Martin Dowd

hyperon software

A definition of arithmetic proof systems for polynomial time equations is given. Say that an equation t = u is hard for the system Π if its bounded instances do not have polynomial length proofs. It is shown that a sufficient condition that a polynomially bounded proof system for the tautologies does not exist, is that every regular arithmetic proo...

It is easy to construct families of polynomial time equations which do not have polynomial length proofs in polynomial time arithmetic. The question is considered, whether such families are of any use in proving that polynomial time arithmetic is not a polynomially bounded proof system for the tautologies.

Recently, arguments have been given in the literature that the continuum hypothesis (CH) might be false. Here, an argument from basic set theory is given, that CH could be true. The existence of this debate illustrates aspects of mathematical knowledge.

Attempts to provide an independent verification of Pfeiffer's method for constructing systems of fundamental sequences have encountered difficulties. Particular questions arise. These can be resolved up to a point, resulting in a system of fundamental sequences of interest.

Russell's paradox raises questions about the nature of the universe of discourse of set theory. These in turn reflect on the question of whether large cardinals exist. Looking at the question from this perspective yields strict criteria for the possible existence of large cardinals.

The "bounded instances" of an arithmetic formula F are the formulas "∧ j |x j | ≤ n ⇒ F ". For a polynomial time equation F Cook defined the "representing formulas", which are propositional formulas that "say the same thing" as the bounded instances. This can be formalized, and the following is true. A "regular arithmetic proof system" Π can be con...

Let LA be polynomial time arithmetic. Let L E 2 A be double exponential time arithmetic. It is shown that Con LA is provable in L E 2 A. Some systems are shown to be equivalent to L E 2 A. It follows that Con LA+Con LA is not provable in L E 2 A. Some questions related to these facts are discussed. It also follows that the bounded instances of Con...

The paper https://www.researchgate.net/publication/ 283750877_New_cases_of_reconstructibility_of_SBT_graphs contains a definition of S-minimal graphs, and an algorithm for enumerating them. The file "fsmg_rel.tgz" provided here contains the source code of an implementation of the algorithm. It also contains data files containing the output graphs,...

The Mahlo rank of any cardinal can be defined. Computing it requires demonstrating how the cardinal is "built up" by iterating the power set operation. The question is considered here, of how the Mahlo rank of a measurable cardinal bears on its Mitchell order. It is shown that the Mitchell order of κ is less than κ ++ , whence supercompact cardinal...

Let LA be polynomial time arithmetic. Let L E 2 A be double exponential time arithmetic. In a previous manuscript the author has given a proof that the consistency of LA is provable in L E 2 A. Here a proof is given that the consistency of LA + Con LA is not provable in L E 2 A. A question remains about this proof.

The Mahlo rank of an inaccessible cardinal is a measure of its "Mahlo-ness". The notion may be considered for Card, the class of cardinals. A discussion is given of properties of this notion.

In recent set theory literature there is much advocacy for a liberal approach to resolving questions of set theory, which are independent of ZFC. In this paper, a conservative approach is considered. Specific questions may be considered, and hypotheses made, based on known behavior of the cumulative hierarchy. The following topics are considered: t...

In 1975 Cook conjectured that the representing formulas of the consistency of PV do not have polynomially bounded extended resolution proofs. In 2020 the author noted that it is a question of interest, even whether it is provable in PV from Con PV that such proofs exist. Here, it is shown that this question has a negative answer. Some related theor...

Some arguments in favor of the continuum hypothesis are given.

Tnis archive contains programs which were used to produce some tables included in the manuscript.

In 1964 H. Pfeiffer defined systems Σ n p of fundamental sequences. To show that Σ n p+1 exists it must be shown that Σ n+1 p has certain properties. This paper represents an effort to provide a new treatment of this problem. Some axioms are defined, and an effort is made to show that Σ n+1 1 satisfies them. Various properties of Σ n+1 1 are shown,...

A form of universal representing formula can be defined in polynomial time arithmetic. Using this, Godel diagonalizations can be performed. It is a question of interest, whether self-referential propositional formulas can be constructed. Preliminary results suggest that this is not straightforward, but the question seems worthy of further considera...

The scheme Σ 2 1 over κ, first defined by Pfeiffer, is used to obtain a lower bound on the smallest repeat point at a measurable cardinal κ.

B-schemes were introduced by the author in a previous manuscript. This paper introduces a new class of schemes, called P-schemes. In 1964 H. Pfeiffer defined schemes Σ n p for n, p ∈ ω. It is a question of interest whether these are P-schemes. It is shown that Σ n 1 is a P-scheme, whence Σ n 2 exists. This is ongoing research, and efforts will be c...

There is a German-English mathematics dictionary at archive.org/details/germanenglishmat00hyma The images are jpx encoded and take a while to render. Also it is not searchable. This version is jpeg encoded, so displays faster, and is searchable. This appears to be a "copyright orphan". The archive.org version has been available since 2012.

