Skip to main content
SpringerLink
Log in

Reasoning and learning with context logic

  • Original Article
  • Published:
Journal of Reliable Intelligent Environments Aims and scope Submit manuscript

Abstract

Context logic (CL), a logical language similar in style to description logics but with a more cognitive motivation as a logical language of cognition, was developed since 2007 to provide a new approach to the symbol grounding problem, a key problem for reliable intelligent environments and other intelligent sensory systems. CL is a three-layered integrated hierarchy of languages: a relational base layer with the expressiveness of propositional logic (CLA), a quantifier-free decidable language (CL0), and an expressive language with full quantification (CL1). As was shown in 2018, the core CLA reasoning can be implemented on a variant of Kanerva’s Vector Symbolic Architecture, the activation bit vector machine (ABVM), shedding new light on the fundamental cognitive faculties of symbol grounding and imagery, but the system raised two questions: first, the core reasoning algorithm was a classical EXPTIME reasoner; second, fundamental aspects for a learning algorithm were sketched but not presented with a full algorithm. This paper addresses those two questions. We present a probabilistic linear time algorithm for reasoning over conjunctive normal form (CNF) CLA formulae together with a dual probabilistic linear time algorithm for learning CLA statements by collecting experienced snapshots in a disjunctive normal form (DNF).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. We, e.g., automatically obtain that Boolean algebras being topologies, although not with particularly interesting properties from the perspective of topology, can serve as models. For automated reasoning, general topologies are unwieldy, but for finite Boolean algebras, a wealth of reasoning frameworks exists, in the SAT reasoning and DL literature [10].

  2. E.g., in the Boolean algebra topology of sets, where the map \(\rightarrow \) is the set inclusion \(\subseteq \), \(\alpha _1\sqsubseteq \alpha _2\) is a model iff \(c(\alpha _1) \subseteq c(\alpha _2)\), i.e., iff the set \(c(\alpha _1)\) is a subset of the set \(c(\alpha _2)\). For a more genuinely topological example, consider the temporally indexed trajectories of moving point objects in the plane as arrows of \({\mathcal {O}}\). In this case, arrows and thus \(\sqsubseteq \) represent a spatiotemporal before between two points locating the same object at different times.

  3. A main benefit of CL is the ability to use the contextualization syntax without, e.g., having to assume distinct world indices as in Hybrid Logics [3]. However, from a point of view of expressiveness, FS2a does not extend the language beyond what FS1 already provides. Alternatively, contextualization could be defined syntactically.

  4. The conventional truth table method provides the set \({\mathcal {M}}\) with the \(c \in {\mathcal {C}}\) the rows of the table; the map \(c(\alpha _1) \rightarrow c(\alpha _2)\) is provided by the propositional logic entailment relation: any row c with the value “true” for \(\alpha _1\) in the table also has the value “true” for \(\alpha _2\).

  5. For a deeper inquiry into the formal reasons why partial orders may be an interesting foundation for decidable reasoning, cf. Kieroński’s discussion of the two-variable guarded fragment with transitive guards [27, 28]. This fragment is sufficient to specify partial orders, since the only axiom for a partial order outside the two-variable fragment is the transitivity requirement.

  6. Project homepage: https://logicallateration.appspot.com.

  7. http://satcompetition.org

References

  1. Baddeley A (1994) The magical number seven: Still magic after all these years? Psychological Review 101(2):353

    Google Scholar 

  2. Bettini C, Brdiczka O, Henricksen K, Indulska J, Nicklas D, Ranganathan A, Riboni D (2010) A survey of context modelling and reasoning techniques. Pervasive and Mobile Computing 6(2):161–180

    Google Scholar 

  3. Blackburn P (2000) Representation, reasoning, and relational structures: a hybrid logic manifesto. Logic Journal of the IGPL 8(3):339

    MathSciNet  MATH  Google Scholar 

  4. Cardelli L, Gordon AD (2000) Mobile ambients. Theoretical Computer Science 240(1):177–213

    MathSciNet  MATH  Google Scholar 

  5. Cohn AG, Bennett B, Gooday J, Gotts NM (1997) Qualitative spatial representation and reasoning with the region connection calculus. GeoInformatica 1(3):275–316

