On Two Extensions of Abstract Categorial Grammars

Categorial Grammar • Type-theoretic grammar formalism for describing natural languages • Based on the implicative fragment of linear logic • Resource sensitivity • Simple but enough expressive • Mildly context-sensitive languages are generated by second-order ACGs (de Groote and Pogodalla 2004) On Two Extensions of ACGs – p.3/45 Abstract Categorial Grammar • Type-theoretic grammar formalism for describing natural languages • Based on the implicative fragment of linear logic • Resource sensitivity • Simple but enough expressive • Mildly context-sensitive languages are generated by second-order ACGs (de Groote and Pogodalla 2004)Categorial Grammar • Type-theoretic grammar formalism for describing natural languages • Based on the implicative fragment of linear logic • Resource sensitivity • Simple but enough expressive • Mildly context-sensitive languages are generated by second-order ACGs (de Groote and Pogodalla 2004) On Two Extensions of ACGs – p.3/45 Type-Theoretic Extensions of ACGs • De Groote and Maarek (2007) • To enable ACGs to, for instance, treat feature structures ACGs with Cartesian product ( &) ACGs with dependent product ( Π) • Those two extensions are Turing-complete formalisms On Two Extensions of ACGs – p.4/45 Type-Theoretic Extensions of ACGs • De Groote and Maarek (2007) • To enable ACGs to, for instance, treat feature structures ACGs with Cartesian product ( &) ACGs with dependent product ( Π) • Those two extensions are Turing-complete formalisms On Two Extensions of ACGs – p.4/45 Type-Theoretic Extensions of ACGs • De Groote and Maarek (2007) • To enable ACGs to, for instance, treat feature structures ACGs with Cartesian product ( &) ACGs with dependent product ( Π) • Those two extensions are Turing-complete formalisms On Two Extensions of ACGs – p.4/45 2. Abstract Categorial Grammars On Two Extensions of ACGs – p.5/45 Types and Terms • A: a set of atomic types TA ::= A | (TA ( TA) α ( β ( γ ( δ abbreviates (α ( (β ( (γ ( δ))). • C: a set of constants ΛC ::= C | x | (λ ◦x.ΛC) | (ΛCΛC) λxyz.stu abbreviates (λx.(λy.(λz.((st)u)))). On Two Extensions of ACGs – p.6/45 Types and Terms • A: a set of atomic types TA ::= A | (TA ( TA) α ( β ( γ ( δ abbreviates (α ( (β ( (γ ( δ))). • C: a set of constants ΛC ::= C | x | (λ ◦x.ΛC) | (ΛCΛC) λxyz.stu abbreviates (λx.(λy.(λz.((st)u)))). On Two Extensions of ACGs – p.6/45 Typing System Σ = 〈A, C, τ〉: a higher-order signature • A: a set of atomic types, • C: a set of constants, • τ : C → T (A). `Σ c : τ(c) for c ∈ C x : α `Σ x : α for α ∈ TA Γ, x : α `Σ t : β Γ `Σ (λx.t) : α ( β x 6∈ dom(Γ) Γ `Σ t : (α ( β) ∆ `Σ u : α Γ, ∆ `Σ (tu) : β dom(Γ) ∩ dom(∆) = ∅ On Two Extensions of ACGs – p.7/45 Lexicon • A lexicon L = 〈σ, θ〉 from Σ1 = 〈A1, C1, τ1〉 to Σ2 = 〈A2, C2, τ2〉: σ : A1 → TA2 (type substitution) θ : C1 → ΛΣ2 (term substitution) `Σ2 θ(c) : σ̂(τ1(c)) is provable for all c ∈ C1, where σ̂ is the homomorphic extension of σ. • We write L (·) for σ̂(·) or θ̂(·), where θ̂ is the homomorphic extension of θ. • x1 : α1, . . . , xm : αm `Σ1 t : α implies x1 : L (α1), . . . , xm : L (αm) `Σ2 L (t) : L (α), On Two Extensions of ACGs – p.8/45 Abstract Categorial Grammars An ACG G = 〈Σ1, Σ2, L , s〉: • Σ1: abstract vocabulary • Σ2: object vocabulary • L : a lexicon from Σ1 to Σ2 • s ∈ A1: the distinguished type Abstract Language: A(G ) = { t ∈ ΛΣ1 |`Σ1 t : s is provable } Object Language: O(G ) = { t ∈ ΛΣ2 | ∃u ∈ A(G ) such that L (u) = t } We work modulo β: (λx.t)u →β t[x := u] Lexicon: `Σ2 L (c) : L (τ1(c))Categorial Grammars An ACG G = 〈Σ1, Σ2, L , s〉: • Σ1: abstract vocabulary • Σ2: object vocabulary • L : a lexicon from Σ1 to Σ2 • s ∈ A1: the distinguished type Abstract Language: A(G ) = { t ∈ ΛΣ1 |`Σ1 t : s is provable } Object Language: O(G ) = { t ∈ ΛΣ2 | ∃u ∈ A(G ) such that L (u) = t } We work modulo β: (λx.t)u →β t[x := u] Lexicon: `Σ2 L (c) : L (τ1(c)) On Two Extensions of ACGs – p.9/45 Abstract Categorial Grammars An ACG G = 〈Σ1, Σ2, L , s〉: • Σ1: abstract vocabulary • Σ2: