Semantics Of Logic Research Articles

Strong Equivalence of Logic Programs with Ordered Disjunction: A Logical Perspective

AbstractLogic Programs with Ordered Disjunction (LPODs) extend classical logic programs with the capability of expressing preferential disjunctions in the heads of program rules. The initial semantics of LPODs, although simple and quite intuitive, is not purely model-theoretic. As a result, certain properties of programs appear non-trivial to formalize in purely logical terms. For example, the current characterization of strong equivalence for LPODs, does not coincide with logical equivalence in some specific logic. This comes in sharp contrast with the well-known characterization of strong equivalence for classical logic programs, which coincides with logical equivalence in the logic of here-and-there. In this paper we obtain a purely logical characterization of strong equivalence for LPODs as logical equivalence in a four-valued logic. Moreover, we provide a new proof of the coNP-completeness of strong equivalence for LPODs, which has an interest in its own right since it relies on the special structure of such programs. Our results are based on the recent logical semantics of LPODs, a fact which we believe indicates that this new semantics may prove to be a useful tool in the further study of LPODs.

Neighbourhood Semantics for Modal Relevant Logics

Neighbourhood Semantics for Modal Relevant Logics

Modeling Abstract Algebra as an OWL Ontology (Student Abstract)

Description logic ontologies serve to model classifications and structural relationships, and to represent and reason about domain knowledge. Modeling the basic classification of abstract algebraic structures as an ontology demonstrates the difficulties presented by their logical semantics, and shed light on the limitations to accurately model further topics in algebra and related mathematical domains.

Finite Entailment of Local Queries in the Z Family of Description Logics

In the last few years the field of logic-based knowledge representation took a lot of inspiration from database theory. A vital example is that the finite model semantics in description logics (DLs) is reconsidered as a desirable alternative to the classical one and that query entailment has replaced knowledge-base satisfiability (KBSat) checking as the key inference problem. However, despite the considerable effort, the overall picture concerning finite query answering in DLs is still incomplete. In this work we study the complexity of finite entailment of local queries (conjunctive queries and positive boolean combinations thereof) in the Z family of DLs, one of the most powerful KR formalisms, lying on the verge of decidability. Our main result is that the DLs ZOQ and ZOI are finitely controllable, i.e. that their finite and unrestricted entailment problems for local queries coincide. This allows us to reuse recently established upper bounds on querying these logics under the classical semantics. While we will not solve finite query entailment for the third main logic in the Z family, ZIQ, we provide a generic reduction from the finite entail- ment problem to the finite KBSat problem, working for ZIQ and some of its sublogics. Our proofs unify and solidify previously established results on finite satisfiability and finite query entailment for many known DLs.

Parametric Triangulation in Forensic Linguistics (on the Example of the Speech Act of Threatening)

The article is devoted to the issues of normalization of methodological tools for forensic linguistic examination of the text. The significance of the study is due to the current state of affairs in the industry, where, despite the successful use of variable procedures of linguistic and legal study of the text in procedural-investigative and judicial practices, there are contradictions in expert opinions, a lack of clarity in the presentation of conclusions. Currently, jurislinguistic theory is objectively unable to offer a single methodological standard of work due to the potential variability of conflict communicative situations, a large number of extralinguistic factors that affect speech interaction. As a solution to this problem, it is proposed to introduce the principles of methodological triangulation into the scientific field of linguistic expertology - the use of several research procedures to solve one problem. The purpose of the scientific work is to substantiate and present a model of the corresponding parametric-triangulation algorithm for forensic linguistic examination of the text on the material of such a type of offense as a threat. The article analyzes the characteristics of the threat presented in the criminal legislation of the Republic of Belarus. The conducted research shows that triangulation can be successfully extrapolated into the methodology of legal language parametrization. This allows 1) to establish the parameters that identify the offense, 2) to determine the methodological procedures for studying them, and 3) to evaluate the obtained research results. A description of the algorithmic methodological model of a forensic linguistic study of a text for the presence of signs of a verbal threat in its content is given. As a result of scientific research, a system of expert procedures is presented, including a number of methods verified in the practice of special linguistic and legal research (automatic interpretation, lexicocentric, gener, logical semantics analysis, etc.). The described methods of expert work make it possible to study three mandatory (subject composition and subject affiliation of an action, addressing and temporal marking) and one optional (action type) parameters of a threat committed verbally.

