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Reasoning in Non-probabilistic Uncertainty: Logic Programming and Neural-Symbolic Computing as Examples

Abstract

This article aims to achieve two goals: to show that probability is not the only way of dealing with uncertainty (and even more, that there are kinds of uncertainty which are for principled reasons not addressable with probabilistic means); and to provide evidence that logic-based methods can well support reasoning with uncertainty. For the latter claim, two paradigmatic examples are presented: logic programming with Kleene semantics for modelling reasoning from information in a discourse, to an interpretation of the state of affairs of the intended model, and a neural-symbolic implementation of input/output logic for dealing with uncertainty in dynamic normative contexts.

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Notes

  1. However, these authors—somewhat paradoxically—in the end come to the view that whatever uncertainty is the topic, probability is the framework for modelling it; cf. Sect. 2.2 for some considerations on the corresponding argument and conclusion.

  2. Later, we shall consider an alternative LP semantics based on Answer Sets (Gelfond and Lifschitz 1991), but we choose Preferred Model Semantics (Shoham 1987) for now because the uniqueness of preferred models is a crucial feature for cognitive processes such as discourse processing.

  3. In terms of concrete examples, the work by Nilsson (1986) comes to mind as a prominent instance falling within the domain of Halpern’s plausibility measures.

  4. Intended model is the psychological notion which corresponds to minimal model. Model is to be read as semantic model. Preferred model is the logical notion used by Shoham (1987). Keeping in mind the terms belonging to different fields, we use intended, minimal and preferred as synonyms.

  5. Abnormality is a technical term for exceptions and should not be taken as having any other overtones. Some terminology: in the conditional \(p \wedge \lnot ab \rightarrow q\), ab is the schematic abnormality clause. A distinct ab is indexed to each conditional, and stands for a disjunction of a list of defeaters for that conditional; CWA\(_{ab}\) is the CWA applied to abnormality clauses; \(\lnot\) is the 3-valued Kleene connective, whereas negation-as-failure is an inference pattern that results in negative conclusions by CWA reasoning from absence of positive evidence; falsum (\(\bot\)) and verum (\(\top\)) are proposition symbols which always take the values false or true respectively; turnstile (\(\vdash\)) and semantic turnstile (\(\models\)) are symbols indicating syntactic and semantic consequence respectively.

  6. Another remark concerning terminology: while in this context the use of the terms ‘head’ and ‘body’ is commonplace in computer science, we will in the following restrict ourselves to ‘antecedent’ and ‘consequent’ in order to maintain homogeneity also with terminology in philosophy and logic.

  7. Cf. (Stenning and van Lambalgen 2008, chapter 2) for the justification, or the Appendix in (Stenning et al. 2017) for a more succinct version.

  8. Here, \(\leftrightarrow\) denotes a classical biconditional in the object language.

  9. Basically, in LP queries can be answered in time linear with the length of the shortest inferential path in the KB.

  10. This direction of reasoning is from effect to cause, from goals to subgoals, or simply backwards in time. The typical backward inferences are modus tollens (\(p \rightarrow q, \lnot q\, \models \lnot p\)) and affirmation of the consequent (\(p \rightarrow q, q\, \models p\)), initiated by the consequent. The psychological findings that these inferences are more difficult might well be a result of the micro-scale of the tasks being used (Sloman and Lagnado 2015), or of the slightly more complex form of the \(\hbox {CWA}_{ab}\) needed (Stenning and van Lambalgen 2008, pp. 176–177).

  11. Nota bene: This stands in harsh contrast to Oaksford and Chater (1998)’s above quoted conventional characterisation of CL as ‘reasoning in certainty’. When this way of conceptualising monotonic CL in contrast to nonmonotonic logics was introduced in the mid-1970s and 1980s—for instance, in the wake of Minsky (1974)’s frames or McCarthy (1980)’s circumscription approach— the underlying concern was not with characterising kinds of uncertainty, but with contrasting two systems on the one specific property of (non)monotonicity.

