Perfect number

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Perfect_number

Illustration of the perfect number status of the number 6

In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.

The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including itself; in symbols, ${\displaystyle {\sigma }_1(n)=2n}$ where ${\displaystyle {\sigma }_1}$ is the sum-of-divisors function. For instance, 28 is perfect as 1 + 2 + 4 + 7 + 14 = 28.

This definition is ancient, appearing as early as Euclid's Elements (VII.22) where it is called τέλειος ἀριθμός (perfect, ideal, or complete number). Euclid also proved a formation rule (IX.36) whereby ${\displaystyle q(q+1)/2}$ is an even perfect number whenever ${\displaystyle q}$ is a prime of the form ${\displaystyle 2^p-1}$ for positive integer ${\displaystyle p}$—what is now called a Mersenne prime. Two millennia later, Leonhard Euler proved that all even perfect numbers are of this form. This is known as the Euclid–Euler theorem.

It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist. The first few perfect numbers are 6, 28, 496 and 8128 (sequence A000396 in the OEIS).

History

In about 300 BC Euclid showed that if 2^p − 1 is prime then 2^p−1(2^p − 1) is perfect. The first four perfect numbers were the only ones known to early Greek mathematics, and the mathematician Nicomachus noted 8128 as early as around AD 100. In modern language, Nicomachus states without proof that every perfect number is of the form ${\displaystyle 2^{n-1}(2^n-1)}$ where ${\displaystyle 2^n-1}$ is prime. He seems to be unaware that n itself has to be prime. He also says (wrongly) that the perfect numbers end in 6 or 8 alternately. (The first 5 perfect numbers end with digits 6, 8, 6, 8, 6; but the sixth also ends in 6.) Philo of Alexandria in his first-century book "On the creation" mentions perfect numbers, claiming that the world was created in 6 days and the moon orbits in 28 days because 6 and 28 are perfect. Philo is followed by Origen, and by Didymus the Blind, who adds the observation that there are only four perfect numbers that are less than 10,000. (Commentary on Genesis 1. 14–19). St Augustine defines perfect numbers in City of God (Book XI, Chapter 30) in the early 5th century AD, repeating the claim that God created the world in 6 days because 6 is the smallest perfect number. The Egyptian mathematician Ismail ibn Fallūs (1194–1252) mentioned the next three perfect numbers (33,550,336; 8,589,869,056; and 137,438,691,328) and listed a few more which are now known to be incorrect. The first known European mention of the fifth perfect number is a manuscript written between 1456 and 1461 by an unknown mathematician. In 1588, the Italian mathematician Pietro Cataldi identified the sixth (8,589,869,056) and the seventh (137,438,691,328) perfect numbers, and also proved that every perfect number obtained from Euclid's rule ends with a 6 or an 8.

Even perfect numbers

See also: Euclid–Euler theorem

Unsolved problem in mathematics:

Are there infinitely many perfect numbers?

Euclid proved that ${\displaystyle 2^{p-1}(2^p-1)}$ is an even perfect number whenever ${\displaystyle 2^p-1}$ is prime (Elements, Prop. IX.36).

For example, the first four perfect numbers are generated by the formula ${\displaystyle 2^{p-1}(2^p-1),}$ with p a prime number, as follows:

$${\displaystyle \begin{array}{rl}p=2& :2^1(2^2-1)=2\times 3=6\\ p=3& :2^2(2^3-1)=4\times 7=28\\ p=5& :2^4(2^5-1)=16\times 31=496\\ p=7& :2^6(2^7-1)=64\times 127=8128.\end{array}}$$

Prime numbers of the form ${\displaystyle 2^p-1}$ are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. For ${\displaystyle 2^p-1}$ to be prime, it is necessary that p itself be prime. However, not all numbers of the form ${\displaystyle 2^p-1}$ with a prime p are prime; for example, 2¹¹ − 1 = 2047 = 23 × 89 is not a prime number. In fact, Mersenne primes are very rare—of the 2,610,944 prime numbers p up to 43,112,609, ${\displaystyle 2^p-1}$ is prime for only 47 of them.

