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CatLog: A Categorial Parser/Theorem-Prover 1 Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya Type Dependency, Type Theory with Records, and Natural-Language Flexibility, London, QMUL June 16th17th 2011 1 The

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CatLog: A Categorial Parser/Theorem-Prover1

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya Type Dependency, Type Theory with Records, and Natural-Language Flexibility, London, QMUL June 16th–17th 2011

1The research reported in the present paper was supported by DGICYT project

SESAAME-BAR (TIN2008-06582-C03-01).

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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CatLog

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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CatLog

A categorial parser/theorem-prover

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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CatLog

A categorial parser/theorem-prover 3000 lines of Prolog

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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CatLog

A categorial parser/theorem-prover 3000 lines of Prolog Implements all the categorial logic and analyses the author has been concerned with to date.

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Concatenation is not enough

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Concatenation is not enough

PSG: CFG ⇒ MCSG

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Concatenation is not enough

PSG: CFG ⇒ MCSG CG: L ⇒ D(b)

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Concatenation is not enough

PSG: CFG ⇒ MCSG CG: L ⇒ D(b) demo: cross serial dependencies

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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More than concatenation

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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More than concatenation

An expression is a well-bracketed string over an alfabet Σ and a placeholder 1:

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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More than concatenation

An expression is a well-bracketed string over an alfabet Σ and a placeholder 1: E ::= ǫ | Σ | 1 | E+E | [E]

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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More than concatenation

An expression is a well-bracketed string over an alfabet Σ and a placeholder 1: E ::= ǫ | Σ | 1 | E+E | [E] The sort σ(s) of an expression s is the number of placeholders 1 which it contains.

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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More than concatenation

An expression is a well-bracketed string over an alfabet Σ and a placeholder 1: E ::= ǫ | Σ | 1 | E+E | [E] The sort σ(s) of an expression s is the number of placeholders 1 which it contains.

◮ concatenation + (linear ordering) Lambek (1958)[5]

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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More than concatenation

An expression is a well-bracketed string over an alfabet Σ and a placeholder 1: E ::= ǫ | Σ | 1 | E+E | [E] The sort σ(s) of an expression s is the number of placeholders 1 which it contains.

◮ concatenation + (linear ordering) Lambek (1958)[5] ◮ bracketing [·] (syntactic domains) Morrill (1992)[9], Moortgat

(1995)[6]

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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More than concatenation

An expression is a well-bracketed string over an alfabet Σ and a placeholder 1: E ::= ǫ | Σ | 1 | E+E | [E] The sort σ(s) of an expression s is the number of placeholders 1 which it contains.

◮ concatenation + (linear ordering) Lambek (1958)[5] ◮ bracketing [·] (syntactic domains) Morrill (1992)[9], Moortgat

(1995)[6]

◮ intercalation × (displacement) Morrill, Valent´

ın & Fadda (2011)[14]

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Semantic representation language

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Semantic representation language

The set T set of semantic types is defined on the basis of a set δ of basic semantic types as follows:

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Semantic representation language

The set T set of semantic types is defined on the basis of a set δ of basic semantic types as follows: T ::= δ | ⊤ | T + T | T &T | T → T | LT

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Semantic frames

A semantic frame comprises a familiy {Dτ}τ∈δ of non-empty basic type domains and a non-empty set W of worlds. This induces a type domain Dτ for each type τ as follows:

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Semantic frames

A semantic frame comprises a familiy {Dτ}τ∈δ of non-empty basic type domains and a non-empty set W of worlds. This induces a type domain Dτ for each type τ as follows:

D⊤ = {∅} singleton set Dτ1+τ2 = Dτ2 ⊎ Dτ1 ({1} × Dτ1) ∪ ({2} × Dτ2) disjoint union Dτ1&τ2 = Dτ1 × Dτ2 {m1, m2| m1 ∈ Dτ1 & m2 ∈ Dτ2} Cartesian product Dτ1→τ2 = D

Dτ1 τ2

the set of all functions from Dτ1 to Dτ2 functional exponentiation DLτ = DW

τ

the set of all functions from W to Dτ functional exponentiation

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Semantic terms

The sets Φτ of terms of type τ for each type τ are defined on the basis

  • f sets Cτ of constants of type τ and denumerably infinite sets Vτ of

variables of type τ for each type τ as follows:

