Many real world problems naturally appear as constraints satisfaction problems (CSP), for which very efficient algorithms are known. Most of these involve the combination of two techniques: some direct propagation of constraints between variables (with the goal of reducing their sets of possible values) and some kind of structured search (depth-first, breadth-first,...). But when such blind search is not possible or not allowed or when one wants a 'constructive' or a 'pattern-based' solution, one must devise more complex propagation rules instead. In this case, one can introduce the notion of a candidate (a 'still possible' value for a variable). Here, we give this intuitive notion a well defined logical status, from which we can define the concepts of a resolution rule and a resolution theory. In order to keep our analysis as concrete as possible, we illustrate each definition with the well known Sudoku example. Part I proposes a general conceptual framework based on first order logic; with the introduction of chains and braids, Part II will give much deeper results.

We present a unified logical framework for representing and reasoning about both quantitative and qualitative preferences in fuzzy answer set programming [Saad, 2010; Saad, 2009; Subrahmanian, 1994], called fuzzy answer set optimization programs. The proposed framework is vital to allow defining quantitative preferences over the possible outcomes of qualitative preferences. We show the application of fuzzy answer set optimization programs to the course scheduling with fuzzy preferences problem described in [Saad, 2010]. To the best of our knowledge, this development is the first to consider a logical framework for reasoning about quantitative preferences, in general, and reasoning about both quantitative and qualitative preferences in particular.

We present a unified logical framework for representing and reasoning about both probability quantitative and qualitative preferences in probability answer set programming [Saad and Pontelli, 2006; Saad, 2006; Saad, 2007a], called probability answer set optimization programs. The proposed framework is vital to allow defining probability quantitative preferences over the possible outcomes of qualitative preferences. We show the application of probability answer set optimization programs to a variant of the well-known nurse restoring problem [Bard and Purnomo, 2005], called the nurse restoring with probability preferences problem. To the best of our knowledge, this development is the first to consider a logical framework for reasoning about probability quantitative preferences, in general, and reasoning about both probability quantitative and qualitative preferences in particular.

Pattern-Based Constraint Satisfaction and Logic Puzzles develops a pure logic, pattern-based perspective of solving the finite Constraint Satisfaction Problem (CSP), with emphasis on finding the "simplest" solution. Different ways of reasoning with the constraints are formalised by various families of "resolution rules", each of them carrying its own notion of simplicity. A large part of the book illustrates the power of the approach by applying it to various popular logic puzzles. It provides a unified view of how to model and solve them, even though they involve very different types of constraints: obvious symmetric ones in Sudoku, non-symmetric but transitive ones (inequalities) in Futoshiki, topological and geometric ones in Map colouring, Numbrix and Hidato, and even much more complex non-binary arithmetic ones in Kakuro (or Cross Sums). It also shows that the most familiar techniques for these puzzles can indeed be understood as mere application-specific presentations of the general rules. Sudoku is used as the main example throughout the book, making it also an advanced level sequel to "The Hidden Logic of Sudoku" (another book by the same author), with: many examples of relationships among different rules and of exceptional situations; comparisons of the resolution potential of various families of rules; detailed statistics of puzzles hardness; analysis of extreme instances.

This paper presents a model-based planner called the Probabilistic Sulu Planner or the p-Sulu Planner, which controls stochastic systems in a goal directed manner within user-specified risk bounds. The objective of the p-Sulu Planner is to allow users to command continuous, stochastic systems, such as unmanned aerial and space vehicles, in a manner that is both intuitive and safe. To this end, we first develop a new plan representation called a chance-constrained qualitative state plan (CCQSP), through which users can specify the desired evolution of the plant state as well as the acceptable level of risk. An example of a CCQSP statement is ``go to A through B within 30 minutes, with less than 0.001% probability of failure." We then develop the p-Sulu Planner, which can tractably solve a CCQSP planning problem. In order to enable CCQSP planning, we develop the following two capabilities in this paper: 1) risk-sensitive planning with risk bounds, and 2) goal-directed planning in a continuous domain with temporal constraints. The first capability is to ensures that the probability of failure is bounded. The second capability is essential for the planner to solve problems with a continuous state space such as vehicle path planning. We demonstrate the capabilities of the p-Sulu Planner by simulations on two real-world scenarios: the path planning and scheduling of a personal aerial vehicle as well as the space rendezvous of an autonomous cargo spacecraft.