Iterated team semantics for a hierarchy of informational types

In this paper, we introduce and study a framework that is inspired by the team semantics for propositional dependence logic but deviates from it in several respects. Most importantly, instead of the two semantic layers used in dependence logic – possible worlds and teams – a whole hierarchy of contexts is introduced and different types of formulas are evaluated at different levels of this hierarchy. This leads to a rich stratification of informational types. In this framework, the dependence operator of dependence logic can be defined by the standard propositional connectives (negation, conjunction, disjunction and implication). We explore the formal aspects of this approach and apply it to a number of puzzling phenomena related to modalities and conditionals.

Operational theories as structural realism

We undertake a reconstruction of the epistemic significance of research on operational theories in quantum foundations. We suggest that the space of operational theories is analogous to the space of possible worlds employed in the possible world semantics for modal logic, so research of this sort can be understood as probing modal structure. Thus we argue that operational axiomatisations of quantum mechanics may be interpreted as a novel form of structural realism; we discuss the consequences of this interpretation for the philosophy of structural realism and the future of operational theories.

Three faces of recursion axioms: the case of constructive dynamic logic of relation changers

Abstract This paper proposes an intuitionistic generalization of van Benthem and Liu’s dynamic logic of relation changers, where relation changers are dynamic operators that rewrite each agent’s accessibility relation. We employ Nishimura’s Kripke semantics for a constructive propositional dynamic logic to define the semantics of relation changers. Strong completeness results of Hilbert-style axiomatizations for constructive dynamic logic of relation changers are established by two types of rewriting strategies via recursion axioms: inside-out and outside-in strategies. To establish the semantic completeness by outside-in strategy, we propose the notion of composition of relation changers and an appropriately defined notion of length of a formula. Moreover, we reveal that recursion axioms have additional two faces: semantic and proof-theoretic ones. As for the semantic face, we introduce an alternative semantics for dynamic logic of relation changers to specify the semantic meaning of recursion axioms used in the inside-out strategy. This leads us to a semantic completeness proof of the axiomatization for the original semantics, which does not require a rewriting strategy based on recursion axioms. As for the proof-theoretic face, we transform each of recursion axioms required in the outside-in strategy into two (left and right) inference rules to provide a sequent calculus for dynamic logic of relation changers. Since the resulting sequent calculus is cut-free but semi-analytic, it still enjoys the Craig interpolation theorem by Maehara’s method.

Semantic ghost imaging based on recurrent-neural-network.

Ghost imaging (GI) illuminates an object with a sequence of light patterns and obtains the corresponding total echo intensities with a bucket detector. The correlation between the patterns and the bucket signals results in the image. Due to such a mechanism different from the traditional imaging methods, GI has received extensive attention during the past two decades. However, this mechanism also makes GI suffer from slow imaging speed and poor imaging quality. In previous work, each sample, including an illumination pattern and its detected bucket signal, was treated independently with each other. The correlation is therefore a linear superposition of the sequential data. Inspired by human's speech, where sequential words are linked with each other by a certain semantic logic and an incomplete sentence could still convey a correct meaning, we here propose a different perspective that there is potentially a non-linear connection between the sequential samples in GI. We therefore built a system based on a recurrent neural network (RNN), called GI-RNN, which enables recovering high-quality images at low sampling rates. The test with MNIST's handwriting numbers shows that, under a sampling rate of 1.28%, GI-RNN have a 12.58 dB higher than the traditional basic correlation algorithm and a 6.61 dB higher than compressed sensing algorithm in image quality. After trained with natural images, GI-RNN exhibits a strong generalization ability. Not only does GI-RNN work well with the standard images such as "cameraman", but also it can recover the natural scenes in reality at the 3% sampling rate while the SSIMs are greater than 0.7.

On Logics and Semantics for Interpretability

AbstractWe study various properties of formalised relativised interpretability. In the central part of this thesis we study for different interpretability logics the following aspects: completeness for modal semantics, decidability and algorithmic complexity.In particular, we study two basic types of relational semantics for interpretability logics. One is the Veltman semantics, which we shall refer to as the regular or ordinary semantics; the other is called generalised Veltman semantics. In the recent years and especially during the writing of this thesis, generalised Veltman semantics was shown to be particularly well-suited as a relational semantics for interpretability logics. In particular, modal completeness results are easier to obtain in some cases; and decidability can be proven via filtration in all known cases. We prove various new and reprove some old completeness results with respect to the generalised semantics. We use the method of filtration to obtain the finite model property for various logics.Apart from results concerning semantics in its own right, we also apply methods from semantics to determine decidability (implied by the finite model property) and complexity of provability (and consistency) problems for certain interpretability logics.From the arithmetical standpoint, we explore three different series of interpretability principles. For two of them, for which arithmetical and modal soundness was already known, we give a new proof of arithmetical soundness. The third series results from our modal considerations. We prove it arithmetically sound and also characterise frame conditions w.r.t. ordinary Veltman semantics. We also prove results concerning the new series and generalised Veltman semantics.Abstract prepared by Luka Mikec.E-mail: luka.mikec1@gmail.comURL: http://hdl.handle.net/2445/177373

