我们为带软亚指数的Lambek演算构建了向量空间语义，将该演算用于为寄生 gap 名词短语以及含回指和省略的话语单元构造组合式向量解释，并在分布性句子相似度任务中对这些构造开展实验。与此前采用带相关模态的Lambek演算的研究不同，本文使用的演算采用了该模态的有界版本，且是可判定的。这种新模态的向量空间语义让我们能够将缩约有意义地定义为投影，并为以往只能通过非线性映射实现的内容提供了线性理论支撑。
英文摘要：
We develop a vector space semantics for Lambek Calculus with Soft Subexponentials, apply the calculus to construct compositional vector interpretations for parasitic gap noun phrases and discourse units with anaphora and ellipsis, and experiment with the constructions in a distributional sentence similarity task. As opposed to previous work, which used Lambek Calculus with a Relevant Modality the calculus used in this paper uses a bounded version of the modality and is decidable. The vector space semantics of this new modality allows us to meaningfully define contraction as projection and provide a linear theory behind what we could previously only achieve via nonlinear maps.
In this paper, we consider the polymodal version of Lambek calculus with subexponential modalities initially introduced by Kanovich, Kuznetsov, Nigam, and Scedrov and its quantale semantics. In our approach, subexponential modalities have an interpretation in terms of quantic conuclei. We show that this extension of Lambek calculus is complete w.r.t quantales with quantic conuclei. Also, we prove a representation theorem for quantales with quantic conuclei and show that Lambek calculus with subexponentials is relationally complete. Finally, we extend this representation theorem to the category…
We give a proof-theoretic and algorithmic complexity analysis for systems introduced by Morrill to serve as the core of the CatLog categorial grammar parser. We consider two recent versions of Morrill's calculi, and focus on their fragments including multiplicative (Lambek) connectives, additive conjunction and disjunction, brackets and bracket modalities, and the ! subexponential modality. For both systems, we resolve issues connected with the cut rule and provide necessary modifications, after which we prove admissibility of cut (cut elimination theorem). We also prove algorithmic undecidabi…
In this project, a rather complete proof-theoretical formalization of Lambek Calculus (non-associative with arbitrary extensions) has been ported from Coq proof assistent to HOL4 theorem prover, with some improvements and new theorems. Three deduction systems (Syntactic Calculus, Natural Deduction and Sequent Calculus) of Lambek Calculus are defined with many related theorems proved. The equivalance between these systems are formally proved. Finally, a formalization of Sequent Calculus proofs (where Coq has built-in supports) has been designed and implemented in HOL4. Some basic results includ…
First-order multiplicative intuitionistic linear logic (MILL1) can be seen as an extension of the Lambek calculus. In addition to the fragment of MILL1 which corresponds to the Lambek calculus (of Moot & Piazza 2001), I will show fragments of MILL1 which generate the multiple context-free languages and which correspond to the Displacement calculus of Morrilll e.a.
CNRS (LaBRI)（法国国家科学研究中心（拉布里实验室））；University of Bordeaux（波尔多大学）
Lambeks Syntactic Calculus, commonly referred to as the Lambek calculus, was innovative in many ways, notably as a precursor of linear logic. But it also showed that we could treat our grammatical framework as a logic (as opposed to a logical theory). However, though it was successful in giving at least a basic treatment of many linguistic phenomena, it was also clear that a slightly more expressive logical calculus was needed for many other cases. Therefore, many extensions and variants of the Lambek calculus have been proposed, since the eighties and up until the present day. As a result, th…
The sequent calculus sL for the Lambek calculus L (lambek 58) has no structural rules. Interestingly, sL is equivalent to a multimodal calculus mL, which consists of the nonassociative Lambek calculus with the structural rule of associativity. This paper proves that the sequent calculus or hypersequent calculus hD of the discontinuous Lambek calculus (Morrill and Valentín), which like sL has no structural rules, is also equivalent to an omega-sorted multimodal calculus mD. More concretely, we present a faithful embedding translation between mD and hD in such a way that it can be said that hD a…
National Research University Higher School of Economics（国立研究型大学高等经济学院）；Steklov Mathematical Institute of RAS（俄罗斯科学院斯捷克洛夫数学研究所）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）；University of Pennsylvania（宾夕法尼亚大学）
Lambek calculus is a logical foundation of categorial grammar, a linguistic paradigm of grammar as logic and parsing as deduction. Pentus (2010) gave a polynomial-time algorithm for determ- ining provability of bounded depth formulas in the Lambek calculus with empty antecedents allowed. Pentus' algorithm is based on tabularisation of proof nets. Lambek calculus with brackets is a conservative extension of Lambek calculus with bracket modalities, suitable for the modeling of syntactical domains. In this paper we give an algorithm for provability the Lambek calculus with brackets allowing empty…
We study an extension of the Distributive Full Non-associative Lambek Calculus with iterative division operators. The iterative operators can be seen as representing iterative composition of linguistic resources or of actions. A complete axiomatization of the logic is provided and decidability is established via a proof of the finite model property.
