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Michael Codish

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14 papers
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14

AAAI Conference 2025 Conference Paper

Complete Symmetry Breaking for Finite Models

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

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.

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AAAI Conference 2024 Conference Paper

SAT-Based Techniques for Lexicographically Smallest Finite Models

  • Mikoláš Janota
  • Choiwah Chow
  • João Araújo
  • Michael Codish
  • Petr Vojtěchovský

This paper proposes SAT-based techniques to calculate a specific normal form of a given finite mathematical structure (model). The normal form is obtained by permuting the domain elements so that the representation of the structure is lexicographically smallest possible. Such a normal form is of interest to mathematicians as it enables easy cataloging of algebraic structures. In particular, two structures are isomorphic precisely when their normal forms are the same. This form is also natural to inspect as mathematicians have been using it routinely for many decades. We develop a novel approach where a SAT solver is used in a black-box fashion to compute the smallest representative. The approach constructs the representative gradually and searches the space of possible isomorphisms, requiring a small number of variables. However, the approach may lead to a large number of SAT calls and therefore we devise propagation techniques to reduce this number. The paper focuses on finite structures with a single binary operation (encompassing groups, semigroups, etc.). However, the approach is generalizable to arbitrary finite structures. We provide an implementation of the proposed algorithm and evaluate it on a variety of algebraic structures.

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AAAI Conference 2020 Conference Paper

Incremental Symmetry Breaking Constraints for Graph Search Problems

  • Avraham Itzhakov
  • Michael Codish

This paper introduces incremental symmetry breaking constraints for graph search problems which are complete and compact. We show that these constraints can be computed incrementally: A symmetry breaking constraint for order n graphs can be extended to one for order n + 1 graphs. Moreover, these constraints induce a special property on their canonical solutions: An order n canonical graph contains a canonical subgraph on the first k vertices for every 1 ≤ k ≤ n. This facilitates a “generate and extend” paradigm for parallel graph search problem solving: To solve a graph search problem ϕ on order n graphs, first generate the canonical graphs of some order k < n. Then, compute canonical solutions for ϕ by extending, in parallel, each canonical order k graph together with suitable symmetry breaking constraints. The contribution is that the proposed symmetry breaking constraints enable us to extend the order k canonical graphs to order n canonical solutions. We demonstrate our approach through its application on two hard graph search problems.

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IJCAI Conference 2017 Conference Paper

The DNA Word Design Problem: A New Constraint Model and New Results

  • Michael Codish
  • Michael Frank
  • Vitaly Lagoon

A fundamental problem in coding theory concerns the computation of the maximum cardinality of a set S of length n code words over an alphabet of size q, such that every pair of code words has Hamming distance at least d, and the set of additional constraints U on S is satisfied. This problem has application in several areas, one of which is the design of DNA codes where q=4 and the alphabet is {A, C, G, T}. We describe a new constraint model for this problem and demonstrate that it improves on previous solutions (computes better lower bounds) for various instances of the problem. Our approach is based on a clustering of DNA words into small sets of words. Solutions are then obtained as the union of such clusters. Our approach is SAT based: we specify constraints on clusters of DNA words and solve these using a Boolean satisfiability solver.

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LOPSTR Conference 2015 Conference Paper

Applying Sorting Networks to Synthesize Optimized Sorting Libraries

  • Michael Codish
  • Luís Cruz-Filipe
  • Markus Nebel
  • Peter Schneider-Kamp

Abstract This paper presents an application of the theory of sorting networks to facilitate the synthesis of optimized general-purpose sorting libraries. Standard sorting libraries are often based on combinations of the classic Quicksort algorithm with insertion sort applied as base case for small, fixed, numbers of inputs. Unrolling the code for the base case by ignoring loop conditions eliminates branching, resulting in code equivalent to a sorting network. This enables further program transformations based on sorting network optimizations, and eventually the synthesis of code from sorting networks. We show that, if considering the number of comparisons and swaps, the theory predicts no real advantage of this approach. However, significant speed-ups are obtained when taking advantage of instruction level parallelism and non-branching conditional assignment instructions, both of which are common in modern CPU architectures. We provide empirical evidence that using code synthesized from efficient sorting networks as the base case for Quicksort libraries results in significant real-world speed-ups.

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SAT Conference 2014 Conference Paper

Simplifying Pseudo-Boolean Constraints in Residual Number Systems

  • Yoav Fekete
  • Michael Codish

Abstract We present an encoding of pseudo-Boolean constraints based on decomposition with respect to a residual number system. We illustrate that careful selection of the base for the residual number system, and when bit-blasting modulo arithmetic, results in a powerful approach when solving hard pseudo-Boolean constraints. We demonstrate, using a range of pseudo-Boolean constraint solvers, that the obtained constraints are often substantially easier to solve.

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IJCAI Conference 2013 Conference Paper

Breaking Symmetries in Graph Representation

  • Michael Codish
  • Alice Miller
  • Patrick Prosser
  • Peter J. Stuckey

There are many complex combinatorial problems which involve searching for an undirected graph satisfying a certain property. These problems are often highly challenging because of the large number of isomorphic representations of a possible solution. In this paper we introduce novel, effective and compact, symmetry breaking constraints for undirected graph search. While incomplete, these prove highly beneficial in pruning the search for a graph. We illustrate the application of symmetry breaking in graph representation to resolve several open instances in extremal graph theory.