This is a toolkit for improving scanned images for OCR. C source code is provided. It was used while creating https://www.researchgate.net/publication/352998687.

In an earlier paper the author defined a proof system Υ Π , and proved facts about it and its relevance to Cook's conjecture. Further facts are proved here. A proof-theoretic equivalent of the full Cook conjecture is given.

TM configurations were defined in [5]. A class of them with properties of interest is constructed. Results of some additional computations on N 1L configurations are given. AMS Subj. Classification: 94B05

Arguments are presented that V = L, indiscernibles do not exist, and certain small large cardinals do exist.

Cook's conjecture may be generalized. The generalized conjecture has specializations of interest. Various facts regarding this topic are presented. A conjecture is made which implies P =NP. Some special cases of this conjecture are stated, one of which is the conjecture of [7],[8] on the specialization for EF. Finally, some further remarks on the t...

An integrated space program may be defined as a coordinated ensemble of components. Such can accomplish space missions in a flexible and cost-effective manner. An example outline is given.

This archive contains output from the computations used to obtain table 1 of "Solutions to the 1-design equations".

This archive contains source code for three optimizations discussed in "Some Mars Trajectory Optimizations"

This archive contains source code for a example implementation of Newtonian orbit reduction, as discussed in appendix 2 of "Some Mars Trajectory Optimizations". An earlier discussion can be found in "Partial derivatives of the universal anomaly with respect to orbital parameters".

This archive contains source code for some computations in "A simple model for genetic algorithm convergence".

This file provides the source code for adding HyperC to gcc-4.1.1.

A propositional modal language is defined wherein statements may refer to their own truth or falsity. A notion of validity is defined, and a complete proof system given. A system similar to both this and Solovay's modal logic of provability is also studied.

It is suggested that lightweight object-oriented extensions of the C programming language have advantages over the C++ language. An example such extension is presented.

Let V denote the sentances in the language of polynomial time which are true in the integers, and let Π 1 V denote the true Π 1 sentances. By combining results of [Ma] and [Do] we show that if P = N P , and if every model of Π 1 V has an end extension which is a model of V , then N P = Co−N P. A question concerning models of V which implies the exi...

Arbitrary well-founded relations are considered, generalizing constructions involving WPS's given in [9]. Some new facts iconcerning function chains and set chains are given, and a new axiom for set theory.

It is shown that the bounded consistency statements of LA have polynomial length LA proofs. It follows that conjectures 2 and 7 of [3] are equivalent. Some new facts relevant to conjecture 7 are proved. A question concerning L E 2 A is raised.

In [1] Cook conjectures that the representing formulas for the consistency of polynomial time arithmetic do not have polynomial length extended resolution proofs. It is known that such proofs do not exist, prov-ably in polynomial time arithmetic. This question can be specialized, to ask whether such proofs can be shown to exist, in even very mild e...

In [11] Williamson shows that it follows in ZFC from a statement which is independent of ZFC, that some instances of the subset sum problem can be solved in polynomial time. He goes on to suggest that P=NP might not be provable in ZFC. Here we argue that as of present knowledge, it doesn't seem any easier to prove that P=NP is unprovable in polynom...

As the title suggests, this book is intended to provide an introduction to modern set theory, to readers with little or no knowledge of mathematical logic. As such, it should be useful to anyone interested in learning about modern set theory.

The purpose of this text is to provide a rigorous treatment of vector calculus, for interested second year undergraduates. Students with an interest in rigor benefit from placing the study of vector calculus on a firm theoretical foundation. This serves as an intellectual exercise, which develops new skills and interests, even in students more inte...

ABSTRACT: Several formal systems related to primitive recursive arithmetic are defined. These are primitive recursive arithmetic in m-adic notation (PRAm), for all m\geq$1; the extension of this containing multiple-digit recursion (PRBm); the extension of the latter containing multiple variable recursion (PRCm); and bounded versions of each of thes...

This text is intended as an introduction to abstract algebra for advanced undergraduates. Much of it is accessible to advanced third year students, and the first nine chapters can be covered even earlier. There are many introductions to abstract algebra, so a new one should have some distinguishing characteristics. The main distinguishing character...

Some new proof theoretic results are presented, concerning PV and propositional representation; and two new simulations between propositional proof systems. It is shown that suitably stated the consistency with PT of the non polynomial boundedness of all propositional systems is equivalent to P = N P. It is also shown that whether extensions of mod...

A variation on Bachmann's restrictions on a system of fundamental sequences is considered. A notation system is given for the closure ordinal of a scheme Σ¯ θ Ω +. Using this a scheme Σ¯ θ¯ ¯ θ Ω ++ is constructed. This is used to obtain an improved lower bound on the smallest repeat point of a measurable cardinal, to that of [4].

