Final Report Summary - MATOMUVI (Mathematical Tools for the Management of Uncertain and Vague Information)
The project MATOMUVI (MAthematical TOols for the Management of Uncertain and Vague Information) is an European project funded under the 2009 IRSES call. It gathers 15 universities in 5 countries (Argentina, Brazil, Czech Republic, Italy, and Spain) and has involved around 40 researchers within these countries. The project spanned across the period March, 15th 2011 — March, 15th 2015.
The scientific objectives of the project were to explore some of the most promising directions outlined by the recent results in many-valued logic, providing a widely applicable, formal and unified setting for reasoning under vagueness and uncertainty.. The project led to a systematisation of the existing knowledge on the subject and extensions of the current tools to cope with more complex problems, where vagueness and uncertainty are entangled, and the reinforcement of the applicability of many-valued logic to real life problems. More specifically, the following objectives were attained during the project.
Objective 1: Systematisation of the study of many-valued logics related to residuated lattices.
The study of many-valued logics has seen in the last decades a vast amount of interest among researchers worldwide. Those studies gave rise to a multitude of logical systems. With our research we gave several contributions to the systematisation of their study, for instance publishing a 3 volume handbook which is, as of today, the most complete and updated source about many-valued and fuzzy logics. Outcomes of our research include a unified algebraic study of prominent many-valued logics; an abstract study of translations and interpretations among logical systems, and the study of systems that arise as formal combinations of these logics.
Objective 2: Formal systems for vagueness and uncertainty.The second objective of our research has been the development of unified formal systems in which vagueness and uncertainty can be handled together. Contrary to classical probability theory, which deals with yes-no events, continuously valued events are modelled by sentences in many-valued logic. The challenge was to extend classical results to many-valued logics, aiming at a general theory of probabilities over continuously valued events. To achieve this objective we investigated the duality theory of the algebras of many-valued logics, providing a geometrical understanding of many-valued events, we advanced in the representation theory of the algebraic semantics of many-valued logic, and an developed new theories that formalise the notion of imprecise probability, combine belief with vagueness and provide frameworks implementable into formal systems geared to applications.
Objective 3: Stable foundations to our collaborations and coordinated further research.Our joint research enterprise has created solid partnerships, both consolidating existing networks of collaborations and generating new connections. One of the more tangible and important outcome has been the exchange of PhD students and Post-Docs that went beyond the project itself. Indeed, Master students from the partner universities are now enrolled in PhD programme in other partner universities, and Post-Docs have been hired under permanent positions in partner countries different from the ones where they carried out their studies.
The final outcome of our project has been twofold:
The unification of the results obtained in the last decades in many-valued logics by virtue of more abstract perspectives and the combination, interpretation, translation of logical systems into other. We are confident that our unified paradigms for reasoning under vagueness and uncertainty will provide useful tools suitable for applications.
The development of new methods and ideas arising from our multi-disciplinary study on the subject involving probability, duality, states and modalities in many-valued logic. The theme affected by these research pertains to the formal management of uncertainty and vagueness; a phenomenon pervading real life.
The milestones reached during the project are the following.
Milestone number 1 — First Conference.At the begin of the project we organised an international conference. This was held in Salerno (Italy) in May 2011. The event served to gather together for the first time the project participants and plan the activities. The conference was open to all researchers interested in the topic of the project and was an important occasion for an up-to-date presentation of the state of art in the field. The conference had roughly 50 participants.Further information can be found at http://logica.dmi.unisa.it/AlgebraicSemantics2011/(se abrirá en una nueva ventana)
Milestone number 2 — Theory of Dualities.During the project we obtained some important results in theory of dualities for many-valued logics. In particular we were able to associate concrete geometrical objects (rational polyhedra) to each finite set of formula of the Lukasiewicz calculus, in a way that completely characterises the logic. Later on, we generalised the framework to a fundamental categorical adjunction holding between any variety of algebras and a corresponding class of topological spaces.
Milestone number 3 — First Workshop.In August 2012, a Conference with title “Philosophy and Mathematics of Uncertainty and Vagueness” was held in Campinas, Brazil. This was a major event in the project, far beyond the small workshop we had in mind at the time of writing the proposal. The conference was preceded by a summer school.Further information can be found at http://bertato.wix.com/pmuv(se abrirá en una nueva ventana)
Milestone number 4 — Unified framework.Since the beginning of the project several participants worked to a Handbook of Fuzzy Logic. This was a major effort of systematisation of the subject. The two books published
- Handbook of Mathematical Fuzzy Logic - Volume 1 and 2. Studies in Logic, Mathematical Logic and Foundations, Petr Cintula, Petr Hájek, Carles Noguera (eds.) , no. 37-38, London, College Publications, pp. 486 - pp. 474, 2011.have already became a standard reference for all researchers in the filed. A third volume is in preparation at the time of writing.