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LOPSTR Conference 2007 Conference Paper

Proving Termination with (Boolean) Satisfaction

  • Michael Codish

Abstract At some point there was the Davis-Putnam-Logemann-Loveland (DPLL) algorithm [6]. Forty five years later, research on Boolean satisfiability (SAT) is still ceaselessly generating even better SAT solvers capable of handling even larger SAT instances. Remarkably, the majority of these tools still bear the hallmark of the DPLL algorithm. In sync with the availability of progressively stronger SAT solvers is an accumulating number of applications which demonstrate that real world problems can often be solved by encoding them into SAT. When successful, this circumvents the need to redevelop complex search algorithms from scratch.

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LPAR Conference 2006 Conference Paper

SAT Solving for Argument Filterings

  • Michael Codish
  • Peter Schneider-Kamp
  • Vitaly Lagoon
  • René Thiemann
  • Jürgen Giesl

Abstract This paper introduces a propositional encoding for lexicographic path orders in connection with dependency pairs. This facilitates the application of SAT solvers for termination analysis of term rewrite systems based on the dependency pair method. We address two main inter-related issues and encode them as satisfiability problems of propositional formulas that can be efficiently handled by SAT solving: (1) the combined search for a lexicographic path order together with an argument filtering to orient a set of inequalities; and (2) how the choice of the argument filtering influences the set of inequalities that have to be oriented. We have implemented our contributions in the termination prover AProVE. Extensive experiments show that by our encoding and the application of SAT solvers one obtains speedups in orders of magnitude as well as increased termination proving power.

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LPAR Conference 2001 Conference Paper

Inferring Termination Conditions for Logic Programs Using Backwards Analysis

  • Samir Genaim
  • Michael Codish

Abstract This paper focuses on the inference of modes for which a logic program is guaranteed to terminate. This generalizes traditional termination analysis where an analyzer tries to verify termination for a specified mode. The contribution is a methodology which combines traditional termination analysis and backwards analysis to obtain termination inference. This leads to a better understanding of termination inference, simplifies its formal justification, and facilitates implementation. We evaluate the application of this approach to enhance an existing termination analyzer to perform also termination inference.

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TCS Journal 2000 Journal Article

Type dependencies for logic programs using ACI-unification

  • Michael Codish
  • Vitaly Lagoon

This paper presents a new notion of typing for logic programs which generalizes the notion of directional types. The generation of type dependencies for a logic program is fully automatic with respect to a given domain of types. The analysis method is based on a novel combination of program abstraction and ACI-unification which is shown to be correct and optimal. Type dependencies are obtained by abstracting programs, replacing concrete terms by their types, and evaluating the meaning of the abstract programs using a standard semantics for logic programs enhanced by ACI-unification. This approach is generic and can be used with any standard semantics. The method is both theoretically clean and easy to implement using general purpose tools. The proposed domain of types is condensing which means that analyses can be carried out in both top-down or bottom-up frameworks with no loss of precision for goal-independent analyses. The proposed method has been fully implemented within a bottom-up approach and the experimental results are promising.

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TCS Journal 1996 Journal Article

Proving implications by algebraic approximation

  • Michael Codish
  • Grigory Mashevitzky

This paper applies techniques of algebraic approximation to provide effective algorithms to determine the validity of universally quantified implications over lattice structures. We generalize the known result which states that any semilattice is approximated in the two element lattice. We show that the validity of a universally quantified implication ψ over a possibly infinite domain can be determined by examining its validity over a simpler domain the size of which is related to the number of constants in ψ. Both the known as well as the new results have high potential in providing practical automated techniques in various areas of application in computer science.

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TCS Journal 1994 Journal Article

Bottom-up abstract interpretation of logic programs

  • Michael Codish
  • Dennis Dams
  • Eyal Yardeni

This paper presents a formal framework for the bottom-up abstract interpretation of logic programs which can be applied to approximate answer substitutions, partial answer substitutions and call patterns for a given program and arbitrary initial goal. The framework is based on a Tp -like semantics defined over a Herbrand universe with variables which has previously been shown to determine the answer substitutions for arbitrary initial goals. The first part of the paper reconstructs this semantics to provide a more adequate initial goals. The first part of the paper reconstructs this semantics to provide a more adequate basis for abstract interpretation. A notion of abstract substitution is introduced and shown to determine an abstract semantic function which for a given program can be applied to approximate the answer substitutions for an arbitrary initial goal. The second part of the paper extends the bottom-up approach to provide approximations of both partial answer substitutions and call patterns. This is achieved by applying Magic Sets and other existing techniques to transform a program in such a way that the answer substitutions of the transformed program correspond to the partial answer substitutions and call patterns of the original program. This facilitates the analysis of concurrent logic programs (ignoring synchronization) and provides a collecting semantics which characterizes both success and call patterns.

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LPAR Conference 1994 Conference Paper

Goal Dependent versus Goal Independent Analysis of Logic Programs

  • Michael Codish
  • Maria Garcia de la Banda
  • Maurice Bruynooghe
  • Manuel V. Hermenegildo

Abstract Goal independent analysis of logic programs is commonly discussed in the context of the bottom-up approach. However, while the literature is rich in descriptions of top-down analysers and their application, practical experience with bottom-up analysis is still in a preliminary stage. Moreover, the practical use of existing top-down frameworks for goal independent analysis has not been addressed in a practical system. We illustrate the efficient use of existing goal dependent, top-down frameworks for abstract interpretation in performing goal independent analyses of logic programs much the same as those usually derived from bottom-up frameworks. We present several optimizations for this flavour of top-down analysis. The approach is fully implemented within an existing top-down framework. Several implementation tradeoffs are discussed as well as the influence of domain characteristics. An experimental evaluation including a comparison with a bottom-up analysis for the domain Prop is presented. We conclude that the technique can offer advantages with respect to standard goal dependent analyses.

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