Every language recognized by the Lambek calculus with brackets is context-free. This is shown by combining an observation by Jäger with an entirely straightforward adaptation of the method Pentus used for the original Lambek calculus. The case of the variant of the calculus allowing sequents with empty antecedents is slightly more complicated, requiring a restricted use of the multiplicative unit.
We present a translation of the Lambek calculus with brackets and the unit constant, $\mathbf{Lb}^{\boldsymbol{*}}_{\mathbf{1}}$, into the Lambek calculus with brackets allowing empty antecedents, but without the unit constant, $\mathbf{Lb}^{\boldsymbol{*}}$. Using this translation, we extend previously known results for $\mathbf{Lb}^{\boldsymbol{*}}$ to $\mathbf{Lb}^{\boldsymbol{*}}_{\mathbf{1}}$: (1) languages generated by categorial grammars based on the Lambek calculus with brackets are context-free (Kanazawa 2017); (2) the polynomial-time algorithm for deciding derivability of bounded dep…
The syntactic calculus of Lambek is a deductive system for the multiplicative fragment of intuitionistic non-commutative linear logic. As a fine-grained calculus of resources, it has many applications, mostly in formal computational investigations of natural languages. This paper introduces a calculus of beta-eta-long normal forms for derivations in the Lambek calculus with multiplicative unit and conjunction among its logical connectives. Reduction to normal form follows the normalization by evaluation (NbE) strategy: (i) evaluate a derivation in a Kripke model of Lambek calculus; (ii) extrac…
Lambek's non-associative syntactic calculus (NL) excels in its resource consciousness: the usual structural rules for weakening, contraction, exchange and even associativity are all dropped. Recently, there have been proposals for conservative extensions dispensing with NL's intuitionistic bias towards sequents with single conclusions: De Groote and Lamarche's classical non-associative Lambek calculus (CNL) and the Lambek-Grishin calculus (LG) of Moortgat and associates. We demonstrate Andreoli's focalization property for said proposals: a normalization result for Cut-free sequent derivations …
This paper shows connections between command injection attacks, continuations, and the Lambek calculus: certain command injections, such as the tautology attack on SQL, are shown to be a form of control effect that can be typed using the Lambek calculus, generalizing the double-negation typing of continuations. Lambek's syntactic calculus is a logic with two implicational connectives taking their arguments from the left and right, respectively. These connectives describe how strings interact with their left and right contexts when building up syntactic structures. The calculus is a form of pro…
We revisit the old work of de Paiva on the models of the Lambek Calculus in dialectica models making sure that the syntactic details that were sketchy on the first version got completed and verified. We extend the Lambek Calculus with a κmodality, inspired by Yetter's work, which makes the calculus commutative. Then we add the of-course modality !, as Girard did, to re-introduce weakening and contraction for all formulas and get back the full power of intuitionistic and classical logic. We also present the categorical semantics, proved sound and complete. Finally we show the traditional proper…
Lambek calculus with a relevant modality $!\mathbf{L^*}$ of arXiv:1601.06303 syntactically resolves parasitic gaps in natural language. It resembles the Lambek calculus with anaphora $\mathbf{LA}$ of (Jäger, 1998) and the Lambek calculus with controlled contraction, $\mathbf{L}_{\Diamond}$, of arXiv:1905.01647v1 which deal with anaphora and ellipsis. What all these calculi add to Lambek calculus is a copying and moving behaviour. Distributional semantics is a subfield of Natural Language Processing that uses vector space semantics for words via co-occurrence statistics in large corpora of data…
In this paper hypergraph Lambek calculus ($\mathrm{HL}$) is presented. This formalism aims to generalize the Lambek calculus ($\mathrm{L}$) to hypergraphs as hyperedge replacement grammars extend context-free grammars. In contrast to the Lambek calculus, $\mathrm{HL}$ deals with hypergraph types and sequents; its axioms and rules naturally generalize those of $\mathrm{L}$. Consequently, certain properties (e.g. the cut elimination) can be lifted from $\mathrm{L}$ to $\mathrm{HL}$. It is shown that $\mathrm{L}$ can be naturally embedded in $\mathrm{HL}$; moreover, a number of its variants ($\ma…
We analyze the complexity of decision problems for Boolean Nonassociative Lambek Calculus admitting empty antecedent of sequents ($\mathsf{BFNL^*}$), and the consequence relation of Distributive Full Nonassociative Lambek Calculus ($\mathsf{DFNL}$). We construct a polynomial reduction from modal logic $\mathsf{K}$ into $\mathsf{BFNL^*}$. As a consequence, we prove that the decision problem for $\mathsf{BFNL^*}$ is PSPACE-hard. We also prove that the same result holds for the consequence relation of DFNL, by reducing $\mathsf{BFNL^*}$ in polynomial time to DFNL enriched with finite set of assum…
National Research University Higher School of Economics（俄罗斯国家研究型高等经济大学）
