Abstract
Logics for unconventional computing is an interdisciplinary research area which brings together computer scientists and engineers dealing with unconventional computing (such as biological, bio-inspired, chemical, physical, etc. computing) with logicians dealing with non-classical logical, algebraic, co-algebraic, and topological methods to initiate the development of novel nature-inspired computation paradigms. This paper is a Preface to the special issue devoted to Logics for unconventional computing.
Graphical Abstract
Logics for unconventional computing [Citation1] initiate the development of new methodologies and architectures for unconventional computations as well as for applications of computational intelligence in the wide and heterogeneous fields of cognitive science, biomechanics, and material science. It can provide to take a critical glance at the design of parallel and emergent computing systems to point out failures and shortcomings of both theoretical and experimental results. This special issue is to examine some recent approaches towards symbolic logics in unconventional computing.
Traditionally, symbolic logic was regarded as foundations of mathematics. In the two-volume work Grundlagen der Mathematik (Foundations of Mathematics) written by David Hilbert and Paul Bernays and originally published in 1934 and 1939, there was proposed an axiomatic system as a basic theory for mathematics. In this system, a mathematical proof was considered a discrete process that can be automatized. Since that work by Hilbert and Bernays, all symbolic-logical systems were based on this discrete treatment of mathematical proofs. Nevertheless, in neuroscience each cognition is treated not only as discrete (concentrated on details), but also as analogue (concentrated on the whole picture). The first mechanism to perceive information is called a lateral inhibition and the second a lateral activation. Andrew Schumann and Alexander V. Kuznetsov show in their paper Foundations of Mathematics under Neuroscience Conditions of Lateral Inhibition and Lateral Activation that mathematicians can deal not only with a logical way of automatic proving from some axioms (the lateral inhibition in mathematics), but also with combining proof trees on tree forests by using some analogies as inference metarules (the lateral activation in mathematics). This means that the conventional foundations of mathematics developed by Hilbert and Bernays are focused just on lateral inhibition effects in proof cognitions, but the new foundations are possible, too, which consider proofs as analogue emergent processes (the lateral activation effects in proof cognitions). Hence, symbolic logics in unconventional computing mean that we can change logics themselves to make them more applicable to simulating natural processes. For example, cognitive processes are not only discrete (i.e. they proceed not only in accordance with a lateral inhibition), but also analogue (i.e. they proceed also in accordance with a lateral activation). As a consequence, an appropriate symbolic logic can be either discrete or analogue, as well.
Another main assumption of Grundlagen der Mathematik that can be broken in unconventional computing was that the mathematical universe is well-founded, i.e. it can be described only by means of inductive definitions. However, in behavioural sciences as well as in cognitive science we ever face some non-well-founded phenomena: an unpredictable behaviour caused by changing pass decisions, loops in cognitions, a self-reference in cognitions, permanent modifications of behavioural scenarios, etc. Thus, the mathematical universe for symbolic logics can be assumed non-well-founded, too. One of the non-well-founded universes is presented by p-adic numbers [Citation2]. These numbers can be infinite. The paper Logics that Formalize p-adic Valued Probability and Their Applications submitted by Angelina Ilić Stepić and Zoran Ognjanović is devoted to different propositional logics for reasoning about probabilities with values on p-adic numbers. Each of these logics is a sound, complete, and decidable extension of classical propositional logic. These logics contain the following probability operators: (i) ‘the probability of the corresponding formula belongs to the p-adic ball with the center r and the radius ’; (ii) ‘the conditional probability of
given
is in the p-adic ball with the center r and the radius
’; (iii) ‘the p-adic distance between the probabilities of
and
is less or equal to
’. The authors propose also some applications of these logics to cognitive science: how can they model the process of thinking.
In unconventional computing there can be applied some conventional symbolic logics, too. So, there exist Petri nets [Citation3], a powerful graphical and formal tool to model dynamic systems, which can be used for programming biological behaviours under the conditions of their complete controlling by attractants and repellents. In the paper Petri Net Models for Physarum Machines Built to Realize Boolean Functions prepared by Krzysztof Pancerz, the author proposes Petri nets to model the behaviour of plasmodia of Physarum polycephalum. These nets realize Boolean functions (i) in their disjunctive canonical (minimal) forms; (ii) in their conjunctive canonical (minimal) forms; (ii) in their particular minterms (primary implicants) forms; and (iv) in their particular maxterms (primary implicents) forms. The realization of any Boolean functions in their canonical forms allows us to represent the plasmodia of Physarum polycephalum as a true biological computer.
The next main assumption of Grundlagen der Mathematik that can be avoided in logics for unconventional computing was that a symbolic logic can be done only as an a priori logic (i.e. a logic before any experience). The paper Model of the Motion of Agents with Memory Based on the Cellular Automaton submitted by Alexander V. Kuznetsov shows that in simulating the group behaviour we can engage an a posteriori system, such as cellular automata [Citation4]. The author considers a simulation of motions of nested groups of agents through unknown landscape and conflicts of agents. This model allows us to examine different levels of the situational awareness of agents and to expect a self-emerging of diverse kinds of tactics. Such a model can simulate a swarm behaviour from the simple one, such as nests of ants, to the advanced one, such as Chimpanzee troops.
The non-aprioristic approach in logics for unconventional computing can help us to change theoretical frameworks in investigations. For instance, on the one hand, the dry sliding friction is a fundamental physical phenomenon which occurs for all classes of materials and material combinations and, on the other hand, the friction force is not considered a fundamental physical force in physics at all. But in unconventional computing we can concentrate on frictions in designing some unconventional computers basing on the friction force. In the paper Logical and Information Aspects in Surface Science: Friction, Capillarity, and Superhydrophobicity written by Michael Nosonovsky, a bubble travelling in a channel is regarded as performing logical control operations including AND/OR/NOT gates. In this computer, the friction force is used as the main cause in dynamics of information.
Another example of the non-aprioristic approach in logics for unconventional computing can be represented by the analysis of reflexion in human cognitions. The paper Reflexion in Mathematical Models of Decision-Making written by Dmitry Novikov, Vsevolod Korepanov, and Alexander Chkhartishvili is devoted to different ways of controlling intelligent behaviour by controlling reflexion – a subjective mechanism allowing actors to evaluate critically the own knowledge and ways of its cognitions as well as the knowledge of other actors. This mechanism can be simple or expanded up to a hierarchy of beliefs. For instance, we can deal with agents’ beliefs about reality, but also with their beliefs about the beliefs of other agents, as well as with their beliefs about the correlations of beliefs of different agents, etc. Reflexive game is defined as a game in which agents make decisions based on the hierarchy of their beliefs. From the point of view of game theory, there are many approaches to explicating the notion of reflexion under conditions of different possible hierarchies of beliefs. So, there are static and dynamic mathematical models of behaviour for a game-theoretic formalization. The authors (i) describe some static reflexive models of noncooperative games with emphasis on informational reflexion and strategic reflexion and (ii) they consider reflexion within a collective behaviour model as well as learning and teaching models. The logic of static models and the logic of dynamic models are too different. The first logic can be reduced to usual algorithmic mathematics and can be presented as multiplicative linear logic [Citation5]. The second logic is based on concurrent calculi and can be so different [Citation6].
Reflexion can be understood as a way of cognitive biases also. For example, in [Citation7], it is shown that a probability of conjunction of two events A and B may be estimated greater than probability of one individual event. It is one of the possible examples of cognitive biases in probabilistic judgments. The paper Quantum Models of Cognition and Decision submitted by Andrzej ᴌukasik explains these biases by involving quantum probability logic. The point is that judgment and decision processes include effects that are typical for quantum mechanics, such as a superposition of beliefs, order effects, and a contextuality of beliefs.
Thus, this special issue is to show how symbolic logics can go beyond the Grundlagen der Mathematik to simulate some natural processes, first of all the processes of cognitions within unconventional computing.
Notes
No potential conflict of interest was reported by the authors.
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