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Unsolved problem in computer science
in polynomial time? More unsolved problems in computer science The P versus NP problem is a major unsolved problem in theoretical computer science. Informally
P_versus_NP_problem
Complexity class
there exist problems in NP that are not in P. The question of whether the classes P and NP are equal or not is known as the P versus NP problem. A consequence
NP-completeness
Complexity class used to classify decision problems
Unsolved problem in computer science P = ? N P {\displaystyle {\mathsf {P\ {\overset {?}{=}}\ NP}}} More unsolved problems in computer science In
NP_(complexity)
Proposition in mathematics that is unproven
unsolved problems; it is also one of the Clay Mathematics Institute Millennium Prize Problems. The P versus NP problem is a major unsolved problem in computer
Conjecture
Boolean satisfiability is NP-complete and therefore that NP-complete problems exist
the P versus NP problem, which is still widely considered the most important unsolved problem in theoretical computer science. The concept of NP-completeness
Cook–Levin_theorem
Seven mathematical problems with a US$1 million prize for each solution
conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Riemann hypothesis, Yang–Mills existence and mass gap, and the Poincaré
Millennium_Prize_Problems
Set of problems in computational complexity theory
answer questions about the fundamental nature of computation. The P versus NP problem, for instance, is directly related to questions of whether nondeterminism
Complexity_class
Inherent difficulty of computational problems
limits on what computers can and cannot do. The P versus NP problem, one of the seven Millennium Prize Problems, is part of the field of computational complexity
Computational complexity theory
Computational_complexity_theory
Problem of determining if a Boolean formula could be made true
polynomial-time algorithm would settle the P versus NP problem - one of the most important open problems in the theory of computing. Nevertheless, heuristic
Boolean satisfiability problem
Boolean_satisfiability_problem
halting problem was first used and formally stated by Martin Davis in his 1958 book Computability and Unsolvability. The P versus NP problem is an unsolved
Philosophy of computer science
Philosophy_of_computer_science
Set of computational problems stated by Richard Karp (1973)
drove interest in the study of NP-completeness and the P versus NP problem. Karp's 21 problems are shown below, many with their original names. The nesting
Karp's 21 NP-complete problems
Karp's_21_NP-complete_problems
Theoretical model of computation
computers. One of the most important open problems in theoretical computer science is the P versus NP problem, which (among other equivalent formulations)
Nondeterministic Turing machine
Nondeterministic_Turing_machine
American computer scientist (born 1963)
the P-NP Puzzler Has Consequences" The New York Times, October 7, 2009(subscription required) - L. Fortnow, "The Status of the P Versus NP Problem", Communications
Lance_Fortnow
Unsolved problem in computational complexity theory
and P ≠ NP, then for many important problems it is not only impossible to get an exact solution in polynomial time (as postulated by the P versus NP problem)
Unique_games_conjecture
Complexity class
#P-complete problem, if it existed, would solve the P versus NP problem by implying that P and NP are equal. No such algorithm is known, nor is a proof
♯P-complete
2012 American film
solve the P versus NP problem, one of the most challenging mathematical problems in history. The title refers to the travelling salesman problem, an optimization
Travelling Salesman (2012 film)
Travelling_Salesman_(2012_film)
Category of mathematical proof
the P versus NP problem. Another technique is the proof of completeness for a complexity class, which provides evidence for the difficulty of problems by
Proof_of_impossibility
Professor of computer science
been working on "geometric complexity theory", an approach to the P versus NP problem through the techniques of algebraic geometry, with Milind Sohoni
Ketan_Mulmuley
List of unsolved computational problems
proposed solutions. P versus NP problem – The P vs NP problem is a major unsolved question in computer science that asks whether every problem whose solution
List of unsolved problems in computer science
List_of_unsolved_problems_in_computer_science
