Modal operator

About: Modal operator is a research topic. Over the lifetime, 1151 publications have been published within this topic receiving 22865 citations. The topic is also known as: modal connective.

Using formal logic to represent sign language phonetics in semi-automatic annotation tasks

Abstract: This thesis presents a formal framework for the representation of Signed Languages (SLs), the languages of Deaf communities, in semi-automatic recognition tasks. SLs are complex visio-gestural communication systems; by using corporal gestures, signers achieve the same level of expressivity held by sound-based languages like English or French. However, unlike these, SL morphemes correspond to complex sequences of highly specific body postures, interleaved with postural changes: during signing, signers use several parts of their body simultaneously in order to combinatorially build phonemes. This situation, paired with an extensive use of the three-dimensional space, make them difficult to represent with tools already existent in Natural Language Processing (NLP) of vocal languages. For this reason, the current work presents the development of a formal representation framework, intended to transform SL video repositories (corpus) into an intermediate representation layer, where automatic recognition algorithms can work under better conditions. The main idea is that corpora can be described with a specialized Labeled Transition System (LTS), which can then be annotated with logic formulae for its study. A multi-modal logic was chosen as the basis of the formal language: the Propositional Dynamic Logic (PDL). This logic was originally created to specify and prove properties on computer programs. In particular, PDL uses the modal operators [a] and to denote necessity and possibility, respectively. For SLs, a particular variant based on the original formalism was developed: the PDL for Sign Language (PDLSL). With the PDLSL, body articulators (like the hands or head) are interpreted as independent agents; each articulator has its own set of valid actions and propositions, and executes them without influence from the others. The simultaneous execution of different actions by several articulators yield distinct situations, which can be searched over an LTS with formulae, by using the semantic rules of the logic. Together, the use of PDLSL and the proposed specialized data structures could help curb some of the current problems in SL study; notably the heterogeneity of corpora and the lack of automatic annotation aids. On the same vein, this may not only increase the size of the available datasets, but even extend previous results to new corpora; the framework inserts an intermediate representation layer which can serve to model any corpus, regardless of its technical limitations. With this, annotations is possible by defining with formulae the characteristics to annotate. Afterwards, a formal verification algorithm may be able to find those features in corpora, as long as they are represented as consistent LTSs. Finally, the development of the formal framework led to the creation of a semi-automatic annotator based on the presented theoretical principles. Broadly, the system receives an untreated corpus video, converts it automatically into a valid LTS (by way of some predefined rules), and then verifies human-created PDLSL formulae over the LTS. The final product, is an automatically generated sub-lexical annotation, which can be later corrected by human annotators for their use in other areas such as linguistics.

Logical Operators for Ontological Modeling

Abstract: We show that logic has more to offer to ontologists than standard first order and modal operators. We first describe some operators of linear logic which we believe are particularly suitable for ontological modeling, and suggest how to interpret them within an ontological framework. After showing how they can coexist with those of classical logic, we analyze three notions of artifact from the literature to conclude that these linear operators allow for reducing the ontological commitment needed for their formalization, and even simplify their logical formulation.

Proofs and Expressiveness in Alethic Modal Logic

Abstract: The above dictionary characterization of alethic modalities states the central notions of alethic modal logic: necessity, and other notions that are usually thought of as being definable in terms of necessity and Boolean negation: impossibility, contingency, and possibility. The syntax of modal propositional logic is inductively defined over a denumerable set of sentence letters p0, p1, p2, . . . as follows: A ::= p | ¬A | (A ∨B) | 2A

Automated reasoning techniques for hybrid logics

Abstract: Hybrid logics augment classical modal logics with machinery for describing and reasoning about identity, which is crucial in many settings. Although modal logics we would today call ``hybrid'' can be traced back to the work of Prior in the 1960's, their systematic study only began in the late 1990's. Part of their interest comes from the fact they fill an important expressivity gap in modal logics. In fact, they are sometimes referred to as ``modal logics with equality''. One of the unifying themes of this thesis is the satisfiability problem for the arguably best-known hybrid logic, H(@,dwn), and some of its sublogics. Satisfiability is the basic problem in automated reasoning. In the case of hybrid logics it has been studied fundamentally using the tableaux method. In this thesis we attempt to complete the picture by investigating satisfiability for hybrid logics using first-order resolution (via translations) and variations of a resolution calculus that operates directly on hybrid formulas. We present firstly several satisfiability-preserving, linear-time translations from H(@,dwn) to first-order logic. These are conceived in a way such that they tend to reduce the search space of a resolution-based theorem prover for first-order logic. We then move our attention to resolution-based calculi that work directly on hybrid formulas. In particular, we will consider the so-called direct resolution calculus. Inspired by first-order logic resolution, we turn this calculus into a calculus of ordered resolution with selection functions and prove that it possesses the reduction property for counterexamples from which it follows its completeness and that it is compatible with the well-known standard redundancy criterion. We also show that certain refinement of this calculus constitutes a decision procedure for H(@), a decidable fragment of H(@,dwn). In the last part of this thesis we investigate certain normal forms for hybrid logics and other extended modal logics. We are interested in normal forms where certain modalities can be guaranteed not to occur under the scope of other modal operators. We will see that these kind of transformations can be exploited in a pre-processing step in order to reduce the number of inferences required by a modal prover. In an attempt to formulate these results in a way that encompasses also other extended modal logics, we arrived at a formulation of modal semantics in terms of a novel type of models that are coinductively defined. Many extended modal logics (such as hybrid logics) can be defined in terms of classes of coinductive models. This way, results that had to be proved separately for each different language (but whose proofs were known to be mere routine) now can be proved in a general way.