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Easy Solutions for a Hard Problem? The Computational Complexity of Reciprocals with Quantificational Antecedents

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Abstract

We report two experiments which tested whether cognitive capacities are limited to those functions that are computationally tractable (PTIME-Cognition Hypothesis). In particular, we investigated the semantic processing of reciprocal sentences with generalized quantifiers, i.e., sentences of the form Q dots are directly connected to each other, where Q stands for a generalized quantifier, e.g. all or most. Sentences of this type are notoriously ambiguous and it has been claimed in the semantic literature that the logically strongest reading is preferred (Strongest Meaning Hypothesis). Depending on the quantifier, the verification of their strongest interpretations is computationally intractable whereas the verification of the weaker readings is tractable. We conducted a picture completion experiment and a picture verification experiment to investigate whether comprehenders shift from an intractable reading to a tractable reading which should be dispreferred according to the Strongest Meaning Hypothesis. The results from the picture completion experiment suggest that intractable readings occur in language comprehension. Their verification, however, rapidly exceeds cognitive capacities in case the verification problem cannot be solved using simple heuristics. In particular, we argue that during verification, guessing strategies are used to reduce computational complexity.

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Notes

  1. A clarification may be in order: when we speak of intractable problems we always refer to NP-hard problems and tacitly assume that \(\mathsf{P}\ne \mathsf{NP}\) (for an introduction to computational complexity see Arora and Barak 2009).

  2. There are different treatments of quantified reciprocals in the literature. Among others, Beck (2001) gave an analysis of reciprocals in terms of plural semantics (see also Langendoen 1978; Heim et al. 1991; Sternefeld 1998, among others). A clear advantage of Beck’s approach is that it is strictly compositional. Compositionality is, however, irrelevant for the present purpose because we are mainly interested in the truth conditions of complete sentences. Note that Beck’s account, for example, predicts truth conditions which are equivalent to the ones in (2).

  3. More precisely, this is the case if there are at least two elements in the quantificational restriction. Dalrymple et al. (1998) stated this condition explicitly in their truth conditions. With upward entailing quantifiers this is, however, often irrelevant, because the quantificational antecedent requires a restriction of at least two elements—either because of its semantics or on pragmatic grounds.

  4. Since complexity theory is concerned with the resource demands that are inherent to a computational problem, but independent of any concrete implementation, it is clearly concerned with Marr’s (1982) computational level.

  5. Strictly speaking, the treatment effect of a factor with three levels corresponds to two fixed effects \(\beta _1,\, \beta _2\) and two random slopes \(S_{1s}, \,S_{2s}\). This will become clear in the discussion of the results below.

  6. In fact, Szymanik (2010) even restricted \(Ram_S,\, Ram_I,\, Ram_W\) to upward entailing quantifiers.

  7. A simple proof of the NP-hardness of MAXIMUM CLIQUE SIZE proceeds via a polynomial-time reduction from 3SAT (see e.g. Garey and Johnson 1979, p. 46). This reduction is completely parallel to the one used by Arora and Barak (2009, p. 51) to proof NP-completeness of INDSET.

  8. The experiment included eight more conditions, i.e. reciprocal sentences with the antecedent most. Originally, we expected most to trigger the scalar implicature not all, which would have allowed us to compare it to all but one and exactly k. The results indicated that this was not the case, or rather, that the implicature could easily be canceled. In the false graph condition with four dots, for instance, we observed more than 60 % acceptance making it impossible to properly analyze most-reciprocals. We therefore decided not to report them in the present paper. For future research it would be interesting to test complex non-monotone quantifiers of the type most but not all which would obviously constitute excellent, albeit rather complex test cases for the PCH.

  9. No interactions could be included into the random slopes of subjects because these models failed to converge. So, the model in (13) was the model with the maximal random effects structure that could be fitted.

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Acknowledgments

This research was funded by the German Research Foundation in the Project B1 Incrementality of Semantic Interpretation of the Collaborative Research Center 833 The Construction of Meaning. The authors would like to thank Fritz Hamm, Nina Gierasimczuk, Janina Radó, Iris van Rooij, Wolfgang Sternefeld, Jakub Szymanik, Rineke Verbrugge and two anonymous reviewers for valuable comments and discussions on earlier drafts of the paper. We would also like to thank our reviewers and audiences at the International Workshop on Computational Semantics 9, the International Conference on Linguistic Evidence 2012 and the ESSLI workshop on Logic and Cognition where we have presented parts of the present work and work closely related to this paper. For their assistance in data collection and preparation of the experimental materials we would like to thank Anna Pryslopska and Aysenur Sarcan. Last but not least, we would like to thank Udo Klein for distributing questionnaires in one of his introductory courses.

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  1. Collaborative Research Center 833, Tübingen University, Nauklerstraße 35, 72074 , Tübingen, Germany

    Fabian Schlotterbeck & Oliver Bott

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Correspondence to Fabian Schlotterbeck.

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Schlotterbeck, F., Bott, O. Easy Solutions for a Hard Problem? The Computational Complexity of Reciprocals with Quantificational Antecedents. J of Log Lang and Inf 22, 363–390 (2013). https://doi.org/10.1007/s10849-013-9181-9

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