Periodic Reporting for period 1 - ClaVa (Clausal forms for Vague information processing)
Berichtszeitraum: 2021-10-01 bis 2023-09-30
The issue being addressed in this project is the understanding of closed classes of formulas in the context of substructural logics, which we denote clausal-form-systems as inherited from the classical logic case. These classes of formulas might offer equivalent computational strength to the full logic for what concerns certain problems (e.g. SAT, MAX-SAT, or other computational questions), but keep a lower complexity or at least, be solvable more efficiently in practical cases.
The importance to society might be long-term, since this is a basic-research project, but it is related to the increase in the exploitation in practical and industrial context of substructural and non-classical logics, which are the better suited to model graded and vague information, or even to be defined ad-hoc according to complex problem management. This increase in potential use is linked to the achievement of lower complexity of classes of formulas of equivalent strength for the problems under study.
The main objective of this project is to study clausal-form systems for real and rational-valued events, facing the questions of their general definition, usage and solvable problems (SAT and optimality) from the point of view of their complexity, algorithmic design and applicability. It is focused on studying in full generality the definition of clausal form in non-classical logics for their use in vague knowledge representation. At the most general level, it is planned to reach a definition based on n-ary- level subsets of literals and using symmetric term-definable operations of arbitrary arity to combine each level, over Mathematical Fuzzy Logics and over some other Substructural Logics. We
* introduce definitions of general clausal forms over some of the above logical systems, study their possible equivalences and their logical properties (axiomatization of fragments, admissible rules, etc);
* design and test efficient solving procedures for three natural questions: SAT (evaluation sending the clause to 1), Max-SAT (evaluation maximizing the number of satisfied clauses) and optimality with respect to an objective (evaluation maximizing/minimizing the clause)
* do complexity classification of some of the clausal form classes, and study their normality (i.e. if any formula is equivalent to one in the desired form for the problems above).
The importance to society might be long-term, since this is a basic-research project, but it is related to the increase in the exploitation in practical and industrial context of substructural and non-classical logics, which are the better suited to model graded and vague information, or even to be defined ad-hoc according to complex problem management. This increase in potential use is linked to the achievement of lower complexity of classes of formulas of equivalent strength for the problems under study.
The main objective of this project is to study clausal-form systems for real and rational-valued events, facing the questions of their general definition, usage and solvable problems (SAT and optimality) from the point of view of their complexity, algorithmic design and applicability. It is focused on studying in full generality the definition of clausal form in non-classical logics for their use in vague knowledge representation. At the most general level, it is planned to reach a definition based on n-ary- level subsets of literals and using symmetric term-definable operations of arbitrary arity to combine each level, over Mathematical Fuzzy Logics and over some other Substructural Logics. We
* introduce definitions of general clausal forms over some of the above logical systems, study their possible equivalences and their logical properties (axiomatization of fragments, admissible rules, etc);
* design and test efficient solving procedures for three natural questions: SAT (evaluation sending the clause to 1), Max-SAT (evaluation maximizing the number of satisfied clauses) and optimality with respect to an objective (evaluation maximizing/minimizing the clause)
* do complexity classification of some of the clausal form classes, and study their normality (i.e. if any formula is equivalent to one in the desired form for the problems above).
* Together with my supervisor, we characterize a new clausal-like form over the infinitely-valued propositional Lukasiewicz logic, LCNF. We prove this class of formulas is normal for SAT: this means that each (arbitrary) formula in the logic has an associated formula in the regular clausal form introduced such that each evaluation satisfies the former one if and only if it satisfies the latter. I have been able to introduce a finite resolution rule which is sound and complete for the infinitely-valued Lukasiewicz logic.
* In work I have done alone, I have proven that the restriction of several FO logics to their modal formulas (a regular form of the predicates logics, among which FO Lukasiewicz, Product, and a large family of others identified with certain conditions) is a fragment which is not axiomatizable (i.e. not recursively enumerable, so not in Sigma_1 from the AH). This is a very strong result that has come as a surprise, since the analogous fragment in classical logic is indeed axiomatizable and decidable. It points to the extreme complexity of these logics.
* Together with the project supervisor and other two researchers, we study reduction to clausal form of non-clausal MaxSAT and MinSAT to clausal MaxSAT and MinSAT. We present three different cost-preserving transformations and we report on an empirical comparison of the performance of the proposed transformations when solved with a state-of-the-art MaxSAT solver.
* Together with my supervisor and a colleague from the Czech Academy of Sciences, where we study different algorithms solving MAX-SAT in the real valued MV-algebra. We later define an alternative analytic method with preprocessing in terms of a Tseitin transformation of the input, followed by a reduction to a system of linear constraints, in analogy to the earlier approaches of Hähnle and Olivetti.
* In a work I have done alone, I have proven that the local modal fragment of the predicate Product logic is indeed decidable (hence in Δ_1 from the Arithmetical Hierarchy) even if its definition as a logic, based on valued Kripke models, requires of infinite models. The reduction of this question to a decidable problem is based on a codification of the infinitary information that I believe can have later uses in related questions.
