Related papers: Comparing and evaluating extended Lambek calculi
Description logics (DLs) are well-known knowledge representation formalisms focused on the representation of terminological knowledge. Due to their first-order semantics, these languages (in their classical form) are not suitable for…
This thesis introduces the "method of structural refinement", which serves as a means of transforming the relational semantics of a modal and/or constructive logic into an 'economical' proof system by connecting two proof-theoretic…
Several formal systems, such as resolution and minimal model semantics, provide a framework for logic programming. In this paper, we will survey the use of structural proof theory as an alternative foundation. Researchers have been using…
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…
The notion of a non-deterministic logical matrix (where connectives are interpreted as multi-functions) extends the traditional semantics for propositional logics based on logical matrices (where connectives are interpreted as functions).…
We study relationship between first order multiplicative linear logic (MLL1), which has been known to provide representations to different categorial grammars, and the recently introduced extended tensor type calculus (ETTC). We identify a…
A natural next step in the evolution of constraint-based grammar formalisms from rewriting formalisms is to abstract fully away from the details of the grammar mechanism---to express syntactic theories purely in terms of the properties of…
We present nested sequent systems for propositional G\"odel-Dummett logic and its first-order extensions with non-constant and constant domains, built atop nested calculi for intuitionistic logics. To obtain nested systems for these…
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 (J\"ager, 1998) and the Lambek calculus…
While context-free grammars are characterized by a simple proof-theoretic grammatical formalism namely categorial grammar and its logic the Lambek calculus, no such characterizations were known for tree-adjoining grammars, and even for any…
Although language models demonstrate remarkable proficiency on mathematical benchmarks, it remains unclear whether this reflects true mathematical reasoning or statistical pattern matching over learning formal syntax. Most existing…
We introduce proper display calculi for basic monotonic modal logic, the conditional logic CK and a number of their axiomatic extensions. These calculi are sound, complete, conservative and enjoy cut elimination and subformula property. Our…
We propose a way to unify two approaches of non-cloning in quantum lambda-calculi: logical and algebraic linearities. The first approach is to forbid duplicating variables, while the second is to consider all lambda-terms as…
A well-known approach to treating syntactic island constraints in the setting of Lambek grammars consists in adding specific bracket modalities to the logic. We adapt this approach to abstract categorial grammars (ACG). Thus we define…
Linear logic was conceived in 1987 by Girard and, in contrast to classical logic, restricts the usage of the structural inference rules of weakening and contraction. With this, atoms of the logic are no longer interpreted as truth, but as…
Most graph query languages are rooted in logic. By contrast, in this paper we consider graph query languages rooted in linear algebra. More specifically, we consider MATLANG, a matrix query language recently introduced, in which some basic…
Answering complex queries over incomplete knowledge graphs (KGs) is a challenging job. Most previous works have focused on learning entity/relation embeddings and simulating first-order logic operators with various neural networks. However,…
The hyperedge replacement grammar (HRG) formalism is a natural and well-known generalization of context-free grammars. HRGs inherit a number of properties of context-free grammars, e.g. the pumping lemma. This lemma turns out to be a strong…
This chapter provides an introduction to the use of diagrammatic language, or perhaps more accurately, diagrammatic calculus, in quantum information and quantum foundations. We illustrate the use of diagrammatic calculus in one particular…
We propose to use Tarski's least fixpoint theorem as a basis to define recursive functions in the calculus of inductive constructions. This widens the class of functions that can be modeled in type-theory based theorem proving tool to…