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Parsing/Theorem-Proving for Logical Grammar CatLog3

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Abstract

\({ CatLog3}\) is a 7000 line Prolog parser/theorem-prover for logical categorial grammar. In such logical categorial grammar syntax is universal and grammar is reduced to logic: an expression is grammatical if and only if an associated logical statement is a theorem of a fixed calculus. Since the syntactic component is invariant, being the logic of the calculus, logical categorial grammar is purely lexicalist and a particular language model is defined by just a lexical dictionary. The foundational logic of continuity was established by Lambek (Am Math Mon 65:154–170, 1958) (the Lambek calculus) while a corresponding extension including also logic of discontinuity was established by Morrill and Valentín (Linguist Anal 36(1–4):167–192, 2010) (the displacement calculus). \({ CatLog3}\) implements a logic including as primitive connectives the continuous (concatenation) and discontinuous (intercalation) connectives of the displacement calculus, additives, 1st order quantifiers, normal modalities, bracket modalities, and universal and existential subexponentials. In this paper we review the rules of inference for these primitive connectives and their linguistic applications, and we survey the principles of Andreoli’s focusing, and of a generalisation of van Benthem’s count-invariance, on the basis of which \({ CatLog3}\) is implemented.

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Notes

  1. Once Cut-elimination/admissibility has been established, the only challenge to decidability here comes from non-linearity: the infinitary left rule of the existential subexponential (Buszkowski 2007; Buszkowski and Palka 2008); and the contraction rule of the universal subexponential. In this connection, linguistically the existential subexponential left rule is not required. But the contraction rule of the universal subexponential is required, for parasitic gaps. The Lambek calculus \(\mathbf Lb! _\mathbf{b }\) with bracket modalities and the universal subexponential bracket conditioned contraction rule of Morrill (2017) is undecidable (Kanovich et al. 2017). However \({ CatLog3}\) uses a different bracket-conditioned contraction rule for the universal subexponential, essentially that of Morrill (2011b).

  2. For example, the semantically inactive additive conjunction \(A{\sqcap }B{:\,}\phi \) abbreviates \( A{ \& }B{:\,}(\phi , \phi )\).

  3. For example, the nondeterministic continuous division \(B{\div }A{:\,}\phi \) abbreviates \((A{\backslash }B){\sqcap }(B/A){:\,}\phi \), which is to say \( (A{\backslash }B){ \& }(B/A){:\,}(\phi , \phi )\).

  4. To anticipate linguistic analysis a little, a hypothetical gap subtype emitted by a relative pronoun corresponding to a long-distance dependency will enter a stoup, percolate in stoups, may contract to create (parasitic) gaps, and finally permutes into a (host) extraction site.

  5. Note the advantages of such polymorphism over assuming empty operators: if say empty determiners were allowed they could a priori occur any number of times in any positions; and they would also overgenerate, for example, the ungrammatical *most or dogs on the pattern of the grammatical most or all dogs.

  6. I thank Max Kanovich for drawing my attention to this design possibility.

  7. The synthetic connectives are: left and right projection and injection {\({\triangleleft ^{-1}}\), \({\triangleright ^{-1}}\), \({\triangleleft }\), \({\triangleright }\)}, (Morrill et al. 2009); split and bridge {\({\check{_{_{}}}}\), \({\hat{_{_{}}}}\)}, (Morrill and Merenciano 1996); and continuous and discontinuous nondeterministic multiplicatives {\({\div }\), \({\times }\), \({\Uparrow }\), \({\Downarrow }\), \({\circledcirc }\)}, (Morrill et al. 2011).

  8. Undefined values in infinitary arithmetic are indicated by \(*\); we could have opted to fail to reject any counts which are undefined, but in fact such cases do not ever occur (Kuznetsov et al. 2017).

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Authors and Affiliations

  1. Department of Computer Science, Universitat Politècnica de Catalunya, Jordi Girona Salgado 1-3, 08034, Barcelona, Spain

    Glyn Morrill

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Correspondence to Glyn Morrill.

