A New Family of Logics for Cognitive Modelling Oliver Kutz

Abstract

We present the key motivations and technical results of a family of weighted descriptions logics, called tooth (or perceptron) logics, which expand on ideas from the paradigm of prototype theory, and which were conceived and designed to provide a cognitively more adequate formal modelling of concepts and concept combinations. The notion of 'concept' has been an elusive one both in cognitive science and psychology as well as in the formal sciences, particularly within logic and knowledge representation. From the cognitivepsychological perspective, a number of empirical phenomena and theoretical perspectives have led to a range of approaches to the definition and modelling of concepts, including most prominently exemplar theory, prototype theory, and theory theory (see [1] for a survey and detailed references).

Keywords

• adding only the 'basic tooth' ∇∇
• looks as follows

-

(︀ 1 : 1, . . . , : )︀ where ∇∇

(︀ 1 : 1, . . . , : )︀ forms a new concept which, in a fixed interpretation , contains exactly those individuals who satisfy ‘enough’ of the concepts such that their accumulative weight reaches the threshold value , formally: ∈ ︀( ∇∇ (︀ 1 : 1, . . . , : ))︀ ⇐⇒

∑︁ ∈{1,...,} { | ∈ } ≥

CEUR Workshop Proceedings

ceur-ws.org ISSN1613-0073

This basic shift of formalisation opens up a surprisingly rich landscape of theoretical and practical developments, and new considerations come into play at the friction point of modelling common sense, defeasible reasoning, and aspects of computational logics such as DLs.

The resulting perceptron logic family, often called tooth logic, has numerous applications, including in concept learning, modelling exceptions, modelling psychological phenomena in the use of concepts, and modelling some of the mechanics of conceptual blending / combination, as well as in concept abstraction. We briefly summarise the main contributions. The first detailed discussion of threshold operators, their formal definition and basic algebraic and logical properties, can be found in [ 5 ]. In [ 6 ], a distinction is made between knowledge-dependent and knowledge-independent tooth expressions, where the core idea of knowledge-dependence is that the features whose weights may be accumulated must provably hold for an individual according to a knowledge base, rather than accidentally in a model. In [ 7 ], summarised also in [ 8 ], the application of tooth expressions to modelling issues with concepts are studied, particularly those stemming from psychological studies, such as the efects of over- or under-extension studied by James Hampton [ 9, 10 ]. For example, an over-extension of a conjunctive concept ⊓ means that we consider more things to enter the conjunctive concept than would be expected when taking a set-theoretic intersection of the extensions of the concepts. For instance, something might be considered ‘a sport that is a game’ without being considered to be a ‘game’ proper. This work is further refined in [ 11 ], where the combination problem is studied in more detail algorithmically by employing a distinction between logically impossible and necessary features, a topic further explored in [ 12 ]. The paper [ 13 ] contains two important contributions. It illustrates on the one hand the applicability of the tooth logic approach to the problem of concept learning from data, and on the other hand it shows that reasoning can be done with of-the-shelf tools by providing an encoding into standard description logics without increasing the computational complexity.

The interpretability of threshold expressions was studied in [ 14 ], providing some evidence that tooth operators are easier to understand by people without formal logic education than equivalent disjunctive normal forms. Regarding further application areas, the core idea of using an accumulation of weighted features was explored in a novel logical approach to exceptions in [ 15 ], where it is exploited to provide a knowledge-dependent definition for the idea that a certain individual is more an than it is a , i.e. providing a notion of conceptual distance that depends on a given knowledge base. The philosophical and conceptual foundations of this approach are further discussed in [ 16 ].

Finally, in the most recent extension of the basic tooth logic, [ 17 ] studies the complexity of reasoning with tooth expressions that include counting perceptrons and illustrates this extension with some modelling examples. Each individual instance of a role successor in DL is considered, and their weights are accumulated. For instance, whilst the original tooth logic might give a certain weight to the concept of ‘having a child’, it does not properly extend the expressivity of ℒ. The counting tooth, on the other hand, can express the concept of ‘having as many daughters as sons’, which indeed extends the expressivity of standard DLs.

Future work is foreseen along the dimensions sketched above. However, we see particular potential in applications of tooth logics to the concept learning problem, and in providing a formal and conceptual bridge between the statistical world of learning approaches and the world of computational logics. Acknowledgments For the general development of the tooth logic I would like to acknowledge my collaborators, in alphabetical order: Pietro Galliani, Claudio Masolo, Daniele Porello, Guendalina Righetti, and Nicolas Troquard. For the application to defeasible reasoning I would like to further acknowledge Gabriele Sacco and Loris Bozzato, for the work on conceptual blending Maria Hedblom, and for the work on interpretability Roberto Confalonieri.

I acknowledge the financial support through the ‘Abstractron’ project funded by the Autonome Provinz Bozen - Südtirol (Autonomous Province of Bolzano-Bozen) through the Research Südtirol/Alto Adige 2022 Call. 1–4

References

[1] G. Righetti , Combining Concepts: Integrating Logical and Cognitive Theories of Concepts , Ph.D. thesis , Free University of Bozen Bolzano, 2023 . URL: https://hdl.handle.net/10863/36798.

[2] C. Masolo , D. Porello , Representing concepts by weighted formulas , in: Formal Ontology in Information Systems - Proceedings of the 10th International Conference, FOIS 2018 , Cape Town , South Africa, 19 -21 September 2018 , 2018 , pp. 55 - 68 .

[3] Rosenblatt , The perceptron: A probabilistic model for information storage and organization in the brain , Psychological Review 65 ( 1958 ) 386 - 408 .

[4] M. Minksy , S. Papert , Perceptrons: An Introduction to Computational Geometry, Cambridge, MA: MIT Press, 1969 .

