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New Logical-Mathematical Methods in Applied Sciences and Modelling of Decision Making

date: August 28th, 2015
The research was financed by the National Science Centre in Poland, based on the decision DEC-2012/07/B/HS1/00263.

The research team: prof. Jan Woleński, dr Andrew Schumann, dr Andrzej Szelc.

Research project objectives/hypothesis:

Limits and possibilities of using logical-mathematical methods in economics and applied researches are studied insufficiently still. The matter is that, on the one hand, traditionally many mathematical methods (probability theory, econometrics, game theory etc.) have been used, on the other hand, all these methods have arisen within the limits of standard mathematics. For example, Aumann’s agreement theorem (the basic proposition in game theory for which Aumann received the Nobel Prize on economy) uses inductive sets as the main construction. These sets are postulated in the set-theoretic axiom of foundation. However, there is a mathematics without this axiom (the so-called non-well-founded mathematics). Within the limits of this mathematics Aumann’s agreement theorem is impossible. How far is it possible to use in game theory non-well-founded mathematics and non-inductive sets? The answer is open still.

In the theory of decision making there are many logical-philosophical assumptions. The most fundamental assumption is that data mining for typical tasks is presented by construction of decision trees which, in turn, are understood as inductive sets. The requirements for sets in data mining are as follows (according to them sets will be inductive):

• Databases should consist of finite number of members;

• All possible relations should be presented as finite loopfree trees.

However, there are cases when databases for decision making contain some insoluble oppositions which hinder setting inductive trees, for example in databases there is an inconsistency which makes it ill-structured: our system should possess a property A, to fulfil a useful function, and should possess a property non-A and we are not able to select either A or non-A. In this case, we cannot refer to usual data mining generally.

Inductive sets are the most fundamental construction in mathematics and logic. The main hypothesis of our research consists in that non-well-founded mathematics and non-inductive sets can be used in decision making more effectively, than methods of standard mathematics. Besides it we are gathered to concentrate on study different logical aspects of decision making such as formal-praxeological, illocutionary, fuzzy, probabilistic, proof-theoretic, etc.

The project is based on interdisciplinary research therefore results of the project may contribute to a variety of different disciplines: decision theory, economics, mathematical logic, probability theory, illocutionary logic, pragmatics, unconventional computing, applied mathematics, etc. This is because

• The project will demonstrate how to employ different logical tools to explicate decision making behaviors. The problem is that most of decision theory is normative because of assuming an ideal decision maker who is fully informed and fully rational. In such theories often just one of the logical aspects is regarded. There are many practical applications of this normative approach, e.g. including software to help people make better decisions (decision support systems). On the other hand, in life (i) there is no ideal decision maker, (ii) we need to take into account how other people in the situation will respond to the decision that is taken. This means that we should consider different logical aspects of decision making simultaneously and may propose massive-parallel theory of decision combining formal-praxeological, illocutionary, fuzzy, probabilistic, proof-theoretic aspects.

• As a result, while in conventional logical decision theory we assume Platonism within the normative approach, where decision maker is fully rational, within the novel approach we aim to propose we avoid Platonism and focus on behaviors of decision making to logically explicate complex and massive-parallel decision making in big companies. Solving this task allows us to build up more powerful logical decision theory and to simulate real decision making procedures.

• We assume to construct massively-parallel decision theory in that (1) we simulate derivations of competing decision makers without using axioms, therefore there is no sense in distinguishing logic and theory (i.e. logical and non-logical axioms), derivable and provable formulas, etc., some derivation traces are circular, i.e. (2) some premises are derivable from themselves, (3) some derivation traces are infinite. This novel proof theory will provide massively-parallel and competing decision making in big companies with logical tools that may describe circularities. The point is that self-organization phenomena in communication within big companies assume circularity and cause-and-effect feedback relations: each component affects the other components, but these components in turn affect the first component. So, in logical simulation of performative systems of massively-parallel and competing decision making we may obtain circular proofs.

• Results of the project will contribute to theoretical foundations of descriptive decision theory explaining behaviors of real decision making and will be interesting for researchers working in relevant topics in decision theory, mathematical logic, economics and engineering because they will open new horizons in the logical simulating of decision making. This is connected to building up a general logical decision theory that is supposed to be obtained.

We assume that this novel theory can be developed within the framework of non-well-founded mathematics and interactivecomputing/concurrency paradigm. One of its means is the co-induction principle instead of the induction one. Notice that coinduction has been well established by R. Milner for reasoning in concurrency theory. Abramsky’s lazy lambda calculus has made co-induction equally important in the theory of functional programming. Later, R. Milner and M. Tofte motivated co-induction through a simple proof about types in a functional language. However, the concepts of co-algebra and co-induction has not yet had much impact in the design or practice of computer algebra systems and computing implementations.

Thus, we aim to develop complete logical tools for analyzing massively parallel and competing decision making, in particular we are going to obtain behavioral logics describing behaviors of decision making, massively-parallel proof theory and a new logical theory of decision. This will allow us to develop novel engineering technologies and to implement decision support systems wider than it was sometime earlier. Algorithms and logical tools that we are going to obtain within this project could be used in massively-parallel computing technologies, in economics, in interactive computations and so on. Socio-economic reasons for carrying out the similar research consist in that we are developing novel technologies which could be applied in different areas of decision making (e.g. in business process management).