While Nicomachus had stated (without proof) that all perfect numbers were of the form ${\displaystyle 2^{n-1}(2^n-1)}$ where ${\displaystyle 2^n-1}$ is prime (though he stated this somewhat differently), Ibn al-Haytham (Alhazen) circa AD 1000 was unwilling to go that far, declaring instead (also without proof) that the formula yielded only every even perfect number. It was not until the 18th century that Leonhard Euler proved that the formula ${\displaystyle 2^{p-1}(2^p-1)}$ will yield all the even perfect numbers. Thus, there is a one-to-one correspondence between even perfect numbers and Mersenne primes; each Mersenne prime generates one even perfect number, and vice versa. This result is often referred to as the Euclid–Euler theorem.

An exhaustive search by the GIMPS distributed computing project has shown that the first 48 even perfect numbers are ${\displaystyle 2^{p-1}(2^p-1)}$ for

p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609 and 57885161 (sequence A000043 in the OEIS).

Three higher perfect numbers have also been discovered, namely those for which p = 74207281, 77232917, and 82589933. Although it is still possible there may be others within this range, initial but exhaustive tests by GIMPS have revealed no other perfect numbers for p below 109332539. As of December 2018, 51 Mersenne primes are known, and therefore 51 even perfect numbers (the largest of which is 2^82589932 × (2^82589933 − 1) with 49,724,095 digits). It is not known whether there are infinitely many perfect numbers, nor whether there are infinitely many Mersenne primes.

As well as having the form ${\displaystyle 2^{p-1}(2^p-1)}$, each even perfect number is the ${\displaystyle (2^p-1)}$-th triangular number (and hence equal to the sum of the integers from 1 to ${\displaystyle 2^p-1}$) and the ${\displaystyle 2^{p-1}}$-th hexagonal number. Furthermore, each even perfect number except for 6 is the ${\displaystyle \frac{2^p+1}{3}}$-th centered nonagonal number and is equal to the sum of the first ${\displaystyle 2^{\frac{p-1}{2}}}$ odd cubes (odd cubes up to the cube of ${\displaystyle 2^{\frac{p+1}{2}}-1}$):

$${\displaystyle \begin{array}{rlrl}6& =2^1(2^2-1)& & =1+2+3,\\ 28& =2^2(2^3-1)& & =1+2+3+4+5+6+7\\ & & & =1^3+3^3\\ 496& =2^4(2^5-1)& & =1+2+3+\cdots +29+30+31\\ & & & =1^3+3^3+5^3+7^3\\ 8128& =2^6(2^7-1)& & =1+2+3+\cdots +125+126+127\\ & & & =1^3+3^3+5^3+7^3+9^3+{11}^3+{13}^3+{15}^3\\ 33550336& =2^{12}(2^{13}-1)& & =1+2+3+\cdots +8189+8190+8191\\ & & & =1^3+3^3+5^3+\cdots +{123}^3+{125}^3+{127}^3\end{array}}$$

Even perfect numbers (except 6) are of the form

$${\displaystyle T_{2^p-1}=1+\frac{(2^p-2)\times (2^p+1)}{2}=1+9\times T_{(2^p-2)/3}}$$

with each resulting triangular number T₇ = 28, T₃₁ = 496, T₁₂₇ = 8128 (after subtracting 1 from the perfect number and dividing the result by 9) ending in 3 or 5, the sequence starting with T₂ = 3, T₁₀ = 55, T₄₂ = 903, T₂₇₃₀ = 3727815, ... It follows that by adding the digits of any even perfect number (except 6), then adding the digits of the resulting number, and repeating this process until a single digit (called the digital root) is obtained, always produces the number 1. For example, the digital root of 8128 is 1, because 8 + 1 + 2 + 8 = 19, 1 + 9 = 10, and 1 + 0 = 1. This works with all perfect numbers ${\displaystyle 2^{p-1}(2^p-1)}$ with odd prime p and, in fact, with all numbers of the form ${\displaystyle 2^{m-1}(2^m-1)}$ for odd integer (not necessarily prime) m.