[5] D. Porello , O. Kutz , G. Righetti, N. Troquard , P. Galliani , C. Masolo , A toothful of concepts: Towards a theory of weighted concept combination , in: Proceedings of the 32nd International Workshop on Description Logics , Oslo, Norway, June 18-21, 2019 , 2019 . URL: http://ceur-ws. org/ Vol- 2373 / paper-24.pdf.

[6] P. Galliani , O. Kutz , D. Porello , G. Righetti, N. Troquard , On knowledge dependence in weighted description logic , in: D. Calvanese , L. Iocchi (Eds.), GCAI 2019. Proceedings of the 5th Global Conference on Artificial Intelligence , Bozen/Bolzano, Italy, 17 -19 September 2019 , volume 65 of EPiC Series in Computing, EasyChair, 2019 , pp. 68 - 80 . URL: https://doi.org/10.29007/hjt1. doi: 10 . 29007/hjt1.

[7] G. Righetti , D. Porello , O. Kutz , N. Troquard , C. Masolo , Pink panthers and toothless tigers: three problems in classification , in: A. Cangelosi , A . Lieto (Eds.), Proceedings of the 7th International Workshop on Artificial Intelligence and Cognition , Manchester, UK, September 10-11 , 2019 , volume 2483 of CEUR Workshop Proceedings, CEUR-WS.org , 2019 , pp. 39 - 53 . URL: https://ceur-ws. org/ Vol- 2483 /paper4.pdf.

[8] G. Righetti , P. Galliani , O. Kutz , D. Porello , C. Masolo , N. Troquard , Weighted Description Logic for Classification Problems , in: D. Calvanese , L. Iocchi (Eds.), GCAI 2019. Proceedings of the 5th Global Conference on Artificial Intelligence , volume 65 of EPiC Series in Computing, EasyChair, 2019 , pp. 108 - 112 . URL: https://easychair.org/publications/paper/S5bV. doi: 10 .29007/vdlq.

[9] J. A. Hampton , Overextension of conjunctive concepts: Evidence for a unitary model of concept typicality and class inclusion ., Journal of Experimental Psychology: Learning, Memory, and Cognition 14 ( 1988 ) 12 - 32 .

[10] J. A. Hampton , Inheritance of attributes in natural concept conjunctions , Memory & Cognition 15 ( 1987 ) 55 - 71 .

[11] G. Righetti , C. Masolo , N. Troquard , O. Kutz , D. Porello , Concept Combination in Weighted Logic, in: E. M. Sanfilippo et al. (Ed.), Proceedings of the Joint Ontology Workshops , CAOS workshop, co -located with the Bolzano Summer of Knowledge (BOSK 2021 ), Virtual & BozenBolzano, Italy, September 10-18 , 2021 , CEUR Workshop Proceedings, CEUR-WS.org, 2021 . URL: http://ceur-ws. org/ Vol- 2708 /foust6.pdf.

[12] G. Righetti , D. Porello , N. Troquard , O. Kutz , M. Hedblom , P. Galliani , Asymmetric Hybrids: Dialogues for Computational Concept Combination , in: B. Brodaric , F. Neuhaus (Eds.), 12th International Conference on Formal Ontology in Information Systems (FOIS 2021 ), volume 344 of Frontiers in Artificial Intelligence and Applications , IOS Press, 2021 . FOIS Best Paper Award.

[13] P. Galliani , G. Righetti , O. Kutz , D. Porello , N. Troquard , Perceptron connectives in knowledge representation , in: C. M. Keet , M. Dumontier (Eds.), Knowledge Engineering and Knowledge Management , 22nd International Conference, EKAW 2020, Bolzano, Italy, September 16-20 , 2020 , Proceedings, volume 12387 of LNCS , Springer, 2020 , pp. 183 - 193 . URL: https://doi.org/10.1007/ 978-3- 030 -61244-3_ 13 . doi: 10 .1007/978-3- 030 -61244-3\_ 13 .

[14] G. Righetti , D. Porello , R. Confalonieri , Evaluating the interpretability of threshold operators , in: Oscar Corcho, L. Hollink , O. Kutz , N. Troquard , F. J. Ekaputra (Eds.), Knowledge Engineering and Knowledge Management , 23rd International Conference, EKAW 2022, Bolzano, Italy, September 26-29 , 2022 , Proceedings, volume 13514 of LNCS , Springer, 2022 , pp. 136 - 151 . URL: https://doi. org/10.1007/978-3- 031 -17105-5_ 10 . doi: 10 .1007/978-3- 031 -17105-5\_ 10 .

[15] G. Sacco , L. Bozzato , O. Kutz , Introducing Weighted Prototypes in Description Logics for Defeasible Reasoning , in: Proceedings of the 38th Italian Conference on Computational Logic (CILC 2023 ), CEUR-WS, 2023 .

[16] G. Sacco , L. Bozzato , O. Kutz , Know Your Exceptions: Towards an Ontology of Exceptions in Knowledge Representation , in: D. Porello , C. Trojahn (Eds.), 14th International Conference on Formal Ontology in Information Systems (FOIS 2024 ), Frontiers in Artificial Intelligence and Applications , IOS Press, 2024 . Nominated for FOIS Best Paper Award .

[17] P. Galliani , O. Kutz , N. Troquard , Succinctness and Complexity of ℒ with Counting Perceptrons , in: Proceedings of the Twentieth International Conference on Principles of Knowledge Representation and Reasoning (KR 2023 ), Rhodes, Greece, September 2- 8 , 2023 . URL: https://doi.org/10.24963/kr.2023/29. doi: 10 .24963/kr.2023/29.