In this thesis, we demonstrate how presheaves of labelled transition systems (LTS) acquire a very natural form of spatially induced independence on their actions when we allow a minimal amount of gluing on selected transitions within such systems. This gluing condition is characterized in the new model of LTS-adapted presheaf, and we also make use of the new model of asynchronous labelled transition system with equivalence (ALTSE) to characterize independence on actions. As such, our main result, the Theorem of Spatially Induced Independence, establishes functors from the categories of LTS-adapted presheaves to the categories of ALTSE-valued presheaves; it is a result that extends a proposition of Malcolm [SSTS] in the context of LTS-valued sheaves on complete Heyting algebras.

Computational learning theory is a branch of theoretical computer science that re-imagines the role of an algorithm from an agent of computation to an agent of learning. The operations of computers become those of the human mind; an important step towards illuminating the limitations of artificial intelligence. The central difference between a learning algorithm and a traditional algorithm is that the learner has access to an oracle who, in constant time, can answer queries about that to be learned. Normally an algorithm would have to discover such information on its own accord. This subtle change in how we model problem solving results in changes in the computational complexity of some classic problems; allowing us to re-examine them in a new light. Specifically two known result are examined: one positive, one negative. It is know that one can efficiently learn Deterministic Finite Automatons with queries, not so of Non-Deterministic Finite Automatons. We generalize these Automatons into Labeled Transition Systems and attempt to learn them using a stronger query.

As software and hardware systems grow more complex and we begin to rely more on their correctness and reliability, it becomes exceedingly important to formally verify certain properties of these systems. If done naïvely, verifying a system can easily require exponentially more work than running it, in order to account for all possible executions. However, there are often symmetries or other properties of a system that can be exploited to reduce the amount of necessary work. In this thesis, we present a number of approaches that do this in the context of the CSP model checker FDR. CSP is named for Communicating Sequential Processes, or parallel combinations of state machines with synchronised communications. In the FDR model, the component processes are typically converted to explicit state machines while their parallel combination is evaluated lazily during model checking. Our contributions are motivated by this model but applicable to other models as well. We first address the scalability of the component machines by proposing a lazy compiler for a subset of CSP_M selected to model parameterised state machines. This is a typical case where the state space explosion can make model checking impractical, since the size of the state space is exponential in the number and size of the parameters. A lazy approach to evaluating these systems allows only the reachable subset of the state space to be explored. As an example, in studying security protocols, it is common to model an intruder parameterised by knowledge of each of a list of facts; even a relatively small 100 facts results in an intractable 2¹⁰⁰ states, but the rest of the system can ensure that only a small number of these states are reachable. Next, we address the scalability of the overall combination by presenting novel algorithms for bisimulation reduction with respect to strong bisimulation, divergence- respecting delay bisimulation, and divergence-respecting weak bisimulation. Since a parallel composition is related to the Cartesian product of its components, performing a relatively time-consuming bisimulation reduction on the components can reduce its size significantly; an efficient bisimulation algorithm is therefore very desirable. This thesis is motivated by practical implementations, and we discuss an implementation of each of the proposed algorithms in FDR. We thoroughly evaluate their performance and demonstrate their effectiveness.