Non-associative Lambek Calculus Research Articles

Sequent Systems for Consequence Relations of Cyclic Linear Logics

Linear Logic is a versatile framework with diverse applications in computer science and mathematics. One intriguing fragment of Linear Logic is Multiplicative-Additive Linear Logic (MALL), which forms the exponential-free component of the larger framework. Modifying MALL, researchers have explored weaker logics such as Noncommutative MALL (Bilinear Logic, BL) and Cyclic MALL (CyMALL) to investigate variations in commutativity. In this paper, we focus on Cyclic Nonassociative Bilinear Logic (CyNBL), a variant that combines noncommutativity and nonassociativity. We introduce a sequent system for CyNBL, which includes an auxiliary system for incorporating nonlogical axioms. Notably, we establish the cut elimination property for CyNBL. Moreover, we establish the strong conservativeness of CyNBL over Full Nonassociative Lambek Calculus (FNL) without additive constants. The paper highlights that all proofs are constructed using syntactic methods, ensuring their constructive nature. We provide insights into constructing cut-free proofs and establishing a logical relationship between CyNBL and FNL.

Residuated Basic Logic

Residuated basic logic (RBL) is the logic of residuated basic algebras, which constitutes a conservative extension of basic propositional logic (BPL). The basic implication is a residual of a non-associative binary operator in RBL. The conservativity is shown by relational semantics. A Gentzen-style sequent calculus GRBL, which is an extension of the distributive full non-associative Lambek calculus, is established for residuated basic logic. The calculus GRBL admits the mix-elimination, subformula, and disjunction properties. Moreover, the class of all residuated basic algebras has the finite embeddability property. The consequence relation of GRBL is decidable.

Serial verb constructions and covert coordinations in Edo – an analysis in Type Logical Grammar

Based on both syntactic and semantic criteria, Stewart (2001) and, following him, Baker and Stewart (1999), distinguish two types of serial verb constructions (SVC) and one type of covert coordination (CC) in Edo. In this article, we present an analysis of these constructions, using Type Logical Grammar (TLG) with an event-based semantic component. We choose as base logic the non-associative Lambek calculus augmented with two unary multiplicative connectives (NL(◊, □)). SVCs and CCs are interpreted as complex event structures. The complex predicates underlying these structures are derived from simple verbs by means of a constructor. SVCs and CCs differ in terms of which part of the complex event structure is denoted. For SVCs, this is the sum of all events in the structure whereas for a CC this is only the first event in the sequence. The two verbs in an SVC and a CC are treated asymmetrically by assuming that the first verb has an extended subcategorization frame. The additional argument is of type vp (possibly modally decorated). Constraints on word order and the realization of arguments are accounted for using structural rules like permutation and contraction. The application of these rules is enforced by making use of the unary connectives.

Lambek Calculus with Conjugates

We study an expansion of the Distributive Non-associative Lambek Calculus with conjugates of the Lambek product operator and residuals of those conjugates. The resulting logic is well-motivated, under-investigated and difficult to tackle. We prove completeness for some of its fragments and establish that it is decidable. Completeness of the logic is an open problem; some difficulties with applying the usual proof method are discussed.

From positive PDL to its non-classical extensions

AbstractWe provide a complete binary implicational axiomatization of the positive fragment of propositional dynamic logic (PDL). The intended application of this result are completeness proofs for non-classical extensions of positive PDL. Two examples are discussed in this article, namely, a paraconsistent extension with modal De Morgan negation and a substructural extension with the residuated operators of the non-associative Lambek calculus. Informal interpretations of these two extensions are outlined.

A proof-theoretic approach to scope ambiguity in compositional vector space models

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 introduce one reading for each quantifier. We combine a polarised focussed sequent calculus for the non-associative Lambek Calculus NL as described in Moortgat and Moot (2011) with a non-compositional ap- proach to quantifier scope in vector space semantics introduced by Hedges and Sadrzadeh (2016) and (Sadrzadeh 2016). In particular, we obtain an automatic and compositional procedure for obtaining quantifier scope ambiguity in vector space models of meaning.

DE MORGAN INTERPRETATION OF THE LAMBEK–GRISHIN CALCULUS

AbstractWe present an embedding of the Lambek–Grishin calculus into an extension of the nonassociative Lambek calculus with negation. The embedding is based on the De Morgan interpretation of the dual Grishin connectives.

