# Polynomial-Space Approximation of No-Signaling Provers

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• Jonas Kvarnström Automated Planning Group Department of Computer and Information Science Linköping University
Jonas Kvarnström Automated Planning Group Department of Computer and Information Science Linköping University What is the complexity of plan generation? . How much time and space (memory) do we need? Vague question – let’s try to be more specific… . Assume an infinite set of problem instances ▪ For example, “all classical planning problems” . Analyze all possible algorithms in terms of asymptotic worst case complexity . What is the lowest worst case complexity we can achieve? . What is asymptotic complexity? ▪ An algorithm is in O(f(n)) if there is an algorithm for which there exists a fixed constant c such that for all n, the time to solve an instance of size n is at most c * f(n) . Example: Sorting is in O(n log n), So sorting is also in O( ), ▪ We can find an algorithm in O( ), and so on. for which there is a fixed constant c such that for all n, If we can do it in “at most ”, the time to sort n elements we can do it in “at most ”. is at most c * n log n Some problem instances might be solved faster! If the list happens to be sorted already, we might finish in linear time: O(n) But for the entire set of problems, our guarantee is O(n log n) So what is “a planning problem of size n”? Real World + current problem Abstraction Language L Approximation Planning Problem Problem Statement Equivalence P = (, s0, Sg) P=(O,s0,g) The input to a planner Planner is a problem statement The size of the problem statement depends on the representation! Plan a1, a2, …, an PDDL: How many characters are there in the domain and problem files? Now: The complexity of PLAN-EXISTENCE .
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• EXPSPACE-Hardness of Behavioural Equivalences of Succinct One
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