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.

Abstract This paper presents thepl-nautylibrary, a Prolog interface to thenautygraph-automorphism tool. Adding the capabilities ofnautyto Prolog combines the strength of the “generate and prune” approach that is commonly used in logic programming and constraint solving, with the ability to reduce symmetries while reasoning over graph objects. Moreover, it enables the integration ofnautyin existing tool-chains, such as SAT-solvers or finite domain constraints compilers which exist for Prolog. The implementation consists of two components:pl-nauty, an interface connectingnauty's C library with Prolog, andpl-gtools, a Prolog framework integrating the software component ofnauty, calledgtools, with Prolog. The complete tool is available as a SWI-Prolog module. We provide a series of usage examples including two that apply to generate Ramsey graphs.

This paper describes a computer-assisted non-existence proof of 9-input sorting networks consisting of 24 comparators, hence showing that the 25-comparator sorting network found by Floyd in 1964 is optimal. As a corollary, we obtain that the 29-comparator network found by Waksman in 1969 is optimal when sorting 10 inputs. This closes the two smallest open instances of the optimal-size sorting network problem, which have been open since the results of Floyd and Knuth from 1966 proving optimality for sorting networks of up to 8 inputs. The proof involves a combination of two methodologies: one based on exploiting the abundance of symmetries in sorting networks, and the other based on an encoding of the problem to that of satisfiability of propositional logic. We illustrate that, while each of these can single-handedly solve smaller instances of the problem, it is their combination that leads to the more efficient solution that scales to handle 9 inputs.