SW-type puzzles and their graphs

In this paper, we present the SW-type of truth-tellers and liars puzzles. We examine the SW-type puzzles where each person can utter a sentence about the person's type and in which he uses only the "and" connective. We make the graphs of these puzzles. The graph of a puzzle has all information about the puzzle if we have no other information to solve the puzzle than the statements given (clear puzzles). We analyze the graphs of the possible puzzles. We give some transformations of graphs based on local information, for instance arrow-adding steps. These local steps are very helpful to solve these puzzles. We show an example that we can solve using these local steps. After this, we examine into the global properties of the graphs. We show a special example when the local steps do not help, but the puzzle is solvable by using global information. Finally we show a graph-algorithm which is a combination of local and global information, and show that it can solve the SW-type puzzles

Logic Programming Applications: What Are the Abstractions and Implementations?

This article presents an overview of applications of logic programming, classifying them based on the abstractions and implementations of logic languages that support the applications. The three key abstractions are join, recursion, and constraint. Their essential implementations are for-loops, fixed points, and backtracking, respectively. The corresponding kinds of applications are database queries, inductive analysis, and combinatorial search, respectively. We also discuss language extensions and programming paradigms, summarize example application problems by application areas, and touch on example systems that support variants of the abstractions with different implementations

Fully Packed Loops in a triangle: matchings, paths and puzzles

Fully Packed Loop configurations in a triangle (TFPLs) first appeared in the study of ordinary Fully Packed Loop configurations (FPLs) on the square grid where they were used to show that the number of FPLs with a given link pattern that has m nested arches is a polynomial function in m. It soon turned out that TFPLs possess a number of other nice properties. For instance, they can be seen as a generalized model of Littlewood-Richardson coefficients. We start our article by introducing oriented versions of TFPLs; their main advantage in comparison with ordinary TFPLs is that they involve only local constraints. Three main contributions are provided. Firstly, we show that the number of ordinary TFPLs can be extracted from a weighted enumeration of oriented TFPLs and thus it suffices to consider the latter. Secondly, we decompose oriented TFPLs into two matchings and use a classical bijection to obtain two families of nonintersecting lattice paths (path tangles). This point of view turns out to be extremely useful for giving easy proofs of previously known conditions on the boundary of TFPLs necessary for them to exist. One example is the inequality d(u)+d(v)<=d(w) where u,v,w are 01-words that encode the boundary conditions of ordinary TFPLs and d(u) is the number of cells in the Ferrers diagram associated with u. In the third part we consider TFPLs with d(w)- d(u)-d(v)=0,1; in the first case their numbers are given by Littlewood-Richardson coefficients, but also in the second case we provide formulas that are in terms of Littlewood-Richardson coefficients. The proofs of these formulas are of a purely combinatorial nature.Comment: 40 pages, 31 figure

Biases in human behavior

The paper shows that biases in individual’s decision-making may result from the process of mental editing by which subjects produce a “representation” of the decision problem. During this process, individuals make systematic use of default classifications in order to reduce the short-term memory load and the complexity of symbolic manipulation. The result is the construction of an imperfect mental representation of the problem that nevertheless has the advantage of being simple, and yielding “satisficing” decisions. The imperfection origins in a trade-off that exists between the simplicity of representation of a strategy and his efficiency. To obtain simplicity, the strategy’s rules have to be memorized and represented with some degree of abstraction, that allow to drastically reduce their number. Raising the level of abstraction with which a strategy’s rule is represented, means to extend the domain of validity of the rule beyond the field in which the rule has been experimented, and may therefore induce to include unintentionally domains in which the rule is inefficient. Therefore the rise of errors in the mental representation of a problem may be the "natural" effect of the categorization and the identification of the building blocks of a strategy. The biases may be persistent and give rise to lock-in effect, in which individuals remain trapped in sub-optimal strategies, as it is proved by experimental results on stability of sub-optimal strategies in games like Target The Two. To understand why sub-optimal strategies, that embody errors, are locally stable, i.e. cannot be improved by small changes in the rules, it is considered Kauffman’ NK model, because, among other properties, it shows that if there are interdependencies among the rules of a system, than the system admits many sub-optimal solutions that are locally stable, i.e. cannot be improved by simple mutations. But the fitness function in NK model is a random one, while in our context it is more reasonable to define the fitness of a strategy as efficiency of the program. If we introduce this kind of fitness, then the stability properties of the NK model do not hold any longer: the paper shows that while the elementary statements of a strategy are interdependent, it is possible to achieve an optimal configuration of the strategy via mutations and in consequence the sub-optimal solutions are not locally stable under mutations. The paper therefore provides a different explanation of the existence and stability of suboptimal strategies, based on the difficulty to redefine the sub-problems that constitute the building blocks of the problem’s representation

Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible

We analyze the computational complexity of the many types of pencil-and-paper-style puzzles featured in the 2016 puzzle video game The Witness. In all puzzles, the goal is to draw a simple path in a rectangular grid graph from a start vertex to a destination vertex. The different puzzle types place different constraints on the path: preventing some edges from being visited (broken edges); forcing some edges or vertices to be visited (hexagons); forcing some cells to have certain numbers of incident path edges (triangles); or forcing the regions formed by the path to be partially monochromatic (squares), have exactly two special cells (stars), or be singly covered by given shapes (polyominoes) and/or negatively counting shapes (antipolyominoes). We show that any one of these clue types (except the first) is enough to make path finding NP-complete ("witnesses exist but are hard to find"), even for rectangular boards. Furthermore, we show that a final clue type (antibody), which necessarily "cancels" the effect of another clue in the same region, makes path finding $\Sigma_2$-complete ("witnesses do not exist"), even with a single antibody (combined with many anti/polyominoes), and the problem gets no harder with many antibodies. On the positive side, we give a polynomial-time algorithm for monomino clues, by reducing to hexagon clues on the boundary of the puzzle, even in the presence of broken edges, and solving "subset Hamiltonian path" for terminals on the boundary of an embedded planar graph in polynomial time.Comment: 72 pages, 59 figures. Revised proof of Lemma 3.5. A short version of this paper appeared at the 9th International Conference on Fun with Algorithms (FUN 2018