  12. Terminology yet again: for the purpose of this article we use law, norm, rule, etc. as synonymous.

  13. Of course this list is non-exhaustive as there are further alternative candidates for a new standard, such as nonmonotonic logic (Horty 1993) or deontic update semantics (van der Torre and Tan 1999).

  14. To give an intuitive example of a CTD, we report the so-called dog-sign example by Prakken and Sergot (1997) already hinted at in the introduction: “Suppose that: there must be no dog around the house, and if there is no dog, there must be no warning sign, but if there is a dog, there must be a warning sign.” Obviously, if there is a dog, the conditional obligation that there must be no sign does not become unconditional, since its condition is not fulfilled. On the other hand, it can also be inferred that if no obligations are violated, there will be no sign (modulo exceptions, of course).

  15. These require much more involved I/O operations, which we shortly discuss in Sect. 3.4 below. Cf. the work by Makinson and van der Torre (2001) and Makinson and van der Torre (2003a) for more detailed treatments.

  16. Makinson and van der Torre (2003a) consider three kinds of permissive norms, namely negative, positive, and static positive permission. In this article, we restrict discussions to the above, and should note that much future work is left to be done when it comes to the provision of connectionist representations for normative and deontic reasoning systems

  17. Given a rule, e.g. \(B \leftarrow A\), input and output vectors are created having ‘1’ in the position corresponding to A in the input vector, and ‘1’ in the position corresponding to B in the output vector.

  18. The presented approach to LP modelling of discourse does not tackle the learning of KB rules, as discourse comprehension generally is assumed to proceed with a mature KB. But an account of learning is nevertheless an important goal for LP models of discourse.

References

  • Alchourrón, C. E., Gärdenfors, P., & Makinson, D. (1985). On the logic of theory change: Partial meet contraction and revision functions. The Journal of Symbolic Logic, 50(02), 510–530.

    MathSciNet  Article  MATH  Google Scholar 

  • Antoniou, G., Billington, D., & Maher, M. (1998). Sceptical logic programming based default reasoning: defeasible logic rehabilitated. In R. Miller, M. Shanahan (Eds.), COMMONSENSE 98, The 4th symposium on logical formalizations of commonsense reasoning, London.

  • Apt, K. R., & Pedreschi, D. (1993). Reasoning about termination of pure prolog programs. Information and Computation, 106, 109–157.

    MathSciNet  Article  MATH  Google Scholar 

  • Baggio, G., Stenning, K., & van Lambalgen, M. (2016). The cognitive interface. In M. Aloni & P. Dekker (Eds.), Cambridge handbook of formal semantics. Cambridge: Cambridge University Press.

    Google Scholar 

  • Boella, G, & van der Torre, L (2005). Permission and authorization in normative multiagent systems. In Procs. of int. conf. on artificial intelligence and law ICAIL (pp. 236–237).

  • Boella, G., & van der Torre, L. (2006). A game theoretic approach to contracts in multiagent systems. IEEE Transactions on Systems, Man, and Cybernetics, Part C, 36(1), 68–79.

    Article  Google Scholar 

  • Boella, G., Pigozzi, G., & van der Torre, L. (2009). Normative framework for normative system change. In 8th Int. joint conf. on autonomous agents and multiagent systems AAMAS 2009, IFAAMAS (pp. 169–176).

  • Bradley, R., & Drechsler, M. (2014). Types of uncertainty. Erkenntnis, 79, 1225–1248.

    MathSciNet  Article  Google Scholar 

  • Doets, K. (1994). From logic to logic programming. Cambridge, MA: MIT Press.

    MATH  Google Scholar 

  • Gabbay, D., Horty, J., Parent, X., van der Meyden, R., & van der Torre, L. (Eds.). (2013). Handbook of deontic logic and normative systems. London: College Publications.

    MATH  Google Scholar 

  • Garcez, A., Broda, K., & Gabbay, D. M. (2001). Symbolic knowledge extraction from trained neural networks: A sound approach. Artificial Intelligence, 125, 155–207.