Millennium Prize Problem
The Navier–Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier–Stokes equations, a system of partial
Navier–Stokes existence and smoothness
Navier–Stokes_existence_and_smoothness
1997 film by Gus Van Sant
Bohman gave them a brief lecture and suggested the computer science P versus NP problem as one that Will could solve. Kleitman and Professor Patrick O'Donnell
Good_Will_Hunting
Complexity class
theory, co-NP is a complexity class. A decision problem X is a member of co-NP if and only if its complement X is in the complexity class NP. The class
Co-NP
Algorithm using holographic reduction
relevant to the P versus NP problem and their impact on computational complexity theory. Although some of the general problems are #P-hard problems, the special
Holographic_algorithm
18 mathematical problems stated in 1998
intractability of Hilbert's Nullstellensatz and an algebraic version of "NP≠P?"". Duke Math. J. 81: 47–54. doi:10.1215/S0012-7094-95-08105-8. Zbl 0882
Smale's_problems
Sequence of operations for a task
complexity can be the fastest algorithm for some problems is an open question known as the P versus NP problem. There are two large classes of such algorithms:
Algorithm
Problem in computer science
complexity P versus NP problem Termination analysis Worst-case execution time Calude, Cristian S. (2021). "Incompleteness and the Halting Problem". Studia
Halting_problem
Classification of computer problems
famous open problem in computer science – whether P = NP – by showing that the complexity class P is not equal to the complexity class NP. The idea behind
Geometric_complexity_theory
Yes/no problem in computer science
functions of an NP-complete problem and its co-NP-complete complement is exactly the same even though the underlying decision problems may not be considered
Decision_problem
Classification of algorithm
would settle the P versus NP problem, considered the most important open problem in computer science and one of the Millennium Prize Problems. An example of
Galactic_algorithm
Impossible task in computing
mathematics and computer science, the Entscheidungsproblem (German for 'decision problem'; pronounced [ɛntˈʃaɪ̯dʊŋspʁoˌbleːm]) is a challenge posed by David Hilbert
Entscheidungsproblem
Method of deriving conclusions
(1) P → ( Q → P ) {\displaystyle P\to (Q\to P)} , (2) ( P → ( Q → R ) ) → ( ( P → Q ) → ( P → R ) ) {\displaystyle (P\to (Q\to R))\to ((P\to Q)\to (P\to
Rule_of_inference
Estimate of time taken for running an algorithm
unsolved P versus NP problem asks if all problems in NP have polynomial-time algorithms. All the best-known algorithms for NP-complete problems like 3SAT
Time_complexity
Algorithm whose behavior and output may depend on the run
more efficient than known deterministic algorithms for many problems. The P versus NP problem encapsulates this conjectured greater efficiency available
Nondeterministic_algorithm
Basic framework of mathematics
this problem by introducing "throws" that form what is presently called a field, in which the cross ratio can be expressed. Apparently, the problem of the
Foundations_of_mathematics
Collection of mathematical objects
characterized by the formula. There are several ways for avoiding the problem. One may prove that the formula defines a set; this is often almost immediate
Set_(mathematics)
Type of infinite structure
dynamical systems and algorithms for tame optimization, and multi-objective problems (PhD thesis). Université Montpellier; Universidad técnica Federico Santa
O-minimal_theory
Computation model defining an abstract machine
ISBN 0-201-02988-X. Centered around the issues of machine-interpretation of "languages", NP-completeness, etc. Hopcroft, John E.; Motwani, Rajeev; Ullman, Jeffrey D.
Turing_machine
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
2nd episode of the 1st season of Numbers
physics concepts, such as the Heisenberg uncertainty principle, P versus NP problem, and Minesweeper game. The episode was directed by Davis Guggenheim
Uncertainty Principle (Numbers)
Uncertainty_Principle_(Numbers)
Paradox in set theory
Gottlob Frege of the paradox in Frege's 1879 Begriffsschrift and framed the problem in terms of both logic and set theory, and in particular in terms of Frege's
Russell's_paradox
Any one of the distinct objects that make up a set in set theory
denoted P(U). Thus the relation ∈ {\displaystyle \in } is a subset of U × P(U). The converse relation ∋ {\displaystyle \ni } is a subset of P(U) × U.