Backpropagation through signal temporal logic specifications: Infusing logical structure into gradient-based methods

This paper presents a technique, named STLCG, to compute the quantitative semantics of Signal Temporal Logic (STL) formulas using computation graphs. STLCG provides a platform which enables the incorporation of logical specifications into robotics problems that benefit from gradient-based solutions. Specifically, STL is a powerful and expressive formal language that can specify spatial and temporal properties of signals generated by both continuous and hybrid systems. The quantitative semantics of STL provide a robustness metric, that is, how much a signal satisfies or violates an STL specification. In this work, we devise a systematic methodology for translating STL robustness formulas into computation graphs. With this representation, and by leveraging off-the-shelf automatic differentiation tools, we are able to efficiently backpropagate through STL robustness formulas and hence enable a natural and easy-to-use integration of STL specifications with many gradient-based approaches used in robotics. Through a number of examples stemming from various robotics applications, we demonstrate that STLCG is versatile, computationally efficient, and capable of incorporating human-domain knowledge into the problem formulation.

Message-Based Logic Semantics and Intentional Linguistic Semantics of Paul Grice.

The paper considers new logical semantics for the classic propositional logic which states that a propositional formula refers to a special structure – “the tree of messages”. This tree of message determines the truth values for the elements of the propositional formula. The source of inspiration for creating a concept of such semantics is a philosophical-linguistic program advanced by H.P. Grice. Grice made a hypothesis according to which the meaning of the speech act is defined more by speaker’s intention than by the “conventional” meaning of utterances. A speaker may have hidden intentions towards his speech act – he is manipulating his audience for political or economic aims, for example. Thus, the true meaning of the utterance the speaker uses may even contradict the “conventional” meanings of those expressions. As a consequence, to understand the true meaning of those utterances you should take the speaker's true intentions into consideration. A message is a textual expression of such intentions. In general, messages can express different meta-rules which specify the context for the propositional formula which refers to the “tree”. Messages can cooperate between each other as well as with the messages from different trees creating specific kinds of “algebras”. By specifying these “algebras” the variety of different alternative propositional logics based on this semantics can be created. By interpreting messages and accordingly modifying the truth values of propositional formulas the situation described by Grice can be emulated. However, it should be admitted that the idea of such logical semantics is wider than just an emulation of the particular linguistic theory.

How to deal with negative preferences in recommender systems: a theoretical framework

Negative information plays an important role in the way we express our preferences and desires. However, it has not received the same attention as positive feedback in recommender systems. Here we show how negative user preferences can be exploited to generate recommendations. We rely on a logical semantics for the recommendation process introduced in a previous paper and this allows us to single out three main conceptual approaches, as well as a set of variations, for dealing with negative user preferences. The formal framework provides a common ground for analysis and comparison. In addition, we show how existing approaches to recommendation correspond to alternatives in our framework.

Linguistic Data Model for Natural Languages and Artificial Intelligence. Part 7. Internal logic 2 Oleg M. Polyakov

Introduction. The article continues a series of publications on the linguistics of relations (hereinafter referred to as R-linguistics) and is devoted to the study of the functioning of logical connectives with nouns and adjectives. The article is the second part of the discussion of internal logic, which examines the use of logical connectives within sentences. This research involves the formation of semantic logic, that is, logic that takes into account the semantics of sentences.Methodology and sources. The results obtained in the previous parts of the series are used as research tools. To develop the necessary mathematical representations in the field of internal logic, the previously formulated semantic concepts and operations are used.Results and discussion. Two types of negation are introduced and their properties are defined. The properties of the linguistic model are formulated. The use of logical connectives with nouns and adjectives is analyzed. It is shown that the connective NOT can be attributed to both external and internal logic. The connectives AND and OR do not have a logical content, but serve to identify the state of a particular concept. Three semantic substitutions are defined. The reasons for the stability of semantic interpretation when changing the state of relations are substantiated.Conclusion. Abandoning the traditional view of natural language logic means abandoning logical operations and inference. This forces us to consider logical operations that now take into account semantics, since they are related to the structure of the linguistic model. Analysis of the functioning of logical connectives with nouns and adjectives shows that they either provide identification of the state of concepts, or can be transferred to external logic, or can operate within the framework of a linguistic model. The formulated substitution rules provide semantically correct substitutions and justify the stability of semantic interpretation in the conditions of changing states of concepts.

