First-Order Definability of Transition Structures Abstract The transition semantics presented in Rumberg (J Log Lang Inf 25(1):77–108, 2016a) constitutes a fine-grained framework for modeling the interrelation of modality and time in branching time structures. In that framework, sentences of the transition language \(\mathcal {L}_\mathsf{t}\) are evaluated on transition structures at pairs consisting of a moment and a set of transitions. In this paper, we provide a class of first-order definable Kripke structures that preserves \(\mathcal {L}_\mathsf{t}\) -validity w.r.t. transition structures. As a consequence, for a certain fragment of \(\mathcal {L}_\mathsf{t}\) , validity w.r.t. transition structures turns out to be axiomatizable. The result is then extended to the entire language \(\mathcal {L}_\mathsf{t}\) by means of a quite natural ‘Henkin move’, i.e. by relaxing the notion of validity to bundled structures.

Supercover Semantics for Deontic Action Logic Abstract The semantics for a deontic action logic based on Boolean algebra is extended with an interpretation of action expressions in terms of sets of alternative actions, intended as a way to model choice. This results in a non-classical interpretation of action expressions, while sentences not in the scope of deontic operators are kept classical. A deontic structure based on Simons’ supercover semantics is used to interpret permission and obligation. It is argued that these constructions provide ways to handle various problems related to free choice permission. The main result is a sound and complete axiomatization of the semantics.

An Application of Peircean Triadic Logic: Modelling Vagueness Abstract Development of decision-support and intelligent agent systems necessitates mathematical descriptions of uncertainty and fuzziness in order to model vagueness. This paper seeks to present an outline of Peirce’s triadic logic as a practical new way to model vagueness in the context of artificial intelligence (AI). Charles Sanders Peirce (1839–1914) was an American scientist–philosopher and a great logician whose triadic logic is a culmination of the study of semiotics and the mathematical study of anti-Cantorean model of continuity and infinitesimals. After presenting Peircean semiotics within AI perspective, a mathematical formulation of a Peircean triadic set is given in relationship with classical and fuzzy sets. Using basic logical operators, all possible respective implication operators, bi-equivalence operators, valid rules of inference, and associative, distributive and commutative logical properties are derived and verified through the truth function approach. In order to suggest practical directions, aggregation operators for Peirce’s triadic logic have been formulated. A mathematical formulation of a medical diagnostic problem and ER diagram of a library management system using Peirce’s triadic relation show potential for further applications of the proposed triadic set and triadic logic. Alongside, a classical AI game—The Wumpus World—is implemented to show practical efficacy in comparison with binary implementation. Besides giving some preliminary formulations for trichotomous set theory and definition of finite automaton, development of hybrid architectures for intelligent agents and evolutionary computations are discussed as potential practical avenues for Peirce’s triadic logic.

Introduction

Combining Machine Learning and Semantic Features in the Classification of Corporate Disclosures Abstract We investigate an approach to improving statistical text classification by combining machine learners with an ontology-based identification of domain-specific topic categories. We apply this approach to ad hoc disclosures by public companies. This form of obligatory publicity concerns all information that might affect the stock price; relevant topic categories are governed by stringent regulations. Our goal is to classify disclosures according to their effect on stock prices (negative, neutral, positive). In the study reported here, we combine natural language parsing with a formal background ontology to recognize disclosures concerning particular topics from a prescribed list. The semantic analysis identifies some of these topics with reasonable accuracy. We then demonstrate that machine learners benefit from the additional ontology-based information when predicting the cumulative abnormal return attributed to the disclosure at hand.

On Involutive Nonassociative Lambek Calculus Abstract 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.

Natural Language Semantics and Computability Abstract This paper is a reflexion on the computability of natural language semantics. It does not contain a new model or new results in the formal semantics of natural language: it is rather a computational analysis, in the context for type-logical grammars, of the logical models and algorithms currently used in natural language semantics, defined as a function from a grammatical sentence to a (non-empty) set of logical formulas—because a statement can be ambiguous, it can correspond to multiple formulas, one for each reading. We argue that as long as we do not explicitly compute the interpretation in terms of possible world models, one can compute the semantic representation(s) of a given statement, including aspects of lexical meaning. This is a very generic process, so the results are, at least in principle, widely applicable. We also discuss the algorithmic complexity of this process.

A Type-Driven Vector Semantics for Ellipsis with Anaphora Using Lambek Calculus with Limited Contraction Abstract 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 (Anaphora and type logical grammar, Springer, Berlin, 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 previous compositional distributional models of relative pronouns, coordination and a restricted account of ellipsis in the DisCoCat framework of Coecke et al. (Mathematical foundations for a compositional distributional model of meaning, 2010. arXiv:1003.4394, Ann Pure Appl Log 164(11):1079–1100, 2013). We show how DisCoCat cannot deal with general forms of ellipsis, which rely on copying of information, and develop a novel way of connecting typelogical grammar to distributional semantics by assigning vector interpretable lambda terms to derivations of LCC in the style of Muskens and Sadrzadeh (in: Amblard, de Groote, Pogodalla, Retoré (eds) Logical aspects of computational linguistics, Springer, Berlin, 2016). What follows is an account of (verb phrase) ellipsis in which word meanings can be copied: the meaning of a sentence is now a program with non-linear access to individual word embeddings. We present the theoretical setting, work out examples, and demonstrate our results with a state of the art distributional model on an extended verb disambiguation dataset.

Construction-Based Compositional Grammar Abstract The paper presents a system for construction classification representing multiple levels of specification, such as grammatical functions, grammatically reflected actants, and lexical semantics, aligned with a compositional system of sign combination mediating between a construction perspective and a valence perspective. The system uses a feature structure formalism based on Head-Driven Phrase Structure Grammar (HPSG) but with essential elements from Lexical Functional Grammar (LFG; cf. Bresnan in Lexical functional syntax. Blackwell, Oxford, 2001), and has as implementation background large scale HPSG grammars. While on the one extreme being able to encode word level selection in multi-word patterns, the system on the other provides a compact format for construction specification, allowing for cross-language comparison both in construction and valence frame inventories. Pivotal in these capacities as well as in sign formalization in general are the grammatical functions. The paper motivates the usefulness of the various functionalities and illustrates the way in which they work together in a formally uniform system.

Parsing/Theorem-Proving for Logical Grammar CatLog3 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.