Element_of_a_set
Number of arguments required by a function
Supplement III. Springer. p. 3. ISBN 978-1-4020-0198-7. Schechter, Eric (1997). Handbook of Analysis and Its Foundations. Academic Press. p. 356. ISBN 978-0-12-622760-4
Arity
Branch of mathematics that studies sets
independent of ZFC, requiring stronger axioms for their proof. A famous problem is the normal Moore space question, a question in general topology that
Set_theory
Set whose elements all belong to another set
displaying short descriptions of redirect targets Subset sum problem – Decision problem in computer science Subsumptive containment – System of elements
Subset
In logic, a statement which is always true
satisfiability problem is NP-complete, and consequently, tautology is co-NP-complete. It is widely believed that (equivalently for all NP-complete problems) no polynomial-time
Tautology_(logic)
Theory of truth in the philosophy of language
ISBN 9780631213260. Retrieved 28 February 2024., p. 326 Parts of section is adapted from Kirkham, 1992. Kemp, Gary. Quine versus Davidson: Truth, Reference, and Meaning
Semantic_theory_of_truth
Branch of computational complexity theory
polynomial in the size of the input. Under the assumption that P ≠ NP, there exist many natural problems that require super-polynomial running time when complexity
Parameterized_complexity
Topics referred to by the same term
an enzyme 4-Nitrophenol or p-nitrophenol Pyridoxine phosphate, a form of vitamin B6 PNP transistor P versus NP problem Plug and play, not requiring
PNP
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
Subfield of mathematics
significant result in this area, Fagin's theorem (1974) established that NP is precisely the set of languages expressible by sentences of existential
Mathematical_logic
Argument whose conclusion must be true if its premises are
humans are animals. (True) Therefore, all humans live on Mars. (False) The problem with the argument is that it is not sound. In order for a deductive argument
Validity_(logic)
Book by Ian Stewart
conjecture Fermat's Last Theorem Three-body problem Riemann hypothesis Poincare conjecture P versus NP problem Navier–Stokes equation Mass gap Birch and
The Great Mathematical Problems
The_Great_Mathematical_Problems
commutativity of a boolean ring Boolean satisfiability problem NP-completeness of the Boolean satisfiability problem Cantor's diagonal argument set is smaller than
List_of_mathematical_proofs
Theorem in geometric topology
upon Richard S. Hamilton's program of using the Ricci flow to solve the problem. By developing a number of new techniques and results in the theory of
Poincaré_conjecture
Study of computable functions and Turing degrees
Gödel on p. 150 (which had also been added to Gödel's reprint in Davis' 1965 compilation).) Church, Alonzo (1936a). "An unsolvable problem of elementary
Computability_theory
Millennium Prize Problem
existence and mass gap problem is an unsolved problem in mathematical physics and mathematics, and one of the seven Millennium Prize Problems defined by the Clay
Yang–Mills existence and mass gap
Yang–Mills_existence_and_mass_gap
Set of all things that may be the input of a mathematical function
open connected subset of R n {\displaystyle \mathbb {R} ^{n}} where a problem is posed, making it both an analysis-style domain and also the domain of
Domain_of_a_function
Conjecture on zeros of the zeta function
Unsolved problem in mathematics Do all non-trivial zeros of the Riemann zeta function have a real part equal to one half? More unsolved problems in mathematics
Riemann_hypothesis
Axioms for the natural numbers
David Hilbert posed the problem of proving their consistency using only finitistic methods as the second of his twenty-three problems. In 1931, Kurt Gödel
Peano_axioms
Function, homomorphism, or morphism
mapping, correspondence, and operator are often used synonymously. Halmos 1970, p. 30. Some authors use the term function with a more restricted meaning, namely
Map_(mathematics)
Field of knowledge
packing were two major problems of discrete mathematics solved in the second half of the 20th century. The P versus NP problem, which remains open to
Mathematics
Reasoning for mathematical statements