    MathSciNet  Article  MATH  Google Scholar 

  • Garcez, A., Broda, K., & Gabbay, D. (2002). Neural-symbolic learning systems: Foundations and applications. Perspectives in neural computing. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Garcez, A., Gabbay, D., & Lamb, L. (2005). Value-based argumentation frameworks as neural-symbolic learning systems. Journal of Logic and Computation, 15(6), 1041–1058.

    MathSciNet  Article  MATH  Google Scholar 

  • Garcez, A., Lamb, L. C., & Gabbay, D. M. (2009). Neural-symbolic cognitive reasoning. Berlin: Springer.

    MATH  Google Scholar 

  • Garcez, A., Besold, T.R., de Raedt, L., Földiak, P., Hitzler, P., Icard, et al. (2015). Neural-symbolic learning and reasoning: Contributions and challenges. In: AAAI Spring 2015 symposium on knowledge representation and reasoning: Integrating symbolic and neural approaches, AAAI technical reports (vol SS-15-03). AAAI Press.

  • Gelfond, M., & Lifschitz, V. (1988). The stable model semantics for logic programming. In Proceedings of the 5th logic programming symposium, MIT Press (pp. 1070–1080).

  • Gelfond, M., & Lifschitz, V. (1991). Classical negation in logic programs and disjunctive databases. New Generation Computing, 9, 365–385.

    Article  MATH  Google Scholar 

  • Gigerenzer, G., Todd, P. M., & The ABC Research Group. (1999). Simple heuristics that make us smart. Oxford: Oxford University Press.

  • Gigerenzer, G., Hertwig, R., & Pachur, T. (2011). Heuristics: The foundations of adaptive behavior. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Graves, A., Mohamed, A., & Hinton, G.E. (2013). Speech recognition with deep recurrent neural networks. CoRR arXiv:abs/1303.5778.

  • Halpern, J. (2005). Reasoning about uncertainty. Cambridge, MA: MIT Press.

    MATH  Google Scholar 

  • Hansen, J. (2006). Deontic logics for prioritized imperatives. Artificial Intelligence and Law, 14(1–2), 1–34.

    Google Scholar 

  • Haykin, S. (1999). Neural networks: A comprehensive foundation. Upper Saddle River: Prentice Hall.

    MATH  Google Scholar 

  • Horty, J. F. (1993). Deontic logic as founded on nonmonotonic logic. Annals of Mathematics and Artificial Intelligence, 9(1–2), 69–91.

    MathSciNet  Article  MATH  Google Scholar 

  • Jörgensen, J. (1937). Imperatives and logic. Erkenntnis, 7, 288–296.

    Google Scholar 

  • Juslin, P., Nilsson, Håkan, & Winman, A. (2009). Probability theory, not the very guide of life. Psychological Review, 116(4), 856–874.

    Article  Google Scholar 

  • Kahneman, D., & Tversky, A. (1982). The concept of probability in psychological experiments. In D. Kahneman, P. Slovic, & A. Tversky (Eds.), The concept of probability in psychological experiments (pp. 509–520). Cambridge: Cambridge University Press.

    Google Scholar 

  • Kern-Isberner, G., & Lukasiewicz, T. (2017). Many facets of reasoning under uncertainty, inconsistency, vagueness, and preferences: A brief survey. Künstliche Intelligenz. doi:10.1007/s13218-016-0480-6.

    Google Scholar 

  • Knight, F. (1921). Risk, uncertainty and profit. New York: Hart, Schaffner and Marx.

    Google Scholar 

  • Kowalski, R. A. (1988). The early years of logic programming. Communications of the ACM, 31, 38–42.

    Article  Google Scholar 

  • Kraus, S., Lehmann, D., & Magidor, M. (1990). Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44(1), 167–207.

    MathSciNet  Article  MATH  Google Scholar 

  • Lindahl, L., & Odelstad, J. (2003). Normative systems and their revision: An algebraic approach. Artificial Intelligence and Law, 11(2–3), 81–104.