object vocabulary • L : a lexicon from Σ1 to Σ2 • s ∈ A1: the distinguished type Abstract Language: A(G ) = { t ∈ ΛΣ1 |`Σ1 t : s is provable } Object Language: O(G ) = { t ∈ ΛΣ2 | ∃u ∈ A(G ) such that L (u) = t } Lexicon: `Σ2 L (c) : L (τ1(c)) On Two Extensions of ACGs – p.9/45Categorial Grammars An ACG G = 〈Σ1, Σ2, L , s〉: • Σ1: abstract vocabulary • Σ2: object vocabulary • L : a lexicon from Σ1 to Σ2 • s ∈ A1: the distinguished type Abstract Language: A(G ) = { t ∈ ΛΣ1 |`Σ1 t : s is provable } Object Language: O(G ) = { t ∈ ΛΣ2 | ∃u ∈ A(G ) such that L (u) = t } Lexicon: `Σ2 L (c) : L (τ1(c)) On Two Extensions of ACGs – p.9/45 Order of an ACG • Order of types: order(p) = 1 for p ∈ A. order(α ( β) = max{order(α) + 1, order(β)} for α ( β ∈ TA. • Order of a higher-order signature Σ = 〈A, C, τ〉: order(Σ) = max{ order(τ(c)) | c ∈ C }. • Order of an ACG G = 〈Σ1, Σ2, L , s〉: order(G ) = order(Σ1). Second-order ACGs generate PTIME languages. (Salvati 2005) Decidability of membership of Third-order ACGs is open. On Two Extensions of ACGs – p.10/45 Order of an ACG • Order of types: order(p) = 1 for p ∈ A. order(α ( β) = max{order(α) + 1, order(β)} for α ( β ∈ TA. • Order of a higher-order signature Σ = 〈A, C, τ〉: order(Σ) = max{ order(τ(c)) | c ∈ C }. • Order of an ACG G = 〈Σ1, Σ2, L , s〉: order(G ) = order(Σ1). Second-order ACGs generate PTIME languages. (Salvati 2005) Decidability of membership of Third-order ACGs is open. On Two Extensions of ACGs – p.10/45 Order of an ACG • Order of types: order(p) = 1 for p ∈ A. order(α ( β) = max{order(α) + 1, order(β)} for α ( β ∈ TA. • Order of a higher-order signature Σ = 〈A, C, τ〉: order(Σ) = max{ order(τ(c)) | c ∈ C }. • Order of an ACG G = 〈Σ1, Σ2, L , s〉: order(G ) = order(Σ1). Second-order ACGs generate PTIME languages. (Salvati 2005) Decidability of membership of Third-order ACGs is open. On Two Extensions of ACGs – p.10/45 Order of an ACG • Order of types: order(p) = 1 for p ∈ A. order(α ( β) = max{order(α) + 1, order(β)} for α ( β ∈ TA. • Order of a higher-order signature Σ = 〈A, C, τ〉: order(Σ) = max{ order(τ(c)) | c ∈ C }. • Order of an ACG G = 〈Σ1, Σ2, L , s〉: order(G ) = order(Σ1). Second-order ACGs generate PTIME languages. (Salvati 2005) Decidability of membership of Third-order ACGs is open. On Two Extensions of ACGs – p.10/45 Strings by Lambda Terms • For an alphabet Ξ, let ΣΞ = 〈{o}, Ξ, τ〉 be the higher-order signature with τ(a) = o ( o for all a ∈ Ξ. • A string a1 . . . ak ∈ Ξ is represented by the term /a1 . . . ak/ = λ ◦z.a1(. . . (akz) . . . ). • The empty string ε is represented as /ε/ = λz.z in particular. • Concatenation is realized by B = λxyz.x(yz). • We write t + u for λz.t(uz). On Two Extensions of ACGs – p.11/45 Example c ∈ C1 τ1(c) L (c) S , NP , Adj type o ( o Pierre NP /Pierre/ Etre Adj ( NP ( S λxy.y + /est/ + x Intelligent Adj /intelligent/ /Pierre/ + /est/ + /intelligent/ /Marie/ + /est/ + /intelligente/ /Marie/ + /et/ + /Pierre/ + /sont/ + /intelligents/ On Two Extensions of ACGs – p.12/45 Example c ∈ C1 τ1(c) L (c) S , NP , Adj type o ( o Pierre NP /Pierre/ Etre Adj ( NP ( S λxy.y + /est/ + x Intelligent Adj /intelligent/ ` Etre Intelligent Pierre : S L (Etre)L (Intelligent)L (Pierre) = (λxy.y + /est/ + x)/intelligent//Pierre/ β /Pierre/ + /est/ + /intelligent/ /Pierre/ + /est/ + /intelligent/ /Marie/ + /est/ + /intelligente/ /Marie/ + /et/ + /Pierre/ + /sont/ + /intelligents/ On Two Extensions of ACGs – p.12/45 Example c ∈ C1 τ1(c) L (c) S , NP , Adj type o ( o Pierre NP /Pierre/ Etre Adj ( NP ( S λxy.y + /est/ + x Intelligent Adj /intelligent/ /Pierre/ + /est/ + /intelligent/ /Marie/ + /est/ + /intelligente/ /Marie/ + /et/ + /Pierre/ + /sont/ + /intelligents/ On Two Extensions of ACGs – p.12/45 3. ACGs with Dependent Product On Two Extensions of ACGs – p.13/45 ACGs with Dependent Product • Dependent products are useful in defining generic syntactic categories, On Two Extensions of ACGs – p.14/45 ACGs with Dependent Product • Dependent products are useful in defining generic syntactic categories, e.g., 