a graph's edges
number and the computational problem of finding it have been studied under many alternative names. Applications of the intersection number include scheduling
Intersection number (graph theory)
Intersection_number_(graph_theory)
Logical principle
profound effect on Hilbert. Reid indicates that Hilbert's second problem (one of Hilbert's problems from the Second International Conference in Paris in 1900)
Law_of_excluded_middle
polyhedron without self-intersections with more than seven faces, all of which share an edge with each other? The Thomson problem – what is the minimum
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Axiom of set theory
Informally put, the axiom of choice says that given any collection of non-empty sets, one can identify another set containing one element chosen from
Axiom_of_choice
Sets with no element in common
element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set. For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while
Disjoint_sets
Process of repeating items in a self-similar way
optimization problem in recursive form. The key result in dynamic programming is the Bellman equation, which writes the value of the optimization problem at an
Recursion
In geometry, set whose intersection with every line is a single line segment
Euclidean space has the following properties: The empty set and the whole space are convex. The intersection of any collection of convex sets is convex. The
Convex_set
Axioms for the natural numbers
set of natural numbers N is defined as the intersection of all sets closed under s that contain the empty set. Each natural number is equal (as a set)
Peano_axioms
Branch of mathematical logic
closed intervals whose lengths tend to zero has a single point in its intersection; the real numbers are not countable).Section II.4 The Baire category
Reverse_mathematics
Mathematical use of "for all"
\exists _{!}S=\exists x.S(x),} which is true if S {\displaystyle S} is not empty, and ∀ ! S = ∀ x . S ( x ) , {\displaystyle \forall _{!}S=\forall x.S(x)
Universal_quantification
Set theory concept
1996 by Oxford University Press, New York]. Set Theory and the Continuum Problem. Dover. ISBN 978-0-486-47484-7. von Neumann, John (1923). "Zur Einführung
Von_Neumann_universe
Measure of similarity and diversity between sets
index measures similarity between finite non-empty sample sets and is defined as the size of the intersection divided by the size of the union of the sample
Jaccard_index
In mathematics, a statement that has been proven
occasion, for example problem when people are not sure whether the statement should be believed to be true. Sometimes the name of a problem in common use does
Theorem
Mathematical logic concept
Yuri Matiyasevich as part of the negative solution to Hilbert's Tenth Problem. Diophantine sets predate recursion theory and are therefore historically
Computably_enumerable_set
few sets in order to avoid an empty intersection. This notion can be used to solve constraints satisfaction problems that are inconsistent by relaxing a
Relaxed_intersection
Ordered listing of items in collection
following table gives the first few values of this enumeration: All (non empty) finite sets are enumerable. Let S be a finite set with n > 0 elements
Enumeration
Sphere that contains a set of objects
their intersection. But an intersection of two non-coinciding spheres of equal radius is contained in a sphere of smaller radius. The problem of computing
Bounding_sphere
Diagram that shows all possible logical relations between a collection of sets
than the equivalent Venn diagram, particularly if the number of non-empty intersections is small. The difference between Euler and Venn diagrams can be
Venn_diagram
In logic, a statement which is always true
period. The problem of determining whether there is any valuation that makes a formula true is the Boolean satisfiability problem; the problem of checking
Tautology_(logic)
Proof in set theory
For example, the conventional proof of the unsolvability of the halting problem is essentially a diagonal argument. Also, diagonalization was originally
Cantor's_diagonal_argument
Model of (first-order) Peano arithmetic that contains non-standard numbers
In mathematical logic, a non-standard model of arithmetic is a model of first-order Peano arithmetic that contains non-standard numbers. The term standard
Non-standard model of arithmetic
Non-standard_model_of_arithmetic