    Google Scholar 

  6. Coronato A, De Pietro G (2011) Formal specification and verification of ubiquitous and pervasive systems. ACM Transactions on Autonomous and Adaptive Systems (TAAS) 6(1):1–6

    Google Scholar 

  7. Cropper A, Dumančić S, Muggleton SH (2020) Turning 30: New ideas in inductive logic programming. arXiv preprint arXiv:200211002

  8. Davidson D (1967) The logical form of action sentences. In: Rescher N (ed) The Logic of Decision and Action. University of Pittsburgh Press, Pittsburgh, Pa., pp 81–95

    Google Scholar 

  9. Ding Y, Laue F, Schmidtke HR, Beigl M (2010) Sensing spaces: light-weight monitoring of industrial facilities. In: Gottfried B, Aghajan H (eds) 5th workshop on behaviour monitoring and interpretation, pp 1–10

  10. Donini FM (2003) Complexity of reasoning. In: Baader F, Calvanese D, McGuinness DL, Nardi D, Patel-Schneider PF (eds) The description logic handbook: theory, implementation, and applications. Cambridge University Press, Cambridge, pp 96–136

    Google Scholar 

  11. Dubois D, Lang J, Prade H (1994) Possibilistic logic. In: Gabbay DM, Hogger CJ, Robinson JA (eds) Handbook of Logic in Artificial Intelligence and Logic Programming, vol 3. Oxford University Press Inc, New York, pp 439–513

    Google Scholar 

  12. Frege G (1879) Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. L. Nebert

  13. Frosst N, Hinton G (2017) Distilling a neural network into a soft decision tree. arXiv preprint arXiv:171109784

  14. Garnelo M, Shanahan M (2019) Reconciling deep learning with symbolic artificial intelligence: representing objects and relations. Current Opinion in Behavioral Sciences 29:17–23. https://doi.org/10.1016/j.cobeha.2018.12.010

    Article  Google Scholar 

  15. Gayler RW (2006) Vector Symbolic Architectures are a viable alternative for Jackendoff’s challenges. Behavioral and Brain Sciences 29(1):78–79

    Google Scholar 

  16. Haarslev V, Lutz C, Möller R (1999) A description logic with concrete domains and a role-forming predicate operator. Journal of Logic and Computation 9(3):351–384

    MathSciNet  MATH  Google Scholar 

  17. Hájek P (1998) Metamathematics of fuzzy logic Trends in Logic, vol 4. Springer Science & Business Media, New York

    MATH  Google Scholar 

  18. Harnad S (1990) The symbol grounding problem. Physica D: Nonlinear Phenomena 42(1–3):335–346

    Google Scholar 

  19. Harnad S (2003) The symbol grounding problem In Encyclopedia of Cognitive Science, Macmillan, New York

    Google Scholar 

  20. Hong D, Schmidtke HR, Woo W (2007) Linking context modelling and contextual reasoning. In: Kofod-Petersen A, Cassens J, Leake DB, Schulz S (eds) 4th International workshop on modeling and reasoning in context (MRC), Roskilde University, pp 37–48

  21. Jackson PC Jr (1992) Proving unsatisfiability for problems with constant cubic sparsity. Artificial intelligence 57(1):125–137

    MathSciNet  Google Scholar 

  22. Jang JSR (1993) ANFIS: adaptive-network-based fuzzy inference system. Systems, Man and Cybernetics, IEEE Transactions on 23(3):665–685

    Google Scholar 

  23. Jerrum MR, Valiant LG, Vazirani VV (1986) Random generation of combinatorial structures from a uniform distribution. Theoretical computer science 43:169–188

    MathSciNet  MATH  Google Scholar 

  24. Kanerva P (1988) Sparse distributed memory. MIT Press, Cambridge

    MATH  Google Scholar 

  25. Kanerva P (2009) Hyperdimensional computing: An introduction to computing in distributed representation with high-dimensional random vectors. Cognitive computation 1(2):139–159

    Google Scholar 

  26. Karunaratne G, Le Gallo M, Cherubini G, Benini L, Rahimi A, Sebastian A (2020) In-memory hyperdimensional computing. Nature. Electronics 3:327–337. https://doi.org/10.1038/s41928-020-0410-3

