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Complete Symmetry Breaking for Finite Models

Marek Dančo, Mikoláš Janota, Michael Codish, João Jorge Araújo

Abstract

This paper introduces a SAT-based technique that calculates a compact and complete symmetry-break for finite model finding, with the focus on structures with a single binary operation (magmas). Classes of algebraic structures are typically described as first-order logic formulas and the concrete algebras are models of these formulas. Such models include an enormous number of isomorphic, i.e. symmetric, algebras. A complete symmetry-break is a formula that has as models, exactly one canonical representative from each equivalence class of algebras. Thus, we enable answering questions about properties of the models so that computation and search are restricted to the set of canonical representations. For instance, we can answer the question: How many non-isomorphic semigroups are there of size $n$? Such questions can be answered by counting the satisfying assignments of a SAT formula, which already filters out non-isomorphic models. The introduced technique enables us calculating numbers of algebraic structures not present in the literature and going beyond the possibilities of pure enumeration approaches.

Complete Symmetry Breaking for Finite Models

Abstract

This paper introduces a SAT-based technique that calculates a compact and complete symmetry-break for finite model finding, with the focus on structures with a single binary operation (magmas). Classes of algebraic structures are typically described as first-order logic formulas and the concrete algebras are models of these formulas. Such models include an enormous number of isomorphic, i.e. symmetric, algebras. A complete symmetry-break is a formula that has as models, exactly one canonical representative from each equivalence class of algebras. Thus, we enable answering questions about properties of the models so that computation and search are restricted to the set of canonical representations. For instance, we can answer the question: How many non-isomorphic semigroups are there of size ? Such questions can be answered by counting the satisfying assignments of a SAT formula, which already filters out non-isomorphic models. The introduced technique enables us calculating numbers of algebraic structures not present in the literature and going beyond the possibilities of pure enumeration approaches.

Paper Structure

This paper contains 14 sections, 2 theorems, 5 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Canonizing Sets Preliminaries
  4. Canonizing Sets for Algebras
  5. Encodings
  6. The first conjunct
  7. The second conjunct
  8. The fourth conjunct
  9. The third conjunct
  10. Complexity Insights
  11. Partial Symmetry Breaks
  12. Experiments
  13. Related Work
  14. Conclusions and Future Work

Key Result

Lemma 7

Algorithm CanSetAlgorithm terminates and returns a canonizing set $\Pi$.

Figures (5)

  • Figure 1: $A$ that satisfies LNH and $\tau(A)$ with $\tau(A) \prec A$.
  • Figure 2: Comparison of the runtime d4 model counter on row-by-row and diagonal ordering.
  • Figure 3: Comparison of the times of canonizing set on row-by-row and diagonal ordering.
  • Figure 4: Sizes of canonizing set on selected algebra classes.
  • Figure 5: Comparison of the sizes of canonizing set on row-by-row and diagonal ordering.

Theorems & Definitions (13)

  • Definition 1: isomorphism
  • Definition 2: isomorphic copy
  • Definition 3: magma vectorization
  • Definition 4: $\preceq$
  • Definition 5: lex-leader
  • Definition 6: canonicity
  • Lemma 7: codish
  • proof : Proof sketch
  • Example 1
  • Definition 9
  • ...and 3 more