    Article  Google Scholar 

  • Lloyd, J. W. (1987). Foundations of logic programming. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Makinson, D., & van der Torre, L. (2000). Input/output logics. Journal of Philosophical Logic, 29(4), 383–408.

    MathSciNet  Article  MATH  Google Scholar 

  • Makinson, D., & van der Torre, L. (2001). Constraints for input-output logics. Journal of Philosophical Logic, 30(2), 155–185.

    MathSciNet  Article  MATH  Google Scholar 

  • Makinson, D., & van der Torre, L. (2003a). Permissions from an input-output perspective. Journal of Philosophical Logic, 32(4), 391–416.

    MathSciNet  Article  MATH  Google Scholar 

  • Makinson, D., & van der, Torre L. (2003b). What is input/output logic? In B. Löwe, W. Malzkorn & T. Räsch (Eds.), Foundations of the formal sciences II: Applications of mathematical logic in philosophy and linguistics, trends in logic (Vol. 17). Kluwer.

  • McCarthy, J. (1980). Circumscription: A form of non-monotonic reasoning. Artificial Intelligence, 13(1), 27–39.

    MathSciNet  Article  MATH  Google Scholar 

  • Minsky, M. (1974). A framework for representing knowledge. Tech. Rep. 306, AI Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA.

  • Mousavi, S., & Gigerenzer, G. (2014). Risk, uncertainty, and heuristics. Journal of Business Research, 67, 1671–1678.

    Article  Google Scholar 

  • Mozina, M., Zabkar, J., & Bratko, I. (2007). Argument based machine learning. Artificial Intelligence, 171(10–15), 922–937.

    MathSciNet  Article  MATH  Google Scholar 

  • Nilsson, N. J. (1986). Probabilistic logic. Artificial intelligence, 28(1), 71–87.

    MathSciNet  Article  MATH  Google Scholar 

  • Nute, D. (1994). Defeasible logic. In D. Gabbay & J. Robinson (Eds.), Handbook of logic in artificial intelligence and logic programming (Vol. 3, pp. 353–396). Oxford: Oxford University Press.

    Google Scholar 

  • Nute, D. (Ed.). (1997). Defeasible deontic logic, synthese library (Vol. 263). Alphen aan den Rijn: Kluwer.

    MATH  Google Scholar 

  • Oaksford, M., & Chater, N. (1998). Rationality in an uncertain world: Essays in the cognitive science of human understanding. Hove: Psychology Press.

    Book  Google Scholar 

  • Pearl, J. (2000). Causality: Models, reasoning, and inferece. Cambridge: Cambridge University Press.

    MATH  Google Scholar 

  • Pijnacker, J., Geurts, B., van Lambalgen, M., Buitelaar, J., & Hagoort, P. (2010). Exceptions and anomalies: An ERP study on context sensitivity in autism. Neuropsychologia, 48, 2940–2951.

    Article  Google Scholar 

  • Pinosio, R. (in prep.) A common core shared by logic programming and probabilistic causal models.

  • Prakken, H., & Sergot, M. (1997). Dyadic deontic logic and contrary-to-duty obligations. In D. Nute (Ed.), Defeasible deontic logic (pp. 223–262). Berlin: Springer.

  • Rumelhart, D. E., Hinton, G. E., & Williams, R. J. (1986). Learning internal representations by error propagation. In D. Rumelhart, J. McClelland & PDP Research Group (Eds.), Parallel Distributed Processing (Vol 1. pp. 318–362). Cambridge: MIT Press.

  • Sen, S., & Airiau, S. (2007). Emergence of norms through social learning. In Procs. of the 20th International Joint Conference on Artificial Intelligence—IJCAI (pp. 1507–1512).

  • Shanahan, M. (2002). Reinventing Shakey. In J. Minker (Ed.), Logic-based artificial intelligence. Dordrecht: Kluwer.

