Abstract
An important paradigm in modeling the complexity of mathematical tasks relies on computational complexity theory, in which complexity is measured through the resources (time, space) taken by a Turing machine to carry out the task. These complexity measures, however, are asymptotic and as such potentially a problematic fit when descriptively modeling mathematical tasks that involve small inputs. In this paper, we argue that empirical data on human arithmetical cognition implies that a more fine-grained complexity measure is needed to accurately study mental arithmetic tasks. We propose a computational model of mental integer addition that is sensitive to the relevant aspects of human arithmetical ability. We show that this model necessitates a two-part complexity measure, since the addition tasks consists of two qualitatively different stages: retrieval of addition facts and the (de)composition of multidigit numbers. Finally, we argue that the two-part complexity measure can be developed into a single response-time measure with the help of empirical study of the two stages.
Highlights
An important paradigm in modeling the complexity of mathematical tasks relies on computational complexity theory, in which complexity is measured through the resources taken by a Turing machine to carry out the task
One approach for introducing such measures is based on the notion of computational complexity taken in theoretical computer science, in which one estimates the resources needed by a Turing machine to solve the problem via a given method
The only thing that truly increases with length of input in the way codified by standard complexity theory is the number of composition steps
Summary
In order to give a more detailed complexity measure for addition problems we need a different setup for the units used in the complexity measure. Standard complexity theory agrees with that intuition, since the input for the first is much longer than that for the second This is reflected in our complexity measure, though only in the number of composition steps. The only thing that truly increases with length of input in the way codified by standard complexity theory is the number of composition steps. This touches on an ongoing debate about the reason carries are more complex By formalizing it as an extra addition step our measure suggests that this is a categorical difference: 18 + 13 is more complex than 18 + 11 not just because the numbers involved are larger (known as the problem-size effect) but because the carry operation itself requires more work. Those effects could combine. 65 + 36 has complexity (4, 3) because it requires both the extra addition 6 + 3 + 1 and an extra composition step
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