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Register a new userParameterized Complexity Results for a Model of Theory of Mind Based on Dynamic Epistemic Logic
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Publication date
2016
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Informatics Engineering
and research's language is
English
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In this paper we introduce a computational-level model of theory of mind (ToM) based on dynamic epistemic logic (DEL), and we analyze its computational complexity. The model is a special case of DEL model checking. We provide a parameterized complexity analysis, considering several aspects of DEL (e.g., number of agents, size of preconditions, etc.) as parameters. We show that model checking for DEL is PSPACE-hard, also when restricted to single-pointed models and S5 relations, thereby solving an open problem in the literature. Our approach is aimed at formalizing current intractability claims in the cognitive science literature regarding computational models of ToM.
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Description logics are knowledge representation languages that have been designed to strike a balance between expressivity and computational tractability. Many different description logics have been developed, and numerous computational problems for these logics have been studied for their computational complexity. However, essentially all complexity analyses of reasoning problems for description logics use the one-dimensional framework of classical complexity theory. The multi-dimensional framework of parameterized complexity theory is able to provide a much more detailed image of the complexity of reasoning problems. In this paper we argue that the framework of parameterized complexity has a lot to offer for the complexity analysis of description logic reasoning problems---when one takes a progressive and forward-looking view on parameterized complexity tools. We substantiate our argument by means of three case studies. The first case study is about the problem of concept satisfiability for the logic ALC with respect to nearly acyclic TBoxes. The second case study concerns concept satisfiability for ALC concepts parameterized by the number of occurrences of union operators and the number of occurrences of full existential quantification. The third case study offers a critical look at data complexity results from a parameterized complexity point of view. These three case studies are representative for the wide range of uses for parameterized complexity methods for description logic problems.
The paper analyzes dynamic epistemic logic from a topological perspective. The main contribution consists of a framework in which dynamic epistemic logic satisfies the requirements for being a topological dynamical system thus interfacing discrete dynamic logics with continuous mappings of dynamical systems. The setting is based on a notion of logical convergence, demonstratively equivalent with convergence in Stone topology. Presented is a flexible, parametrized family of metrics inducing the latter, used as an analytical aid. We show maps induced by action model transformations continuous with respect to the Stone topology and present results on the recurrent behavior of said maps.
The standard semantics of multi-agent epistemic logic $S5$ is based on Kripke models whose accessibility relations are reflexive, symmetric and transitive. This one dimensional structure contains implicit higher-dimensional information beyond pairwise interactions, that has been formalized as pure simplicial models in previous work from the authors. Here we extend the theory to encompass all simplicial models - including the ones that are not pure. The corresponding Kripke models are those where the accessibility relation is symmetric and transitive, but might not be reflexive. This yields the epistemic logic $KB4$ which can reason about situations where some of the agents may die.
In this paper we study the solvability of the equality negation task in a simple wait-free model where processes communicate by reading and writing shared variables or exchanging messages. In this task, two processes start with a private input value in the set {0,1,2}, and after communicating, each one must decide a binary output value, so that the outputs of the processes are the same if and only if the input values of the processes are different. This task is already known to be unsolvable; our goal here is to prove this result using the dynamic epistemic logic (DEL) approach introduced by Goubault, Ledent and Rajsbaum in GandALF 2018. We show that in fact, there is no epistemic logic formula that explains why the task is unsolvable. We fix this issue by extending the language of our DEL framework, which allows us to construct such a formula, and discuss its utility.
The elimination distance to some target graph property P is a general graph modification parameter introduced by Bulian and Dawar. We initiate the study of elimination distances to graph properties expressible in first-order logic. We delimit the problems fixed-parameter tractability by identifying sufficient and necessary conditions on the structure of prefixes of first-order logic formulas. Our main result is the following meta-theorem: for every graph property P expressible by a first order-logic formula phiin Sigma_3, that is, of the form phi=exists x_1exists x_2cdots exists x_r forall y_1forall y_2cdots forall y_s exists z_1exists z_2cdots exists z_t psi, where psi is a quantifier-free first-order formula, checking whether the elimination distance of a graph to P does not exceed k, is fixed-parameter tractable parameterized by k. Properties of graphs expressible by formulas from Sigma_3 include being of bounded degree, excluding a forbidden subgraph, or containing a bounded dominating set. We complement this theorem by showing that such a general statement does not hold for formulas with even slightly more expressive prefix structure: there are formulas phiin Pi_3, for which computing elimination distance is W[2]-hard.
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