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Semantic terms

The sets Φτ of terms of type τ for each type τ are defined on the basis

  • f sets Cτ of constants of type τ and denumerably infinite sets Vτ of

variables of type τ for each type τ as follows: Φτ ::= Cτ Φτ ::= Vτ Φ⊤ ::= d Φτ ::= Φτ1+τ2 → Vτ1.Φτ; Vτ2.Φτ case statement Φτ+τ ′ ::= ι1Φτ first injection Φτ ′+τ ::= ι2Φτ second injection Φτ ::= π1Φτ&τ ′ first projection Φτ ::= π2Φτ ′&τ second projection Φτ&τ ′ ::= (Φτ, Φτ ′)

  • rdered pair formation

Φτ ::= (Φτ ′→τ Φτ ′) functional application Φτ→τ ′ ::= λVτΦτ ′ functional abstraction Φτ ::= ˇΦLτ extensionalization ΦLτ ::= ˆΦτ intensionalization

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Valuations

Given a semantic frame, a valuation f mapping each constant of type τ into an element of Dτ, an assignment g mapping each variable of type τ into an element of Dτ, and a world i ∈ W , each term φ of type τ receives an interpretation [φ]g,i ∈ Dτ as follows:

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Valuations

Given a semantic frame, a valuation f mapping each constant of type τ into an element of Dτ, an assignment g mapping each variable of type τ into an element of Dτ, and a world i ∈ W , each term φ of type τ receives an interpretation [φ]g,i ∈ Dτ as follows: [a]g,i = f (a) for constant a ∈ Cτ [x]g,i = g(x) for variable x ∈ Vτ [d]g,i = ∅ [φ → x.ψ; y.ψ]g,i = [ψ](g−{(x,g(x))})∪{(x,d)},i if [φ]g,i = 1, d [χ](g−{(y,g(y))})∪{(y,d)},i if [φ]g,i = 2, d [ι1φ]g,i = 1, [φ]g,i [ι2φ]g,i = 2, [φ]g,i [π1φ]g,i = fst([φ]g,i) [π2φ]g,i = snd([φ]g,i) [(φ, ψ)]g,i = [φ]g,i, [ψ]g,i [(φ ψ)]g,i = [φ]g,i([ψ]g,i) [λxφ]g,i = d → [φ](g−{(x,g(x))})∪{(x,d)},i [ˇφ]g,i = [φ]g,i(i) [ˆφ]g,i = j → [φ]g,j

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Free substitution

An occurrence of a variable x in a term is called free if and only if it does not fall within any part of the term of the form x.· or λx·; otherwise it is bound (by the closest x. or λx within the scope of which it falls). The result φ{ψ/x} of substituting term ψ (of type τ) for variable x (of type τ) in a term φ is the result of replacing by ψ every free occurrence of x in φ. We say that ψ is free for x in φ if and only if no variable in ψ becomes bound in φ{ψ/x}. We say that a term is modally closed if and only if every occurrence of ˇ occurs within the scope of an ˆ. A modally closed term is denotationally invariant across worlds. We say that a term ψ is modally free for x in φ if and only if either ψ is modally closed, or no free

  • ccurrence of x in φ is within the scope of an ˆ.

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Semantic conversion laws

φ → y.ψ; z.χ = φ → x.(ψ{x/y}); z.χ if x is not free in ψ and is free for y in ψ φ → y.ψ; z.χ = φ → y.ψ; x.(χ{x/z}) if x is not free in χ and is free for z in χ λyφ = λx(φ{x/y}) if x is not free in φ and is free for y in φ α-conversion ι1φ → y.ψ; z.χ = ψ{φ/y} if φ is free for y in ψ and modally free for y in ψ ι2φ → y.ψ; z.χ = χ{φ/z} if φ is free for z in χ and modally free for z in χ π1(φ, ψ) = φ π2(φ, ψ) = ψ (λxφ ψ) = φ{ψ/x} if ψ is free for x in φ, and modally free for x in φ ˇˆφ = φ β-conversion (π1φ, π2φ) = φ λx(φ x) = φ if x is not free in φ ˆˇφ = φ if φ is modally closed η-conversion

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Categorial Logic

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Categorial Logic

The family of types is {Fi}i∈N

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Categorial Logic

The family of types is {Fi}i∈N where Fi is the set of types of sort i,

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Categorial Logic

The family of types is {Fi}i∈N where Fi is the set of types of sort i, i.e. which are interpreted as sets of expressions of sort i.

