Jamie Frost – Franks Society MT10

What is language?

What is a language? Wikipedia “A set of symbols of communication and the elements used to manipulate them.” OED “The system of spoken or written communication used by a particular country, people, community, etc., typically consisting of words used within a regular grammatical and syntactic structure.”

What is a language The possible symbols that each ‘unit’ in the language can take. For human languages, the alphabet may be at a character level. Or we could choose it to be at a word level...

What is a language Σ 2 = Σ × Σ gives us all the possible pairs of symbols. Σ * = { λ } ∪ Σ ∪ (Σ × Σ) ∪ (Σ × Σ × Σ) ∪... is known as the Kleene Star, and gives us all possible strings, i.e. containing any combination of symbols and of any length.

What is a language Any sensible language doesn’t allow a unrestricted combination of symbols. Human languages are bounded by some grammatical structure.

Mint.

Grammars So how do we define a grammar? Grammars limit our possible strings to certain forms. These vary in expressiveness – the more expressible they are, the harder it is to do certain common tasks with them. Expressiveness Task Complexity Tasks might include: “finding the grammatical structure of a string given a grammar”, or “does a string satisfy a given grammar?”. “are two grammars equivalent?”

The Chomsky Hierarchy In 1956, Noam Chomsky characterised languages according to how ‘complex’ or expressible they are. This is known as the Chomsky Hierarchy. A language that satisfies a given type is also a instance of all the grammars above it. GrammarLanguage Type-0Recursively Enumerable Type-1Context Sensitive Type-2Context Free Type-3Regular

A Formal Grammar Consists of: 1.S ⟶ i love T 2.T ⟶ T and T 3.T ⟶ smurfs 4.T ⟶ smurfettes Terminal symbols (i.e. our alphabet Σ) Non-terminal symbols (N) A start symbol (S ϵ N) Production rules

S ⟶ i love T ⟶ i love T and T ⟶ i love smurfs and T ⟶ i love smurfs and T and T ⟶ i love smurfs and smurfs and T ⟶ i love smurfs and smurfs and smurfettes A Formal Grammar 1.S ⟶ i love T 2.T ⟶ T and T 3.T ⟶ smurfs 4.T ⟶ smurfettes C) We’re not allowed to ‘finish’ until we only have terminal symbols. A) Start with start symbol B) We can use the production rules the replace things. Think of it as a game...

Regular Grammars The most restrictive. The LHS of the production rules can only be a single non-terminal. The RHS of the production rules can be one of (a) a single terminal symbol (b) a single non-terminal symbol (c) a terminal followed by a non-terminal or (d) the empty symbol. The idea is that you don’t have ‘memory’ of the symbols you’ve previously emitted in the string.

Regular Grammars Example Example Generation: S aS aaS aaaT aaabT aaabb Notice we’re always generating at the end of the string.

Regular Grammar a a b This kind of diagram is known as a ‘nondeterministic finite automaton’ or NFA.

Regular Grammar We can use this picture to work out the regular grammar: a a b

It’s Voting Time... The language of palindromes, i.e. strings which are the same when reversed, e.g. “madam”, “acrobats stab orca”. { a n b n | n ≥ 1 } i.e. ab, aabb, aaabbb, aaaabbbb,... Neither are. The problem is that we cannot ‘remember’ the symbols already emitted. We can use something called the pumping lemma to check if a language is regular.

Context Free Grammars The restriction on the RHS of the production rules is now loosened; we can have any combination of non- terminals and terminals. We still restrict the LHS however to a single non- terminal. This is why the grammar is known as “context free”, since the production is not dependent on the context in which it occurs. While generating a string:... abXd abyd The production rule which allows the X to become a y is not contingent on the context, i.e. The preceding b or the proceeding d.

Context Free Grammars Examples Example Generation: S aSa acSca acbca

Examples Context Free Grammars Example generation: S aSb aaSbb aaaSbbb aaabbb

It’s Voting Time... { a n b n c n | n ≥ 1 } i.e. abc, aabbcc, aaabbbccc,... Nope. A bit harder to see this time. Can use a variant of the Pumping Lemma called the Bar-Hillel Lemma to show it isn’t. (But informal explanation as such: We could have non-terminals at the a-b and b-c boundary generating these pairs, but since our language is context free these non-terminals expand independently of each other, thus we can only ensure a and b have the same count, or b and c. And we can’t have a rule of the form S-> X abc Y because then we can’t subsequently increase the number of b’s.)

Context-Sensitive Grammars Now an expansion of a non-terminal is dependent on the context it appears in. Example generation: S aSBC aaBCBC aaBHBC aaBBCC aabBCC aabbCC aabbcC aabbcc i.e. a ‘C’ can change into a ‘c’ only when preceded by another ‘c’. Note that this context (i.e. this preceding ‘c’) must remain unchanged. Preservation of context is the only restriction in CSGs.

