Mathematical logic is commonly regarded as a mathematical discipline that has applications only inside mathematics. The past few decades have shown, however, that logic - in particular various non-classical and many-valued logical systems - do have applications outside mathematics, too. These applications are besides, of course, computer sciences in 'real world' problems rising e.g. from engineering, natural sciences and medicine.
The amount of researchers in the world studying non-classical logic and, in particular, many-valued logic is several hundreds, however, in Finland this research does mostly not have any tradition. Thus, one of the objectives on my research is to increase the familiarity and rise the standard of research on many-valued logic in Finland.
My research in based on my dissertation A Mathematical Study of Fuzzy Logic: an Algebraic Approach (1994) which I prepared in Czechoslovakia and in Italy. In it I showed e.g. that Zadeh's fuzzy logic and fuzzy set theory presented in the 1960s can be regarded as a well-defined mathematical many-valued logic and algebraically corresponding to an injective MV-algebra.
As a reward for positive international feedback of my Ph.D. thesis, an international publishing house (Physica-Verlag - a Springer Verlag Company) invited me to write a monograph Mathematics Behind Fuzzy Logic; it was published in 1999. I regard this monograph as my best scientific contribution till now. It is used as a text book at least at the University of Erlangen (Germany), Technical Universities of Ostrava and Prague (Czech Republic), Universities of Salerno and Naples (Italy), University of Ghent (Belgium) and at the Polish Academy of Science. Moreover, based on that monograph, I have given courses of lectures at University of Joensuu (1996 and 1998, 28+14h), Lappeenranta University of Technology (1999, 28+14h), University of Technology (2000, 14h) and Tampere University of Technology (2001, 24+14h).
After writing the monograph I officiated in 1999-2001, besides lecturer in mathematics at Tampere University of Technology, as a part time researcher at the University of Technology (HUT), laboratory of Transportation. One of my scientific discoveries many-valued equivalence (fuzzy similarity) dates from this HUT period.
Based on many-valued equivalence relations, I was developing adaptive traffic signal systems , predicting models for traffic flow, models for real-time reservoir operation and sports medicine application.
In addition, I have continued research on theoretical foundations of many-valued logic (in spite of heavy teaching load and responsibilities of a father of two little ones). These results have been published in leading international journals.
Since 1960s, computer hardware has grown in power double for constant cost roughly once per every two years. Thus, conventional approaches to the fabrication of computer technology are beginning to run up against fundamental difficulties of size. Quantum mechanical effects are beginning to interfere in the functioning of electronic devices as they are made smaller and smaller.
One possible solution to the problem is provided by the theory of quantum computation, which is based on the idea of using quantum mechanics to perform computations, instead of classical physics. Quantum computation and quantum information is the study of the information processing tasks that can be accomplished using quantum mechanical systems.
If quantum computers can be physically constructed they will offer an essential speed advantage over classical computers. This speed advantage is so significant that many researchers believe that no conceivable amount of progress in classical computation would be able to overcome the gap between the power of a classical computer and the power of a quantum computer.
Changes occurring to a quantum state can be described using the language of quantum computation. Analogous to the way a classical computer is built from an electrical circuit containing wires and logic gates (AND, OR, NOT, NAND, etc), a quantum computer is built from a quantum circuit containing wires and elementary quantum gates to carry around and manipulate the quantum information.
Due to quantum effects, the logic of quantum computation over quantum circuits is not classical Boolean logic but some partly unknown many-valued logic. Obviously, this logic is related to what is called quantum logic and well known for decades. However, it seems that the logic of quantum computation is not precisely quantum logic.
My intention is to create the logic of quantum computation. It is remarkable that quantum logic and Lukasiewicz many-valued logic are closely related. Indeed, the corresponding algebraic structures differ from each other only with respect to one axiom! Thus, the logic of quantum computation should be related to Lukasiewicz many-valued logic.
The study will be started by introducing the language, semantics and syntax of this logic. In the beginning I will work with zero order logic. Being aware of the fact that forcing (introduced by Cohen) in not a suitable method to prove completeness theorems in many-valued logic, the intention is to construct a Lindenbaum algebra corresponding to this logic. This construction (possibly) offers an elegant way to prove the completeness theorem of this logic and study it's further properties via deductive systems (filters). If this scheme turns out to be realisable then the next step is to create first order logic of quantum computation.
The research will be done in close co-operation with professor Keijo Ruohonen (TUT, department of mathematics) who initiated me into the foundations of this topic.
The topic is suitable for a masters/doctoral thesis, too.
