Indistinguishable objects often occur when modelling problems in constraint programming, as well as in other related paradigms. They occur when objects can be viewed as being drawn from a set of unlabelled objects, and the only operation allowed on them is equality testing. For example, the golfers in the social golfer problem are indistinguishable. If we do label the golfers, then any relabelling of the golfers in one solution gives another valid solution. Therefore, we can regard the symmetric group of size $n$ as acting on a set of $n$ indistinguishable objects. In this paper, we show how we can break the symmetries resulting from indistinguishable objects. We show how symmetries on indistinguishable objects can be defined properly in complex types, for example in a matrix indexed by indistinguishable objects. We then show how the resulting symmetries can be broken correctly. In Essence, a high-level modelling language, indistinguishable objects are encapsulated in "unnamed types". We provide an implementation of complete symmetry breaking for unnamed types in Essence.

Local search is a common method for solving combinatorial optimisation problems. We focus on general-purpose local search solvers that accept as input a constraint model -- a declarative description of a problem consisting of a set of decision variables under a set of constraints. Existing approaches typically take as input models written in solver-independent constraint modelling languages like MiniZinc. The Athanor solver we describe herein differs in that it begins from a specification of a problem in the abstract constraint specification language Essence, which allows problems to be described without commitment to low-level modelling decisions through its support for a rich set of abstract types. The advantage of proceeding from Essence is that the structure apparent in a concise, abstract specification of a problem can be exploited to generate high quality neighbourhoods automatically, avoiding the difficult task of identifying that structure in an equivalent constraint model. Based on the twin benefits of neighbourhoods derived from high level types and the scalability derived by searching directly over those types, our empirical results demonstrate strong performance in practice relative to existing solution methods.

The Essence language allows a user to specify a constraint problem at a level of abstraction above that at which constraint modelling decisions are made. Essence specifications are refined into constraint models using the Conjure automated modelling tool, which employs a suite of refinement rules. However, Essence is a rich language in which there are many equivalent ways to specify a given problem. A user may therefore omit the use of domain attributes or abstract types, resulting in fewer refinement rules being applicable and therefore a reduced set of output models from which to select. This paper addresses the problem of recovering this information automatically to increase the robustness of the quality of the output constraint models in the face of variation in the input Essence specification. We present reformulation rules that can change the type of a decision variable or add attributes that shrink its domain. We demonstrate the efficacy of this approach in terms of the quantity and quality of models Conjure can produce from the transformed specification compared with the original.

Pen and paper puzzles like Sudoku, Futoshiki and Skyscrapers are hugely popular. Solving such puzzles can be a trivial task for modern AI systems. However, most AI systems solve problems using a form of backtracking, while people try to avoid backtracking as much as possible. This means that existing AI systems do not output explanations about their reasoning that are meaningful to people. We present Demystify, a tool which allows puzzles to be expressed in a high-level constraint programming language and uses MUSes to allow us to produce descriptions of steps in the puzzle solving. We give several improvements to the existing techniques for solving puzzles with MUSes, which allow us to solve a range of significantly more complex puzzles and give higher quality explanations. We demonstrate the effectiveness and generality of Demystify by comparing its results to documented strategies for solving a range of pen and paper puzzles by hand, showing that our technique can find many of the same explanations.

The n-Queens problem is to place n chess queens on an n by n chessboard so that no two queens are on the same row, column or diagonal. The n-Queens Completion problem is a variant, dating to 1850, in which some queens are already placed and the solver is asked to place the rest, if possible. We show that n-Queens Completion is both NP-Complete and #P-Complete. A corollary is that any non-attacking arrangement of queens can be included as a part of a solution to a larger n-Queens problem. We introduce generators of random instances for n-Queens Completion and the closely related Blocked n-Queens and Excluded Diagonals Problem. We describe three solvers for these problems, and empirically analyse the hardness of randomly generated instances. For Blocked n-Queens and the Excluded Diagonals Problem, we show the existence of a phase transition associated with hard instances as has been seen in other NP-Complete problems, but a natural generator for n-Queens Completion did not generate consistently hard instances. The significance of this work is that the n-Queens problem has been very widely used as a benchmark in Artificial Intelligence, but conclusions on it are often disputable because of the simple complexity of the decision problem. Our results give alternative benchmarks which are hard theoretically and empirically, but for which solving techniques designed for n-Queens need minimal or no change.

Constraint propagation is one of the key techniques in constraint programming, and a large body of work has built up around it. Special-purpose constraint propagation algorithms frequently make implicit use of short supports — by examining a subset of the variables, they can infer support (a justification that a variable-value pair still forms part of a solution to the constraint) for all other variables and values and save substantial work. Recently short supports have been used in general purpose propagators, and (when the constraint is amenable to short supports) speed ups of more than three orders of magnitude have been demonstrated. In this paper we present ShortSTR2, a development of the Simple Tabular Reduction algorithm STR2+. We show that ShortSTR2 is complementary to the existing algorithms ShortGAC and HaggisGAC that exploit short supports, while being much simpler. When a constraint is amenable to short supports, the short support set can be exponentially smaller than the full-length support set. Therefore ShortSTR2 can efficiently propagate many constraints that STR2+ cannot even load into memory. We also show that ShortSTR2 can be combined with a simple algorithm to identify short supports from full-length supports, to provide a superior drop-in replacement for STR2+.