Thesis on the nature of computability
the sake of argument (i.e. a "thesis")? In the course of studying the problem, Church and his student Stephen Kleene introduced the notion of λ-definable
Church–Turing_thesis
Pair of logical equivalences
the union of two sets is the same as the intersection of their complements The complement of the intersection of two sets is the same as the union of their
De_Morgan's_laws
Infinite cardinal number
partition Set operations: intersection union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton
Aleph_number
Mathematical function that can be computed by a program
complexity theory, the problem of computing the value of a function is known as a function problem, by contrast to decision problems whose results are either
Computable_function
Symbolic description of a mathematical object
addition (+), multiplication (×), or set operations like union (∪), or intersection (∩). (Functions can be understood as unary operations) Brackets ( ) With
Expression_(mathematics)
Measure of algorithmic complexity
It is also possible to show the non-computability of K by reduction from the non-computability of the halting problem H, since K and H are Turing-equivalent
Kolmogorov_complexity
Structure of a formal language
p. 233, ISBN 9781466513457. For more on this subject, see undecidable problem. Chomsky, Noam (Sep 1956). "Three models for the description of language"
Formal_grammar
Mathematical theory of data types
Between 1902 and 1908, Bertrand Russell proposed various solutions to this problem. By 1908, Russell arrived at a ramified theory of types together with an
Type_theory
Abstract mathematics problem
Sheldon Ross in his 1988 book A First Course in Probability. The problem starts with an empty vase and an infinite supply of balls. An infinite number of steps
Ross–Littlewood_paradox
Form of mathematical proof
induction and the first and fourth axioms. Proof. Suppose there exists a non-empty set, S, of natural numbers that has no least element. Let P(n) be the
Mathematical_induction
Basic framework of mathematics
philosophical problems, the main one being that before this discovery, the parallel postulate and all its consequences were considered as true. So, the non-Euclidean
Foundations_of_mathematics
Subfield of automated reasoning and mathematical logic
logic, the problem of deciding the validity of a formula varies from trivial to impossible. For the common case of propositional logic, the problem is decidable
Automated_theorem_proving
Area of mathematical logic
signature and let Φ {\displaystyle \Phi } be a countable set of non-isolated types over the empty set. Then there is a model M {\displaystyle {\mathcal {M}}}
Model_theory
Existence and cardinality of models of logical theories
logicians in the early 20th century, as the distinction between first-order and non-first-order properties was not yet understood. One such consequence is the
Löwenheim–Skolem_theorem
Mathematical set that can be enumerated
{\displaystyle \mathbb {N} } a surjection g i {\displaystyle g_{i}} from the non-empty collection of surjections from N {\displaystyle \mathbb {N} } to A i {\displaystyle
Countable_set
Theory that allows sets to be elements of themselves
Non-well-founded set theories (sometimes unhyphenated, as nonwellfounded; or poorly founded) are variants of axiomatic set theory that allow sets to be
Non-well-founded_set_theory
Elements in exactly one of two sets
the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets { 1 , 2 , 3 } {\displaystyle
Symmetric_difference
Unrelated vertices in graphs
geometric intersection graph is just a set of disjoint (non-overlapping) shapes. The problem of finding maximum independent sets in geometric intersection graphs
Independent set (graph theory)
Independent_set_(graph_theory)
Infinite set that is not countable
_{1}} . In 1900, David Hilbert posed this question as the first of his 23 problems. The statement that ℵ 1 = ℶ 1 {\displaystyle \aleph _{1}=\beth _{1}} is
Uncountable_set
Standard system of axiomatic set theory
{\displaystyle y} have the same elements, then they belong to the same sets. Every non-empty set x {\displaystyle x} contains a member y {\displaystyle y} such that
Zermelo–Fraenkel_set_theory
Symbol representing a property or relation in logic
In logic, a predicate is a non-logical symbol that represents a property or a relation, though, formally, does not need to represent anything at all.