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Primitive Types

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Primitive Types

We assume terms composed from (feature) variables, constants and function letters.

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Primitive Types

We assume terms composed from (feature) variables, constants and function letters. There is a family of sorted and ranked primitive types {Pi,n}i,n∈N

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Primitive Types

We assume terms composed from (feature) variables, constants and function letters. There is a family of sorted and ranked primitive types {Pi,n}i,n∈N Atomic types: Ai ::= Pi,nt1 . . . tn

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Primitive Types

We assume terms composed from (feature) variables, constants and function letters. There is a family of sorted and ranked primitive types {Pi,n}i,n∈N Atomic types: Ai ::= Pi,nt1 . . . tn are types: Fi ::= Ai

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Primitive Types

We assume terms composed from (feature) variables, constants and function letters. There is a family of sorted and ranked primitive types {Pi,n}i,n∈N Atomic types: Ai ::= Pi,nt1 . . . tn are types: Fi ::= Ai The homomorphic type map T maps primitive syntactic types into (not necessarily primitive) semantic types.

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Multiplicative Basis: Disp. Calculus with Brackets (Db)

Spirit of Lambek (1958)[5]. Brackets: Morrill (1992)[9], Moortgat (1995)[6]. Displacement: Morrill, Valent´ ın & Fadda (2011)[14].

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Multiplicative Basis: Disp. Calculus with Brackets (Db)

Spirit of Lambek (1958)[5]. Brackets: Morrill (1992)[9], Moortgat (1995)[6]. Displacement: Morrill, Valent´ ın & Fadda (2011)[14].

Fi ::= [ ]−1Fi T([ ]−1A) = T(A) [[ ]−1A] = {s| [s] ∈ [A]} antibracket Fi ::= Fi T( A) = T(A) [ A] = {[s]| s ∈ [A]} bracket

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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SLIDE 40

Multiplicative Basis: Disp. Calculus with Brackets (Db)

Spirit of Lambek (1958)[5]. Brackets: Morrill (1992)[9], Moortgat (1995)[6]. Displacement: Morrill, Valent´ ın & Fadda (2011)[14].

Fi ::= [ ]−1Fi T([ ]−1A) = T(A) [[ ]−1A] = {s| [s] ∈ [A]} antibracket Fi ::= Fi T( A) = T(A) [ A] = {[s]| s ∈ [A]} bracket Fj ::= Fi\Fi+j T(A\C) = T(A) → T(C) [A\C] = {s2| ∀s1 ∈ [A], s1+s2 ∈ [C]} under Fi ::= Fi+j/Fj T(C/B) = T(B) → T(C) [C/B] = {s1| ∀s2 ∈ [B], s1+s2 ∈ [C]}

  • ver

Fi+j ::= Fi·Fj T(A·B) = T(A)&T(B) [A·B] = {s1+s2| s1 ∈ [A] & s2 ∈ [B]} product F0 ::= I T(I) = ⊤ [I] = {0} product unit

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Multiplicative Basis: Disp. Calculus with Brackets (Db)

Spirit of Lambek (1958)[5]. Brackets: Morrill (1992)[9], Moortgat (1995)[6]. Displacement: Morrill, Valent´ ın & Fadda (2011)[14].