We consider tensor grammars, which are an example of \commutative" grammars, based on the classical (rather than intuitionistic) linear logic. They can be seen as a surface representation of abstract categorial grammars ACG in the sense that derivations of ACG translate to derivations of tensor grammars and this translation is isomorphic on the level of string languages. The basic ingredient are tensor terms, which can be seen as encoding and generalizing proof-nets. Using tensor terms makes the syntax extremely simple and a direct geometric meaning becomes transparent. Then we address the pro…
We develop a categorical compositional distributional semantics for Lambek Calculus with a Relevant Modality, which has a limited version of the contraction and permutation rules. The categorical part of the semantics is a monoidal biclosed category with a coalgebra modality as defined on Differential Categories. We instantiate this category to finite dimensional vector spaces and linear maps via quantisation functors and work with three concrete interpretations of the coalgebra modality. We apply the model to construct categorical and concrete semantic interpretations for the motivating examp…
We investigate language interpretations of two extensions of the Lambek calculus: with additive conjunction and disjunction and with additive conjunction and the unit constant. For extensions with additive connectives, we show that conjunction and disjunction behave differently. Adding both of them leads to incompleteness due to the distributivity law. We show that with conjunction only no issues with distributivity arise. In contrast, there exists a corollary of the distributivity law in the language with disjunction only which is not derivable in the non-distributive system. Moreover, this d…
In this paper we introduce Commutative/Non-Commutative Logic (CNC logic) and two categorical models for CNC logic. This work abstracts Benton's Linear/Non-Linear Logic by removing the existence of the exchange structural rule. One should view this logic as composed of two logics; one sitting to the left of the other. On the left, there is intuitionistic linear logic, and on the right is a mixed commutative/non-commutative formalization of the Lambek calculus. Then both of these logics are connected via a pair of monoidal adjoint functors. An exchange modality is then derivable within the logic…
\emph{Focused sequent calculi} are a refinement of sequent calculi, where additional side-conditions on the applicability of inference rules force the implementation of a proof search strategy. Focused cut-free proofs exhibit a special normal form that is used for defining identity of sequent calculi proofs. We introduce a novel focused display calculus fD.LG and a fully polarized algebraic semantics FP.LG for Lambek-Grishin logic by generalizing the theory of \emph{multi-type calculi} and their algebraic semantics with \emph{heterogenous consequence relations}. The calculus fD.LG has \emph{st…
This paper establishes the normalisation of natural deduction or lambda calculus formulation of Intuitionistic Non Commutative Logic --- which involves both commutative and non commutative connectives. This calculus first introduced by de Groote and as opposed to the classical version by Abrusci and Ruet admits a full entropy which allow order to be relaxed into any suborder. Our result also includes, as a special case, the normalisation of natural deduction the Lambek calculus with product, which is unsurprising but yet unproved. Regarding Intuitionistic Non Commutative Logic with full entrop…
We develop a vector space semantics for verb phrase ellipsis with anaphora using type-driven compositional distributional semantics based on the Lambek calculus with limited contraction (LCC) of Jäger (2006). Distributional semantics has a lot to say about the statistical collocation-based meanings of content words, but provides little guidance on how to treat function words. Formal semantics on the other hand, has powerful mechanisms for dealing with relative pronouns, coordinators, and the like. Type-driven compositional distributional semantics brings these two models together. We review pr…
We study how to relate well-known hypergraph grammars based on the double pushout (DPO) approach and grammars over the hypergraph Lambek calculus HL (called HL-grammars). It turns out that DPO rules can be naturally encoded by types of HL using methods similar to those used by Kanazawa for multiplicative-exponential linear logic. In order to generalize his reasonings we extend the hypergraph Lambek calculus by adding the exponential modality, which results in a new calculus HMEL0; then we prove that any DPO grammar can be converted into an equivalent HMEL0-grammar. We also define the conjuncti…
This paper introduces a natural deduction calculus for intuitionistic logic of belief $\mathsf{IEL}^{-}$ which is easily turned into a modal $λ$-calculus giving a computational semantics for deductions in $\mathsf{IEL}^{-}$. By using that interpretation, it is also proved that $\mathsf{IEL}^{-}$ has good proof-theoretic properties. The correspondence between deductions and typed terms is then extended to a categorical semantics for identity of proofs in $\mathsf{IEL}^{-}$ showing the general structure of such a modality for belief in an intuitionistic framework.