Implication in finite posets with pseudocomplemented sections

It is well-known that relatively pseudocomplemented lattices can serve as an algebraic semantics of intuitionistic logic. To extend the concept of relative pseudocomplementation to non-distributive lattices, the first author introduced so-called sectionally pseudocomplemented lattices, i.e. lattices with top element 1 where for every element y the interval [y, 1], the so called section, is pseudocomplemented. We extend this concept to posets with top element. Our goal is to show that such a poset can be considered as an algebraic semantics for a certain kind of more general intuitionistic logic provided an implication is introduced as shown in the paper. We prove some properties of such an implication. This implication is “unsharp” in the sense that the value for given entries need not be a unique element, but may be a subset of the poset in question. Using this implication we show that we can even recover the order of the original poset. Further, a new “unsharp” operator odot of conjunction can be introduced which is adjoint to “unsharp” implication and hence we obtain an “unsharp” residuated poset.

Non-Normal Super-Strict Implications

This paper introduces the logics of super-strict implications that are based on C.I. Lewis' non-normal modal logics S2 and S3. The semantics of these logics is based on Kripke's semantics for non-normal modal logics. This solves a question we left open in a previous paper by showing that these logics are weakly connexive.

Paraconsistent da Costa Weakening of Intuitionistic Negation: What does it mean?

In this paper we consider the systems of weakening of intuitionistic negation logic mZ, introduced in [1], [2], which are developed in the spirit of da Costa's approach. We take a particular attention on the philosophical considerations of the paraconsistent mZ logic w.r.t. the constructive semantics of the intuitionistic logic, and we show that mZ is a subintuitionistic logic. Hence, we present the relationship between intuitionistic and paraconsistent subintuitionistic negation used in mZ. Then we present a significant number of examples for this subintuitionistic and paraconsistent mZ logics: Logic Programming with Fiting's fixpoint semantics for paraconsistent weakening of 3-valued Kleene's and 4-valued Belnap's logics. Moreover, we provide a canonical construction of infinitary-valued mZ logics and, in particular, the paraconsistent weakening of standard Zadeh's fuzzy logic and of the Godel-Dummet t-norm intermediate logics.

Actualizing distributed knowledge in bounded groups

Abstract The idealizations resulting from the use of Kripke semantics in Epistemic Logic are inherited by formalizations of group epistemic notions. For example, distributed knowledge is often taken to reflect the potential knowledge of a group: what agents would know if they had unbounded means of communication and deductive ability. However, this does not specify whether/how this potential can be actualized, especially since real people are not unbounded reasoners. Inspired by experiments on group reasoning, we identify two dimensions of actualizing distributed knowledge: communication and inference. We build a dynamic framework with effortful actions accounting for both, combining impossible-worlds semantics and action models, and we provide a method for extracting a sound and complete axiomatization.

Linguistic Data Model for Natural Languages and Artificial Intelligence. Part 7. Internal Logic 1

Introduction. The article continues a series of publications on the linguistics of relations (hereinafter referred to as R-linguistics) and is devoted to the study of the functioning of logical connectives with verbs. The article is the first part of the discussion of internal logic, which examines the use of logical connectives within sentences. This research involves the formation of semantic logic, that is, logic that takes into account the semantics of sentences.Methodology and sources. The results obtained in the previous parts of the series are used as research tools. To develop the necessary mathematical representations in the field of internal logic, the previously formulated semantic concepts and operations are used.Results and discussion. The use of logical connectives with verbs is analyzed. It is shown that these connectives actually refer to external logic, although in some cases it is necessary to adjust part of sentence, taking into account the semantics of the linguistic model. The concept of semantic substitution is defined and the first rule of substitution for verbs is formulated and justified.Conclusion. Abandoning the traditional view of natural language logic means abandoning logical operations and logical inference. This forces us to consider logical operations that now take into account semantics, since they are related to the structure of the linguistic model. Analysis of the functioning of logical connectives with verbs shows that they are related to the linguistic model, which leads to the need for various semantic transformations of the text when such connectives are used. In particular, the use of logical connectives can lead to the loss of the meaning of the text. The rejection of logical inference is compensated by the appearance of semantic substitution rules, one of which is considered in this paper.

Proof Theory of Riesz Spaces and Modal Riesz Spaces

We design hypersequent calculus proof systems for the theories of Riesz spaces and modal Riesz spaces and prove the key theorems: soundness, completeness and cut elimination. These are then used to obtain completely syntactic proofs of some interesting results concerning the two theories. Most notably, we prove a novel result: the theory of modal Riesz spaces is decidable. This work has applications in the field of logics of probabilistic programs since modal Riesz spaces provide the algebraic semantics of the Riesz modal logic underlying the probabilistic mu-calculus.