Owing to their form, ${\displaystyle 2^{p-1}(2^p-1),}$ every even perfect number is represented in binary form as p ones followed by p − 1 zeros; for example:

$${\displaystyle \begin{array}{rcl}{6}_{10}=& 2^2+2^1& ={110}_2\\ {28}_{10}=& 2^4+2^3+2^2& ={11100}_2\\ {496}_{10}=& 2^8+2^7+2^6+2^3+2^4& ={111110000}_2\\ {8128}_{10}=& 2^{12}+2^{11}+2^{10}+2^9+2^8+2^7+2^6& ={1111111000000}_2\end{array}}$$

Thus every even perfect number is a pernicious number.

Every even perfect number is also a practical number (cf. Related concepts).

Odd perfect numbers

Unsolved problem in mathematics:

Are there any odd perfect numbers?

(more unsolved problems in mathematics)

It is unknown whether any odd perfect numbers exist, though various results have been obtained. In 1496, Jacques Lefèvre stated that Euclid's rule gives all perfect numbers, thus implying that no odd perfect number exists. Euler stated: "Whether ... there are any odd perfect numbers is a most difficult question". More recently, Carl Pomerance has presented a heuristic argument suggesting that indeed no odd perfect number should exist. All perfect numbers are also harmonic divisor numbers, and it has been conjectured as well that there are no odd harmonic divisor numbers other than 1. Many of the properties proved about odd perfect numbers also apply to Descartes numbers, and Pace Nielsen has suggested that sufficient study of those numbers may lead to a proof that no odd perfect numbers exist.

Any odd perfect number N must satisfy the following conditions:

• N > 10¹⁵⁰⁰.
• N is not divisible by 105.
• N is of the form N ≡ 1 (mod 12) or N ≡ 117 (mod 468) or N ≡ 81 (mod 324).
• N is of the form

${\displaystyle N=q^{\alpha }{p}_1^{2e_1}\cdots p_k^{2e_k},}$

where:

• q, p₁, ..., p_k are distinct odd primes (Euler).
• q ≡ α ≡ 1 (mod 4) (Euler).
• The smallest prime factor of N is at most ${\displaystyle \frac{k-1}{2}.}$
• Either q^α > 10⁶², or p_j^2e_j  > 10⁶² for some j.
• ${\displaystyle N<2^{(4^{k+1}-2^{k+1})}}$
• ${\displaystyle \alpha +2e_1+2e_2+2e_3+\cdots +2e_k\ge \frac{66k-191}{25}}$.
• ${\displaystyle q{p}_1{p}_2{p}_3\cdots p_k<2N^{\frac{17}{26}}}$.

• The largest prime factor of N is greater than 10⁸ and less than ${\displaystyle \sqrt[3]{3N}.}$ 
• The second largest prime factor is greater than 10⁴, and is less than ${\displaystyle \sqrt[5]{2N}}$.
• The third largest prime factor is greater than 100, and less than ${\displaystyle \sqrt[6]{2N}.}$
• N has at least 101 prime factors and at least 10 distinct prime factors. If 3 is not one of the factors of N, then N has at least 12 distinct prime factors.

Furthermore, several minor results are known about the exponents e₁, ..., e_k.

• Not all e_i ≡ 1 (mod 3).
• Not all e_i ≡ 2 (mod 5).
• If all e_i ≡ 1 (mod 3) or 2 (mod 5), then the smallest prime factor of N must lie between 10⁸ and 10¹⁰⁰⁰.
• More generally, if all 2e_i+1 have a prime factor in a given finite set S, then the smallest prime factor of N must be smaller than an effectively computable constant depending only on S.
• If (e₁, ..., e_k) =  (1, ..., 1, 2, ..., 2) with t ones and u twos, then ${\displaystyle (t-1)/4\le u\le 2t+\sqrt{\alpha }}$.
• (e₁, ..., e_k) ≠ (1, ..., 1, 3), (1, ..., 1, 5), (1, ..., 1, 6).
• If e₁ = ... = e_k = e, then
  • e cannot be 3, 5, 24, 6, 8, 11, 14 or 18.
  • ${\displaystyle k\le 2e^2+8e+2}$ and ${\displaystyle N<2^{4^{2e^2+8e+3}}}$.

In 1888, Sylvester stated:

... a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number]—its escape, so to say, from the complex web of conditions which hem it in on all sides—would be little short of a miracle.