Fi ::= [ ]−1Fi T([ ]−1A) = T(A) [[ ]−1A] = {s| [s] ∈ [A]} antibracket Fi ::= Fi T( A) = T(A) [ A] = {[s]| s ∈ [A]} bracket Fj ::= Fi\Fi+j T(A\C) = T(A) → T(C) [A\C] = {s2| ∀s1 ∈ [A], s1+s2 ∈ [C]} under Fi ::= Fi+j/Fj T(C/B) = T(B) → T(C) [C/B] = {s1| ∀s2 ∈ [B], s1+s2 ∈ [C]}

  • ver

Fi+j ::= Fi·Fj T(A·B) = T(A)&T(B) [A·B] = {s1+s2| s1 ∈ [A] & s2 ∈ [B]} product F0 ::= I T(I) = ⊤ [I] = {0} product unit Fj ::= Fi+1↓kFi+j T(A↓kC) = T(A) → T(B) [A↓kC] = {s2| ∀s1 ∈ [A], s1×ks2 ∈ [C]} infix Fi+1 ::= Fi+j↑kFj T(C↑kB) = T(B) → T(C) [C↑kB] = {s1| ∀s2 ∈ [B], s1×ks2 ∈ [C]} extract Fi+j ::= Fi+1⊙kFj T(A⊙B) = T(A)&T(B) [A⊙kB] = {s1×ks2| s1 ∈ [A] & s2 ∈ [B]} wrap F1 ::= J T(J) = ⊤ [J] = {1} wrap unit

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Configurations

The set O of configurations is defined as follows, where Λ is the empty string and [ ] is the metalinguistic placeholder: O ::= Λ | [ ] | F0 | Fi+1{O : . . . : O

  • i+1 O’s

} | O, O | [O] The sort of a configuration Γ is the number of (metalinguistic) placeholders it contains. Where ∆ is a configuration of sort at least 1 and Γ is a configuration, ∆|>Γ (∆|<Γ) is the configuration resulting from replacing the leftmost (rightmost) placeholder in ∆ by Γ. The figure − → A

  • f a type A is defined by:

(1) − → A =      A if sA = 0 A{[ ] : . . . : [ ]

  • sA [ ]’s

} if sA > 0

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Configurations (cont.)

The usual configuration distinguished occurrence notation ∆(Γ) signifies a configuration ∆ with a distinguished subconfiguration Γ, i.e. a configuration occurrence Γ with (external) context ∆. In the hypersequent calculus the distinguished hyperoccurrence notation ∆Γ signifies a configuration hyperoccurrence Γ with external and internal context ∆ as follows: where Γ is a configuration of sort i and ∆1, . . . , ∆i are configurations, the fold Γ ⊗ ∆1, . . . , ∆i is the result of replacing the successive placeholders in Γ by ∆1, . . . , ∆i respectively; the distinguished hyperoccurrence notation ∆Γ represents ∆0(Γ ⊗ ∆1, . . . , ∆i). A sequent Γ ⇒ A comprises an antecedent configuration Γ of sort i and a succedent type A of sort i. The types which are allowed to enter into the antecedent are the input (•) types and the types which are allowed to enter into the succedent are the output (o) types.

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Db Calculus, pt. 1/3

id − → A : x ⇒ A: x ∆− → A : x ⇒ B: ψ [ ]−1L ∆[ − − − → [ ]−1A: x] ⇒ B: ψ [Γ] ⇒ A: φ [ ]−1R Γ ⇒ [ ]−1A: φ ∆[− → A : x] ⇒ B: ψ L ∆− − → A: x ⇒ B: ψ Γ ⇒ A: φ R [Γ] ⇒ A: φ

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Db Calculus, pt. 2/3

Γ ⇒ A: φ ∆− → C : z ⇒ D: ω \L ∆Γ, − − → A\C: y ⇒ D: ω[(y φ)/z] − → A : x, Γ ⇒ C: χ \R Γ ⇒ A\C: λxχ Γ ⇒ B: ψ ∆− → C : z ⇒ D: ω /L ∆− − → C/B: x, Γ ⇒ D: ω[(x ψ)/z] Γ, − → B : y ⇒ C: χ /R Γ ⇒ C/B: λyχ ∆− → A : x, − → B : y ⇒ D: ω ·L ∆− − → A·B: z ⇒ D: ω[π1z/x, π2z/y] Γ1 ⇒ A: φ Γ2 ⇒ B: ψ ·R Γ1, Γ2 ⇒ A·B: (φ, ψ) ∆Λ ⇒ A: φ IL ∆− → I : t ⇒ A: φ IR Λ ⇒ I: d