On Involutive Nonassociative Lambek Calculus

Involutive Nonassociative Lambek Calculus (InNL) is a nonassociative version of Noncommutative Multiplicative Linear Logic (MLL) (Abrusci in J Symb Log 56:1403–1451, 1991), but the multiplicative constants are not admitted. InNL adds two linear negations to Nonassociative Lambek Calculus (NL); it is a strongly conservative extension of NL (Buszkowski in Amblard, de Groote, Pogodalla, Retoré (eds) Logical aspects of computational linguistics. LNCS, vol 10054. Springer, Berlin, pp 68–84, 2016). Here we also add unary modalities satisfying the residuation law and De Morgan laws. For the resulting logic InNLm, we define and study phase spaces (some frame models, typical for linear logics). We use them to prove the cut elimination theorem for a one-sided sequent system for InNLm, introduced here. Phase spaces are also employed in studying auxiliary systems InNLm(k), assuming the k-cyclic law for negation. The latter behave similarly as Classical Nonassociative Lambek Calculus, studied in de Groote and Lamarche (Stud Log 71(3):355–388, 2002) and Buszkowski (2016). We reduce the provability in InNLm to that in InNLm(k). This yields the equivalence of type grammars based on InNLm with (epsilon -free) context-free grammars and the PTIME complexity of InNLm.

Involutive Nonassociative Lambek Calculus: Sequent Systems and Complexity

In [5] we study Nonassociative Lambek Calculus (NL) augmented with De Morgan negation, satisfying the double negation and contraposition laws. This logic, introduced by de Grooté and Lamarche [10], is called Classical Non-Associative Lambek Calculus (CNL). Here we study a weaker logic InNL, i.e. NL with two involutive negations. We present a one-sided sequent system for InNL, admitting cut elimination. We also prove that InNL is PTIME.

Unified correspondence and proof theory for strict implication

The unified correspondence theory for distributive lattice expansion logics (DLE-logics) is specialized to strict implication logics. As a consequence of a general semantic consevativity result, a wide range of strict implication logics can be conservatively extended to Lambek Calculi over the bounded distributive full non-associative Lambek calculus (BDFNL). Many strict implication sequents can be transformed into analytic rules employing one of the main tools of unified correspondence theory, namely (a suitably modified version of) the Ackermann lemma based algorithm $\msf{ALBA}$. Gentzen-style cut-free sequent calculi for BDFNL and its extensions with analytic rules which are transformed from strict implication sequents, are developed.

Dynamic epistemic logic in update logic

We show that dynamic epistemic logic (DEL) is a substructural logic and that it is an extension of the update logic introduced in the companion article [12]. We identify axioms and inference rules that completely characterize the DEL product update, and we provide a sequent calculus for DEL. Finally, we show that DEL with a finite number of atomic events is as expressive as epistemic logic. In parallel, we provide a sequent calculus for update logic which turns out to be a generalization of the non-associative Lambek calculus.

Displaying updates in logic

Abstract We generalize the language of substructural logics interpreted over the ternary relational semantics. This is motivated by our intention to capture within the logical framework of substructural logics various logic-based formalisms dealing with common sense reasoning and logical dynamics. This initiative is based on the key observation that an update can be represented abstractly by the ternary relation of the substructural framework: the first argument of the ternary relation represents an initial situation, the second an informative event and the third the resulting situation after the occurrence of the informative event. Refining the language of substructural logics may allow us to account for the fine-grained reasoning present in common sense reasoning. Thus, we introduce three triples of connectives that are interconnected by means of cyclic permutations. The usual fusion, implication and co-implication connectives form one of these triples. We define a cut-free display calculus for our language that generalizes the display calculus for modal logic. We do not resort to structural connectives for truth constants and we show how we can obtain our display rules using Gaggle Theory. We prove the soundness and strong completeness of our display calculus via a Henkin-style construction. Using correspondence results from substructural logics, we also obtain sound and complete display calculi for a wide variety of classical and substructural logics. Then, we define dual substructural connectives and we provide a display calculus for our language extended with these dual connectives. Finally, we focus on the specific case of bi-intuitionistic logic for which we provide a novel sound and strongly complete display calculus. In a companion article (Aucher, 2016, J. Logic Comput.), we provide a sequent calculus for update logic that generalizes the non-associative Lambek calculus and a sequent calculus for dynamic epistemic logic that extends this sequent calculus for update logic.