    Article  Google Scholar 

  27. Kieroński E (2006) On the complexity of the two-variable guarded fragment with transitive guards. Information and Computation 204(11):1663–1703. https://doi.org/10.1016/j.ic.2006.08.001

    Article  MathSciNet  MATH  Google Scholar 

  28. Kieronski E, Tendera L (2016) Finite satisfiability of the two-variable guarded fragment with transitive guards and related variants. arXiv:e-prints1611.03267

  29. Kleyko D, Osipov E, Gayler RW, Khan AI, Dyer AG (2015) Imitation of honey bees’ concept learning processes using vector symbolic architectures. Biologically Inspired Cognitive Architectures 14:57–72

    Google Scholar 

  30. Kleyko D, Osipov E, Senior A, Khan AI, Şekerciogğlu YA (2016) Holographic graph neuron: A bioinspired architecture for pattern processing. IEEE transactions on neural networks and learning systems 28(6):1250–1262

    MathSciNet  Google Scholar 

  31. Knauff M, Kassubek J, Mulack T, Greenlee MW (2000) Cortical activation evoked by visual mental imagery as measured by fMRI. Neuroreport 11(18), 3957–3962

    Google Scholar 

  32. Kosslyn S (1980) Image and Mind. The MIT Press, Cambridge, MA

    Google Scholar 

  33. Kosslyn S (1994) Image and Brain: the Resolution of the Imagery Debate. The MIT Press, Cambridge, MA

    Google Scholar 

  34. Kralik JD, Lee JH, Rosenbloom PS, Jackson PC, Epstein SL, Romero OJ, Sanz R, Larue O, Schmidtke HR, Lee SW, McGreggor K (2018) Metacognition for a common model of cognition. Procedia Computer Science 145:730–739

    Google Scholar 

  35. McCarthy J (1980) Circumscription—a form of nonmonotonic reasoning. In: Stanford University, Technical report

  36. McCarthy J (1986) Applications of circumscription to formalizing common-sense knowledge. Artificial Intelligence 28(1):89–116

    MathSciNet  Google Scholar 

  37. McCarthy J, Hayes PJ (1981) Some philosophical problems from the standpoint of artificial intelligence In Readings in artificial intelligence, Elsevier, Amsterdam, pp 431–450

    Google Scholar 

  38. Miller GA (1956) The magical number seven, plus or minus two: Some limits on our capacity for processing information. Psychological review 63(2):81

    Google Scholar 

  39. Mitrokhin A, Sutor P, Summers-Stay D, Fermüller C, Aloimonos Y (2020) Symbolic representation and learning with hyperdimensional computing. Front Robot AI 7

  40. Monk JD (2018) The mathematics of Boolean algebra. In: Zalta EN (ed) The Stanford encyclopedia of philosophy, fall, 2018th edn. Stanford University, Metaphysics Research Lab

  41. Muggleton S, De Raedt L (1994) Inductive logic programming: Theory and methods. The Journal of Logic Programming 19:629–679

    MathSciNet  MATH  Google Scholar 

  42. Newell A (1994) Unified theories of cognition. Harvard University Press,

    Google Scholar 

  43. Nolt J (2018) Free logic. In: Zalta EN (ed) The Stanford encyclopedia of philosophy, fall, 2018th edn. Stanford University, Metaphysics Research Lab

  44. Safavian SR, Landgrebe D (1991) A survey of decision tree classifier methodology. IEEE transactions on systems, man, and cybernetics 21(3):660–674

    MathSciNet  Google Scholar 

  45. Schmidhuber J (2015) Deep learning in neural networks: An overview. Neural networks 61:85–117

    Google Scholar 

  46. Schmidtke HR (2016) Granular mereogeometry. In: Ferrario R, Kuhn W (eds) Formal ontology in information systems: proceedings of the 9th international conference (FOIS 2016), IOS Press, Frontiers in artificial intelligence and applications, vol 283, pp 81–94

  47. Schmidtke HR (2018a) A canvas for thought. Procedia Computer Science 145:805–812

    Google Scholar 

  48. Schmidtke HR (2018b) Logical lateration - a cognitive systems experiment towards a new approach to the grounding problem. Cognitive Systems Research 52:896–908. https://doi.org/10.1016/j.cogsys.2018.09.008