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Db Calculus, pt. 3/3

Γ ⇒ A: φ ∆− → C : z ⇒ D: ω ↓kL ∆Γ|k − − − → A↓kC: y ⇒ D: ω[(y φ)/z] − → A : x|kΓ ⇒ C: χ ↓kR Γ ⇒ A↓kC: λxχ Γ ⇒ B: ψ ∆− → C : z ⇒ D: ω ↑kL ∆− − − → C↑kB: x|kΓ ⇒ D: ω[(x ψ)/z] Γ|k − → B : y ⇒ C: χ ↑kR Γ ⇒ C↑kB: λyχ ∆− → A : x|k − → B : y ⇒ D: ω ⊙kL ∆− − − → A⊙kB: z ⇒ D: ω[π1z/x, π2z/y] Γ1 ⇒ A: φ Γ2 ⇒ B: ψ ⊙kR Γ1|kΓ2 ⇒ A⊙kB: (φ, ψ) ∆[ ] ⇒ A: φ JL ∆− → J : t ⇒ A: φ JR [ ] ⇒ J: d

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Additives

Lambek (1961)[4], Girard (1987)[2], van Benthem (1991)[16], Morrill (1990; 1994, ch. 6)[7][15], Kanazawa (1992)[3]. Fp

i

::= Fp

i &Fp i

T(A&B) = T(A)&T(B) [A&B] ≈ [A] ∩ [B] additive conjunction Fp

i

::= Fp

i +Fp i

T(A+B) = T(A) + T(B) [A+B] ≈ [A] ∪ [B] additive disjunction

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Additives (cont.)

Γ− → A : x ⇒ D: ω &L1 Γ− − − → A&B: z ⇒ D: ω[π1z/x] Γ− → B : y ⇒ D: ω &L2 Γ− − − → A&B: z ⇒ D: ω[π2z/y] Γ ⇒ A: φ Γ ⇒ B: ψ &R Γ ⇒ A&B: (φ, ψ) Γ− → A : x ⇒ D: φ Γ− → B : y ⇒ D: ψ +L Γ− − − → A+B: z ⇒ D: z → x.φ; y.ψ Γ ⇒ A: φ +R1 Γ ⇒ A+B: ι1φ Γ ⇒ B: ψ +R2 Γ ⇒ A+B: ι2ψ

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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Semantically inert additives

Morrill (1994, ch. 6)[15] Fp

i

::= Fp

i ⊓ Fp i

T(A ⊓ B) = T(A) = T(B) [A ⊓ B] ≈ [A] ∩ [B] semantically inert additive conjunction Fp

i

::= Fp

i ⊔ Fp i

T(A ⊔ B) = T(A) = T(B) [A ⊔ B] ≈ [A] ∪ [B] semantically inert additive disjunction

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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SLIDE 50

Semantically inert additives (cont.)

Γ− → A : z ⇒ D: ω ⊓L1 Γ− − − → A ⊓ B: z ⇒ D: ω Γ− → B : z ⇒ D: ω ⊓L2 Γ− − − → A ⊓ B: z ⇒ D: ω Γ ⇒ A: χ Γ ⇒ B: χ ⊓R Γ ⇒ A ⊓ B: χ Γ− → A : z ⇒ D: χ Γ− → B : z ⇒ D: χ ⊔L Γ− − − → A ⊔ B: z ⇒ D: χ Γ ⇒ A: χ ⊔R1 Γ ⇒ A ⊔ B: χ Γ ⇒ B: χ ⊔R2 Γ ⇒ A ⊔ B: χ

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

slide-51
SLIDE 51

Difference

  • Cf. Morrill & Valent´

ın (2010)[12] Fi o ::= Fi o − Fi o T(A − B) = T(A) Γ ⇒ A: φ ⊢ Γ ⇒ B: −R Γ ⇒ A − B: φ

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

slide-52
SLIDE 52

Modalities

Morrill (1990)[8] Fp

i

::= Fp

i

T(A) = LA modality Fp

i

::= Fp

i

T(A) = LA modality (rigid designator) Γ− → A : x ⇒ C: χ L Γ− → A: y ⇒ C: χ[ˇy/x] Γ ⇒ A: φ R Γ ⇒ A: ˆφ Γ− → A : x ⇒ C: χ L Γ− → A: x ⇒ C: χ Γ ⇒ A: φ R Γ ⇒ A: φ

