Abstract
The recognition that human minds/brains are finite systems with limited resources for computation has led researchers in cognitive science to advance the Tractable Cognition thesis: Human cognitive capacities are constrained by computational tractability. As also human-level AI in its attempt to recreate intelligence and capacities inspired by the human mind is dealing with finite systems, transferring this thesis and adapting it accordingly may give rise to insights that can help in progressing towards meeting the classical goal of AI in creating machines equipped with capacities rivaling human intelligence. Therefore, we develop the “Tractable Artificial and General Intelligence Thesis” and corresponding formal models usable for guiding the development of cognitive systems and models by applying notions from parameterized complexity theory and hardness of approximation to a general AI framework. In this chapter we provide an overview of our work, putting special emphasis on connections and correspondences to the heuristics framework as recent development within cognitive science and cognitive psychology.
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The corresponding proofs of the respective results can be found in Robere and Besold (2012). Moreover, in the theorem statements W[1] refers to the class of problems solvable by constant depth combinatorial circuits with at most 1 gate with unbounded fan-in on any path from an input gate to an output gate. In parameterized complexity, the assumption W[1] ≠ FPT can be seen as analogous to P ≠ NP.
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On the other hand, considering more restrictive notions than APX as, for instance, PTAS (the class of problems for which there exists a polynomial-time approximation scheme, i.e., an algorithm which takes an instance of a optimization problem and a parameter ε > 0 and, in polynomial time, solves the problem within a factor 1 +ε of the optimal solution) does not seem meaningful to us either, as also human satisficing does not approximate optimal solutions up to an arbitrary degree but in experiments normally yields rather clearly defined cut-off points at a certain approximation level.
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For reasons unclear to the authors this perspective seems to be more widespread and far deeper rooted in AI and cognitive systems research than in (theoretical) cognitive science and cognitive modeling where complexity analysis and formal computational analysis in general by now have gained a solid foothold.
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Here we presuppose that cognitive capacities can be seen as information processing systems. Still, this seems to be a fairly unproblematic claim, as it simply aligns cognitive processes with computations processing incoming information (e.g., from sensory input) and resulting in a certain output (e.g., a certain behavioral or mental reaction) dependent on the input.
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Fortunately, this way of conceptualizing a cognitive capacity naturally links to research in artificial cognitive systems. When trying to build a system modeling one or several selected cognitive capacities, we consider a general set of inputs (namely all scenarios in which a manifestation of the cognitive capacity can occur) which we necessarily formally characterize—although maybe only implicitly—in order to make the input parsable for the system, hypothesize a function mapping inputs onto outputs (namely the computations we have the system apply to the inputs) and finally obtain a well-characterized set of outputs (namely all the outputs our system can produce given its programming and the set of inputs).
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In discussions with researchers working in AI and cognitive systems very occasionally critical feedback relating to the choice of FPT, APX, and FPA as reference classes has been given, as these have (curiously enough) been perceived as too less restrictive. Harshly contrasting with the previously discussed criticism it was argued that human-level cognitive processing should be of linear complexity or less. Still, we do not see a problem here: Neither are we fundamentalist about this precise choice of upper boundaries, nor do we claim that these are the only meaningfully applicable ones. Nonetheless, we decided for them because they can quite straightforwardly be justified and are backed up by close correspondences with other relevant notions from theoretical and practical studies in cognitive science and AI.
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Of course this also explicitly includes the case in which the considered classes are conceptually not restricted to the rather coarse-grained hierarchy used in “traditional” complexity theory, but if also the significantly finer and more subtle possibilities of class definition and differentiation introduced by parametrized complexity theory and other recent developments are taken into account.
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Besold, T.R., Robere, R. (2016). When Thinking Never Comes to a Halt: Using Formal Methods in Making Sure Your AI Gets the Job Done Good Enough. In: Müller, V.C. (eds) Fundamental Issues of Artificial Intelligence. Synthese Library, vol 376. Springer, Cham. https://doi.org/10.1007/978-3-319-26485-1_4
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