## Abstract

The plausibility of so-called ‘rational explanations’ in cognitive science is often contested on the grounds of computational intractability. Some have argued that intractability is a pseudoproblem, however, because cognizers do not actually perform the rational calculations posited by rational models; rather, they only behave *as if* they do. Whether or not the problem of intractability is dissolved by this gambit critically depends, inter alia, on the semantics of the ‘as if’ connective. First, this paper examines the five most sensible explications in the literature, and concludes that none of them actually circumvents the problem. Hence, rational ‘as if’ explanations must obey the minimal computational constraint of tractability. Second, this paper describes how rational explanations could satisfy the tractability constraint. Our approach suggests a computationally unproblematic interpretation of ‘as if’ that is compatible with the original conception of rational analysis.

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## Notes

- 1.
Note that Marr saw ‘what’-questions as intimately tied to ‘why’-questions: e.g., why is \(f\) the appropriate function for \(\phi \) to realize? An answer to such questions would presuppose a specification of the conditions for appropriateness. We’ll later return to this point when addressing rational explanation.

- 2.
Such cases of underdetermination of theory by evidence themselves do not determine any stronger antirealist conclusion. An alternative lesson to be drawn is just that cognitive scientists taking top-down explanatory approaches may benefit from theoretical constraints on computational-level theories (van Rooij 2008).

- 3.
A common presumption is that such \(f\)s not only characterize the ‘what’ of a capacity \(\phi \), i.e., \(f\) itself, but also provide the grounds for understanding why \(f\) is what characterizes \(\phi \). Of course, to presume that \(f\) is what \(\phi \) does because \(f\) is optimal for the goals \(G\) of \(\phi \) in environment \(E\) is not to claim that \(f\) itself explains why \(f\) is the function that characterizes \(\phi \). Rather, the ‘why’ explanation is a narrative that \(f\) is as it is because of the principle of rationality and the assumptions that \(\phi \) has goals \(G\) and the environment of adaptation is \(E\). Moreover, to genuinely be a ‘why’ explanation, rational analysts should not only show that \(f\) is optimal for \(G\) and \(E\), but also that \(f\) is as it is

*because*it is optimal for \(G\) and \(E\) (Danks 2008). Because this difference between \(f\) as a computational-level description of \(\phi \) and the rational narrative surrounding and motivating \(f\) is seldomly made explicit, cognitive scientists are sometimes led to the mistaken idea that \(f\)s derived via rational analysis are rational explanations in the sense of explaining the ‘why’ of \(f\). In our view, the distinction should be made explicit and the ‘why’ explanation should be differentiated from the computational-level \(f\) itself (cf. Danks 2013). After all, even if the rational narrative were falsified, \(f\) could still correctly characterize \(\phi \); the correctness of that characterization is independent of the truth-value of the rational story about how \(f\) came to be. - 4.
NP-hard functions owe their name to being as hard as any function in the class NP, where ‘NP’ abbreviates ‘

*N*ondeterministic*P*olynomial time’. - 5.
- 6.
This point cannot be overemphasized. The classification of a function \(f\) as intractable does not come lightly. For even were there to exist a proof that some such tractable algorithm exists, then even if we could know nothing else about it, we would be led to the classification of \(f\) as tractable.

- 7.
Note that, here, Chater and Oaksford do use the term ‘calculation’ to refer to the computational process involved in determining the output of some \(\phi \), in this case perceptual organization. Perhaps ‘calculation’ is intended in a broad sense, meaning ‘computation’ and not being synonymous with ‘rational calculation’.

- 8.
- 9.
For instance, it may be believed that \(f(i)\) and \(f_H(i)\) are the same for many inputs \(i \in I = I_H\), or it may be conjectured that the difference between \(f(i)\) and \(f_H(i)\) is small for many inputs \(i \in I\).

- 10.
- 11.
We thank an anonymous reviewer for raising this possibility: even if it is \(f_H\) rather than \(f\) that accurately characterizes the capacity of interest \(\phi \), in practice \(f_H\) may be unknown; so couldn’t rational analysts contend that the appeal to ‘as if’ merely serves as a sort of ‘promissory note’? That is, until we determine what \(f_H\) is, \(f\) can serve as a (instrumentalist) working hypothesis that allows research to productively continue. This may be the case. We don’t contest that intractable \(f\)s can instrumentally lead to important results on occasion. For instance, postulating intractable \(f\)s raises scientifically fruitful questions about the conditions under which those functions may be tractable, and answering those questions may lead to (realist) hypotheses about \(f_H\). What we do contest is the claim that the intractability of \(f\) is rendered permanently irrelevant by an appeal to ‘as if’. Whatever instrumentalist commitments are invoked, it’s still the case that, in order for the computational-level theory to be computationally plausible and explanatory, at some point in time—sooner or later—\(f_H\) needs to be determined.

- 12.
- 13.
A function \(f\) is self-paddable if and only if a set of instances \(i_1, i_2,\ldots , i_m\) of \(f\) can be embedded in a single instance \(i_E\) of \(f\) such that \(f(i_1), f(i_2),\ldots ,f(i_m)\) can be derived from \(f(i_E)\). For more details, see Definition 6 in van Rooij and Wareham (2012).

- 14.
The formal tools for putting this type of revisionary approach into practice have been extensively described by van Rooij (2008) (see also Blokpoel et al. 2013; van Rooij and Wareham 2008), and builds on the the mathematical theory of parameterized complexity (Downey and Fellows 1999). Using proof techniques from this mathematical theory, it can be shown that some intractable (NP-hard) functions \(f: I \rightarrow O\) can be computed in fixed-parameter (fp-) tractable time \(O(g(K)|i|^c)\), i.e., where \(g\) can be any function of the parameters \(k_1, k_2,\ldots , k_m\) in set \(K = \{k_1, k_2,\ldots , k_m\}\), \(|i|\) denotes the input size, and \(c\) is a constant. Note that in such event, the intractable \(f\) can be computed efficiently (in polynomial-time), even for large inputs, provided the assumption that \(f\) operates only on inputs in which the parameters in \(K\) are restricted to relatively small values (each \(k << |i|\)). If rational analysts were to have theoretical and/or empirical reasons for this assumption, then revising \(f\) to \(f'\)—where \(f'\) is \(f\) restricted to inputs with small values for parameter \(k_1, k_2,\ldots , k_m\)—would yield a tractable function \(f'\) that will be rational according to the rational analysis that yielded it as \(f\).

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van Rooij, I., Wright, C.D., Kwisthout, J. *et al.* Rational analysis, intractability, and the prospects of ‘as if’-explanations.
*Synthese* **195, **491–510 (2018). https://doi.org/10.1007/s11229-014-0532-0

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### Keywords

- Psychological explanation
- Rational analysis
- Computational-level theory
- Intractability
- NP-hard
- Approximation