Interpreting tractable versus intractable reciprocal sentences

In three experiments, we investigated the computational complexity of German reciprocal sentences with different quantificational antecedents. Building upon the tractable cognition thesis (van Rooij, 2008) and its application to the verification of quantifiers (Szymanik, 2010) we predicted complexity differences among these sentences. Reciprocals with all-antecedents are expected to preferably receive a strong interpretation (Dalrymple et al., 1998), but reciprocals with proportional or numerical quantifier antecedents should be interpreted weakly. Experiment 1, where participants completed pictures according to their preferred interpretation, provides evidence for these predictions. Experiment 2 was a picture verification task. The results show that the strong interpretation was in fact possible for tractable all but one-reciprocals, but not for exactly n. The last experiment manipulated monotonicity of the quantifier antecedents. Formal semantics hasn’t paid much attention to issues of computational complexity when the meaning of an expression is derived. However, when it comes to semantic processing in humans (and computers) with limited processing resources, computational tractability becomes one of the most important constraints a cognitively realistic semantics must face. Two consequences come to mind immediately. If there is a choice between algorithms, we should choose tractable ones over intractable ones. And secondly, meanings which cannot be effectively computed shouldn’t be posited for natural language expressions. In this paper we present three psycholinguistic experiments investigating the latter aspect. Following traditions in computer science, a number of cognitive scientists have defined computational tractability as polynomial-time-computability (for an overview see van Rooij, 2008) leading to the P-Cognition Hypothesis (PCH): cognitive capacities are limited to those functions that can be computed in polynomial time. These functions are input-output functions in the sense of Marr (1982)’s first level. One objection against the PCH is that computational complexity is defined in terms of limit behavior as the input increases. In practice, however, the input may be rather small. van Rooij (2008) points out that the input size can be parametrized turning a problem that is intractable for a large input size into a tractable one for small inputs. We manipulated the input size in an experiment to test this more refined version of the PCH. An interesting test case for the PCH are quantified sentences containing reciprocal expressions of the form Q of the As R each other. Consider (1-a) – (1-c).