    Google Scholar 

  • Shoham, Y. (1987). A semantical approach to non-monotonic logics. In Proceedings of the tenth international joint conference on artificial intelligence (IJCAI) (pp. 388–392).

  • Shoham, Y., & Tennenholtz, M. (1997). On the emergence of social conventions: Modeling, analysis, and simulations. Artificial Intelligence, 94(1–2), 139–166.

    Article  MATH  Google Scholar 

  • Sloman, S., & Lagnado, D. (2015). Causality in thought. The Annual Review of Psychology, 66, 1–25.

    Article  Google Scholar 

  • Stenning, K., & van Lambalgen, M. (2008). Human reasoning and Cognitive Science. Cambridge, MA: MIT Press.

    Google Scholar 

  • Stenning, K., & van Lambalgen, M. (2010). The logical response to a noisy world. In M. Oaksford (Ed.), Cognition and conditionals: Probability and logic in human thought (pp. 85–102). Oxford: Oxford University Press.

    Chapter  Google Scholar 

  • Stenning, K., & Varga, A. (2016). Many logics for the many things that people do in reasoning. In L. Ball & V. Thompson (Eds.), International Handbook of Thinking and Reasoning. Abingdon-on-Thames: Psychology Press.

    Google Scholar 

  • Stenning, K., Martignon, L., & Varga, A. (2017). Adaptive reasoning: integrating fast and frugal heuristics with a logic of interpretation. Decision.

  • Tosatto, S. C., Boella, G., van der Torre, L., & Villata, S. (2012). Abstract normative systems: Semantics and proof theory. In G. Brewka, T. Eiter, & S. A. McIlraith (Eds.), Principles of knowledge representation and reasoning: Proceedings of the thirteenth international conference. AAAI Press.

  • Towell, G. G., & Shavlik, J. W. (1994). Knowledge-based artificial neural networks. Artificial Intelligence, 70(1), 119–165.

    Article  MATH  Google Scholar 

  • Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185(4157), 1124–1131.

    Article  Google Scholar 

  • Tversky, A., & Kahneman, D. (1983). Extensional versus intuitive reasoning: The conjunction fallacy in probability judgment. Psychological Review, 90(4), 293.

    Article  Google Scholar 

  • van der Torre, L. (1997). Reasoning about obligations. PhD thesis, Erasmus University Rotterdam.

  • van der Torre, L., & Tan, Y. (1999). Deontic update semantics. In P. McNamara & H. Prakken (Eds.), Norms, logics and information systems. new studies on deontic logic and computer science. Amsterdam: IOS Press.

    Google Scholar 

  • van der Torre, L. (2010). Deontic redundancy: A fundamental challenge for deontic logic. In Deontic Logic in Computer Science, 10th International Conference ( \(\Delta\) EON 2010).

  • van Lambalgen, M., & Hamm, F. (2004). The proper treatment of events. Oxford: Blackwell.

    Google Scholar 

  • Varga, A. (2013). A formal model of infants’ acquisition of practical knowledge from observation. PhD thesis, Central European University, Budapest.

  • von Wright, G. H. (1951). Deontic logic. Mind, 60, 1–15.

    Article  Google Scholar 

  • Weston, J., Chopra, S., & Bordes, A. (2014). Memory networks. CoRR arXiv:abs/1410.3916.

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Acknowledgements

We want to thank the following people for their indispensable contributions to different parts of the work reported in this article: Guido Boella, Silvano Colombo Tosatto, Valerio Genovese, Laura Martignon, Alan Perotti, and Alexandra Varga.

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Correspondence to Tarek R. Besold.

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Besold, T.R., Garcez, A.d., Stenning, K. et al. Reasoning in Non-probabilistic Uncertainty: Logic Programming and Neural-Symbolic Computing as Examples. Minds & Machines 27, 37–77 (2017). https://doi.org/10.1007/s11023-017-9428-3

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Keywords

  • Uncertainty in reasoning
  • Interpretation
  • Logic programming
  • Dynamic norms
  • Neural-symbolic integration