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The present article is a revised version of Morrill (2017) ‘Parsing Logical Grammar: CatLog3’, in Loukanova and Liefke (eds.) Proceedings of the Workshop on Logic and Algorithms in Computational Linguistics 2017, LACompLing2017, DiVA, Stockholm University. The research was partially supported by MINECO Project TIN2017-89244-R. I thank three anonymous JLLI reviewers for their thoughtful comments, and Oriol Valentín for many related discussions.

Appendix: Relativisation examples

Appendix: Relativisation examples

In this appendix we illustrate further by presenting the \({ CatLog3}\) LaTeX output for relativisation extraction and parasitic extraction examples. This comprises output which is unedited, except for the resizing of derivations into sideways figures; the details can be zoomed online.

(eac(rel(9))) \(\mathbf{man }{+}[[\mathbf{that }{+}[\mathbf{ mary }]{+}\mathbf{likes }{+}\mathbf{today }]]: { CN}{} { s(m)}\)

$$\begin{aligned}&{\square }{} { CN}{} { s(m)}: { man}, [[{\blacksquare }{\forall }n({[]^{-1}}{[]^{-1}}({ CN}{ n}\backslash { CN}{ n})/{\blacksquare }(({\langle \rangle }Nt(n){\sqcap }!{\blacksquare }Nt(n))\backslash Sf)): \\&\lambda A\lambda B\lambda C[({ B}\ { C})\wedge ({ A}\ { C})], [{\blacksquare }Nt(s(f)): { m}], {\square }(({\langle \rangle }{\exists }gNt(s(g))\backslash Sf)/{\exists }aNa): \\&{\hat{\ }}\lambda D\lambda E({ Pres}\ (({\check{\ }}{} { like}\ { D})\ { E})), {\square }{\forall }a{\forall }f(({\langle \rangle }Na\backslash Sf)\backslash ({\langle \rangle }Na\backslash Sf)): {\hat{\ }}\lambda F\lambda G({\check{\ }}{} { today}\ ({ F}\ { G}))]]\ \\&\Rightarrow \ { CN}{} { s(m)} \end{aligned}$$
Fig. 22
figure 22

Derivation of man that Mary likes today

Full size image

See Fig. 22.

\(\lambda C[({\check{\ }}{} { man}\ { C})\wedge ({\check{\ }}{ today}\ ({ Pres}\ (({\check{\ }}{} { like}\ { C})\ { m})))]\)

(eac(rel(10))) \(\mathbf{man }{+}\mathbf{that }{+}[\mathbf{ the }{+}\mathbf{friends }{+}\mathbf{of }]{+}\mathbf{walk }: { CN}{} { s(m)}\)

$$ \begin{aligned}&{\square }{} { CN}{} { s(m)}: { man}, {\blacksquare }{\forall }n({[]^{-1}}{[]^{-1}}({ CN}{ n}\backslash { CN}{ n})/{\blacksquare }(({\langle \rangle }Nt(n){\sqcap }!{\blacksquare }Nt(n))\backslash Sf)): \\&\lambda A\lambda B\lambda C[({ B}\ { C})\wedge ({ A}\ { C})], [{\blacksquare }{\forall }n(Nt(n)/{ CN}{} { n}): \iota , \\&{\square }({ CN}{} { p}/{ PP}{} { of}): { friends}, {\square }(({\forall }n({ CN}{} { n}\backslash { CN}{} { n})/{\blacksquare }{\exists }bNb){ \& }({ PP}{ of}/{\exists }aNa)):\\&{\hat{\ }}({\check{\ }}{} { of}, \lambda D{ D})], {\square }({\langle \rangle }({\exists }aNa{-}{\exists }gNt(s(g)))\backslash Sf): {\hat{\ }}\lambda E({ Pres}\ ({\check{\ }}{} { walk}\ { E}))\ \Rightarrow \ { CN}{} { s(m)} \end{aligned}$$
$$ \begin{aligned}&{\square }{} { CN}{} { s(m)}: { man}, {\blacksquare }{\forall }n({[]^{-1}}{[]^{-1}}({ CN}{ n}\backslash { CN}{ n})/{\blacksquare }(({\langle \rangle }Nt(n){\sqcap }!{\blacksquare }Nt(n))\backslash Sf)):\\&\lambda A\lambda B\lambda C[({ B}\ { C})\wedge ({ A}\ { C})], [{\blacksquare }{\forall }n(Nt(n)/{ CN}{} { n}): \iota ,\\&{\square }({ CN}{} { p}/{ PP}{} { of}): { friends}, {\square }(({\forall }n({ CN}{} { n}\backslash { CN}{} { n})/{\blacksquare }{\exists }bNb){ \& }({ PP}{ of}/{\exists }aNa)): \\&{\hat{\ }}({\check{\ }}{} { of}, \lambda D{ D})], {\square }({\langle \rangle }{\exists }aNa\backslash Sb): {\hat{\ }}\lambda E({\check{\ }}{} { walk}\ { E})\ \Rightarrow \ { CN}{} { s(m)} \end{aligned}$$

