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FEATERLESS HPSG & HPSG : featureless or unprincipled? 2 papers by Andrew Moshier Comparative Introduction to lexicalist syntactic theories WS 2005/2006 Antske Fokkens antske@gmail.com Today’s presentation • Mostly based on the article Featureless HPSG. • Why all this trouble? – Two main questions the model of HPSG leads to.. – Three ways to approach HPSG – The engineering approach: importance of modularity – The interaction of feature geometry and principles: a modularity problem – Towards a solution • Categorial model for HPSG • Definitions for a model of HPSG Two main questions • The HPSG model leads to two interesting questions: (1) What should the model look like? (2) What should the role of a model be? • While the second question logically comes first, only the first has been addressed so far…. The second question • ….so what is the answer to the second question? Role of the Model • A model – Firstly is there to ensure coherence in the theory (“or so logicians tell us”) – But more importantly a model: – Allows for independency, – Non-entailment and – Relative coherence • These two properties are also mentioned by Pollard and Sag (1987) “…HPSG shares a concern about matters of modularity of design and the deduction of particular facts from the complex interaction of general principles.” The problem with HPSG • Lack of modularity this can be seen when • Comparing the different definitions of the Head Feature Principle • Organizing agreement features • From a theoretical point of view, it is difficult to proof that certain definitions are equal, but there is also another side of the story…. Three approaches to HPSG • HPSG is often looked at from a linguistics point of view, or • A theoretic point of view, but • Can also be looked at from an engineering point of view • Looking at HPSG from the engineering point of view, there are a couple of things one needs to take into account…. Implementing a grammar… • Is a task that is too complex for one person to complete by him/herself. • Involves many revisions/refinements of rules at different places in the grammar • Therefore, the notion of modularity is important. The importance of Modularity • Several engineers need to be able to work on the grammar at the same time • without changes in the area of one engineer causing problems in a domain another engineer is working on… • Modularity allows the engineer to make changes in one place, without it causing many problematic changes in another part of the grammar. How to improve modularity in HPSG? • Moshier’s proposal : the way HPSG is designed, it reminds of a system of category theoretic constraints…therefore it can be reanalyzed using only category- theoretic properties. Category theory • A category is a model of the first order theory of composition. • Every arrow has a specific domain and codomain • If f : A → B , then there is an arrow f with domain A and codomain B • Arrows compose iff their codomain and domain match correctly • If f : A → B and g : B → C then there is an arrow g○f : A → C and • If g○f : A → C , then there is a domain B for which f : A → B and g : B → C • Entities that can serve as domains are called objects. Category theory (cont) • Composition is associative f○ (g ○ h) = (f ○ g) ○ h • Every object has an identity arrow, so if B is an object, -then there exists an arrow idB : B → B for which • h ○ idB = h for all h : B → C’ and • j = idB ○ j for all j : A’ → B Category theory –denoting objects and arrows • For any category A, let |A| denote the collection of objects of the category • And for A,B є |A|, let A(A,B) denote the collection of arrows from A to B. Functor • A functor from category A to B is a structure preserving map from A to B. The structure of a category corresponds to the composition of the (identity) arrows. • A functor F from A to B maps arrows in A to arrows in B, in a way that – (i) there exists an identity arrow F(idA) – (ii) F(f ○ g) = F(f) ○ F(g) wherever f ○ g is defined Diagram • For any category A an A-diagram is a triple D = (G,A,f) where • G is a directed graph • A labels vertices and f edges, so that each vertex v is labeled by an object Av є |A| and each edge v to v’ is