Milestone number 5 — Generalised Probability.In our study of probability over many-valued events, we obtained a number of mathematical results that strongly support our perspective on the subject. Among the others, we proved that a Dutch Book argument can be used to characterise coherence for imprecise probability and we gave an interpretation in terms of bets of conditional probability over many-valued events. Further, we studied the algebraic and Kripke models of logical systems in which uncertain and vagueness can be jointly formally handled.
Milestone number 6 — Second Workshop.
A second workshop was held in Rio de Janeiro in April 2013, as a satellite event to the international conference “Universal Logic”, an event that gathered around 300 participants. The workshop was a great occasion of discussion among the project participants and showcased to project to a wide audience of practitioners.
Further information can be found at http://www.uni-log.org/ss4-MVL.html(se abrirá en una nueva ventana)
Milestone number 7 — Theory of Internal States.
We tackled and solved an important problem in the theory of MV-algebras with internal states: we provided a complete characterisation of the subdirectly irreducible algebras in the vairety. This allowed us to also classify subdirectly irreducible state morphism MV-algebras. We also studied algebraic correspondents of coherence criteria à la de Finetti. We singled out different properties of states on MV-algebras and expanded de Finetti’s coherence criterion to conditional probability, introducing a stronger rationality criterion called stable coherence. Finally we provided a logical and algebraic characterisation of coherence for assignments of partially undetermined events.
Milestone number 8 — Applications.
Generalising the notion of probability in MV-algebras, suitable extensions to non-classical events of other classical uncertainty measures (belief functions, necessity measures) have been introduced and the corresponding formal systems have been studied. These results pave the way for both flexible and powerful formalisms for knowledge representation. They provide formal models and effective reasoning tools that can cope with different kinds of imperfect information: vagueness, uncertainty and/or approximations.
Milestone number 9 — Final Conference
The final event of project took place in Buenos Aires (Argentina), on February 2015. The main achievements of the project were showcased and discussed with international researchers.
Further information can be found at http://www-2.dc.uba.ar/congresos/matomuvi2015/?lang=en(se abrirá en una nueva ventana)
Notably, among the new strategies of cooperation discussed at the conference, the project participants decided to apply for a RISE project together with other universities form all around the world, thus trying to expand the network of collaborations built during the project to a world-wide scale. The project was successfully submitted under the RISE call 2015.
The scientific objectives of the project were to explore some of the most promising directions outlined by the recent results in many-valued logic, providing a widely applicable, formal and unified setting for reasoning under vagueness and uncertainty.. The project led to a systematisation of the existing knowledge on the subject and extensions of the current tools to cope with more complex problems, where vagueness and uncertainty are entangled, and the reinforcement of the applicability of many-valued logic to real life problems. More specifically, the following objectives were attained during the project.
Objective 1: Systematisation of the study of many-valued logics related to residuated lattices.
The study of many-valued logics has seen in the last decades a vast amount of interest among researchers worldwide. Those studies gave rise to a multitude of logical systems. With our research we gave several contributions to the systematisation of their study, for instance publishing a 3 volume handbook which is, as of today, the most complete and updated source about many-valued and fuzzy logics. Outcomes of our research include a unified algebraic study of prominent many-valued logics; an abstract study of translations and interpretations among logical systems, and the study of systems that arise as formal combinations of these logics.
Objective 2: Formal systems for vagueness and uncertainty.The second objective of our research has been the development of unified formal systems in which vagueness and uncertainty can be handled together. Contrary to classical probability theory, which deals with yes-no events, continuously valued events are modelled by sentences in many-valued logic. The challenge was to extend classical results to many-valued logics, aiming at a general theory of probabilities over continuously valued events. To achieve this objective we investigated the duality theory of the algebras of many-valued logics, providing a geometrical understanding of many-valued events, we advanced in the representation theory of the algebraic semantics of many-valued logic, and an developed new theories that formalise the notion of imprecise probability, combine belief with vagueness and provide frameworks implementable into formal systems geared to applications.
Objective 3: Stable foundations to our collaborations and coordinated further research.Our joint research enterprise has created solid partnerships, both consolidating existing networks of collaborations and generating new connections. One of the more tangible and important outcome has been the exchange of PhD students and Post-Docs that went beyond the project itself. Indeed, Master students from the partner universities are now enrolled in PhD programme in other partner universities, and Post-Docs have been hired under permanent positions in partner countries different from the ones where they carried out their studies.