ISBN 978-0-08-053318-6. See in particular p. 3: "The study of Proof Theory is traditionally motivated by the problem of formalizing mathematical proofs; the
Mathematical_proof
Problem in applied mathematics
polynomial-time solutions to NP problems (see P versus NP problem), then there is no generic solution to the sign problem. This leaves open the possibility
Numerical_sign_problem
Consistency of the axioms of arithmetic
In mathematics, Hilbert's second problem was posed by David Hilbert in 1900 as one of his 23 problems. It asks for a proof that arithmetic is consistent
Hilbert's_second_problem
Standard system of axiomatic set theory
Z 0 , P ( Z 0 ) , P ( P ( Z 0 ) ) , P ( P ( P ( Z 0 ) ) ) , . . . } , {\displaystyle \{Z_{0},{\mathcal {P}}(Z_{0}),{\mathcal {P}}({\mathcal {P}}(Z_{0}))
Zermelo–Fraenkel_set_theory
Sequence of words formed by specific rules
early example of an undecidable problem. Post would later use this paper as the basis for a 1947 proof "that the word problem for semigroups was recursively
Formal_language
British mathematician
other problems such as the P versus NP problem. He has also developed an interest, in joint work with Mohan Ganesalingam, in automated problem solving
Timothy_Gowers
Mathematical set of all subsets of a set
powerset of S is variously denoted as P(S), 𝒫(S), P(S), P ( S ) {\displaystyle \mathbb {P} (S)} , or 2S. Any subset of P(S) is called a family of sets over
Power_set
Target set of a mathematical function
1970, p. 76 Bourbaki 1970, p. 77 Forster 2003, pp. 10–11 Eccles 1997, p. 91 (quote 1, quote 2); Mac Lane 1998, p. 8; Mac Lane, in Scott & Jech 1967, p. 232;
Codomain
Unproved conjecture in mathematics
curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. It is named after
Birch and Swinnerton-Dyer conjecture
Birch_and_Swinnerton-Dyer_conjecture
Additional mathematical object
enumerable Computable function Computable set Decision problem decidable undecidable P NP P versus NP problem Kolmogorov complexity Lambda calculus Primitive
Mathematical_structure
Theorem for proving more complex theorems
Merriam-Webster. Loewen, Nathan R. B. (March 12, 2018), Beyond the Problem of Evil, Lexington Books, p. 47, ISBN 9781498555739 Higham, Nicholas J. (1998). Handbook
Lemma_(mathematics)
Infinite cardinal number
and authors (updated ed.). Providence, RI: American Mathematical Society. p. 16. ISBN 0-8218-0053-1. MR 0553111. Miller, Jeff. "Earliest uses of symbols
Aleph_number
Symbol representing a property or relation in logic
Maksimova, Larisa (2003). Problems in Set Theory, Mathematical Logic, and the Theory of Algorithms. New York: Springer. p. 52. ISBN 0306477122. Introduction
Predicate_(logic)
Algebraic manipulation of "true" and "false"
Boolean satisfiability problem (SAT), and is of importance to theoretical computer science, being the first problem shown to be NP-complete. The closely
Boolean_algebra
Axiom of set theory
the statement that P = NP, the Riemann hypothesis, and many other unsolved mathematical problems. When attempting to solve problems in this class, it makes
Axiom_of_choice
Fragment of first-order logic
likewise be formulated in term logic. On the other hand, a modern view of the problem of multiple generality in traditional logic concludes that quantifiers
Monadic_predicate_calculus
Measure of algorithmic complexity
halting problem. In particular, no program P computing a lower bound for each text's Kolmogorov complexity can return a value essentially larger than P's own
Kolmogorov_complexity
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
Formal system of logic
Goldfarb (1981). "The Undecidability of the Second-Order Unification Problem" (PDF). Theoretical Computer Science. 13 (2): 225–230. doi:10.1016/0304-3975(81)90040-2
Higher-order_logic
Term in logic and deductive reasoning
proof) Type soundness Smith, Peter (2010). "Types of proof system" (PDF). p. 5. Gensler, Harry J., 1945- (January 6, 2017). Introduction to logic (Third ed
Soundness
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
Set of elements common to all of some sets
enumerable Computable function Computable set Decision problem decidable undecidable P NP P versus NP problem Kolmogorov complexity Lambda calculus Primitive