Utrecht University（乌得勒支大学）；Shandong University（山东大学）；Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）；University of Johannesburg（约翰内斯堡大学）
In recent years, the compositional distributional approach in computational linguistics has opened the way for an integration of the \emph{lexical} aspects of meaning into Lambek's type-logical grammar program. This approach is based on the observation that a sound semantics for the associative, commutative and unital Lambek calculus can be based on vector spaces by interpreting fusion as the tensor product of vector spaces. In this paper, we build on this observation and extend it to a `vector space semantics' for the \emph{general} Lambek calculus, based on \emph{algebras over a field} $\mat…
Institute for Theoretical Physics（理论物理研究所）；Center for Complex Systems Studies（复杂系统研究中心）；Utrecht University（乌得勒支大学）；Utrecht Institute of Linguistics OTS（乌得勒支语言学研究所OTS）
Extended versions of the Lambek Calculus currently used in computational linguistics rely on unary modalities to allow for the controlled application of structural rules affecting word order and phrase structure. These controlled structural operations give rise to derivational ambiguities that are missed by the original Lambek Calculus or its pregroup simplification. Proposals for compositional interpretation of extended Lambek Calculus in the compact closed category of FVect and linear maps have been made, but in these proposals the syntax-semantics mapping ignores the control modalities, eff…
We present a variant of the calculus of deductive systems developed in (Lambek 1972, 1974), and give a generalization of the Curry-Howard-Lambek theorem giving an equivalence between the category of typed lambda-calculi and the category of cartesian closed categories and exponential-preserving morphisms that leverages the theory of generalized categories (Schoenbaum 2016). We discuss potential applications and extensions.
We investigate the extent to which compositional vector space models can be used to account for scope ambiguity in quantified sentences (of the form "Every man loves some woman"). Such sentences containing two quantifiers introduce two readings, a direct scope reading and an inverse scope reading. This ambiguity has been treated in a vector space model using bialgebras by (Hedges and Sadrzadeh, 2016) and (Sadrzadeh, 2016), though without an explanation of the mechanism by which the ambiguity arises. We combine a polarised focussed sequent calculus for the non-associative Lambek calculus NL, as…
We prove that all axiomatic extensions of the full Lambek calculus with exchange can be axiomatized by formulas on the $\mathcal N_3$ level of the substructural hierarchy.
We use the Lambek Calculus with soft sub-exponential modalities to model and reason about discourse relations such as anaphora and ellipsis. A semantics for this logic is obtained by using truncated Fock spaces, developed in our previous work. We depict these semantic computations via a new string diagram. The Fock Space semantics has the advantage that its terms are learnable from large corpora of data using machine learning and they can be experimented with on mainstream natural language tasks. Further, and thanks to an existing translation from vector spaces to quantum circuits, we can also…
Oxford University Computing Laboratory（牛津大学计算实验室）；University of Cambridge Computer Laboratory（剑桥大学计算机实验室）
We propose a mathematical framework for a unification of the distributional theory of meaning in terms of vector space models, and a compositional theory for grammatical types, for which we rely on the algebra of Pregroups, introduced by Lambek. This mathematical framework enables us to compute the meaning of a well-typed sentence from the meanings of its constituents. Concretely, the type reductions of Pregroups are `lifted' to morphisms in a category, a procedure that transforms meanings of constituents into a meaning of the (well-typed) whole. Importantly, meanings of whole sentences live i…