This paper resolves a common complexity issue in the Bethe approximation of statistical physics and the Belief Propagation (BP) algorithm of artificial intelligence. The Bethe approximation and the BP algorithm are heuristic methods for estimating the partition function and marginal probabilities in graphical models, respectively. The computational complexity of the Bethe approximation is decided by the number of operations required to solve a set of non-linear equations, the so-called Bethe equation. Although the BP algorithm was inspired and developed independently, Yedidia, Freeman and Weiss (2004) showed that the BP algorithm solves the Bethe equation if it converges (however, it often does not). This naturally motivates the following question to understand limitations and empirical successes of the Bethe and BP methods: is the Bethe equation computationally easy to solve? We present a message-passing algorithm solving the Bethe equation in a polynomial number of operations for general binary graphical models of n variables where the maximum degree in the underlying graph is O(log n). Our algorithm can be used as an alternative to BP fixing its convergence issue and is the first fully polynomial-time approximation scheme for the BP fixed-point computation in such a large class of graphical models, while the approximate fixed-point computation is known to be (PPAD-)hard in general. We believe that our technique is of broader interest to understand the computational complexity of the cavity method in statistical physics.

The universal-algebraic approach has proved a powerful tool in the study of the computational complexity of constraint satisfaction problems (CSPs). This approach has previously been applied to the study of CSPs with finite or (infinite) ω-categorical templates, and relies on two facts. The first is that in finite or ω-categorical structures A, a relation is primitive positive definable if and only if it is preserved by the polymorphisms of A. The second is that every finite or ω-categorical structure is homomorphically equivalent to a core structure. In this paper, we present generalizations of these facts to infinite structures that are not necessarily ω-categorical. Specifically, we prove that every CSP can be formulated with a template A such that a relation is primitive positive definable in A if and only if it is firstorder definable in A and preserved by the infinitary polymorphisms of A. Using existential positive closure we rederive and extend known results about cores, presenting the new notions of core theories (models of which will be cores) and core companions (which are defined analogously to model companions in model theory). We prove a uniqueness result for core companions that yields the uniqueness of model-complete cores of ω-categorical structures. Existential positive closure is also the crucial concept to give an exact characterization of those CSPs that can be formulated with (a finite or) an ω-categorical template.

We are dealing with the problem of space layout planning here. We present an architectural conceptual CAD approach. Starting with design specifications in terms of constraints over spaces, a specific enumeration heuristics leads to a complete set of consistent conceptual design solutions named topological solutions. These topological solutions which do not presume any precise definitive dimension correspond to the sketching step that an architect carries out from the Design specifications on a preliminary design phase in architecture.

Many AI synthesis problems such as planning or scheduling may be modelized as constraint satisfaction problems (CSP). A CSP is typically defined as the problem of finding any consistent labeling for a fixed set of variables satisfying all given constraints between these variables. However, for many real tasks such as job-shop scheduling, time-table scheduling, design?, all these constraints have not the same significance and have not to be necessarily satisfied. A first distinction can be made between hard constraints, which every solution should satisfy and soft constraints, whose satisfaction has not to be certain. In this paper, we formalize the notion of possibilistic constraint satisfaction problems that allows the modeling of uncertainly satisfied constraints. We use a possibility distribution over labelings to represent respective possibilities of each labeling. Necessity-valued constraints allow a simple expression of the respective certainty degrees of each constraint. The main advantage of our approach is its integration in the CSP technical framework. Most classical techniques, such as Backtracking (BT), arcconsistency enforcing (AC) or Forward Checking have been extended to handle possibilistics CSP and are effectively implemented. The utility of our approach is demonstrated on a simple design problem.

We consider the problem of creating fair course timetables in the setting of a university. Our motivation is to improve the overall satisfaction of individuals concerned (students, teachers, etc.) by providing a fair timetable to them. The central idea is that undesirable arrangements in the course timetable, i.e., violations of soft constraints, should be distributed in a fair way among the individuals. We propose two formulations for the fair course timetabling problem that are based on max-min fairness and Jain's fairness index, respectively. Furthermore, we present and experimentally evaluate an optimization algorithm based on simulated annealing for solving max-min fair course timetabling problems. The new contribution is concerned with measuring the energy difference between two timetables, i.e., how much worse a timetable is compared to another timetable with respect to max-min fairness. We introduce three different energy difference measures and evaluate their impact on the overall algorithm performance. The second proposed problem formulation focuses on the tradeoff between fairness and the total amount of soft constraint violations. Our experimental evaluation shows that the known best solutions to the ITC2007 curriculum-based course timetabling instances are quite fair with respect to Jain's fairness index. However, the experiments also show that the fairness can be improved further for only a rather small increase in the total amount of soft constraint violations.