Analogy and similarity play an important role in scientific reasoning. Following Leibniz, the founder of mathematical logic, the identity of two objects A and B means that they share all their properties, hence, objects A and B may be said to be partially identical if they share some (or most) of their properties. According to professor Ilkka Niiniluoto (1988), The real challenge is that we have to extend our treatment from simple analogy to multiple analogy. In 1998, I proposed a solution to this challenge by generalising the classical (i.e. Boolean) equivalence relation on the basis of Lukasiewicz' many-valued logic, and applied the induced many-valued similarity on Zadeh's fuzzy sets. In traffic signal control, for example, the idea in constructing control mechanisms is to express an expert's knowledge by simple rules of inference of style: IF the pedestrian waiting time is short and there are some vehicles approaching, THEN extend vehicles' green signal. Here short and some are typical fuzzy set. Now, any such set generates a many-valued similarity relation, moreover, combining the partial similarities on the basis of Lukasiewicz's logic, the result is again a many-valued similarity. In practice, an actual traffic situation is compared to several ideal situations, and the most similar case is fired.
The aim in this research is to deal more extensively with many-valued similarity. The research in based, besides my own earlier works, on a fundamental paper Fuzziness and Fuzzy Equality (1982) by professor Ales Pultr, where it is proved that fuzzy similarity can be naturally represented by subsets of generalised metric spaces, moreover, a connection to Hausdoff super space construction is established. These results offer topological tools to study many-valued similarity, an approach not utilised till now. A new open field of research are partial similarity relations that are only reflective and transitive but not symmetric, or reflective and symmetric but not transitive. Such many-valued relations have several applications in literature.
The topic is suitable for a masters thesis, too.
Mathematical theory related to the question 'Can computers formulate and justify scientific hypotheses?' was developed in the frame of research related to the GUHA method and a logical calculus was defined and studied by professor Hajek et al. in the 1960s. This calculus is further studied and developed. Various practically important and theoretically interesting results are achieved. These results are closely related to classes of association rules. There are e.g. classes of implication rules, double implication rules of equivalency rules. My current concerns further properties of these classes of association rules.
The GUHA method is based on a first-order, finite-model, non-classical logic utilising non-standard quantifiers. The goal of the GUHA method is to offer all interesting facts following from the analysed data to the given problem. GUHA is realised by GUHA-procedures. GUHA-procedure is a computer program, the input of which consists of the data to be analysed and of a few parameters de-fining a very large set of potentially interesting hypothe-ses. GUHA procedure automatically gene-rates each particular hypothesis from the given set and tests if it is supported by data. The output of the pro-cedure consists of all hypotheses supported by the given data. Development of the GUHA method started in 1966 and now the GUHA can be considered as a method of data mining. This means that among others also the association rules corresponding to statistical hypotheses tests are mined. A 4ft-Miner procedure is one of contemporary implementations of the GUHA.
Part of the research activity during the five years period is concerning development of new data mining procedures including procedures for relational data mining. This research activity concerns various types of support of users of new procedures including automatic chaining of data mining procedures to reach the given analytical goal. An other research activity concerns presentation of results of data mining in natural language, e.g. translating the found results automatically into Finnish (this is part of miss Elimia Ylirinne's masters thesis).
In 1962 C. C. Chang introduced in Helsinki another many-valued logical system, a [-1,1]-valued logic and MV*-algebras. It seems, however, that the international research community has forgotten this valuable work. While acting as a referee in international journals I have seen several new attempts to introduce something similar to what Chang did already 40 years ago. Unfortunately, the inventors of these new logical systems are not informed of Chang's results and their results have been quite poor. However, real world applications e.g. in decision making processes seem to apply such logical systems, thus, it seems that Chang's [-1,1]-logic is worth a closer consideration.
Today we know that MV-algebras are particular BL-algebras and various many-valued logical systems can be studied algebraically by means of deductive systems (filters) of these algebras. This rises a natural question of a possible BL*-valued logic as a generalisation of Chang's [-1,1]-valued logic. Another direction of generalisation and research are various MV*-valued similarity relations.
Many of the properties valid in MV-algebras or BL-algebras can be translated into the theory of MV*-algebras and BL*-algebras. However, the relation of these algebraic structures is not simple. This is due to the 'negative part' of MV*- and BL*-structures that does not exist in MV- and BL-structures at all. Answers to these questions have, besides theoretical interest, many applications, too.
The topic is suitable for a masters thesis, too.
Non-linear problems in other than logical variables are possible to handle, too. This what is called Mixed Integer Non-linear Programming (MINLP) is quite a new topic of research. In the technology program 'Process Integration' of TEKES there is a joint project concerning industrial applications of MI(N)LP, and lecturer Risto Silvennoinen (TUT, department of mathematics) has been participating in it since the year 2000.
The idea of my research is to study if a more general approach is possible for modelling many-valued logical conditions. The first relevant questions to solve sound 'Will modelling become more natural if more alternatives in answers than just 'yes' or 'no' are permitted? What kind of logic algebra would take the role of the binary variable linear algebra? Will a many-valued approach introduce effectiveness into computation algorithms?'
The research will be done in close co-operation with lecturer Silvennoinen. The topic is suitable for a masters/doctoral thesis, too.