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A Logic-Based Approach to Incremental Reasoning on Multi-agent Systems

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Modeling, Dynamics, Optimization and Bioeconomics IV (ICABR 2017, DGS 2018)

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Abstract

We introduce the notion of strongly distributed multi-agent systems and present a uniform approach to incremental automated reasoning on them. The approach is based on systematic use of two logical reduction techniques: Feferman-Vaught reductions and syntactically defined translation schemes. The distributed systems are presented as logical structures \(\mathcal{A}\)’s. We propose a uniform template for methods, which allow for certain cost evaluation of formulae of logic \({\mathcal L}\) over \(\mathcal{A}\) from values of formulae over its components and values of formulae over the index structure \(\mathcal{I}\). Given logic \(\mathcal{L}\), structure \(\mathcal{A}\) as a composition of structures \(\mathcal{A}_i, i \in I\), index structure \(\mathcal{I}\) and formula \(\phi \) of the logic to be evaluated on \(\mathcal{A}\), the question is: what is the reduction sequence for \(\phi \) if any. We show that if we may prove preservation theorems for \(\mathcal{L}\) as well as if \(\mathcal{A}\) is a strongly distributed composition of its components then the corresponding reduction sequence for \(\mathcal{A}\) may be effectively computed. We show that the approach works for lots of extensions of FOL but not all. The considered extensions of FOL are suitable candidates for modeling languages for components and services, used in incremental automated reasoning, data mining, decision making, planning and scheduling. A short complexity analysis of the method is also provided.

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Notes

  1. 1.

    The first element of the First stream satisfy the condition: First element of a fragment is labeled by \(\otimes \). However, its fragment does not satisfy the condition: The fragments are succeeding.

  2. 2.

    The prefix order means here the prefix order on strings, which is a special kind of substring relation. \(p\prec _1 p^{\prime }\) iff \(p\prec p^{\prime }\) and \(p^{\prime }\) has exactly one more transition than p.

  3. 3.

    In the definition, the prefix order \(\prec _1\) generalizes the intuitive concept of a tree by introducing the possibility of continuous progress and continuous branching.

  4. 4.

    \(n_{\imath }\) is the size of the coding of \(T_{\imath }\).

  5. 5.

    However, in practical applications, the complexity, as a rule, is simply exponential.

  6. 6.

    Note that this includes reduction sequences for disjoint pairs and disjoint multiples.

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Authors and Affiliations

  1. Software Engineering Department, Ort Braude College, Rehov Snunit 51, POB 78, Karmiel, 2161002, Israel

    Elena V. Ravve & Zeev Volkovich

  2. Faculty of Engineering Management, Poznan University of Technology, ul. Jacka Rychlewskiego 2, 60-965, Poznan, Poland

    Gerhard-Wilhelm Weber

  3. IAM, METU, 06800, Ankara, Turkey

    Gerhard-Wilhelm Weber

Authors
  1. Elena V. Ravve
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  2. Zeev Volkovich
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  3. Gerhard-Wilhelm Weber
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Correspondence to Elena V. Ravve .

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  1. Department of Mathematics and LIAAD—INESC TEC, Faculty of Science, University of Porto, Porto, Portugal

    Alberto Pinto

  2. Department of Agricultural and Resource Economics, University of California, Berkeley, CA, USA

    David Zilberman

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Ravve, E.V., Volkovich, Z., Weber, GW. (2021). A Logic-Based Approach to Incremental Reasoning on Multi-agent Systems. In: Pinto, A., Zilberman, D. (eds) Modeling, Dynamics, Optimization and Bioeconomics IV. ICABR DGS 2017 2018. Springer Proceedings in Mathematics & Statistics, vol 365. Springer, Cham. https://doi.org/10.1007/978-3-030-78163-7_18

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