1. How can optimality of a found plan be verified in classical planning?
In classical planning, the optimality of a found plan can be verified using certifying algorithms. These algorithms provide a machine-readable certificate proving the validity of the output. The validity of the triple <input, output, certificate> can be checked by an independent verifier. The most commonly used proof formalisms for certifying algorithms are based on reverse asymmetric tautology, such as DRAT and LRAT. For classical planning, the concept of certifying algorithms is applied to verify the optimality of a found plan by ensuring that the cost of the plan is a lower bound for the plan cost. Two types of certificates are presented for verifying lower bounds: one reduces optimality to unsolvability, allowing the use of existing unsolvability certificates, and the other constructs native optimality certificates that directly reason about lower bounds. Certificates should have properties such as soundness, completeness, efficient generation and verification, and general applicability to commonly used planning techniques. Theoretical analysis, empirical studies, and verification of optimal plans are conducted to support these approaches.
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2. What is the STRIPS formalism and how is it defined?
The STRIPS formalism, introduced by Fikes and Nilsson in 1971, is a planning approach used to define tasks in the context of artificial intelligence. A task is represented as a tuple P = <V, A, I, G>, where V is a finite set of propositional variables, A is a finite set of actions, I is the initial state, and G is the goal description. The set of all states in a planning task P is the power set of V, denoted as S. The goal description G V defines the set of goal states as S G = {s S | G s}. An action in A is defined by preconditions, add effects, delete effects, and cost. The maximal action cost in A is denoted as c max (A), and actions with cost i are defined as A A|i = {a A | cost(a) = i}. An action a is applicable in state s if its preconditions are satisfied in s. Applying an action a to state s yields its successor state s a = (s \ del (a)) add (a). A sequence of actions, or path, p = <a 1, . . ., a n>, leads to state s p = s a 1 . . .a n, where each action a i must be applicable in the state s a 1 . . .a i-1. A state s' is reachable from state s if there exists a path p such that s p = s'. A plan for state s is a sequence of actions that leads to the goal state G s p. A plan is optimal if there is no plan with a lower cost. Proof systems facilitate the construction of formal proofs by defining how initial knowledge can be acquired and expanded. The STRIPS formalism is used in proof systems to reason about sets of states and actions, and to derive knowledge about the deadness of state sets and subset relations.
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3. What is the g value in blind search algorithms?
In blind search algorithms, the g value annotates each state s seen with the cost of the cheapest path from I to s. It is used to expand states in ascending order of their g value. The algorithm terminates upon expanding a goal state, at which point it is guaranteed that the found solution has optimal cost c(P) and that all seen states with g(s) < c(P) were expanded. While blind search does not reason about cost from state s to the goal, we can infer a lower bound based on the g value and the optimal plan cost, namely gc(s) >= c(P) - g(s).
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4. What are the performance differences between compilation-based and optimality proof system-based approaches in generating certificates?
The performance differences between compilation-based and optimality proof system-based approaches in generating certificates can be observed in terms of success rates and resource constraints. Compilation-based approaches, such as u-h max and u-h M&S, perform significantly worse compared to optimality proof system-based approaches like o-blind and o-h max. The success rates for creating certificates are 88% and 82% for o-blind and o-h max respectively, while compilation-based approaches have lower success rates. Additionally, compilation-based approaches have more forgiving resource constraints, but they still experience a significant drop in success rates. The optimality proof system-based configurations can create certificates for a higher percentage of tasks solved by base. In terms of verification, o-h max only verifies 86% of created certificates, while o-blind reaches 98%. The certificate size and complexity play a role in these results, with o-h max requiring sub-certificies for heuristic values, while compilation-based approaches only need them for dead-ends. Overall, the optimality proof system-based approaches show better performance in generating and verifying certificates compared to compilation-based approaches.
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