Intersection_(set_theory)
Mathematical set containing no elements
property P holds. Conversely, if for some property P and some set V, the following two statements hold: For every element of V the property P holds There
Empty_set
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
Theory of machine learning
assumptions,[citation needed] such as: Computational complexity – P ≠ NP (the P versus NP problem); Cryptographic – One-way functions exist. There are several
Computational_learning_theory
Academic subfield of computer science
discussed further at Complexity classes P and NP, and P versus NP problem is one of the seven Millennium Prize Problems stated by the Clay Mathematics Institute
Theory_of_computation
Unsolved problem in geometry
In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular
Hodge_conjecture
Logical incompatibility between two or more propositions
of postulates] a new definition must be given. Post's solution to the problem is described in the demonstration "An Example of a Successful Absolute
Contradiction
Mathematical term; concerning axioms used to derive theorems
23 unsolved problems in mathematics from 1900, Hilbert made the continuum hypothesis the first problem on the list. Hilbert's sixth problem asked for "axiomatization
Axiomatic_system
Glock, H.J. (2008). What is Analytic Philosophy?. Cambridge University Press. p. 1. ISBN 978-0-521-87267-6. Retrieved 2023-08-28. Weir, Alan (2024), "Formalism
Mathematical_object
Austrian mathematician and computer scientist (1964–2022)
September 2016, Woeginger maintained a list of attempts to settle the P versus NP problem on his website. As of its last update, the list tallied 116 attempts
Gerhard_J._Woeginger
Mathematical-logic system
computable function can decide the question. This was historically the first problem for which undecidability could be proven. As usual for such a proof, computable
Lambda_calculus
Diagram that shows all possible logical relations between a collection of sets
supplemental rules for the standard Venn diagram, in order to account for certain problem cases. For instance, regarding the issue of representing singular statements
Venn_diagram
Collection of sets in mathematics that can be defined based on a property of its members
Smullyan, Raymond M.; Fitting, Melvin (2010), Set Theory And The Continuum Problem, Dover Publications, ISBN 978-0-486-47484-7 Monk, Donald J. (1969), Introduction
Class_(set_theory)
Class of problems solvable in polynomial time
(the P ⊊ N P {\displaystyle {\mathsf {P}}\subsetneq {\mathsf {NP}}} hypothesis) remains unproven. Another open problem is whether NP = co-NP; since P = co-P
P_(complexity)
Mathematical proof expressed visually
8, American Mathematical Society, p. 527, doi:10.1090/S0002-9904-1937-06588-8 Gallery of Proofs, Art of Problem Solving, retrieved 2015-05-28 Gallery
Proof_without_words
Subset of a function's codomain
surjection Essential range Hungerford 1974, p. 3; Childs 2009, p. 140. Dummit & Foote 2004, p. 2. Rudin 1991, p. 99. Weisstein, Eric W. "Range". mathworld
Range_of_a_function
Subfield of automated reasoning and mathematical logic
For the common case of propositional logic, the problem is decidable but co-NP-complete, and hence only exponential-time algorithms are believed to exist
Automated_theorem_proving
Mathematical logic concept
Business Media. p. 23. ISBN 978-0-387-68441-3. Murty, M. Ram; Fodden, Brandon. "Chapter 5: Hilbert's Tenth Problem". Hilbert's tenth problem: an introduction
Computably_enumerable_set
P VERSUS-NP-PROBLEM
P VERSUS-NP-PROBLEM
Boy/Male
Biblical
That cuts or divides; a nail; a gryphon; a horseman.
Boy/Male
Biblical Greek Latin
A lamp, new-tilled land.
Male
Greek
(ΠεÏσεÏÏ‚) Greek myth name of the founder of Mycenae and the hero who killed the half-mortal gorgon Medousa. If Greek, the first element of the name might have derived from the word pertho, PERSEUS means "to sack, to destroy." And according to Carl Daling Buck in his Comparative Grammar of Greek and Latin, the -eus suffix found in so many Greek names is typically used to form an agent noun. If so, Perseus was a "destroyer" by profession, i.e. a "soldier," which is a fitting name for this legendary hero.Â
Boy/Male
Greek
King of Thrace.