Predicate_(logic)
Theorem in set theory
Klein; Walther von Dyck; David Hilbert; Otto Blumenthal (eds.), "Über das Problem der Wohlordnung", Mathematische Annalen (in German), 76 (4): 438–443, doi:10
Schröder–Bernstein_theorem
Function, homomorphism, or morphism
partition Set operations: intersection union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton
Map_(mathematics)
Every set is smaller than its power set
Another way to think of the proof is that B {\displaystyle B} , empty or non-empty, is always in the power set of A {\displaystyle A} . For f {\displaystyle
Cantor's_theorem
Mathematical function such that every output has at least one input
second kind. A non-injective surjective function (surjection, not a bijection) An injective surjective function (bijection) An injective non-surjective function
Surjective_function
Target set of a mathematical function
partition Set operations: intersection union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton
Codomain
Mathematical concept
infinite-dimensional) vector space can be created by starting with the empty set and for each ordinal α > 0 choosing a vector that is not in the span
Transfinite_induction
Mathematical set containing all objects
proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory include a universal set. Many set theories
Universal_set
defined by André Bouchet in 1987. Algorithms for matroid intersection and the matroid parity problem can be extended to some cases of delta-matroids. Delta-matroids
Delta-matroid
Class of formal logics
mathematical functions. It was also the first logic capable of dealing with the problem of multiple generality, for which Aristotle's system was impotent. Frege
Classical_logic
Whether a decision problem has an effective method to derive the answer
In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Logical systems are decidable
Decidability_(logic)
about whether the set intersection of two given sets is non-empty. The input to the problem is n finite sets. The sum of the sizes of all sets is N (which
Set_intersection_oracle
Limitative results in mathematical logic
and Turing's theorem that there is no algorithm to solve the halting problem. The incompleteness theorems apply to formal systems that are of sufficient
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Form of logic that allows quantification over predicates
in various ways—to what is now called first-order logic—eliminated this problem: sets and properties cannot be quantified over in first-order logic alone
Second-order_logic
Unsolved problem in computational complexity theory
the permutation group isomorphism problem and the permutation group intersection problem. For the latter two problems, Babai, Kantor & Luks (1983) obtained
Graph_isomorphism_problem
Axiomatic set theories based on the principles of mathematical constructivism
here the property of emptiness does not partition the set theoretical domain of discourse into two decidable parts. For any such non-trivial property, the
Constructive_set_theory
Yes/no problem in computer science
decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding
Decision_problem
Category of mathematical proof
1882, which showed that the problem of squaring the circle cannot be solved because the number π is transcendental (i.e., non-algebraic), and that only
Proof_of_impossibility
Theorem about the intersections of d-dimensional convex sets
finite intersection property characterization of compactness: a collection of closed subsets of a compact space has a non-empty intersection if and only
Helly's_theorem
1979 conjecture in combinatorics
Unsolved problem in mathematics If any two sets in some finite family of sets have a union that also belongs to the family, must some element belong to
Union-closed_sets_conjecture
School of thought in philosophy of mathematics
proclaimed all classes are useful fictions he solved the problem of the "unit" class, but the overall problem did not go away; rather, it arrived in a new form:
Logicism
Method of deriving conclusions
for deciding what is true and false. Paraconsistent logics solve this problem by modifying the rules of inference in such a way that the principle of
Rule_of_inference
INTERSECTION NON-EMPTINESS-PROBLEM
INTERSECTION NON-EMPTINESS-PROBLEM
Male
English
 English short form of Spanish Alonso, LON means "noble and ready." Compare with another form of Lon.
Boy/Male
Greek
Son of Apollo.
Male
Scandinavian
 Scandinavian form of Icelandic Jóhann, JON means "God is gracious." Compare with other forms of Jon.
Male
French
French form of Greek Noe, NOÉ means "rest."
Female
Vietnamese
Vietnamese name NGON means "good communication."
Biblical
same as Non
Girl/Female
Biblical
Posterity, a fish, eternal.