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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SLIDE 53

Exponentials, pt. 1/2

F0

  • ::=

!A0 T(!A) = T(A) structural modality Girard (1987)[2], Barry et al. (1991)[1 F0

  • ::=

F0

  • +

T(A+) = list(T(A)) Kleene plus Morrill (1994)[15]

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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SLIDE 54

Exponentials, pt. 2/2

Γ(A: x) ⇒ B: ψ !L Γ(!A: x) ⇒ B: ψ ∆(!A: x, Γ) ⇒ B: ψ !P ∆(Γ, !A: x) ⇒ B: ψ ∆(Γ, !A: x) ⇒ B: ψ !P ∆(!A: x, Γ) ⇒ B: ψ ∆(!A: x, [!A: x, Γ]) ⇒ B: ψ !C ∆(!A: x, Γ) ⇒ B: ψ ∆([Γ, !A: x], !A: x) ⇒ B: ψ !C ∆(Γ, !A: x) ⇒ B: ψ Γ ⇒ A: φ

+R

Γ ⇒ A+: (φ, d) Γ ⇒ A: φ ∆ ⇒ A+: ψ +R Γ, ∆ ⇒ A+: (φ, ψ)

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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SLIDE 55

(Semantically Inert) First-order quantification

Fp

i

::= ∀XFp

i

T(∀xA) = T(A) 1st order univ. qu. Morrill (1994)[15] Fp

i

::= ∃XFi T(∃xA) = T(A) 1st order exist. qu. Morrill (1994)[15] Γ− − − − → A[t/x]: z ⇒ B: ψ ∀L Γ− − → ∀xA: z ⇒ B: ψ Γ ⇒ A[a/x]: φ ∀R∗ Γ ⇒ ∀xA: φ Γ− − − − → A[a/x]: z ⇒ B: φ ∃L∗ Γ− − → ∃xA: z ⇒ B: φ Γ ⇒ A[t/x]: φ ∃R Γ ⇒ ∃xA: φ

∗ no a in conclusion

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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SLIDE 56

Synthetic Connectives

In addition to the primitive connectives we may define synthetic connectives which do not extend expressivity, but which permit

  • abbreviations. Unary synthetic connectives are:2

⊲−1A =df J\A {s| 1+s ∈ A} T(⊲−1A) = T(A) right projection Morrill, Valent´ ın & Fadda (2009)[13] ⊳−1A =df A/J {s|s+1 ∈ A} T(⊳−1A) = T(A) left projection Morrill, Valent´ ın & Fadda (2009)[13] ⊲A =df J·A {1+s| s ∈ A} T(⊲A) = T(A) right injection Morrill, Valent´ ın & Fadda (2009)[13] ⊳A =df A·J {s+1| s ∈ A} T(⊳A) = T(A) left injection Morrill, Valent´ ın & Fadda (2009)[13] ˇk A =df A↑k I {s| s×k 0 ∈ A} T(ˇk A) = T(A) split Morrill & Merenciano (1996)[10] ˆk A =df A⊙k I {s×k 0| s ∈ A} T(ˆk A) = T(A) bridge Morrill & Merenciano (1996)[10]

2By unicity of the insertion point, when sA = 0, ˇ>A = ˇ<A and we write just ˇA,

and when sA = 1, ˆ>A = ˆ<A and we write just ˆA.

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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SLIDE 57

Synthetic connectives (cont.)

Continuous and discontinuous nondeterministic connectives (Morrill & Valent´ ın 2010[11]) which are binary synthetic connectives are thus, where +(s1, s2, s3) if and only if s3 = s1+s2 or s3 = s2+s1, and ×(s1, s2, s3) if and only if s3 = s1 ×> s2 or s3 = s2 ×< s1:

B A (A\B) ⊓ (B/A) {s| ∀s′ ∈ A, s3, +(s, s′, s3) ⇒ s3 ∈ B} T( B A ) = T(A) → T(B)

  • nondet. d

A ⊗ B (A·B) ⊔ (B·A) {s3| ∃s1 ∈ A, s2 ∈ B, +(s1, s2, s3)} T(A ⊗ B) = T(A)&T(B)