The Chomsky Hierarchy Gram mar LanguageAutomatonRules Type-0Recursively enumerable Turing Machine α ⟶ βα ⟶ β Type-1Context sensitive Linear bounded Non- deterministic Turing Machine α A β ⟶ αγβ Type-2Context Free Non-deterministic Pushdown Automaton A ⟶ γ Type-3RegularFinite State Automaton A ⟶ a and A ⟶ aB The picture with circles and arrows we saw earlier.

English as a CFG Before we get on to classifying English according to the Chomsky Hierarchy, let’s see how English might be represented as a CFG. Our starting non-terminal S is a sentence. Since sentences operate independently syntactically, it’s sufficient to examine grammar on a sentence level. Our terminals/alphabet Σ is just a dictionary in the literal sense. Σ = { a, aardvark,...., zebra, zoology, zyzzyva }

English as a CFG Our non-terminals are ‘constituents’, such as noun phrases, verb phrases, verbs, determiners, prepositional phrases, etc. These can be subdivided into further constituents (e.g. NP = noun phrase), or generate a terminals (e.g. V = verb) Non-TerminalNameExample NPNoun Phrasethe cat VPVerb Phrasechastised the politician PPPrepositional Phrasewith the broccoli CONJConjunctionand VVerbchundered ADVAdverbeverywhere

English as a CFG Can use an American style ‘top-down’ generative form of grammar. S NP VP NP DT N NP PN VP VP PP NP ⟶ NP PP PP P NP NP NP CONJ NP S NP VP NP DT N NP PN VP VP PP NP ⟶ NP PP PP P NP NP NP CONJ NP DT the DT a N monkey N student N ⟶ telescope PN Corey P with P over CONJ and CONJ or V ⟶ saw V ⟶ ate DT the DT a N monkey N student N ⟶ telescope PN Corey P with P over CONJ and CONJ or V ⟶ saw V ⟶ ate

Ambiguity Curiously, it’s possible to generate a sentence in multiple ways! S ⟶ NP VP ⟶ PN VP ⟶ Corey VP ⟶ Corey VP PP ⟶ Corey V NP PP ⟶ Corey saw NP PP ⟶... ⟶ Corey saw the monkey with the telescope. S ⟶ NP VP ⟶ PN VP ⟶ Corey VP ⟶ Corey V NP ⟶ Corey saw NP ⟶ Corey saw NP PP ⟶... ⟶ Corey saw the monkey with the telescope.

Ambiguity We say that a formal grammar that can yield the same string from multiple derivations is ‘ambiguous’.

So what kind of language...

Embedded Structures (Yngve 60) The cat likes tuna fish. The cat the dog chased likes tuna fish. The cat the dog the rat bit chased likes tuna fish. The cat the dog the rat the elephant admired bit chased likes tuna fish.

Embedded Structures The cat the dog the rat the elephant admired bit chased likes tuna fish. If we let A = { the dog, the rat, the elephant } and B = { admired, bit, chased } then we represent centre-embedding as such: But we already know from earlier that a n b n is not regular!

So what kind of language...

Swiss German A number of languages, such as Dutch and Swiss German, allow for cross-serial dependencies...mer d’chind em Hans es huus haend wele laa halfe aastriiche...we the children/ACC Hans/DAT the house/ACC have wanted to let help paint. DAT = Dative noun: the indirect object of a verb (e.g. John gave Mary the book”). ACC = Accusative noun: the direct object of a verb (e.g. John gave Mary the book”).

Swiss German Shieber (1985) notes that among such sentences, those with all accusative NPs preceding all dative NPs, and all accusative-subcategorising verbs preceding all dative-subcategorising verbs are acceptable. The number of verbs requiring dative objects (halfe) must equal the number of dative NPs (em Hans) and similarly for accusatives....mer d’chind em Hans es huus haend wele laa halfe aastriiche...we the children/ACC Hans/DAT the house/ACC have wanted to let help paint.

Swiss German...mer d’chind em Hans es huus haend wele laa halfe aastriiche...we the children/ACC Hans/DAT the house/ACC have wanted to let help paint.

Summary The Chomsky Hierarchy brings together different grammar formalisms, listing them in increasing order of expressiveness. Languages don’t have to be human ones: they allow us to generate strings with some given alphabet Σ, subject to some grammatical constraints. The least expressive languages are Regular Grammars, which are insufficient to represent the English language. But Context Free Grammars are insufficient to represent languages with cross-serial dependencies, such as Swiss German.

Questions?