UNDECIDABILITY OF CONSEQUENCE RELATION IN FULL NON-ASSOCIATIVE LAMBEK CALCULUS

AbstractWe prove that the consequence relation in the Full Non-associative Lambek Calculus is undecidable. An encoding of the halting problem for 2-tag systems using finitely many sequents in the language {⋅,∨} is presented. Therefore already the consequence relation in this fragment is undecidable. Moreover, the construction works even when the structural rules of exchange and contraction are added.

Non-associative Lambek calculus with modalities: interpolation, complexity and FEP

Non-associative Lambek calculus with modalities: interpolation, complexity and FEP

Two weak Lambek style calculi: DNL and DNL−∗

The calculus DNL results from the non-associative Lambek calculus NL by splitting the product functor into the right (⊲) and left (⊳) product interacting respectively with the right (/) and left (\) residuation. Unlike NL, sequent antecedents in the Gentzen-style axiomatics of DNL are not phrase structures (i.e., bracketed strings) but functor-argument structures. DNL− is a weaker variant of DNL restricted to fa-structures of order ≤ 1. When axiomatized by means of introduction/elimination rules for / and \, it shows a perfect analogy to NL which DNL lacks.

Relational Semantics of the Lambek Calculus Extended with Classical Propositional Logic

We show that the relational semantics of the Lambek calculus, both nonassociative and associative, is also sound and complete for its extension with classical propositional logic. Then, using filtrations, we obtain the finite model property for the nonassociative Lambek calculus extended with classical propositional logic.

Epistemic closure and commutative, nonassociative residuated structures

K-axiom-based epistemic closure for explicit knowledge is rejected for even the most trivial cases of deductive inferential reasoning on account of the fact that the closure axiom does not extend beyond a raw consequence relation. The recognition that deductive inference concerns interaction as much as it concerns consequence allows for perspectives from logics of multi-agent information flow to be refocused onto mono-agent deductive reasoning. Instead of modeling the information flow between different agents in a communicative or announcement setting, we model the information flow between different states of a single agent as that agent reasons deductively. The resource management of the database of agent states for the deductive reasoning fragment in question is covered by the residuated structure that encodes the nonassociative Lambek Calculus with permutation, bottom, and identity: NLP01.

On the Complexity of Nonassociative Lambek Calculus with Unit

Nonassociative Lambek Calculus (NL) is a syntactic calculus of types introduced by Lambek [8]. The polynomial time decidability of NL was established by de Groote and Lamarche [4]. Buszkowski [3] showed that systems of NL with finitely many assumptions are decidable in polynomial time and generate context-free languages; actually the P-TIME complexity is established for the consequence relation of NL. Adapting the method of Buszkowski [3] we prove an analogous result for Nonassociative Lambek Calculus with unit (NL1). Moreover, we show that any Lambek grammar based on NL1 (with assumptions) can be transformed into an equivalent context-free grammar in polynomial time.

S4 enriched multimodal categorial grammars are context-free

Bar-Hillel et al. [Y. Bar-Hillel, C. Gaifman, E. Shamir, On categorial and phrase structure grammars, Bulletin of the Research Council of Israel F (9) (1960) 1–16] prove that applicative categorial grammars weakly recognize the context-free languages. Buszkowski [W. Buszkowski, Generative capacity of non-associative Lambek calculus, Bulletin of the Polish Academy of Sciences: Mathematics 34 (1986) 507–518] proves that grammars based on the product-free fragment of the non-associative Lambek calculus NL recognize exactly the context-free languages. Kandulski [M. Kandulski, The equivalence of non-associative Lambek categorial grammars and context-free grammars, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 34 (1988) 41–52] furthers this result by proving that grammars based on NL also recognize exactly the context-free languages. Jäger [G. Jäger, Residuation, structural rules and context freeness, Journal of Logic, Language and Information 13 (2004) 47–59] proves that categorial grammars based on NL ♢ , the non-associative Lambek calculus enriched with residuated modalities, weakly recognize exactly the context-free languages. We extend this result, proving that categorial grammars based on NL S 4 , the enrichment of NL ♢ by the axioms 4 and T , weakly recognize exactly the context-free languages.

On Reduction Systems Equivalent to the Non-associative Lambek Calculus with the Empty String

On Reduction Systems Equivalent to the Non-associative Lambek Calculus with the Empty String