OVERVIEW OF EXPLOITATION/DISSEMINATION
* 2 indexed already published journal paper publications (one of them in the top journal in general logic), 1 submitted journal publication under revision since June 2023 (https://arxiv.org/abs/2306.13903) 1 manuscript in latter stages of development to be submitted to a journal.
* 1 published article in indexed proceedings
* Participation as INVITED SPEAKER in 5 international conferences. Participation as contributed speaker (with peer review) in 3 international conferences.
* In work I have done alone, I have proven that the restriction of several FO logics to their modal formulas (a regular form of the predicates logics, among which FO Lukasiewicz, Product, and a large family of others identified with certain conditions) is a fragment which is not axiomatizable (i.e. not recursively enumerable, so not in Sigma_1 from the AH). This is a very strong result that has come as a surprise, since the analogous fragment in classical logic is indeed axiomatizable and decidable. It points to the extreme complexity of these logics.
* Together with the project supervisor and other two researchers, we study reduction to clausal form of non-clausal MaxSAT and MinSAT to clausal MaxSAT and MinSAT. We present three different cost-preserving transformations and we report on an empirical comparison of the performance of the proposed transformations when solved with a state-of-the-art MaxSAT solver.
* Together with my supervisor and a colleague from the Czech Academy of Sciences, where we study different algorithms solving MAX-SAT in the real valued MV-algebra. We later define an alternative analytic method with preprocessing in terms of a Tseitin transformation of the input, followed by a reduction to a system of linear constraints, in analogy to the earlier approaches of Hähnle and Olivetti.
* In a work I have done alone, I have proven that the local modal fragment of the predicate Product logic is indeed decidable (hence in Δ_1 from the Arithmetical Hierarchy) even if its definition as a logic, based on valued Kripke models, requires of infinite models. The reduction of this question to a decidable problem is based on a codification of the infinitary information that I believe can have later uses in related questions.
OVERVIEW OF EXPLOITATION/DISSEMINATION
* 2 indexed already published journal paper publications (one of them in the top journal in general logic), 1 submitted journal publication under revision since June 2023 (https://arxiv.org/abs/2306.13903) 1 manuscript in latter stages of development to be submitted to a journal.
* 1 published article in indexed proceedings
* Participation as INVITED SPEAKER in 5 international conferences. Participation as contributed speaker (with peer review) in 3 international conferences.
All the results detailed in the previous point go beyond the state of the art, since they introducing new logical structures, theorems and experiments in the literature.
The potential impact is related to the results in the following way:
1) The usage of (infinitely valued) Lukasiewicz logic in applied grounds where the problem under study is SAT, relying on the clausal form I have developed and in the resolution methods proven complete for them. This could avoid the usage of numerical tools, or reduce them to MIP or other very efficient tools, instead of having to rely in gneralistic tools that have a slower performance.
2) More efficient approach to the MAX-SAT question in many-valued grounds, and the possibility of working with non-clasusal MAX-SAT issues. This is a question appearing often in industry (finding configurations that maximize/minimize costs, or number of elements satisfied).
2) The potential use of local product modal logics in applications, after I have seen they are decidable. Their complexity might still be very high, making this a challenge, but at least it has been seen that an algorithmic procedure can solve them, in contrast to the next case.
3) The warning behind possible uses of the global entailments on modal logics over many-valued algebras for modeling practical questions. We have seen these logics, in full generality, are not only undecidable but not even RE, so even if they might be tempting to explot as systems to model complex situations given their expressivity, for this same reason they should be avoided and smaller fragments (of lower complexity) would need to be identified if these structures are to be applied in practice.
Since the project has consisted in mathematical results and theoretical studies, and given my area of research, it is hard for me to establish the socio-economic impact and societal implications. I believe AI and other related technological tools can eventually benefit from my results, both in society-related applications and in industrial ones related to society in a second-degree, but I cannot really assess any particular case.
The potential impact is related to the results in the following way:
1) The usage of (infinitely valued) Lukasiewicz logic in applied grounds where the problem under study is SAT, relying on the clausal form I have developed and in the resolution methods proven complete for them. This could avoid the usage of numerical tools, or reduce them to MIP or other very efficient tools, instead of having to rely in gneralistic tools that have a slower performance.
2) More efficient approach to the MAX-SAT question in many-valued grounds, and the possibility of working with non-clasusal MAX-SAT issues. This is a question appearing often in industry (finding configurations that maximize/minimize costs, or number of elements satisfied).
2) The potential use of local product modal logics in applications, after I have seen they are decidable. Their complexity might still be very high, making this a challenge, but at least it has been seen that an algorithmic procedure can solve them, in contrast to the next case.
3) The warning behind possible uses of the global entailments on modal logics over many-valued algebras for modeling practical questions. We have seen these logics, in full generality, are not only undecidable but not even RE, so even if they might be tempting to explot as systems to model complex situations given their expressivity, for this same reason they should be avoided and smaller fragments (of lower complexity) would need to be identified if these structures are to be applied in practice.
Since the project has consisted in mathematical results and theoretical studies, and given my area of research, it is hard for me to establish the socio-economic impact and societal implications. I believe AI and other related technological tools can eventually benefit from my results, both in society-related applications and in industrial ones related to society in a second-degree, but I cannot really assess any particular case.