Minor results

All even perfect numbers have a very precise form; odd perfect numbers either do not exist or are rare. There are a number of results on perfect numbers that are actually quite easy to prove but nevertheless superficially impressive; some of them also come under Richard Guy's strong law of small numbers:

• The only even perfect number of the form n³ + 1 is 28 (Makowski 1962).
• 28 is also the only even perfect number that is a sum of two positive cubes of integers (Gallardo 2010).
• The reciprocals of the divisors of a perfect number N must add up to 2 (to get this, take the definition of a perfect number, ${\displaystyle {\sigma }_1(n)=2n}$, and divide both sides by n):
  • For 6, we have ${\displaystyle 1/6+1/3+1/2+1/1=2}$;
  • For 28, we have ${\displaystyle 1/28+1/14+1/7+1/4+1/2+1/1=2}$, etc.
• The number of divisors of a perfect number (whether even or odd) must be even, because N cannot be a perfect square.
  • From these two results it follows that every perfect number is an Ore's harmonic number.
• The even perfect numbers are not trapezoidal numbers; that is, they cannot be represented as the difference of two positive non-consecutive triangular numbers. There are only three types of non-trapezoidal numbers: even perfect numbers, powers of two, and the numbers of the form ${\displaystyle 2^{n-1}(2^n+1)}$ formed as the product of a Fermat prime ${\displaystyle 2^n+1}$ with a power of two in a similar way to the construction of even perfect numbers from Mersenne primes.
• The number of perfect numbers less than n is less than ${\displaystyle c\sqrt{n}}$, where c > 0 is a constant. In fact it is ${\displaystyle o(\sqrt{n})}$, using little-o notation.
• Every even perfect number ends in 6 or 28, base ten; and, with the only exception of 6, ends in 1 in base 9. Therefore, in particular the digital root of every even perfect number other than 6 is 1.
• The only square-free perfect number is 6.

Related concepts

Euler diagram of numbers under 100:

   Abundant

   Primitive abundant

   Highly abundant

   Superabundant and highly composite

   Colossally abundant and superior highly composite

   Weird

   Perfect

   Composite

   Deficient

The sum of proper divisors gives various other kinds of numbers. Numbers where the sum is less than the number itself are called deficient, and where it is greater than the number, abundant. These terms, together with perfect itself, come from Greek numerology. A pair of numbers which are the sum of each other's proper divisors are called amicable, and larger cycles of numbers are called sociable. A positive integer such that every smaller positive integer is a sum of distinct divisors of it is a practical number.

By definition, a perfect number is a fixed point of the restricted divisor function s(n) = σ(n) − n, and the aliquot sequence associated with a perfect number is a constant sequence. All perfect numbers are also ${\displaystyle \mathcal{S}}$-perfect numbers, or Granville numbers.

A semiperfect number is a natural number that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. Most abundant numbers are also semiperfect; abundant numbers which are not semiperfect are called weird numbers.

Syntax

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Syntax

Major levels of linguistic structure. Syntax is shown encompassed by semantics, and encompassing morphology.

In linguistics, syntax (/ˈsɪntæks/ SIN-taks) is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure (constituency), agreement, the nature of crosslinguistic variation, and the relationship between form and meaning (semantics). There are numerous approaches to syntax that differ in their central assumptions and goals.

Etymology

The word syntax comes from Ancient Greek roots: σύνταξις "coordination", which consists of σύν syn, "together", and τάξις táxis, "ordering".

Topics

The field of syntax contains a number of various topics that a syntactic theory is often designed to handle. The relation between the topics is treated differently in different theories, and some of them may not be considered to be distinct but instead to be derived from one another (i.e. word order can be seen as the result of movement rules derived from grammatical relations).

Sequencing of subject, verb, and object

"Subject and object" redirects here. For philosophical terms, see Subject (philosophy) and Object (philosophy).

One basic description of a language's syntax is the sequence in which the subject (S), verb (V), and object (O) usually appear in sentences. Over 85% of languages usually place the subject first, either in the sequence SVO or the sequence SOV. The other possible sequences are VSO, VOS, OVS, and OSV, the last three of which are rare. In most generative theories of syntax, the surface differences arise from a more complex clausal phrase structure, and each order may be compatible with multiple derivations. However, word order can also reflect the semantics or function of the ordered elements.