(eac(rel(11))) \(\mathbf{man }{+}[[\mathbf{that }{+}[[\mathbf{ the }{+}\mathbf{friends }{+}\mathbf{of }]]{+}\mathbf{admire }]]: { CN}{ s(m)}\)

$$ \begin{aligned}&{\square }{} { CN}{} { s(m)}: { man}, [[{\blacksquare }{\forall }n({[]^{-1}}{[]^{-1}}({ CN}{ n}\backslash { CN}{ n})/{\blacksquare }(({\langle \rangle }Nt(n){\sqcap }!{\blacksquare }Nt(n))\backslash Sf)): \\&\lambda A\lambda B\lambda C[({ B}\ { C})\wedge ({ A}\ { C})], [[{\blacksquare }{\forall }n(Nt(n)/{ CN}{} { n}): \iota , \\&{\square }({ CN}{} { p}/{ PP}{} { of}): { friends}, {\square }(({\forall }n({ CN}{} { n}\backslash { CN}{ n})/{\blacksquare }{\exists }bNb){ \& }({ PP}{} { of}/{\exists }aNa)): {\hat{\ }}({\check{\ }}{} { of}, \lambda D{ D})]], \\&{\square }(({\langle \rangle }({\exists }aNa{-}{\exists }gNt(s(g)))\backslash Sf)/{\exists }aNa): {\hat{\ }}\lambda E\lambda F({ Pres}\ (({\check{\ }}{} { admire}\ { E})\ { F}))]]\ \Rightarrow \ { CN}{} { s(m)} \end{aligned}$$
Fig. 23
figure 23

Derivation of man that the friends of admire

Full size image

See Fig. 23.

\(\lambda C[({\check{\ }}{} { man}\ { C})\wedge ({ Pres}\ (({\check{\ }}{} { admire}\ { C})\ (\iota \ ({\check{\ }}{ friends}\ { C}))))]\)

Fig. 24
figure 24

Derivation of paper that John filed without reading

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Fig. 25
figure 25

Derivation of paper that the editor of filed without reading

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(eac(rel(12))) \(\mathbf{paper }{+}[[\mathbf{that }{+}[\mathbf{ john }]{+}\mathbf{filed }{+}[[\mathbf{without }{+}\mathbf{reading }]]]]: { CN}{} { s(n)}\)

