Historically 3D game environments have almost always been immutable. Mutable environments are a technical challenge that will affect performance. For games of the future to continue approaching realism, mutable environments are an essential step. Popularized by the game title Minecraft (2009), the use of voxel engines in games has become increasingly common. However, by the nature of the discrete position of voxels, the method is limited in representing arbitrary polyhedral shapes like angled slopes. It also prohibits smooth mutations including proper movement and rotation of objects within the voxelization. This is generally mitigable with more voxels. However, this paper proposes a more precise solution to the problem. A tetrahedral mesh engine (tetra engine). By altering tetrahedral topology, vertex positions, and material of individual tetrahedrons, tetras are intended to solve the issue of arbitrary polyhedral shapes for voxels. Additionally, the tetrahedral mesh shows other promises such as providing a robust collision detection method and as an acceleration structure for e.g. raycasting. The research question can be summarized as investigating the feasibility of a tetra engine as a mutable game environment. 3 sub-research questions are given: The first regarding performance, the second regarding robustness, and the third different types mutations. The research questions are addressed along with a proof of concept (POC). The POC intends to investigate a proposal for the most efficient robust solution possible for the most basic mutation type known as edits. Because of time constraints, it does not cover all parts of the research questions but works as a bottom-up approach to understand what is required to realize a full-fledged tetra engine. Thus, a large part of the research question is answered theoretically both hypothetically with grounds from the POC and through previous work. The result shows a much more critical robustness consideration than expected and suggests relying on slower but more robust algorithms that are known to work. In conclusion, nothing suggests that a scalable future-proof tetra engine is impossible, but the algorithms required are much less efficient than those for voxels and robustness is an issue to overcome. However, numerous hypothetical advantages particularly regarding deformation and fluid simulation are still recognized and it is not obvious that future mutable environments would not benefit from a tetra engine rather than voxels. / Historiskt sätt har 3D-datorspelmiljöer nästan uteslutande varit oföränderliga. Det finns goda anledningar till detta. Föränderliga miljöer är en teknisk utmaning som påverkar speleffektiviteten. Om man däremot vill fortsätta utveckla mer realistiska datorspel är föränderliga spelmiljöer ett naturligt steg. Spelet Minecraft (2009) populariserade användandet av voxlar i datorspel vars material kan ändras för att förändra miljön. Sedan dess har voxlar i datorspel blivit mer och mer populära. Dock, på grund av voxlars diskreta positioner har de begränsningar i dess förmåga att representera godtyckliga polyedriska former så som arbiträrt vinklade plan. Detta omöjliggör även mjuka rörelser inklusive rörelse och rotering inom voxeliseringen. Det här problemet kan generellt dämpas med hjälp av mindre och fler voxlar. Men, för att lösa problemet ordentligt föreslår den här rapporten en annan lösning, nämligen en tetrahedrisk-mesh-motor (tetramotor). Genom att ändra tetraedrisk topologi, hörn positioner, samt material av individuella tetraedrar ska tetrorna kunna forma arbiträra polyedriska former och lösa problemet med voxlar. Utöver detta visar en tetraedrisk mesh andra intressanta möjligheter likt en metod för robust kollisions hantering och som en accelereringsstruktur för till exempel ray-casting. Forskningsfrågan kan sammanfattas som att utforska tillämpbarheten av en tetramotor för föränderliga datorspelsmiljöer. Tre underfrågor ges: Den första handlar om beräkningseffektivitet, den andra angående robusthet och den tredje oliky typer av mutationer. Forskningsfrågorna angrips med hjälp av en proof of concept lösning (POC). POC:en är menad att utforska ett förslag på en effektiv och robust metod för direkta förändringar. På grund av tidsbegränsningar så täcker inte POC:en alla delar av frågeställningarna men är menad som en botten upp lösning för att inse och förstå kraven för en komplett tetramotor. På grund av detta är en stor del av forskningsfrågorna svarade teoretiskt, både hypotetiskt baserat på insikterna från POC:en och med grund från tidigare forskning. Resultatet visar en mycket mer kritisk hänsyn till robusthet än förväntat och föreslår att man använder sig av långsammare men fungerande existerande lösningar. Slutsatsen är att inget tyder på att en skalbar framtidssäker tetramotor är en omöjlighet, men att algoritmerna är mycket långsammare än de som krävs för voxlar. Däremot finns potentieally fördelar med en tetra motor, främst med hänsyn till deformering och flödessimulering. Det är därför inte självklart att framtida muterbara spelmijöer inte skulle ha fördelar av en tetramotor istället för voxlar.

A fin de diseñar e implementar lenguajes de programación que tengan en cuenta el paradigma borroso modificaremos el lambda cálculo clásico, adjuntando a cada término un grado, y redefiniendo la beta-reducción, obteniendo que para que el nuevo cálculo verifique la propiedad de Church-Rosser la transmisión de los grados debe hacerse por medio de una función que sea una t-norma o s-conorma. Utilizando esta nueva herramienta diseñamos un lenguaje no determinista que satisface los requerimientos de la programación con datos borrosos. / With the aim of designing and implementing programming languages that take into account the fuzzy paradigm we will modify the classical lambda calculus by adding a degree to each term and by redefining the b-reduction. Thus, for the new calculus to verify the Church-Rosser property, the degree computed with can be made through a function that is a t-norm or an s-conorm. With this new tool we design a nondeterminist language that satisfies fuzzy dataprogramming requirements, and an example of its behaviour is shown.