On the basis of the new logical approach to decision making it would be possible to project an appropriate architecture of decision support system that can be used for supporting implementation and projecting of economic firms.

Published Results:

Monographs:

[1] Andrew Schumann, Unconventional Logics in Decision Making. Rzeszów: WSIiZ, 2014.

Edited Volumes:

[1] Andrzej Dąbrowski, Andrew Schumann, Jan Woleński (eds), Podejmowanie decyzji. Pojęcia, teorie, kontrowersje. Kraków: Copernicus Center Press, 2015.

Proceedings:

[1] Andrew Schumann, Krzysztof Pancerz, Martin Grube, and Andrew Adamatzky, Context-Based Games and Physarum Policephalum as Simulation Model, In: Proceedings of Unconventional Computation & Natural Computation 2014, July 14-18, 2014, Ontario, Canada

[2] Andrew Schumann, p-Adic Valued Fuzzyness and Experiments with Physarum Polycephalum, In: 11th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD 2014), August 19-21, 2014, IEEE

[3] Andrew Schumann, Physarum Syllogistic L-Systems, FUTURE COMPUTING 2014 : The Sixth International Conference on Future Computational Technologies and Applications, May 25-29, 2014,

[4] Andrew Schumann, Reversible Logic Gates on Physarum Polycephalum, In: International Conference of Numerical Analysis and Applied Mathematics, September 22-28, 2014,

[5] Andrew Schumann, Bayesian Networks on Transition Systems, In: 5th World Congress and School on Universal Logic, June 20–30, 2015,

[6] Andrew Schumann, On the meanings of performative propositions used by traders, In: Fourth International Conference on Philosophy of Language and Linguistics (PhiLang2015), 14-16 May 2015

[7] Andrew Schumann, Reflexive games in e-university, In: Science and Information Conference (SAI), July 28-30, 2015, IEEE.

Chapters in Books:

[1] Andrew Schumann, Podejmowanie decyzji na zbiorach nie-ufundowanych, In: Andrzej Dąbrowski, Andrew Schumann, Jan Woleński (eds), Podejmowanie decyzji. Pojęcia, teorie, kontrowersje. Kraków: Copernicus Center Press, 2015.

[2] Jan Woleński, Operator decycyjny i negacja, In: Andrzej Dąbrowski, Andrew Schumann, Jan Woleński (eds), Podejmowanie decyzji. Pojęcia, teorie, kontrowersje. Kraków: Copernicus Center Press, 2015.

[3] Andrew Schumann, Andrzej Szelc, Nowe granice symulacji logicznej w inteligencji biznesowej, In: Andrzej Dąbrowski, Andrew Schumann, Jan Woleński (eds), Podejmowanie decyzji. Pojęcia, teorie, kontrowersje. Kraków: Copernicus Center Press, 2015.

[4] Andrew Schumann, Logika illokucyjna dla eksplikacji kognitywnych aspektów interakcji ludzkiej, In: Andrzej Dąbrowski, Jan Woleński (eds), Metodologiczne i teoretyczne problemy kognitywistyki. Kraków: Copernicus Center Press, 2014, pp. 287-308.

Articles in Journals:

[1] Andrew Schumann, Jan Woleński, Two Squares of Oppositions and Their Applications in Pairwise Comparisons Analysis, Fundamenta Informaticae, 2016 (accepted).

[2] Andrew Schumann, Jan Woleński, Decisions on Databases, Fuzzy Databases and Codatabases, Operations Research and Decisions, 25(3), 2015.

[3] Andrew Schumann, Towards context-based concurrent formal theories, Parallel Processing Letters, 25, 2015, 1540008.

[4] Andrew Schumann, Ludmila Akimova, Syllogistic System for the Propagation of Parasites. The Case of Schistosomatidae (Trematoda: Digenea), Studies in Logic, Grammar and Rhetoric, 40 (53), 2015, 303-319.

[5] Jan Woleński, Teoria decyzji w świetle filozofii, Kwartalnik Filozoficzny, 42 (3), 2014, 5–30.

[6] Andrew Schumann, Andrzej Szelc, Towards New Probabilistic Assumptions in Business Intelligence, Studia Humana, 3 (4), 2014, 11–21.

[7] Andrew Schumann, Reflexive Games and Non-Archimedean Probabilities, p-Adic Numbers, Ultrametric Analysis and Applications, 6 (1), 2014, 66–79.

[8] Andrew Schumann, Probabilities on streams and reflexive games, Operations Research and Decisions, 24 (1), 2014, 71-96.

[9] Andrei Khrennikov, Andrew Schumann, Quantum Non-Objectivity from Performativity of Quantum Phenomena, PHYSICA SCRIPTA, T163, 2014, 014020.

[10] Andrew Schumann, Payoff Cellular Automata and Reflexive Games, Journal of Cellular Automata, 9 (4), 2014, 287-313.

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