Grammatical relations

Another description of a language considers the set of possible grammatical relations in a language or in general and how they behave in relation to one another in the morphosyntactic alignment of the language. The description of grammatical relations can also reflect transitivity, passivization, and head-dependent-marking or other agreement. Languages have different criteria for grammatical relations. For example, subjecthood criteria may have implications for how the subject is referred to from a relative clause or coreferential with an element in an infinite clause.

Constituency

Constituency is the feature of being a constituent and how words can work together to form a constituent (or phrase). Constituents are often moved as units, and the constituent can be the domain of agreement. Some languages allow discontinuous phrases in which words belonging to the same constituent are not immediately adjacent but are broken up by other constituents. Constituents may be recursive, as they may consist of other constituents, potentially of the same type.

Early history

The Aṣṭādhyāyī of Pāṇini, from c. 4th century BC in Ancient India, is often cited as an example of a premodern work that approaches the sophistication of a modern syntactic theory since works on grammar had been written long before modern syntax came about. In the West, the school of thought that came to be known as "traditional grammar" began with the work of Dionysius Thrax.

For centuries, a framework known as grammaire générale, first expounded in 1660 by Antoine Arnauld and Claude Lancelot in a book of the same title, dominated work in syntax: as its basic premise the assumption that language is a direct reflection of thought processes and so there is a single most natural way to express a thought.

However, in the 19th century, with the development of historical-comparative linguistics, linguists began to realize the sheer diversity of human language and to question fundamental assumptions about the relationship between language and logic. It became apparent that there was no such thing as the most natural way to express a thought and so logic could no longer be relied upon as a basis for studying the structure of language.

The Port-Royal grammar modeled the study of syntax upon that of logic. (Indeed, large parts of Port-Royal Logic were copied or adapted from the Grammaire générale.) Syntactic categories were identified with logical ones, and all sentences were analyzed in terms of "subject – copula – predicate". Initially, that view was adopted even by the early comparative linguists such as Franz Bopp.

The central role of syntax within theoretical linguistics became clear only in the 20th century, which could reasonably be called the "century of syntactic theory" as far as linguistics is concerned. (For a detailed and critical survey of the history of syntax in the last two centuries, see the monumental work by Giorgio Graffi (2001).)

Theories

See also: Theory of language

There are a number of theoretical approaches to the discipline of syntax. One school of thought, founded in the works of Derek Bickerton, sees syntax as a branch of biology, since it conceives of syntax as the study of linguistic knowledge as embodied in the human mind. Other linguists (e.g., Gerald Gazdar) take a more Platonistic view since they regard syntax to be the study of an abstract formal system. Yet others (e.g., Joseph Greenberg) consider syntax a taxonomical device to reach broad generalizations across languages.

Syntacticians have attempted to explain the causes of word-order variation within individual languages and cross-linguistically. Much of such work has been done within the framework of generative grammar, which holds that syntax depends on a genetic endowment common to the human species. In that framework and in others, linguistic typology and universals have been primary explicanda.

Alternative explanations, such as those by functional linguists, have been sought in language processing. It is suggested that the brain finds it easier to parse syntactic patterns that are either right- or left-branching but not mixed. The most-widely held approach is the performance–grammar correspondence hypothesis by John A. Hawkins, who suggests that language is a non-innate adaptation to innate cognitive mechanisms. Cross-linguistic tendencies are considered as being based on language users' preference for grammars that are organized efficiently and on their avoidance of word orderings that cause processing difficulty. Some languages, however, exhibit regular inefficient patterning such as the VO languages Chinese, with the adpositional phrase before the verb, and Finnish, which has postpositions, but there are few other profoundly exceptional languages. More recently, it is suggested that the left- versus right-branching patterns are cross-linguistically related only to the place of role-marking connectives (adpositions and subordinators), which links the phenomena with the semantic mapping of sentences.

