A Crash Course on Generalized Possibilistic Logic

An active set is a unifying space being able to act as a “bridge” for transferring information, ideas and results between distinct types of uncertainties and different types of applications. An active set is a set of agents who independently deliver true or false values for a given proposition. It is not a simple vector of logic values for different propositions, the results are a vector but the set is not. The difference between an ordinary set and active set is that the ordinary set has passive elements with values of the attributes defined by an external agent, in the active set any element is an agent that internally defines the value of a given attribute for a passive element. So agents in many cases are in a logic conflict and this generates semantic uncertainty on the evaluation. Criteria and agents are the two variables by which we give different logic values to the same attribute or proposition. In modal logic, given a proposition, we can evaluate the proposition only when we know the worlds where the proposition is located. In epistemic logic any world is an agent that knows that the proposition is true or false. The active set is a set of agents as in epistemic logic but the difference with modal logic is that all the agents (worlds) are not separate but are joined in the evaluation of the given proposition. In active set for one agent and one criteria we have one logic value but for many agents and criteria the evaluation is not true or false but is a matrix of true and false. This matrix is not only a logic evaluation as in the modal logic but give us the conflicting structure of the active set evaluation. The agent multi dimensional space to evaluate active set also includes the Hilbert multidimensional space where is possible to simulate quantum logic gate. New logic operation are possible as fuzzy gate operations and more complex operations as conflicting solving, consensus operations, syntactic inconsistency, semantic inconsistency and knowledge integration. In the space of the agents evaluations morphotronic geometric operations are a new frontier to model new types of computers. In this paper we show the connection between classical, fuzzy, evidence theory and active sets.

Modal logics for qualitative possibility theory

This paper introduces possibilistic functional dependencies. These dependencies are associated with a particular possibility distribution over possible worlds of a classical database. The possibility distribution reflects a layered view of the database. The highest layer of the (classical) database consists of those tuples that certainly belong to it, while the other layers add tuples that only possibly belong to the database, with different levels of possibility. The relation between the confidence levels associated with the tuples and the possibility distribution over possible database worlds is discussed in detail in the setting of possibility theory. A possibilistic functional dependency is a classical functional dependency associated with a certainty level that reflects the highest confidence level where the functional dependency no longer holds in the layered database. Moreover, the relationship between possibilistic functional dependencies and possibilistic logic formulas is established. Related work is reviewed, and the intended use of possibilistic functional dependencies is discussed in the conclusion.

This paper proposes a conceptually simple but expressive framework for handling propositional information stemming from several sources, namely a two-tiered propositional logic augmented with classical modal axioms (BC-logic), a fragment of the non-normal modal logic EMN, whose semantics is expressed in terms of two-valued monotonic set-functions called Boolean capacities. We present a theorem-preserving translation of Belnap logic in this setting. As special cases, we can recover previous translations of three-valued logics such as Kleene and Priest logics. Our translation bridges the gap between Belnap logic, epistemic logic, and theories of uncertainty like possibility theory or belief functions, and paves the way to a unified approach to various inconsistency handling methods.

In this study, the design of digital logic gates and circuits in ternary logic is presented. The ternary logic is observed to be a better alternative to the traditional binary logic because it offers faster computations, smaller chip area, and lesser interconnects. Thus, it allows designing the low‐complex, high‐speed, and energy‐efficient circuits in future digital design. A novel technique is proposed to design the ternary logic gates using multi‐threshold graphene nanoribbon field‐effect transistors (GNRFETs). The GNRFET threshold voltage is controlled by the width of the graphene nanoribbon, which is defined by the dimer lines number. Three different inverters are designed namely standard, positive, and negative inverters along with the basic and universal logic gates. Additionally, the ternary half adder and full adder are proposed that further helps to design the complex arithmetic circuits. All the proposed ternary logic circuits have been extensively simulated in SPICE for functional verification and performance analysis. The performance of the proposed ternary logic circuits is compared with the existing designs presented in the literature. The comparison results show that the propagation delay and circuit area of GNRFET‐based circuits are reduced with an average of 41.3 and 64%, respectively, than the existing ternary circuits.

The study is to show the similarities between J. Fodor's Language of Thought hypothesis and epistemic modal logic. According to the J. Fodor's hypothesis there is the language of thought (also called Mentalese) that is the meta-language in which mental representations of attitudes of organism to propositions expressed in object-language (for example: knowledge, belief, hope, desire, statement) are formulated. These attitudes are called attitudes. In the hypothesis propositional attitudes are thoughts and relations between organism and proposition. Propositional attitudes are of interest for epistemic modal logics. In epistemic logics propositional attitudes are relations considered as modalities, i.e. propositions with modal operators. On certain conditions J. Fodor's Language of Thought hypothesis may be considered as a sort of epistemic modal logic, only it would require certain updating as a formal logical system. The similarities between J. Fodor's Language of Thought and epistemic modal logics give two opportunities. First opportunity is to make special version of epistemic modal logic as an instrument of research for cognitive psychology based on J. Fodor's Language of Thought hypothesis. Second opportunity is to accept the idea of J. Fodor's Language of Thought as an innate language and to attain comprehension that epistemic modal logics are not especially constructed formal languages (formal systems) but innate cognitive programs. It might enable to extrapolate the idea of innate languages onto all formal languages including logical languages and to attempt verifying that idea experimentally within cognitive sciences.