Boy/Male
Greek
Son of Danae.
Boy/Male
Shakespearean
Much Ado About Nothing' A Headborough.
Boy/Male
Irish
Derived from fear “â€manâ€â€ and gus “â€strengthâ€â€ and signifies “â€a strong warrior, virile.â€â€ According to the legend of the Cattle Raid of Cooley (read the legend) Fergus was the king of Ulster and his lover, the cunning Nessa, duped him into letting her son Conchobhar rule in his place for a year so that in years to come her son could be called “â€the son of a king.â€â€ Fergus consented but after the year Conchobhar refused to relinquish the throne and so Fergus joined Maebh in her battle against Ulster, his native province.
Boy/Male
Irish
Name of a saint.
Male
Hungarian
Hungarian form of English Philip, FÜLÖP means "lover of horses."
Male
English
Anglicized form of Greek Bariesou, BAR-JESUS means "son of Jesus." In the New Testament bible, this is the name of a false prophet.
Boy/Male
Greek
Son of Danae.
Boy/Male
Celtic Irish Gaelic Scottish
Manly.
Biblical
savior; deliverer, The Greek form of the name Joshua or Jeshua, a contraction of Jehoshua, that is, help of Jehovah or saviour. Latin: Jesus, Iesus, Iesu, Josue. Greek: Ieous from Hebrew Yeshua. Also means safety, victory and who's help is Jehovah or it may be from the verb "Yasha", "to save," and = Jehovah Savior, or simply Savior; a late form of Hebrew "yehosua", the Jesus means of which is "YHWH is salvation" or "YHWH saves/has saved." Online definition of "savior." Latin term drove out Old English "hæland" which means "healer" as the preferred descriptive term for Jesus.
Boy/Male
Latin
Youthful.
Boy/Male
American, Australian, Chinese, English, French, German, Greek, Hawaiian, Hebrew, Latin, Spanish
God will Help; The Lord is Salvation; Named for Jesus
Female
Greek
(Î ÎÏσις) Greek name PERSIS means "Persian woman." In the bible, this is the name of a Christian woman mentioned by Paul in his epistle to the Romans.
Male
English
Irish and Scottish Anglicized form of Gaelic Fearghus, FERGUS means "strong-man." In Irish mythology, this was the name an Ulster hero.
Girl/Female
Greek Latin
From Persia.
Boy/Male
Greek Latin
Portal to Hades.
Girl/Female
Biblical
Winged, feathered.
P VERSUS-NP-PROBLEM
P VERSUS-NP-PROBLEM
Boy/Male
Indian
Mountain
Surname or Lastname
English
English : metonymic occupational name for a maker of ecclesiastical copes, from Old French chape (see Chapel).
Girl/Female
Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi
Goddess Saraswati
Boy/Male
Hindu, Indian, Kannada
Forehead Eyes Shiva
Boy/Male
Tamil
One who conquered the mind
Boy/Male
English American French
Form of Rufus: Red-haired.
Boy/Male
German, Scottish
From the Stockade
Boy/Male
Indian
Graceful, Elegant, Connoisseur
Boy/Male
Tamil
Girl/Female
Indian
Shining, Goddess
P VERSUS-NP-PROBLEM
P VERSUS-NP-PROBLEM
P VERSUS-NP-PROBLEM
P VERSUS-NP-PROBLEM
P VERSUS-NP-PROBLEM
imp. & p. p.
of Verse
a.
Of or pertaining to a vein or veins; as, the venous circulation of the blood.
v. t.
To tell in verse, or poetry.
a.
Of or pertaining to a verse.
a.
Thin; watery; like serum; as the serous fluids.
a.
Of or pertaining to serum; as, the serous glands, membranes, layers. See Serum.
prep.
Against; as, John Doe versus Richard Roe; -- chiefly used in legal language, and abbreviated to v. or vs.
n.
A stanza; a stave; as, a hymn of four verses.
n.
A verse.
v. i.
To make verses; to versify.
a.
Marked with veins; veined; as, a venous leaf.
n. sing. & pl.
A verse or verses. See Verse.