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We present a theory, and its computer implementation, of how mental simulations underlie the abductions of informal algorithms and deductions from these algorithms. Three experiments tested the theory's predictions, using an environment of a single railway track and a siding. This environment is akin to a universal Turing machine, but it is simple enough for nonprogrammers to use. Participants solved problems that required use of the siding to rearrange the order of cars in a train (experiment 1). Participants abduced and described in their own words algorithms that solved such problems for trains of any length, and, as the use of simulation predicts, they favored "while-loops" over "for-loops" in their descriptions (experiment 2). Given descriptions of loops of procedures, participants deduced the consequences for given trains of six cars, doing so without access to the railway environment (experiment 3). As the theory predicts, difficulty in rearranging trains depends on the numbers of moves and cars to be moved, whereas in formulating an algorithm and deducing its consequences, it depends on the Kolmogorov complexity of the algorithm. Overall, the results corroborated the use of a kinematic mental model in creating and testing informal algorithms and showed that individuals differ reliably in the ability to carry out these tasks.

This article reports investigations of inferences that depend both on connectives between clauses, such as or else, and on relations between entities, such as in the same place as. Participants made more valid inferences from biconditionals--for instance, Ann is taller than Beth if and only if Beth is taller than Cath--than from exclusive disjunctions (Exp. 1). They made more valid transitive inferences from a biconditional when a categorical premise affirmed rather than denied one of its clauses, but they made more valid transitive inferences from an exclusive disjunction when a categorical premise denied rather than affirmed one of its clauses (Exp. 2). From exclusive disjunctions, such as either Ann is not in the same place as Beth or else Beth is not in the same place as Cath, individuals tended to infer that all three individuals could be in different places, whereas in fact this was impossible (Exps. 3a and 3b). The theory of mental models predicts all of these results.

People who have no experience with programming can create informal programs to rearrange the order of cars in trains. To find out whether they rely on kinematic mental simulations, the current studies examined participants' eye movements in two experiments in which participants performed various moves and rearrangements on a railway consisting of a main track running from left to right and a siding entered from and exited to the left track. In Experiment 1, they had to imagine different sorts of single moves of cars on the railway. The sequences of their fixations resembled iconic gestures: they tended to look at the starting location of the imagined move, and then at its final location. In Experiment 2, the task was to create descriptions of how to solve four sorts of rearrangements that differ in their Kolmogorov complexity. It predicted the time to find the correct solution and the relative number and duration of fixations recorded during the description of each move for rearrangements of different complexity. Participants were more likely to fixate on the symbols on the cars than anything else, and they fixated longer when the rearrangement was more difficult. They also tended to fixate regions of the tracks where a car's movement began or ended, as if they were imagining a car moving along the tracks. The results suggest that humans rely on a kinematic mental simulation when creating informal algorithms.