    Article  Google Scholar 

  49. Schmidtke HR (2018c) A survey on verification strategies for intelligent transportation systems. Journal of Reliable Intelligent Environments 4(4):211–224. https://doi.org/10.1007/s40860-018-0070-5

    Article  Google Scholar 

  50. Schmidtke HR (2020a) Logical rotation with the activation bit vector machine. Proced Comput Sci 169:568–577

  51. Schmidtke HR (2020b) Textmap: a general purpose visualization system. Cogn Syst Res 59:27–36

  52. Schmidtke HR (2020c) The TextMap general purpose visualization system: core mechanism and case study. In: Samsonovich A (ed) Biologically inspired cognitive architectures. Advances in intelligent systems and computing, vol 948. Springer, Cham, pp 455–464

  53. Schmidtke HR, Woo W (2007) A size-based qualitative approach to the representation of spatial granularity. In: Veloso MM (ed) Twentieth international joint conference on artificial intelligence, pp 563–568

  54. Schmidtke HR, Woo W (2009) Towards ontology-based formal verification methods for context aware systems. In: Tokuda H, Beigl M, Brush A, Friday A, Tobe Y (eds) Pervasive. Springer, Cham, pp 309–326

    Google Scholar 

  55. Schmidtke HR, Hong D, Woo W (2008) Reasoning about models of context: A context-oriented logical language for knowledge-based context-aware applications. Revue d’Intelligence Artificielle 22(5):589–608

    Google Scholar 

  56. Schmidtke HR, Yu H, Masomo P, Kinai A, Shema A (2014) Contextual reasoning in an intelligent electronic patient leaflet system. In: Gonzalez A, Brézillon P (eds) Context in computing, a cross-disciplinary approach for modeling the real world. Springer, Cham, pp 557–573

    Google Scholar 

  57. Stallings W (2005) Wireless communications and networks. Pearson Education, New York

    Google Scholar 

  58. Steels L (2008) The symbol grounding problem has been solved so what’s next Symbols and embodiment: Debates on meaning and cognition. Springer, New York, pp 223–244

    Google Scholar 

  59. Taddeo M, Floridi L (2005) Solving the symbol grounding problem: a critical review of fifteen years of research. Journal of Experimental & Theoretical Artificial Intelligence 17(4), 419–445

    Google Scholar 

  60. Taylor HA, Tversky B (1992) Spatial mental models derived from survey and route descriptions. Journal of Memory and language 31(2):261–292

    Google Scholar 

  61. Tversky B (1993) Cognitive maps, cognitive collages, and spatial mental models. In: European conference on spatial information theory, Springer, pp 14–24

  62. Tversky B, Lee P (1999) On pictorial and verbal tools for conveying routes. In: Freksa C, Mark D (eds) Spatial Information Theory. Springer, Berlin, pp 51–64

    Google Scholar 

  63. Valiant LG (1979) The complexity of computing the permanent. Theoretical computer science 8(2):189–201

    MathSciNet  MATH  Google Scholar 

  64. Van Heijenoort J (1967) From Frege to Gödel: a source book in mathematical logic, 1879–1932. Harvard University Press

    MATH  Google Scholar 

  65. Vogt P (2002) The physical symbol grounding problem. Cognitive Systems Research 3(3), 429–457

    Google Scholar 

  66. Wessel M (2001) Obstacles on the way to qualitative spatial reasoning with description logics: some undecidability results. Descr Logics 49

  67. Zadeh LA (1975) Fuzzy logic and approximate reasoning. Synthese 30(3), 407–428

    MATH  Google Scholar 

Download references

Acknowledgements

I am grateful to Christian Freksa and the Bremen Spatial Cognition Center (BSCC) at the University of Bremen for comments and infrastructure support and to the Hanse-Wissenschaftskolleg (HWK) for funding.

Author information

Authors and Affiliations

  1. 19061, Schwerin, Germany

    H. R. Schmidtke

Authors
  1. H. R. Schmidtke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. R. Schmidtke.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Schmidtke, H.R. Reasoning and learning with context logic. J Reliable Intell Environ 7, 171–185 (2021). https://doi.org/10.1007/s40860-020-00121-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40860-020-00121-2

Keywords

Search

Navigation