The final outcome of our project has been twofold:
The unification of the results obtained in the last decades in many-valued logics by virtue of more abstract perspectives and the combination, interpretation, translation of logical systems into other. We are confident that our unified paradigms for reasoning under vagueness and uncertainty will provide useful tools suitable for applications.
The development of new methods and ideas arising from our multi-disciplinary study on the subject involving probability, duality, states and modalities in many-valued logic. The theme affected by these research pertains to the formal management of uncertainty and vagueness; a phenomenon pervading real life.
The milestones reached during the project are the following.
Milestone number 1 — First Conference.At the begin of the project we organised an international conference. This was held in Salerno (Italy) in May 2011. The event served to gather together for the first time the project participants and plan the activities. The conference was open to all researchers interested in the topic of the project and was an important occasion for an up-to-date presentation of the state of art in the field. The conference had roughly 50 participants.Further information can be found at http://logica.dmi.unisa.it/AlgebraicSemantics2011/(se abrirá en una nueva ventana)
Milestone number 2 — Theory of Dualities.During the project we obtained some important results in theory of dualities for many-valued logics. In particular we were able to associate concrete geometrical objects (rational polyhedra) to each finite set of formula of the Lukasiewicz calculus, in a way that completely characterises the logic. Later on, we generalised the framework to a fundamental categorical adjunction holding between any variety of algebras and a corresponding class of topological spaces.
Milestone number 3 — First Workshop.In August 2012, a Conference with title “Philosophy and Mathematics of Uncertainty and Vagueness” was held in Campinas, Brazil. This was a major event in the project, far beyond the small workshop we had in mind at the time of writing the proposal. The conference was preceded by a summer school.Further information can be found at http://bertato.wix.com/pmuv(se abrirá en una nueva ventana)
Milestone number 4 — Unified framework.Since the beginning of the project several participants worked to a Handbook of Fuzzy Logic. This was a major effort of systematisation of the subject. The two books published
- Handbook of Mathematical Fuzzy Logic - Volume 1 and 2. Studies in Logic, Mathematical Logic and Foundations, Petr Cintula, Petr Hájek, Carles Noguera (eds.) , no. 37-38, London, College Publications, pp. 486 - pp. 474, 2011.have already became a standard reference for all researchers in the filed. A third volume is in preparation at the time of writing.
Milestone number 5 — Generalised Probability.In our study of probability over many-valued events, we obtained a number of mathematical results that strongly support our perspective on the subject. Among the others, we proved that a Dutch Book argument can be used to characterise coherence for imprecise probability and we gave an interpretation in terms of bets of conditional probability over many-valued events. Further, we studied the algebraic and Kripke models of logical systems in which uncertain and vagueness can be jointly formally handled.
Milestone number 6 — Second Workshop.
A second workshop was held in Rio de Janeiro in April 2013, as a satellite event to the international conference “Universal Logic”, an event that gathered around 300 participants. The workshop was a great occasion of discussion among the project participants and showcased to project to a wide audience of practitioners.
Further information can be found at http://www.uni-log.org/ss4-MVL.html(se abrirá en una nueva ventana)
Milestone number 7 — Theory of Internal States.
We tackled and solved an important problem in the theory of MV-algebras with internal states: we provided a complete characterisation of the subdirectly irreducible algebras in the vairety. This allowed us to also classify subdirectly irreducible state morphism MV-algebras. We also studied algebraic correspondents of coherence criteria à la de Finetti. We singled out different properties of states on MV-algebras and expanded de Finetti’s coherence criterion to conditional probability, introducing a stronger rationality criterion called stable coherence. Finally we provided a logical and algebraic characterisation of coherence for assignments of partially undetermined events.
Milestone number 8 — Applications.
Generalising the notion of probability in MV-algebras, suitable extensions to non-classical events of other classical uncertainty measures (belief functions, necessity measures) have been introduced and the corresponding formal systems have been studied. These results pave the way for both flexible and powerful formalisms for knowledge representation. They provide formal models and effective reasoning tools that can cope with different kinds of imperfect information: vagueness, uncertainty and/or approximations.
Milestone number 9 — Final Conference
The final event of project took place in Buenos Aires (Argentina), on February 2015. The main achievements of the project were showcased and discussed with international researchers.
Further information can be found at http://www-2.dc.uba.ar/congresos/matomuvi2015/?lang=en(se abrirá en una nueva ventana)
Notably, among the new strategies of cooperation discussed at the conference, the project participants decided to apply for a RISE project together with other universities form all around the world, thus trying to expand the network of collaborations built during the project to a world-wide scale. The project was successfully submitted under the RISE call 2015.