labeled by an arrow fe є A (Av, Av’). • .v e .v‘ label label . Av fe . Av’ Diagram (cont) .v e .v‘ e . v’’ label label . Av fe . Av’ . Av’’ • Let E(v, v’) denote the set of edges in G from v to v’ • Let G(v, v’) denote the set of paths in G from vertex v to vertex v’ • The labeling of edges in G determines, for each p є G(v, v’), an arrow fp : Av → Av’ • If <e0,…,en-1> є G(v, v’) and en є E(v’,v’’), then f<e ,…,en> = fen ○ f<e ,..,en-1> 0 0 Directed graph G • Determines [G] with vertices as objects and paths as arrows. • Composition is defined as concatenation of paths .v p .v‘ F .v p . v‘ L:[G] L:[G] . Av fp . Av’ F . B v gp . Bv’ – L : [G] A –F:A B – (G, B, g) is the diagram determined by F ○ L Commutativity • An A-diagram (G, A, f) commutes iff • For each to vertices v and v’ and • For every two paths p,q є G(v, v’) fp = fq A model of HPSG • A functor Μ : H → A • ‘H is a formally defined category of the form [H] for the directed graph H in which both vertices and edges are certain strings over a fixed alphabet…H takes the place of a formal language for HPSG’. • If A is a Set (and we will suppose it is for reasons of simplification), then a model of HPSG is a Set-diagram • M sends objects of H to sets and arrows of H to functions. H • Objects of H correspond to sign and synsem • Arrows correspond to paths of features (f.i SYNSEM|LOC|CAT) The Codomain • Is the “semantic category” • M picks out sets as the interpretations of sort names and functions as the interpretations of feature paths H FEATURE-PATH sign M M A Interpretation of sign Interpretation of feature path To model HPSG… • “For each sort σ there must be corresponding sorts that represent lists of σ and sets of σ.” • This means that – H must be a rich enough vocabulary to say things about lists and sets. – A must have suitable interpretations of lists and sets – M must carry the vocabulary of H over to interpretations in A correctly interpreting lists and sets. • Each of these points will be discussed in detail. Formal categories for HPSG • H should be defined formally, as a formal language • H should have a vocabulary which is rich enough • A particular H should be determined by something analogous to HPSG appropriateness conditions Sets S and L • S and L • Let S and L be disjoint sets, a formal category over alphabets S and L is of the form [H], for a directed graph H satisfying that – the vertices of H are drawn from S* – the edges of H are drawn from L* Basic Objects and Arrows and their symbols • Basic Objects (BO) and Basic Arrows (BA) are disjoint infinite sets. – BO includes objects like sign, – BA includes arrows like PHON. • Symbols for Objects are {1, x, L, P, M, [, ]} • Symbols for Arrows are {id, ◊, , , <,>, π,π’,nil, cons, Λ,[,]} H = [H] is a formal category over BO U SO and BA U SA Then H has: • Formal Identities (FI) • Formal Terminals (FT) • Formal Products (FPr) • Formal Lists (FL) • Formal Powers (FPo) H is definite provided that canonical names are used only occur as required by the definition of formal HPSG categories (if FI, FT, FPr, FL and FPo are defined) Formal identities • H has formal identities iff id : σ → σ is an edge in H for all σ є |H | • Defined: If id : σ → τ is an edge, then σ = τ Formal Terminals • H has formal terminals iff 1. 1 є |H |; and 2. σ є |H | implies ◊ : σ → 1 is an edge • Defined: If ◊ : σ → τ is an edge, then τ = 1 Formal Products • H has formal products iff 1. σ, τ є |H| implies that [σ x τ ] є |H|, and that π: [σ x τ] → σ and π’ [σ x τ] → τ are edges; and 2. If p : ρ → σ and q: ρ → τ are paths, then <p,q> : ρ → [σ x τ] is an edge. • Defined : If [ω x ω’] є |H|, then ω, ω’ є |H| if π : ρ → σ is an edge then ρ = [σ x τ] for some τ; and if π’ : ρ → τ is an edge, then ρ = [σ x τ] for some σ. Also if <p,q> : ρ → ρ’ is an edge, the ρ’ = [σ x τ], and p: ρ → σ and q : ρ → τ are paths for some σ and τ. Formal Lists • H has formal lists, iff it has formal products and 1. σ є |H| implies that L[σ] є |H| and that nil : 1 → L[σ] and cons : [σ x L[σ]] → L[σ] are edges; and 2. If p : 1 → τ and q : [σ x τ] → τ are paths, then fold[p,q] : L[σ] → τ is an edge. • defined L[ω] є |H| , then ω є |H|. If nil : ρ → τ is an edge then ρ = 1 and τ = L[σ] for some σ. If cons : ρ → τ, then ρ = [σ x L[σ]] and τ = L[σ] for some σ. If fold[p,q] : ρ → τ is an edge, then for some σ є |H|, ρ = L[σ], and p : 1 → τ and q : [σ x τ] → τ are paths. Formal Powers • H has formal powers iff 1. σ є |H| implies that P[σ], M[σ] є |H| and that leg : M[σ] → P[σ] and leg’ M[σ] → σ are edges 2. If l : ρ →τ and r : ρ → σ are paths Λ[l,r] : τ → P[σ] is an edge • Defined P[ω] є |H| iff M[ω] є |H| . If P[ω] є |H| then ω є |H|. If leg : ρ → τ is an edge, then ρ = M[σ] and τ = P[σ] for some σ. If leg’ : ρ → τ is an edge then ρ = M[τ]. If Λ[l,r] : τ → u is an edge, then there exists σ,ρ є |H| so that u = P[σ] and r : ρ → σ and l: ρ → τ are paths. Appropriateness conditions • How to specify this formal definite H ? • In HPSG specifications are introduced via appropriateness conditions specifying the existence of a feature f for a sort σ. • Moshier looks for a formal counterpart for appropriateness conditions. Feature structure • In HPSG: f0 τ0 PHON L(phon-string) s: sign : fn-1 τn-1 SYNSEM synsem • S is a basic sort name, f are feature names and τ are intended to denote other sorts. Subsorts • Another specification in HPSG is that sorts are regarded as partially ordered. • In a formal HPSG category, arrows can be used to define subsorts, f.i., if σ is a subsort of τ, then there is an arrow σ → τ • This arrow (with certain additional properties (?!)) can be seen as an appropriate condition that is running backwards… • In both cases there is a basic sort and a set of pairs (f, τ) Appropriateness condition for HPSG • ‘Define an appropriateness condition for a formal category H to be a triple α = (s, φ, d) where 1. s є BO 2. φ is a finite set of pairs in BA x |H| 3. And d є {f,b}, here d is intended to indicate the direction that arrows are supposed to run’ • If d = f, then α is called a forward appropriateness condition, else it is called a backward appropriateness condition Approp. condition (cont.) • The extent of α = (s,φ, d) , denoted by E(α) is defined: • E(s,φ,d) = {(s,f,τ) | (f,τ) є φ} for d = f {(τ,f,s) | (f,τ) є φ} for d = b. • The focus of α, denoted F(α) is defined as {s}. H+α • If H is a formal HPSG category and α is an appropriate condition for H, then – Let H + α denote the least formal HPSG category so that – F(α) |H + α| and – For all (σ, f,ρ) є E(α), f: σ → ρ is an edge in H + α • H + α is guaranteed to exist (formal categories of HPSG are inductive, and • When triples E(α) already correspond to edges in H, then H + α = H Interpretations of formal HPSG categories • An interpretation of a formal HPSG category is a functor I : H → A, for which the following properties hold: Interpretations • Formal identity: I(id : σ → σ) is the identity for I(σ) • Formal Terminal: I(1) is a terminal object. • Formal Products : I((σ x τ)) is a product of I(σ) and I(τ). I(:→) and I(π’: [σ x τ] → τ) are corresponding projections for p : ρ → σ and q: ρ → τ, I(p : ρ → σ) = I(<p,q>)| π : ρ → σ) I(q: ρ → τ) = I(<p,q>)| π’ : ρ → τ) Interpretations (cont.) • Formal lists I(nil : 1 → L[σ]) and I(cons : [σ x L[σ]] → L[σ]) define lists for I(σ ). For p : 1 → τ and q : (σ x τ) → τ, I(p : 1 → τ) = I(nil|fold[p,q] : 1 → τ) I([id x fold[p,q]]|q : (σ x L[σ]] → τ) = I(cons|fold[p,q] : (σ x L[σ]] → τ) Interpretations (cont) • Formal Powers I(leg : M[σ] → P[σ]) and I(leg’ M[σ] → σ ) constitute a universal relation to I(σ). For p : ρ →τ and q : ρ → σ holds; • A g I(M[σ]) f I(leg) I(τ) I(Λ[p,q]) I(P[σ]) is a pullback of I(Λ[p,q] : τ → P[σ]) and I(M[σ] → P[σ]), then for every irreducible table g’ B I(σ) f’ I(τ) If there is an arrow h : I(ρ) → B so that I(p : ρ →τ ) = f’h and I(q : ρ → σ ) = g’h, then there is also an arrow h’ : A → B, so that f = f’h’ and g = g’h’. Summary • Today we have seen – How a formal model of HPSG can be developed. – A precisely defined vocabulary of sort names and arrow names has been defined, in which HPSG can be formalized. – How such a category can be interpreted in Set. Other parts we will see next week… • Constraints on Interpretations • The principles • Equivalent theories… Next week • Please read the articles again ;-)…

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posted: | 9/9/2012 |

language: | English |

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