This thesis focuses mainly on the Maximum Independent Set (MIS) problem. Some related graph theoretical combinatorial problems are also considered. As these problems are generally NP-hard, we study their complexity in hereditary graph classes, i.e. graph classes defined by a set F of forbidden induced subgraphs. We revise the literature about the issue, for example complexity results, applications, and techniques tackling the problem. Through considering some general approach, we exhibit several cases where the problem admits a polynomial-time solution. More specifically, we present polynomial-time algorithms for the MIS problem in: + some subclasses of $S_{2;j;k}$-free graphs (thus generalizing the classical result for $S_{1;2;k}$-free graphs); + some subclasses of $tree_{k}$-free graphs (thus generalizing the classical results for subclasses of P5-free graphs); + some subclasses of $P_{7}$-free graphs and $S_{2;2;2}$-free graphs; and various subclasses of graphs of bounded maximum degree, for example subcubic graphs. Our algorithms are based on various approaches. In particular, we characterize augmenting graphs in a subclass of $S_{2;k;k}$-free graphs and a subclass of $S_{2;2;5}$-free graphs. These characterizations are partly based on extensions of the concept of redundant set [125]. We also propose methods finding augmenting chains, an extension of the method in [99], and finding augmenting trees, an extension of the methods in [125]. We apply the augmenting vertex technique, originally used for $P_{5}$-free graphs or banner-free graphs, for some more general graph classes. We consider a general graph theoretical combinatorial problem, the so-called Maximum -Set problem. Two special cases of this problem, the so-called Maximum F-(Strongly) Independent Subgraph and Maximum F-Induced Subgraph, where F is a connected graph set, are considered. The complexity of the Maximum F-(Strongly) Independent Subgraph problem is revised and the NP-hardness of the Maximum F-Induced Subgraph problem is proved. We also extend the augmenting approach to apply it for the general Maximum Π -Set problem. We revise on classical graph transformations and give two unified views based on pseudo-boolean functions and αff-redundant vertex. We also make extensive uses of α-redundant vertices, originally mainly used for $P_{5}$-free graphs, to give polynomial solutions for some subclasses of $S_{2;2;2}$-free graphs and $tree_{k}$-free graphs. We consider some classical sequential greedy heuristic methods. We also combine classical algorithms with αff-redundant vertices to have new strategies of choosing the next vertex in greedy methods. Some aspects of the algorithms, for example forbidden induced subgraph sets and worst case results, are also considered. Finally, we restrict our attention on graphs of bounded maximum degree and subcubic graphs. Then by using some techniques, for example ff-redundant vertex, clique separator, and arguments based on distance, we general these results for some subclasses of $S_{i;j;k}$-free subcubic graphs.

This thesis focuses mainly on the Maximum Independent Set (MIS) problem. Some related graph theoretical combinatorial problems are also considered. As these problems are generally NP-hard, we study their complexity in hereditary graph classes, i.e. graph classes defined by a set F of forbidden induced subgraphs. We revise the literature about the issue, for example complexity results, applications, and techniques tackling the problem. Through considering some general approach, we exhibit several cases where the problem admits a polynomial-time solution. More specifically, we present polynomial-time algorithms for the MIS problem in: + some subclasses of $S_{2;j;k}$-free graphs (thus generalizing the classical result for $S_{1;2;k}$-free graphs); + some subclasses of $tree_{k}$-free graphs (thus generalizing the classical results for subclasses of P5-free graphs); + some subclasses of $P_{7}$-free graphs and $S_{2;2;2}$-free graphs; and various subclasses of graphs of bounded maximum degree, for example subcubic graphs. Our algorithms are based on various approaches. In particular, we characterize augmenting graphs in a subclass of $S_{2;k;k}$-free graphs and a subclass of $S_{2;2;5}$-free graphs. These characterizations are partly based on extensions of the concept of redundant set [125]. We also propose methods finding augmenting chains, an extension of the method in [99], and finding augmenting trees, an extension of the methods in [125]. We apply the augmenting vertex technique, originally used for $P_{5}$-free graphs or banner-free graphs, for some more general graph classes. We consider a general graph theoretical combinatorial problem, the so-called Maximum -Set problem. Two special cases of this problem, the so-called Maximum F-(Strongly) Independent Subgraph and Maximum F-Induced Subgraph, where F is a connected graph set, are considered. The complexity of the Maximum F-(Strongly) Independent Subgraph problem is revised and the NP-hardness of the Maximum F-Induced Subgraph problem is proved. We also extend the augmenting approach to apply it for the general Maximum Π -Set problem. We revise on classical graph transformations and give two unified views based on pseudo-boolean functions and αff-redundant vertex. We also make extensive uses of α-redundant vertices, originally mainly used for $P_{5}$-free graphs, to give polynomial solutions for some subclasses of $S_{2;2;2}$-free graphs and $tree_{k}$-free graphs. We consider some classical sequential greedy heuristic methods. We also combine classical algorithms with αff-redundant vertices to have new strategies of choosing the next vertex in greedy methods. Some aspects of the algorithms, for example forbidden induced subgraph sets and worst case results, are also considered. Finally, we restrict our attention on graphs of bounded maximum degree and subcubic graphs. Then by using some techniques, for example ff-redundant vertex, clique separator, and arguments based on distance, we general these results for some subclasses of $S_{i;j;k}$-free subcubic graphs.