  • nondet. pro

A⇓C (A↓>C) ⊓ (A↓>C) {s2| ∀s1 ∈ A, s3, ×(s1, s2, s3) ⇒ s3 ∈ C} T(A⇓C) = T(A) → T(C)

  • nondet. in

C⇑B (C↑>B) ⊓ (C↑<B) {s1| ∀s2 ∈ B, s3, ×(s1, s2, s3) ⇒ s3 ∈ C} T(C⇑B) = T(B) → T(C)

  • nondet. ext

A⊚B (A⊙>B) ⊔ (A⊙<B) {s3| ∃s1 ∈ A, s2 ∈ B, ×(s1, s2, s3)} T(A⊚B) = T(A)&T(B)

  • nondet. disc

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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SLIDE 58

Synthetic Rules, pt. 1/4

Γ− → A : x ⇒ B: ψ ⊳−1L Γ − − − → ⊳−1A: x, [ ] ⇒ B: ψ Γ, [ ] ⇒ A: φ ⊳−1R Γ ⇒ ⊳−1A: φ Γ− → A : x, [ ] ⇒ B: ψ ⊳L Γ− → ⊳A: x ⇒ B: ψ Γ ⇒ A: φ ⊳R Γ, [ ] ⇒ ⊳A: φ Γ− → A : x ⇒ B: ψ ⊲−1L Γ[ ], − − − → ⊲−1A: x ⇒ B: ψ [ ], Γ ⇒ A: φ ⊲−1R Γ ⇒ ⊲−1A: φ Γ[ ], − → A : x ⇒ B: ψ ⊲L Γ− → ⊲A: x ⇒ B: ψ Γ ⇒ A: φ ⊲R [ ], Γ ⇒ ⊲A: φ

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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SLIDE 59

Synthetic Rules, pt. 2/4

∆− → B : y ⇒ C: χ ˇkL ∆− − → ˇkB: y|kΛ ⇒ C: χ ∆|kΛ ⇒ B: ψ ˇkR ∆ ⇒ ˇkB: ψ ∆− → B : y|kΛ ⇒ C: χ ˆkL ∆− − → ˆkB: y ⇒ C: χ ∆ ⇒ B: ψ ˆkR ∆|kΛ ⇒ ˆkB: ψ

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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SLIDE 60

Synthetic Rules, pt. 3/4

Γ ⇒ A: φ ∆− → C : z ⇒ D: ω L1 ∆Γ, − → C A : y ⇒ D: ω[(y φ)/z] Γ ⇒ A: ψ ∆− → C : z ⇒ D: ω L2 ∆ − → C A : x, Γ ⇒ D: ω[(x ψ)/z] − → A : x, Γ ⇒ C: χ Γ, − → A : x ⇒ C: χ R Γ ⇒ C A : λxχ ∆− → A : x, − → B : y ⇒ D: ω ∆− → B : y, − → A : x ⇒ D: ω ⊗L ∆− − − → A ⊗ B: z ⇒ D: ω[π1z/x, π2z/y] Γ1 ⇒ A: φ Γ2 ⇒ B: ψ ⊗R1 Γ1, Γ2 ⇒ A ⊗ B: (φ, ψ) Γ1 ⇒ B: ψ Γ2 ⇒ A: φ ⊗R2 Γ1, Γ2 ⇒ A ⊗ B: (φ, ψ)

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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SLIDE 61

Synthetic Rules, pt. 4/4

Γ ⇒ A: φ ∆− → C : z ⇒ D: ω ⇓L ∆Γ|k − − → A⇓C: y ⇒ D: ω[(y φ)/z] − → A : x|>Γ ⇒ C: χ − → A : x|<Γ ⇒ C: χ ⇓R Γ ⇒ A⇓C: λxχ Γ ⇒ B: ψ ∆− → C : z ⇒ D: ω ⇑L ∆− − → C⇑B: x|kΓ ⇒ D: ω[(x ψ)/z] Γ|> − → B : y ⇒ C: χ Γ|< − → B : y ⇒ C: χ ⇑R Γ ⇒ C⇑B: λyχ ∆− → A : x|> − → B : y ⇒ D: ω ∆− → A : x|< − → B : y ⇒ D: ω ⊚L ∆− − − → A⊚B: z ⇒ D: ω[π1z/x, π2z/y] Γ1 ⇒ A: φ Γ2 ⇒ B: ψ ⊚ Γ1|kΓ2 ⇒ A⊚B: (φ, ψ)

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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SLIDE 62

Guy Barry, Mark Hepple, Neil Leslie, and Glyn Morrill. Proof Figures and Structural Operators for Categorial Grammar. In Proceedings of the Fifth Conference of the European Chapter of the Association for Computational Linguistics, Berlin, 1991. Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1–102, 1987.