Theoretical syntactic models

Dependency grammar

Main article: Dependency grammar

Dependency grammar is an approach to sentence structure in which syntactic units are arranged according to the dependency relation, as opposed to the constituency relation of phrase structure grammars. Dependencies are directed links between words. The (finite) verb is seen as the root of all clause structure and all the other words in the clause are either directly or indirectly dependent on the root. Some prominent dependency-based theories of syntax are the following:

• Recursive categorical syntax, or algebraic syntax
• Functional generative description
• Meaning–text theory
• Operator grammar
• Word grammar

Lucien Tesnière (1893–1954) is widely seen as the father of modern dependency-based theories of syntax and grammar. He argued vehemently against the binary division of the clause into subject and predicate that is associated with the grammars of his day (S → NP VP) and remains at the core of most phrase structure grammars. In the place of that division, he positioned the verb as the root of all clause structure.

Categorial grammar

Main article: Categorial grammar

Categorial grammar is an approach in which constituents combine as function and argument, according to combinatory possibilities specified in their syntactic categories. For example, other approaches might posit a rule that combines a noun phrase (NP) and a verb phrase (VP), but CG would posit a syntactic category NP and another NP\S, read as "a category that searches to the left (indicated by \) for an NP (the element on the left) and outputs a sentence (the element on the right)." Thus, the syntactic category for an intransitive verb is a complex formula representing the fact that the verb acts as a function word requiring an NP as an input and produces a sentence level structure as an output. The complex category is notated as (NP\S) instead of V. The category of transitive verb is defined as an element that requires two NPs (its subject and its direct object) to form a sentence. That is notated as (NP/(NP\S)), which means, "A category that searches to the right (indicated by /) for an NP (the object) and generates a function (equivalent to the VP) which is (NP\S), which in turn represents a function that searches to the left for an NP and produces a sentence."

Tree-adjoining grammar is a categorial grammar that adds in partial tree structures to the categories.

Stochastic/probabilistic grammars/network theories

Theoretical approaches to syntax that are based upon probability theory are known as stochastic grammars. One common implementation of such an approach makes use of a neural network or connectionism.

Functional grammars

Main article: Functional theories of grammar

Functionalist models of grammar study the form–function interaction by performing a structural and a functional analysis.

• Functional discourse grammar (Dik)
• Prague linguistic circle
• Role and reference grammar (RRG)
• Systemic functional grammar

Generative syntax

Generative syntax is the study of syntax within the overarching framework of generative grammar. Generative theories of syntax typically propose analyses of grammatical patterns using formal tools such as phrase structure grammars augmented with additional operations such as syntactic movement. Their goal in analyzing a particular language is to specify rules which generate all and only the expressions which are well-formed in that language. In doing so, they seek to identify innate domain-specific principles of linguistic cognition, in line with the wider goals of the generative enterprise. Generative syntax is among the approaches that adopt the principle of the autonomy of syntax by assuming that meaning and communicative intent is determined by the syntax, rather than the other way around.

Generative syntax was proposed in the late 1950s by Noam Chomsky, building on earlier work by Zellig Harris, Louis Hjelmslev, and others. Since then, numerous theories have been proposed under its umbrella:

• Transformational grammar (TG) (Original theory of generative syntax laid out by Chomsky in Syntactic Structures in 1957)
• Government and binding theory (GB) (revised theory in the tradition of TG developed mainly by Chomsky in the 1970s and 1980s)
• Minimalist program (MP) (a reworking of the theory out of the GB framework published by Chomsky in 1995)

Other theories that find their origin in the generative paradigm are:

• Arc pair grammar
• Generalized phrase structure grammar (GPSG)
• Generative semantics
• Head-driven phrase structure grammar (HPSG)
• Lexical functional grammar (LFG)
• Nanosyntax
• Relational grammar (RG)
• Harmonic grammar (HG)

Cognitive and usage-based grammars

Main article: Cognitive Linguistics

The Cognitive Linguistics framework stems from generative grammar but adheres to evolutionary, rather than Chomskyan, linguistics. Cognitive models often recognise the generative assumption that the object belongs to the verb phrase. Cognitive frameworks include the following:

• Cognitive grammar
• Construction grammar (CxG)
• Emergent grammar