$$\begin{aligned}&{\square }{} { CN}{} { s(n)}: { paper}, [[{\blacksquare }{\forall }n({[]^{-1}}{[]^{-1}}({ CN}{ n}\backslash { CN}{ n})/{\blacksquare }(({\langle \rangle }Nt(n){\sqcap }!{\blacksquare }Nt(n))\backslash Sf)): \\&\lambda A\lambda B\lambda C[({ B}\ { C})\wedge ({ A}\ { C})], [{\blacksquare }Nt(s(m)): { j}], {\square }(({\langle \rangle }{\exists }gNt(s(g))\backslash Sf)/{\exists }aNa): \\&{\hat{\ }}\lambda D\lambda E({ Past}\ (({\check{\ }}{} { file}\ { D})\ { E})), [[{\blacksquare }{\forall }a{\forall }f({[]^{-1}}(({\langle \rangle }Na\backslash Sf)\backslash ({\langle \rangle }Na\backslash Sf))/({\langle \rangle }Na\backslash Spsp)):\\&\lambda F\lambda G\lambda H[({ G}\ { H})\wedge \lnot ({ F}\ { H})], {\square }(({\langle \rangle }{\exists }aNa\backslash Spsp)/{\exists }aNa): {\hat{\ }}\lambda I\lambda J(({\check{\ }}{ read}\ { I})\ { J})]]]]\ \\&\Rightarrow \ { CN}{} { s(n)} \end{aligned}$$

See Fig. 24.

\(\lambda C[({\check{\ }}{} { paper}\ { C})\wedge [({ Past}\ (({\check{\ }}{} { file}\ { C})\ { j}))\wedge \lnot (({\check{\ }}{} { read}\ { C})\ { j})]]\)

(eac(rel(13))) \(\mathbf{paper }{+}[[\mathbf{that }{+}[[\mathbf{ the }{+}\mathbf{editor }{+}\mathbf{of }]]{+}\mathbf{filed }{+}[[\mathbf{without }{+}\mathbf{ reading }]]]]: { CN}{} { s(n)}\)

$$ \begin{aligned}&{\square }{} { CN}{} { s(n)}: { paper}, [[{\blacksquare }{\forall }n({[]^{-1}}{[]^{-1}}({ CN}{ n}\backslash { CN}{ n})/{\blacksquare }(({\langle \rangle }Nt(n){\sqcap }!{\blacksquare }Nt(n))\backslash Sf)):\\&\lambda A\lambda B\lambda C[({ B}\ { C})\wedge ({ A}\ { C})], [[{\blacksquare }{\forall }n(Nt(n)/{ CN}{} { n}): \iota , {\square }({\forall }g{ CN}{} { s(g)}/{ PP}{} { of}): { editor}, \\&{\square }(({\forall }n({ CN}{} { n}\backslash { CN}{} { n})/{\blacksquare }{\exists }bNb){ \& }({ PP}{ of}/{\exists }aNa)): {\hat{\ }}({\check{\ }}{} { of}, \lambda D{ D})]], \\&{\square }(({\langle \rangle }{\exists }gNt(s(g))\backslash Sf)/{\exists }aNa): {\hat{\ }}\lambda E\lambda F({ Past}\ (({\check{\ }}{} { file}\ { E})\ { F})),\\&{} [[{\blacksquare }{\forall }a{\forall }f({[]^{-1}}(({\langle \rangle }Na\backslash Sf)\backslash ({\langle \rangle }Na\backslash Sf))/({\langle \rangle }Na\backslash Spsp)): \lambda G\lambda H\lambda I[({ H}\ { I})\wedge \lnot ({ G}\ { I})],\\&{\square }(({\langle \rangle }{\exists }aNa\backslash Spsp)/{\exists }aNa): {\hat{\ }}\lambda J\lambda K(({\check{\ }}{ read}\ { J})\ { K})]]]]\ \Rightarrow \ { CN}{} { s(n)} \end{aligned}$$

See Fig. 25.

\(\lambda C[({\check{\ }}{} { paper}\ { C})\wedge [({ Past}\ (({\check{\ }}{} { file}\ { C})\ (\iota \ ({\check{\ }}{ editor}\ { C}))))\wedge \lnot (({\check{\ }}{} { read}\ { C})\ (\iota \ ({\check{\ }}{} { editor}\ { C})))]]\)

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Morrill, G. Parsing/Theorem-Proving for Logical Grammar CatLog3. J of Log Lang and Inf 28, 183–216 (2019). https://doi.org/10.1007/s10849-018-09277-w

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