Count(q) versus the Pigeon-Hole Principle

For each

A Fractal which violates the Axiom of Determinacy

By use of the axiom of choice I construct a symmetrical and self-similar subset A \subseteq [0,1] \subseteq R. Then by an elementary strategy stealing argument it is shown that A is not determined. The (possible) existence of fractals like A clarifies the status of the controversial Axiom of Determinacy

Finitisation in Bounded Arithmetic

I prove various results concerning un-decidability in weak fragments of Arithmetic. All results are concerned with S^{1}_{2} \subseteq T^{1}_{2} \subseteq S^{2}_{2} \subseteq T^{2}_{2} \subseteq.... a hierarchy of theories which have already been intensively studied in the literature. Ideally one would like to separate these systems. However this is generally expected to be a very deep problem, closely related to some of the most famous and open problems in complexity theory. In order to throw some light on the separation problems, I consider the case where the underlying language is enriched by extra relation and function symbols. The paper introduces a new type of results. These state that the first three levels in the hierarchy (i.e. S^{1}_{2}, T^{1}_{2} and S^{2}_{2}) are never able to distinguish (in a precise sense) the "finite'' from the "infinite''. The fourth level (i.e. T^{2}_{2}) in some cases can make such a distinction. More precisely, elementary principles from finitistical combinatorics (when expressed solely by the extra relation and function symbols) are only provable on the first three levels if they are valid when considered as principles of general (infinitistical) combinatorics. I show that this does not hold for the fourth level. All results are proved by forcing

Bootstrapping the Primitive Recursive Functions by 47 Colors

I construct a concrete coloring of the 3 element subsets of N. It has the property that each homogeneous set {s_0, s_1, s_2, ..., s_r}, r >= s_0 - 1 has its elements spread so much apart that F_{omega}(s_i) < s_{i+1} for i = 1, 2, ..., r -1. It uses only 47 colors. This is more economical than the approximately 160000 colors used by Ketonen and Solovay

A Complexity Gap for Tree-Resolution

It is shown that any sequence  psi_n of tautologies which expresses thevalidity of a fixed combinatorial principle either is "easy" i.e. has polynomialsize tree-resolution proofs or is "difficult" i.e requires exponentialsize tree-resolution proofs. It is shown that the class of tautologies whichare hard (for tree-resolution) is identical to the class of tautologies whichare based on combinatorial principles which are violated for infinite sets.Actually it is shown that the gap-phenomena is valid for tautologies basedon infinite mathematical theories (i.e. not just based on a single proposition).We clarify the link between translating combinatorial principles (ormore general statements from predicate logic) and the recent idea of using the symmetrical group to generate problems of propositional logic.Finally, we show that it is undecidable whether a sequence  psi_n (of thekind we consider) has polynomial size tree-resolution proofs or requiresexponential size tree-resolution proofs. Also we show that the degree ofthe polynomial in the polynomial size (in case it exists) is non-recursive,but semi-decidable.Keywords: Logical aspects of Complexity, Propositional proof complexity,Resolution proofs.

Hidden Markov models and neural networks for speech recognition

The Hidden Markov Model (HMMs) is one of the most successful modeling approaches for acoustic events in speech recognition, and more recently it has proven useful for several problems in biological sequence analysis. Although the HMM is good at capturing the temporal nature of processes such as speech, it has a very limited capacity for recognizing complex patterns involving more than first order dependencies in the observed data sequences. This is due to the first order state process and the assumption of state conditional independence between observations. Artificial Neural Networks (NNs) are almost the opposite: they cannot model dynamic, temporally extended phenomena very well, but are good at static classification and regression tasks. Combining the two frameworks in a sensible way can therefore lead to a more powerful model with better classification abilities. The overall aim of this work has been to develop a probabilistic hybrid of hidden Markov models and neural networks and ..

New Record-Breaking Condorcet Domains on 10 and 11 Alternatives

The study of large Condorcet domains (CD) has been a significant area of interest in voting theory. In this paper, our goal was to search for large CDs. With a straightforward combinatorial definition, searching for large CDs is naturally suited for algorithmic techniques and optimisations. For each value of $n>2$, one can ask for the largest CD, and we suggest that finding new record-sized CDs provides an important benchmark for heuristic-based combinatorial optimisation algorithms. Despite extensive research over the past three decades, the CD sizes identified in 1996 remain the best known for many values of n. When $n>8$, conducting an exhaustive search becomes computationally unfeasible, thereby prompting the use of heuristic methods. To address this, we introduce a novel heuristic search algorithm in which a specially designed heuristic function, backed by a lookup database, directs the search towards beneficial branches in the search tree. We report new records of sizes 1082 (surpassing the previous record of 1069) for $n=10$, and 2349 (improving the previous 2324) for $n=11$. Notably, these discoveries exhibit characteristics distinct from those of known CDs. We examine the structure of these newfound Condorcet domains. Our results underscore the potential of AI-driven and inspired techniques in pushing the boundaries of mathematical research and tackling challenging combinatorial optimisation tasks

A Tough Nut for Tree Resolution

One of the earliest proposed hard problems for theorem provers isa propositional version of the Mutilated Chessboard problem. It is wellknown from recreational mathematics: Given a chessboard having twodiagonally opposite squares removed, prove that it cannot be covered withdominoes. In Proof Complexity, we consider not ordinary, but 2n * 2nmutilated chessboard. In the paper, we show a 2^Omega(n) lower bound for tree resolution