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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

  1. 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 16th–17th 2011 1 The 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

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

  3. CatLog A categorial parser/theorem-prover Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

  4. 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

  5. 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

  6. Concatenation is not enough Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

  7. Concatenation is not enough PSG: CFG ⇒ MCSG Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

  8. 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

  9. 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

  10. More than concatenation Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. Semantic representation language Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

  18. 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

  19. 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 | L T Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

  20. Semantic frames A semantic frame comprises a familiy { D τ } τ ∈ δ of non-empty basic type domains and a non-empty set W of worlds. This induces a type domain D τ for each type τ as follows: Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

  21. Semantic frames A semantic frame comprises a familiy { D τ } τ ∈ δ of non-empty basic type domains and a non-empty set W of worlds. This induces a type domain D τ for each type τ as follows: = {∅} singleton set D ⊤ D τ 1+ τ 2 = D τ 2 ⊎ D τ 1 ( { 1 } × D τ 1 ) ∪ ( { 2 } × D τ 2 ) disjoint union = D τ 1 × D τ 2 {� m 1 , m 2 �| m 1 ∈ D τ 1 & m 2 ∈ D τ 2 } Cartesian product D τ 1& τ 2 D τ 1 = the set of all functions from D τ 1 to D τ 2 functional exponentiation D τ 1 → τ 2 D τ 2 D W D L τ = the set of all functions from W to D τ functional exponentiation τ Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

  22. Semantic terms The sets Φ τ of terms of type τ for each type τ are defined on the basis of sets C τ of constants of type τ and denumerably infinite sets V τ 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

  23. Semantic terms The sets Φ τ of terms of type τ for each type τ are defined on the basis of sets C τ of constants of type τ and denumerably infinite sets V τ of variables of type τ for each type τ as follows: Φ τ ::= C τ Φ τ ::= V τ Φ ⊤ ::= d Φ τ ::= Φ τ 1 + τ 2 → V τ 1 . Φ τ ; V τ 2 . Φ τ case statement Φ τ + τ ′ ::= ι 1 Φ τ first injection Φ τ ′ + τ ::= ι 2 Φ τ second injection Φ τ ::= π 1 Φ τ & τ ′ first projection Φ τ ::= π 2 Φ τ ′ & τ second projection Φ τ & τ ′ ::= (Φ τ , Φ τ ′ ) ordered pair formation Φ τ ::= (Φ τ ′ → τ Φ τ ′ ) functional application Φ τ → τ ′ ::= λ V τ Φ τ ′ functional abstraction Φ τ ::= ˇΦ L τ extensionalization Φ L τ ::= ˆΦ τ intensionalization Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

  24. Valuations Given a semantic frame, a valuation f mapping each constant of type τ into an element of D τ , an assignment g mapping each variable of type τ into an element of D τ , and a world i ∈ W , each term φ of type τ receives an interpretation [ φ ] g , i ∈ D τ as follows: Glyn Morrill Departament de LSI Universitat Polit` ecnica de Catalunya CatLog: A Categorial Parser/Theorem-Prover

  25. Valuations Given a semantic frame, a valuation f mapping each constant of type τ into an element of D τ , an assignment g mapping each variable of type τ into an element of D τ , and a world i ∈ W , each term φ of type τ receives an interpretation [ φ ] g , i ∈ D τ as follows: [ a ] g , i f ( a ) for constant a ∈ C τ = [ x ] g , i g ( x ) for variable x ∈ V τ = [ d ] g , i ∅ = � [ ψ ] ( g −{ ( x , g ( x )) } ) ∪{ ( x , d ) } , i if [ φ ] g , i = � 1 , d � [ φ → x .ψ ; y .ψ ] g , i = if [ φ ] g , i = � 2 , d � [ χ ] ( g −{ ( y , g ( y )) } ) ∪{ ( y , d ) } , i [ ι 1 φ ] g , i � 1 , [ φ ] g , i � = [ ι 2 φ ] g , i � 2 , [ φ ] g , i � = [ π 1 φ ] g , i fst ([ φ ] g , i ) = [ π 2 φ ] 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 ) = [ λ x φ ] g , i d �→ [ φ ] ( g −{ ( x , g ( x )) } ) ∪{ ( x , d ) } , i = [ˇ φ ] g , i [ φ ] g , i ( i ) = [ˆ φ ] 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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