  • M. Kanazawa.

The Lambek calculus enriched with additional connectives. Journal of Logic, Language and Information, 1:141–171, 1992.

  • J. Lambek.

On the Calculus of Syntactic Types. In Roman Jakobson, editor, Structure of Language and its Mathematical Aspects, Proceedings of the Symposia in Applied Mathematics XII, pages 166–178. American Mathematical Society, Providence, Rhode Island, 1961. Joachim Lambek. The mathematics of sentence structure. American Mathematical Monthly, 65:154–170, 1958.

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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SLIDE 63

Reprinted in Buszkowski, Wojciech, Wojciech Marciszewski, and Johan van Benthem, editors, 1988, Categorial Grammar, Linguistic & Literary Studies in Eastern Europe volume 25, John Benjamins, Amsterdam, 153–172. Michael Moortgat. Multimodal linguistic inference. Journal of Logic, Language and Information, 5:349–385, 1995. Also in Bulletin of the IGPL, 3(2,3):371–401, 1995.

  • G. Morrill.

Grammar and Logical Types. In Martin Stockhof and Leen Torenvliet, editors, Proceedings of the Seventh Amsterdam Colloquium, pages 429–450, 1990. Also in G. Barry and G. Morrill, editors, Studies in Categorial Grammar, Edinburgh Working Papers in Cognitive Science, Volume 5, pages 127–148: 1990. Revised version published as Grammar and Logic, Theoria, LXII, 3:260–293, 1996. Glyn Morrill. Intensionality and Boundedness. Linguistics and Philosophy, 13(6):699–726, 1990.

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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SLIDE 64

Glyn Morrill. Categorial Formalisation of Relativisation: Pied Piping, Islands, and Extraction Sites. Technical Report LSI-92-23-R, Departament de Llenguatges i Sistemes Inform` atics, Universitat Polit` ecnica de Catalunya, 1992. Glyn Morrill and Josep-Maria Merenciano. Generalising discontinuity. traitement automatique des langues, 37(2):119–143, 1996. Glyn Morrill and Oriol Valent´ ın. Generalized Discontinuity. Formal Grammar 2010, Copenhagen, 2010. Glyn Morrill and Oriol Valent´ ın. On Anaphora and the Binding Principles in Categorial Grammar. In Anuj Dawar and Ruy de Queiroz, editors, Proceedings of the 17th International Workshop on Logic, Language, Information and Computation, WoLLIC 2010, Bras´ ılia, number LNAI 6188 in Lecture Notes in Artificial Intelligence, pages 176–190, Berlin, 2010. Springer. Glyn Morrill, Oriol Valent´ ın, and Mario Fadda. Dutch Grammar and Processing: A Case Study in TLG.

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

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SLIDE 65

In Peter Bosch, David Gabelaia, and J´ erˆ

  • me Lang, editors, Logic,

Language, and Computation: 7th International Tbilisi Symposium, Revised Selected Papers, number 5422 in Lecture Notes in Artificial Intelligence, pages 272–286, Berlin, 2009. Springer. Glyn Morrill, Oriol Valent´ ın, and Mario Fadda. The Displacement Calculus. Journal of Logic, Language and Information, 20(1):1–48, 2011. doi 10.1007/s10849-010-9129-2. Glyn V. Morrill. Type Logical Grammar: Categorial Logic of Signs. Kluwer Academic Publishers, Dordrecht, 1994.

  • J. van Benthem.

Language in Action: Categories, Lambdas, and Dynamic Logic. Number 130 in Studies in Logic and the Foundations of

  • Mathematics. North-Holland, Amsterdam, 1991.

Revised student edition printed in 1995 by the MIT Press.

Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover