System Theory

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Publication citations are given where a version of the report has been published. Often material has been edited in publication and the cited version differs to some extent.

Adaptive control theory: the structural and behavioural properties of adaptive controllers, Brian R Gaines, In A. R. Meetham & R. A. Hudson (Eds.), Encyclopaedia of Linguistics, Information & Control, 1-9, London: Pergamon Press, 1969. PDF.

The term 'adaptive' has long been applied in the biological sciences to denote the plasticity of behaviour shown by an organism in its struggle to survive in a novel or changeable environment. More recently control engineers have designed controllers with a similar ability to modify their behavioural strategies in the face of unpredictable changes in the controlled plant or its input, and they have used the term 'adaptive' to qualify both this type of controller and its behaviour. The aptness of a common description for biological and synthetic systems remains to be demonstrated by future studies coupling the disciplines of biology and controi-engineering, but research on 'adaptive control' exists in its own right as a body of experimental, practical and theoretical work central to modern automatic control, and it is the foundations of this work which form the subject of this review.

Memory minimisation in control with stochastic automata, Brian R Gaines, Electronics Letters, 7(24), 710-711, 1971. PDF.

Stochastic automata have been shown to require less states than deterministic automata in the solution of certain recognition and hypothesis-testing problems. This letter extends the result to a class of control problems involving the regulation of a discrete dynamical system.

The role of randomness in cybernetic systems, Brian R Gaines, Proceedings of Cybernetics Society Conference on Recent Topics in Cybernetics, London 20th September 1974. PDF.

This paper demonstrates that random phenomena, although most often treated in the context of system malfunction, can play major constructive roles in the philosophy, theory and application of cybernetic systems. A control-theoretic example is given to show that a simple stochastic automaton can solve a regulator problem otherwise requiring a recursive automaton, and insoluble for finite-state automata. An identification-theoretic example is given to show that the assumption of causality in modelling a simple acausal system can lead to models which grow in size, on average, precisely at the rate of acquisition of observations. The significance and applications of these results are discussed and illustrated. Finally, it is suggested that the interpretation of stochastic automata theory in terms of system malfunction is far too restrictive in its stimulation of theoretical developments. A wider view of the role of random phenomena might aid the development of results and techniques which actually deliver what automata theory has always seemed to promise.

A calculus of possibility, eventuality and probability, Brian R Gaines, Technical Reprt EES-MMS-FUZI-75, Department of Electrical Engineering Science, University of Essex, Colchester, UK, 1975. PDF.

Three distinct explicata of uncertaIn system behaviour are developed and it Is shown that they each give rise to different phenomena that are confounded If only probability theory is used to represent them. It is also shown that a conventional binary representation of possible transitions in non-deterministic automata cannot support certain legitimate arguments about the resultant behavIour. A weakened logic of probabilIty is developed as a precise explicatum of all that may be inferred about non-determInistic, but also non-probabilistic, behaviour. This is extended to cover all three forms of uncertain behaviour, and their combinations, leading to a rigorous calculus of possibility, eventuality and probabiltty.

Stochastic and fuzzy logics, Brian R Gaines, Electronics Letters 11, 188-189, 1975. PDF.

It is shown that it is possible to regard stochastic and fuzzy logics as being derived from two different constraints on a probability logic: statistical independence (stochastic) and logical implication (fuzzy). To contrast the merits of the two logics, some published data on a fuzzy-logic controller is reanalysed using stochastic logic and it is shown that no significant difference results in the control policy.

Control engineering and artificial intelligence, Brian R Gaines, Lecture Notes of AISB Summer School on Artificial Intelligence, 52-60, Cambridge, UK: AISB, 1975. PDF.

The relatlionship between control engtneering and artificial intteligence is significant for researchers in both disciplines

Multivalued logics and fuzzy reasoning, Brian R Gaines, Lecture Notes of AISB Summer School on Artificial Intelligence, 100-112, Cambridge, UK: AISB, 1975. PDF.

These notes are concerned with recent developments in multivalued logic, particularly in fuzzy logic and its status as a model for human linguistic reasoning. This first section discusses the status of formal logic and the need for logics of approximate reasoning with vague data. The following sections present a basic account of fuzzy sets theory; fuzzy logics; Zadeh's model of linguistic hedges and fuzzy reasoning and finally a bibliography of all Zadeh's papers and other selected references.

The logic of automata, Brian R Gaines & L J Kohout, International Journal of General Systems 2(1), 191-208, 1975. PDF.

Automata are the prime example of general system over discrete state spaces and yet the theory of automata is fragmentary and is not clear what makes a general structure an automaton. This paper investigates the logical foundations of automata relating it to the semanrics of our notions of uncertainty, state and state-determined. A single framework is established for the conventional spectrum of automata: deterministic, probabilistic, fuzzy and non-deterministic, which shows this set to be, in some sense, complete. Counter-examples are then developed to show that this spectrum alone is indequate to describe the behaviour of certain forms uncertain systems. Finally ageneral formu!ation is developed based on the fundamental semantics of our notion of a state that shows that the logical structure of an automaton must be at least a positive ordered semiring. The role of probability logic, its relationship to fuzzy logic, the roles of topological models of automata, and the symmetry between inputs and outputs in hyperstate/hyperinput-determined systems are also discussed.

Posible automata, Brian R Gaines & L J Kohout, Proceedings 1975 International Symposium Multiple-Valued Logic, 183-196, 1975. PDF.

This paper is concerned with the widest class of automaton structures whose semantics is compatible with our notions of state and automaton. It is first shown that the conventional spectrum of deterministic, stochastic, fuzzy and non-deterministic can be fitted into a single framework which is complete. In particular new results on the normalization of fuzzy automata and the relationship between fuzzy and stochastic automata are derived. A practical counter-example is then developed that does not fit into this spectrum, showing it to be inadequate. This is based on the richer interpretation of the notion of possibility that is required in the analysis of system stability and reliability. Finally, basic arguments are advanced to show that the structure must be at least an ordered semiring and at most a commutative ordered semiring.

Discrete Systems & Fuzzy Reasoning, Ebrahim H. Mamdani & Brian R Gaines (Eds), 1976. PDF.

The Fuzzy Workshop held at Queen Mary College on Friday 9th January. 1976 was aimed at bringing together the several groups working on fuzzy reasoning in the U.K. It echoes the similar workshops held in the U.S.A., Japan and Europe, in attempting to consolidate and cross-fertilize this rapidly growing field.

Why fuzzy reasoning?, Brian R Gaines, In E. H. Mamdani & B. R. Gaines (Eds.), Discrete Systems and Fuzzy Reasoning, 39-44, Colchester, UK: University of Essex., 1976. PDF.

The topics of "fuzzy logic" and "fuzzy reasoning" are not clear-cut subject areas with well-defined results and track records. Instead they represent a wealth of recent activity on an international front that may be seen to have its technical roots in philosophical and mathematical studies of "multi-valued logics" and "vague reasoning", but which owes much of its present impetus to engineering interest from those concerned with "information systems." Much of the current literature on fuzzy logic is neither precise in its objectives nor accurate in its conclusions. Much of the current effort duplicates activities taking place, or having taken place, elsewhere. This seminar is intended to introduce this area, relate it to other subject areas concerned with reasoning and decision-making, and give pointers to the most useful literature and areas of development.

Research notes on fuzzy reasoning, Brian R Gaines, In E. H. Mamdani & B. R. Gaines (Eds.), Discrete Systems and Fuzzy Reasoning, 45-49, Colchester, UK: University of Essex, 1976. PDF.

These notes were based on discussions with Lotfi Zadeh at Berkeley in May 1975

General fuzzy logics, Brian R Gaines, Proceedings of the Third European Meeting on Cybernetics and Systems Research, 270-275, 1976. PDF.

There is no one logical system that stands out clearly as the logic of vagueness, uncertainty or fuzzy reasoning. It has been shown that a non-functional basic probability logic provides a formal foundation for a general logic of uncertainty encompassing both fuzzy and probability logics. Classical probability logic is obtained by adding the law of the excluded middle. The fuzzy logic L1 is obtained by demanding strong truth-functionality.

Foundations of fuzzy reasoning, Brian R Gaines, International Journal of Man-Machine Studies 8(6), 623-668, 1976. PDF.

This paper gives an overview of the theory of fuzzy sets and fuzzy reasoning as proposed and developed by Lotfi Zadeh. In particular it reviews the philosophical and logical antecedents and foundations for this theory and its applications. The problem of borderline cases in set theory and the two classical approaches ofpreeisifying them out, or admitting them as a third case, are discussed, leading to Zadeh's suggestion of continuous degrees of set membership. The extension of basic set operations to such fuzzy sets, and the relationship to other multivalued logics for set theory, are then outlined. Thefuzzification of mathematical structures leads naturally to the concepts of fuzzy logics and inference, and consideration of implication suggests Lukasiewicz infinite-valued logic as a base logic for fuzzy reasoning. The paradoxes of the barber, and of sorites, are then analysed to illustrate fuzzy reasoning in action and lead naturally to Zadeh's theory of linguistic hedges and truth. Finally, the logical, modeltheoretic and psychological derivations of numeric values in fuzzy reasoning are discussed, and the rationale behind interest in fuzzy reasoning is summarized.

Fuzzy reasoning and the logics of uncertainty, Brian R Gaines, Proceedings 6th International Symposium Multiple-Valued Logic, IEEE 76CH1111-4C, 179-188, 1976. PDF.

This paper is concerned with the foundations of fuzzy reasoning and its relationships with other logics of uncertainty. The definitions of fuzzy logics are first examined and the role of fuzzification discussed. It is shown that fuzzification of PC gives a known multivalued logic but with inappropriate semantics of implication and various alternative forms of implication are discussed. In the main section the discussion is broadened to other logics of uncertainty and it is argued that there are close links, both formal and semantic, between fuzzy logic and probability logics. A basic multivalued logic is developed in terms of a truth function over a lattice of propositions that encompasses a wide range of logics of uncertainty. Various degrees of truth functionality are then defined and used to derive specific logics including probability logic and Lukasiewicz infinitely valued logic. Quantification and modal operators over the basic logic are introduced. Finally, a semantics for the basic logic is introduced in terms of a population (of events, or people, or neurons) and the semantic significance of the constraints giving rise to different logics is discussed.

The fuzzy decade: a bibliography of fuzzy systems and closely related topics, Brian R Gaines & L J Kohout, International Journal Man-Machine Studies 9, 1-68, 1977. PDF.

The main part of the paper consists of a bibliography of some 1150 items, each keyword-indexed with some 750 being classified as concerned with fuzzy system theory and its applications. The remaining items are concerned with closely related topics in many-valued logic, linguistics, the philosophy of vagueness, etc. These background references are annotated in an initial section that outlines the relationship of fuzzy system theory to other developments and provides pointers to various possible fruitful interrelationships. Topics covered include: the philosophy and logic of imprecision and vagueness; other non-standard logics; foundations of set theory; probability theory; fuzzification of mathematical systems; linguistics and psychology; and applications.

Decision: theory and pactice, Brian R Gaines, Proceedings 3rd Conference on Decision Making in Complex Systems, G. Pask (Ed.), Richmond, Surrey, UK: Systems Researc, 1978. PDF.

In these notes I have attempted to bring together In uneasy synthesis several strands of my own studies. They are confluent but It would be premature to present them as an integral whole. They are best regarded scaffolding for an architecture of decision - taken with other contributions and the discussion of this conference, they may yield further glimpses of the structures for which we are all striving.

Fuzzy and probability uncertainty logics, Brian R Gaines, Information and Control 38(2), 154-169, 1978. PDF.

Probability theory and fuzzy logic have been presented as quite distinct theoretical foundations for reasoning and decision making in situations of uncertainty. This paper establishes a common basis for both forms of logic of uncertainty in which a basic uncertainty logic is defined in terms of a valuation on a lattice of propositions. The (non-truth-functional) connectives for conjunction, disjunction, equivalence, implication, and negation are defined in terms which closely resemble those of probability theory. Addition of the axiom of the excluded middle to the basic logic gives a standard probability logic. Alternatively, addition of a requirement for strong truth-functionality (truth-value of connective determined by truth-value of constituents) gives a fuzzy logic with connectives, including implication, as in Lukasiewicz' infinitely valued logic. A common semantics for all such variants is given in terms of binary responses from a population. The type of population, e.g., physical events, people, or neurons, determines whether the model is of physical probability, subjective belief, or human decision-making. The formal theory and the semantics together illustrate clearly the precise similarities and differences between fuzzy and probability logics.

Progress in general systems research, Brian R Gaines, In Klir, G.J., Ed. Applied General Systems Research. pp. 3-28. New York, USA: Plenum Press, 1978. PDF.

This paper argues that it is the "systems approach" rather than any "general systems theory," real or imagined, attained or sought, that is the coherent theme of general systems research. In this we are as much followers as leaders -- the systems approach permeates modern scientific thought, even that which specifically denies the relevance of general systems theory -- the pioneers predicted and recognized a trend as much as they created and motivated it.

Decision: foundation and pactice II, Brian R Gaines & Mildred L G Shaw, In G. Pask (Ed.), Proceedings 4th Conference on Decision Making in Complex Systems. Richmond, Surrey, UK: Systems Research PDF.

Our overall objective is to create man-computer symbiotic systems for decision-making that utilize the full capabilities of all the sub-systems involved. To do this involves both practical work on actual decision-making systems and foundational work on the logics and system theories underlying decision making. We see a strong convergence between theories based on Spencer Brown's calculus of distinctions and practice based on Kelly's (1955) personal construct theory.

General systems research: quo vadis?, Brian R Gaines, General Systems: Yearbook of the Society for General Systems Research, Vol.24, 1979, pp.1-9. PDF.

The first section is concerned with what we mean by a system and gives a definition of the term which itself demonstrates some aspects of systems thinking. The following two sections consider the role of social concern and ideology in the systems approach and give a systems model for it. The final sections show the significance of presuppositions in systems thinking and give some indication of recent progress.

Analysing analogy, Brian R Gaines & Mildred L G Shaw, In R. Trappl, L. Ricciardi & G. Pask (Eds.), Progress in Cybernetics and Systems Research. Vol. IX, 379-386, Washington: Hemisphere, 1982. PDF.

Analogical reasoning is one of the most important techniques used by people, yet it has proved very difficult to represent the process in formal terms amenable to operational implementation in computer algorithms. In this paper we analyse the concept of an analogy and its application to reasoning processes. An analogy is seen to be a partial correspondence between two systems. In representing it in map-theoretic terms a third system naturally arises which may be injected into each of the others to represent the partial correspondence. This system may be called an 'analogy' system capturing the particular analogy under consideration. We then go on to consider multiple analogies between two systems and their inter-relationships and show that these form a semi-lattice with a truth-system as the minimal element. Because analogies are between systems with structure they have to capture the transformations that define the structure rather than just the elements of the systems. Hence the systems themselves are most simply represented as categories and the mappings as faithful functors between them. In this paper we give not only the formal theory but a number of systemic and programming examples to illustrate our analysis.

Is there a knowledge environment?, Brian R Gaines and Mildred L G Shaw, In Lasker, G., Ed. The Relation Between Major World Problems and Systems Learning. pp. 35-41. Society for General Systems Research, 1983. PDF.

The analogy between the physical world and that of knowledge enables us to speak of the "knowledge environment" and discuss various environmental problems of great significance. However, analogies can be misleading if they are not based on systemic relationships. In this paper we take the three worlds, physical, mental, and world 3, as defined by Popper, and analyze them in a common systemic framework. In particular we discuss the use of the term "living organism" in relation to the three worlds. We conclude that the metaphor of a knowledge environment does have an underlying systemic model which can be formalized as required. The analysis also applies to the world of subjective experience and gives a common foundation for ecological notions such as pollution in the worlds of biology, the mind and ideas.

Precise past--fuzzy future, Brian R Gaines, International Journal of Man-Machine Studies 19(1), 117-134, 1983. PDF.

This paper examines the motivation and foundations of fuzzy sets theory, now some 20 years old, particularly possible misconceptions about possible operators and relations to probability theory. It presents a Standard uncertainty logic (SUL) that subsumes Standard propositional, fuzzy and probability logies, and shows how many key results may be derived within SUL without further constraints. These include resolutions of Standard paradoxes such as those of the bald man and of the barber, decision rules used in pattern recognition and control, the derivation of numeric truth values from the axiomatic form of the SUL, and the derivation of operators such as the arithmetic mean. The addition of the constraint of truth-functionality to a SUL is shown to give fuzzy, or Lukasiewicz infinitely-valued, logic. The addition of the constraint of the law of the excluded middle to a SUL is shown to give probability, or modal S5, logic. An example is given of the use of the two logies in combination to give a possibility vectoR when modelling sequential behaviour with uncertain observations. This paper is based on the banquet address with the same title given at NAFIP-1, the First North American Fuzzy Information Processing Group Workshop, held at Utah State University, May 1982.

Methodology in the large: modeling all there is, Brian R Gaines, Systems Research, 1(2), 91-103, 1984. PDF.

The human mind has been inventive of a multitude of methodologies to explain observed phenomena, predict possible worlds, determine which ones should be made real, and bring this about. In recent years the computer has enhanced our capability to project the detail of possible worlds and widen our vision of the consequences of our actions. We bring together the physical and human variables, political and economic policies, and constraints of resources, and expect increasing model realism. With advances in computer technology we have begun to see the possibility of large-scale methodologies that eventually cope with all there is. This paper is concerned with placing our endeavours for model realism in the socio-economic sphere within the much broader context of humankind's overall endeavours for model realism. It expresses the main philosophical problems underlying our situation in a parable, and then goes on to sketch some of the solutions proposed for them in the past. It concludes with a model for the role of the computer as a vehicle to explore Popper's World 3, and the use of his notion of three worlds to express the different styles and motivations of schools of modeling and concepts of realism.

Hierarchies of distinctions as generators of system theories, Brian R Gaines & Mildred L G Shaw, In Smith, A.W., Ed. Proceedings of the Society for General Systems Research International Conference. pp. 559-566. California: Intersystems Publications, 1984. PDF.

This paper generates a variety of general-systems theoretic concepts and a range of systems theories using a minimal set of primitives. The notion of a distinction is taken as primitive and the natural hierarchy of distinctions becomes the primitive structure. A number of system theories are then analysed in terms of these primitives: Zadeh's fuzzy sets theory; Klir's epistemological hierarchy of system modeling; Popper's 3 worlds theory of system types; Pask's conversation theory of system interaction; and Checkland's soft systems theory. A small number of types of distinction are shown to underly these theories and, taken in various combinations, to generate them.

Three world views and system philosophies, Brian R Gaines & Mildred L G Shaw, In Banathy, B.H., Ed. Systems Inquiring: Theory, Philosophy, Methodology. pp. 244-252. Seaside, California: Intersystems Publications, 1985. PDF.

General systems theory (GST) should be neutral with regard to major philosophical issues in order to be able to encompass all philosophies. This paper develops GST based on the primitive notion of a distinction as a basis for knowledge science and technology. It analyses the development of systems of distinction and how different philosophies arise through different basic distinctions generating world hypotheses. It is suggested that any situation can be analyzed in terms of the distinctions being made, whether they are ascribed to necessity or choice, and the inferences possible between them.

From fuzzy logic to expert systems, Brian R Gaines & M L G Shaw, Information Sciences 36(1-2), 5-16, 1983. PDF.

The history and achievements of fuzzy set theory (FST) are considered in relation to developments in expert systems (ESs). An overview is given of early ES developments and the role of linguistic rules and metarules. It is suggested that FST and ESs are not just mathematical and technological advances but also represent major paradigm shifts in system theory. The main shift is away from the normative application of technology to change the world to be theoretically tractable, and towards increasing model realism. The limitations of classical system theory when applied to natural systems were the impetus behind the development of FST.

Aether as a referring term, Brian R Gaines, 2006. PDF.

The aether has been used in the philosophy of science literature as a stereotypical example of a theoretical construct that played a major role in scientific development but was abandoned when it was found to be non-referring. This is generally ascribed to Einstein’s development of special relativity but, in practice, the term and concept continued to be used by Einstein, and many others to the present day. In recent years the theory of the quantum vacuum has come to be seen as capturing the essence of the aether, accounting for phenomena of electromagnetism and gravitation, and having observable and useful consequences through the Casimir effect. This paper discusses the implications for the philosophy of science literature of the aether becoming accepted as a referring term, firstly in the theory of natural kinds, and secondly in studies that use it as an example of a non-referring term.

Living in an uncertain universe, Brian R Gaines, In Rudolf Seising, Eric Trillas, Claudio Termini (eds.) On Fuzziness: A Homage to Lotfi Zadeh, Springer, 2012. PDF.

It has been interesting to revisit the era when my path crossed with those of John Andreae, Joe Goguen, GeorgeKlir, Ladislav Kohout, Abe Mamdani, Gordon Pask, Ted Poppelbaum and Lotfi Zadeh, all precious friends and colleagues, some of whom are, unfortunately, no longer with us; their ideas live on and flourish as a continuing inspiration. I was tempted to entitle this article after Gurdieff, Meetings with Remarkable People, and that will be its focus, particularly the background to the initial development of fuzzy controllers. There have been studies analyzing their history and the basis of their success—this article provides some details from my personal experience.

CPCS 5-Apr-2012