Surname or Lastname
English, German, Dutch, French (Noé, Noë), Spanish (Noé), Catalan (Noè)
English, German, Dutch, French (Noé, Noë), Spanish (Noé), Catalan (Noè) : from the Biblical personal name Noach ‘Noah’, which means ‘comfort’ in Hebrew. According to the Book of Genesis, Noah, having been forewarned by God, built an ark into which he took his family and representatives of every species of animal, and so was saved from the flood that God sent to destroy the world because of human wickedness. The personal name was not common among non-Jews in the Middle Ages, but the Biblical story was an extremely popular subject for miracle plays. In many cases, therefore, the surname probably derives from a nickname referring to someone who had played the part of Noah in a miracle play or pageant, rather than from a personal name.
Biblical
posterity; a fish; eternal
Female
English
Variant form of Old English Nona, NONI means "ninth."
Female
English
Short form of English Nancy, NAN means "favor; grace."
Female
Russian
(Ðона) Russian name derived from Greek enatos, NONA means "ninth." Compare with another form of Nona.
Male
Norwegian
Danish and Norwegian form of Old Norse Hákon, HÅKON means "high son."
Female
Hawaiian
Hawaiian name NOE means "mist; misty rain."
Female
English
(רï‹×Ÿ) Hebrew unisex name RON means "joy, song." Compare with strictly masculine Ron.
Male
English
 Pet form of English Jonathan, JON means "God has given." Compare with other forms of Jon.
Boy/Male
American, Australian
Little Son
Male
English
 Short form of English/Scottish Ronald, RON means "wise ruler." Compare with another form of Ron.
Female
English
Variant spelling of English Noah, NOA means "motion."Â
Male
Hebrew
(רï‹×Ÿ) Hebrew unisex name RON means "joy, song." Compare with another form of Ron.
INTERSECTION NON-EMPTINESS-PROBLEM
INTERSECTION NON-EMPTINESS-PROBLEM
Boy/Male
Muslim
Strong, Solid, Firm, Sharp
Boy/Male
Arabic, Muslim
Servant of the Firm / Strong (Allah)
Boy/Male
American, Anglo, Arabic, Australian, British, Christian, English, German, Irish, Jamaican, Teutonic
Bard; Surname; Guardian; Watchman
Girl/Female
Hindu, Indian, Malayalam, Marathi, Tamil
Garland of Flowers
Female
French
Feminine form of French Olivier, probably OLIVIE means "elf army."
Boy/Male
Greek
Defender; protector of mankind. Famous Bearer: Alexander the Great.
Girl/Female
Arabic
Sweetheart; Beloved
Girl/Female
Arabic, Muslim
Princess
Girl/Female
English American
Darling. From the Old English 'dearling'.
Surname or Lastname
English
English : variant of Shillito.
INTERSECTION NON-EMPTINESS-PROBLEM
INTERSECTION NON-EMPTINESS-PROBLEM
INTERSECTION NON-EMPTINESS-PROBLEM
INTERSECTION NON-EMPTINESS-PROBLEM
INTERSECTION NON-EMPTINESS-PROBLEM
n.
Insufficiency; emptiness.
a.
No; not. See No, a.
n.
Inanition; void space; vacuity; emptiness.
a.
Pertaining to, or formed by, intersections.
n.
The act of intercepting; as, interception of a letter; interception of the enemy.
n.
Want of solidity or substance; unsatisfactoriness; inability to satisfy desire; vacuity; hollowness; the emptiness of earthly glory.
n.
The state of being empty; absence of contents; void space; vacuum; as, the emptiness of a vessel; emptiness of the stomach.
n.
Intervention; interposition.
n.
Intervention; interposition.
n.
A line of division or intersection; as, the tendinous inscriptions, or intersections, of a muscle.
n.
The act, state, or place of intersecting.
n.
Mutual or reciprocal action or influence; as, the interaction of the heart and lungs on each other.
n.
Interposition; intervention.
n.
Vanity; emptiness; -- now used only in the phrase in vain.
n.
The point or line in which one line or surface cuts another.
a.
Not any; not one; none.
p. pr. & vb. n.
of Non-pros
n.
Want of knowledge; lack of sense; vacuity of mind.