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Formally Verified Algorithms for Upper-Bounding State Space Diameters

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Abstract

A completeness threshold is required to guarantee the completeness of planning as satisfiability, and bounded model checking of safety properties. We investigate completeness thresholds related to the diameter of the underlying transition system. A valid threshold, the diameter is the maximum element in the set of lengths of all shortest paths between pairs of states. The diameter is not calculated exactly in our setting, where the transition system is succinctly described using a (propositionally) factored representation. Rather, an upper bound on the diameter is calculated compositionally, by bounding the diameters of small abstract subsystems, and then composing those. We describe our formal verification in HOL4 of compositional algorithms for computing a relatively tight upper bound on the system diameter. Existing compositional algorithms are characterised in terms of the problem structures they exploit, including acyclicity in state-variable dependencies, and acyclicity in the state space. Such algorithms are further distinguished by: (1) whether the bound calculated for abstractions is the diameter, sublist diameter or recurrence diameter, and (2) the “direction” of traversal of the compositional structure, either top-down or bottom-up. As a supplement, we publish our library—now over 14k lines—of HOL4 proof scripts about transition systems. That shall be of use to future related mechanisation efforts, and is carefully designed for compatibility with hybrid systems.

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Notes

  1. An abstraction’s state space corresponds to a minor of the digraph modelling the state space of the concrete system. A minor is derived by a sequence of vertex contractions, as well as edge and vertex deletions. Vertex contraction of two vertices is an operation that replaces the two vertices with one new vertex that inherits their incoming and outgoing edges.

  2. This representation is equivalent to representations commonly used in the model checking and AI planning communities (e.g. STRIPS by Fikes and Nilsson [29] and SMV by McMillan [40]). Whereas conventional expositions in the planning and model-checking literature would also define initial conditions and goal/safety criteria, here we omit those features from discussion. Our bounds and the different topological properties we consider are independent of those features.

  3. Our definition is equivalent to those in [35, 55] in the context of AI planning.

  4. The definition of \( {\rightarrow } \) has an implicit \( \delta \) parameter; however, we hide it using the ad hoc overloading ability in HOL.

  5. \(\textsf {N}\) has \( \delta \) and parameters, but we hide them with HOL’s ad hoc overloading ability.

  6. Note that we use HOL4’s overloading capacity to hide the \(\delta \) relative to whose domain the complement of \( vs \) is taken, i.e. \({\mathcal {D}}(\delta ) \setminus vs \) is written as \(\overline{ vs }\).

  7. \(\textsf {S}\) has \( vs \), \( \delta \) and \( \overset{\rightarrow }{x} \) parameters, but we hide them with HOL’s ad hoc overloading ability.

  8. In the rest of this proof we omit the \( vs \) and/or \(\delta \) arguments from \(\partial (,,)\) and \(\textsf {S}\) as they do not change.

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Acknowledgements

We thank Daniel Jackson for suggesting applying diameter upper-bounding on the hotel key protocol verification. We also thank Dr. Alban Grastien and Dr. Patrik Haslum for their the very helpful discussions and insightful feedback which they gave us through the entire project. Additionally, we thank the anonymous reviewers for their detailed and helpful reviews. Lastly, we would like to note that the first author is supported by the DFG Koselleck Grant NI 491/16-1.

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Funding was provided by Commonwealth Scientific and Industrial Research Organisation.

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Authors and Affiliations

  1. Technical University of Munich, Munich, Germany

    Mohammad Abdulaziz

  2. Australian National University, Canberra, Australia

    Michael Norrish & Charles Gretton

  3. Data61 Canberra Research Laboratory, Canberra, Australia

    Michael Norrish

  4. Griffith University, Brisbane, Australia

    Charles Gretton

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  1. Mohammad Abdulaziz
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Correspondence to Mohammad Abdulaziz.

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Abdulaziz, M., Norrish, M. & Gretton, C. Formally Verified Algorithms for Upper-Bounding State Space Diameters. J Autom Reasoning 61, 485–520 (2018). https://doi.org/10.1007/s10817-018-9450-z

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