The paper deals with a single-item production planning problem with uncertain demands modeled by fuzzy intervals whose membership functions are possibility distributions for the values of the uncertain demands. Optimization criteria, in the setting of possibility theory, that lead to choose robust production plans under fuzzy demands are given. Algorithms for determining optimal robust production plans with respect to the proposed criteria are provided and some computational experiments are presented.

Modal Logics for Qualitative Possibility and Beliefs

Tremendous amounts of data are being generated and should be processed during the massive employment of artificial intelligence and edge/cloud computing, necessitating computers with higher capabilities. However, as Moore's Law approaches its limit, enhancing computing capability through transistor miniaturization is becoming less feasible. New strategies for “more-than-Moore” are highly preferred to drive future technological advancements. Multiple-valued logic, especially ternary logic with its superior data density compared to traditional binary logic, offers a viable route to augment computing capability while reducing chip interconnections and dynamic power consumption. Despite the growing interest in ternary logic within academia and industry, its practical application remains limited due to the increased transistor count and connections required by earlier logic gate implementations. In this thesis, we present efficient implementation methods of basic ternary logic gates based on emerging nano-electronics including two-dimensional material-based transistors. The fabricated ternary devices show high performance in DC gain and low power consumption. Additionally, compact models of the devices have been established and calibrated, facilitating the circuit design and simulation in Electronic Design Automation (EDA) tools. At the circuit level, this thesis introduces novel ternary arithmetic circuits, including adders and multipliers, that are optimized for low power consumption and minimal delay. Finally, we showcase the practical benefits of ternary logic systems in data processing through the implementation of carry-free arithmetic circuits, underscoring their potential as a transformative computing paradigm.

Local properties in modal logic

This note corrects a claim made in the above-mentioned paper about the exact representation of a conditional preference network by means of a possibilistic logic base with partially ordered symbolic weights. We provide a counter-example that shows that the possibilistic logic representation is indeed not always exact. This is the basis of a short discussion on the difficulty of obtaining an exact representation.

In this paper we propose an explicit form of knowledge-based programming. Our initial motivation is the distributed implementation of game-theoretical algorithms, but we abstract away from the game-theoretical details and describe a general scenario, where a group of agents each have some initially private bits of information which they can then communicate to each other. We draw on existing literature to give a formal model using modal logic to represent the knowledge of the agents as well as how that knowledge changes as they communicate. We sketch an implementation which enables processes in a distributed system to explicitly evaluate knowledge formulae. Then we prove that the implementation captures the formal model, and therefore correctly reflects the general scenario. Finally we look at how our approach lends itself to generalisations, and discuss application perspectives.

This paper presents a logic of questions developed as an extension of (S5) epistemic logic. We discuss many features that are important for erotetic logic (formalization and semantics of questions, answerhood conditions, and inferential structures with questions). The aim is to introduce an erotetic system which corresponds well with epistemic terms and can form an appropriate background for dynamic approaches in epistemic logic.

21 Modal logic and philosophy

We introduce KBE, a modal epistemic logic for reasoning about Knowledge, Belief and Estimation, three attitudes involved in an agent's decision-making process. In our logic, Knowledge and Belief are captured by S4.2, a modal logic holding a distinguished position among the epistemic logics investigated in AI and Philosophy. The Estimation operator of KBE is a kind of generalized 'many' or 'most' quantifier, whose origins go back to the work of J. Burgess and A. Herzig, but its model-theoretic incarnation ('weak filters') has been introduced by K. Schlechta and V. Jauregui. We work with complete weak filters ('weak ultrafilters') as we are interested in situations where an estimation can be always reached. The axiomatization of KBE comprises 'bridge' axioms which reflect the intuitive relationship of 'estimation' to 'knowledge' and 'belief', several introspective properties are shown to hold and it comes out that believing i¾? can be equivalently defined in KBE as 'estimating that i¾? is known', an interesting fact and an indication of the intuitive correctness of the introduced estimation operator. The model theory of KBE comprises a class of frames combining relational Kripke frames with Scott-Montague semantics, in which neighborhoods are collections of 'large' sets of possible worlds. Soundness and completeness is mentioned and a tableaux proof procedure is sketched.