Schemes may be used to define systems of ordinal functions. Such systems may be used to obtain lower bounds on the smallest repeat point. Using a scheme Σ CTT , it is shown that the smallest ordinal θ T such that there is no T-separating set at θ T is at least an ordinal which is larger than the analog of the Bachmann-Howard ordinal for the Bachman...

An ordinal larger than the Bachmann-Howard ordinal is defined, as the closure ordinals of a scheme longer than ǫ Ω 0. This gives an example for considering general properties of schemes. For example, the notion of "built-up" "CNF-standard" schemes is considered. An improvement to the lower bound of [4] on the smallest repeat point is given.

In previous papers the author has considered set chains mod the club filter. Here they are considered mod the Π 1 n-enforce-able filter. Remarks are made on repeat points, including sufficient conditions for their non-existence.

In a previous paper the author used methods of Witzany to give a lower bound for the smallest repeat point of a coherent sequence. Here the notion of a T-separating set is introduced, and the lower bound is improved.

The question is considered, whether for some limit ordinal α, Lα has an infinite set of indiscernibles. This is true if α is an ω-Erdos cardinal. Whether the hypothesis can be weakened is a question of interest.

N1L configurations occur in linear double error correcting codes. Bounding their length is a question of interest. A conjecture is given which would yield a lower bound.

New axioms for set theory have already been given by the author making use of ∑11 well-founded relations, whose definition "reflects" in a club (i.e., defines a well-founded relation on a club of cardinals). The method is generalized by requiring only that the definition reflect in a stationary set, for some specific stationary sets.

A variant of reconstructibility of colored graphs is defined, and some facts proved. Some computational facts from an earlier paper are revised.

Downsets over a regular cardinal κ arise from Π11 normal form. For inaccessible κ downsets correspond to the trees of generalized descriptive set theory. Basic properties of downsets, trees, and the correspondence are considered. A new characterization of weak compactness is given. Some facts about Σ11 WPS's are proved.

In earlier papers the author showed that all graphs which are not single-block trunk (SBT) graphs are reconstructible, and that two families of SBT graphs are reconstructible. Here some further families of SBT graphs are shown to be reconstructible.

In a previous paper the author found some minimal 1-(v,3) designs with large b, by exhaustive search. Here some further such are found, by more ad-hoc methods. A construction is given for v = 3t+1 for t ≥ 2, where b grows quadratically with v.

A normal form is given for ∏(Formula presented.) formulas over Vκ for a regular cardinal κ. Using it, a new characterization of weakly compact cardinals is given.

In an earlier paper the author has shown that a graph is reconstructible, except possibly if it has a single-block trunk. Here this result is shown for colored graphs. Some other results on colored graphs are given. In particular it is shown that reconstructibility is false for edge colored graphs.

Appropriate restrictions may be placed on a Σ11well-order on Vκwhere κ is an inaccessible cardinal, so that it gives rise to a function chain when κ is a Mahlo cardinal. Set chains may be defined for all second order formulas. Postulating that the sets in the resulting set chain are stationary yields a powerful new axiom.

The author has shown that a separable graph with more than one block which is not an edge is reconstructible. Here reconstructibility is proved for a single such block, in two particular cases.

Function and set chains can be constructed using ordinal notation systems. Here, earlier results are improved, and longer chains constructed. In particular, a simpler method is given for constructing set chains.

It is shown that the multiset of reconstruction trees of the connected components of a graph is strongly reconstructible. It is then shown, that an annotated version of the block-cutpoint tree is strongly reconstructible. A refinement of this result is given. Some cases of reconstructibility of separable graphs are given. A conjecture which implies...

The 1-designs with given values for the parameters v and k can be characterized as nonnegative integer solutions to a system of linear equations with integer coefficients. Some facts about these equations are proved.

In earlier papers the author has used schemes to establish some bounds in small large cardinal theory. Here the methods are improved, yielding improved bounds.

In previous papers by the author, schemes were used to specify some new axioms for set theory, to give a lower bound on the Mahlo rank of a weakly compact cardinal, and to give a chain in the Galvin-Hajnal order with properties of interest. Here, various improvements are made.

The Mahlo rank of a cardinal is a measure of how often the Mahlo operation may be iterated, starting with the inaccessible cardinals, while still retaining a stationary set. It is a question of considerable interest, what the least value of this can be for a weakly compact cardinal. A lower bound is shown, of a right associated exponential stack of...

It is suggested that new axioms can be added to ZFC by proceeding cautiously. Some examples are given, together with justifications. A discussion of V = L is also given.

It is straightforward that, if GCH holds, then the Mahlo rank of an inaccessible cardinal κ is less than κ ++ . If it could be shown that the Mitchell order of a measurable cardinal is at most its Mahlo rank, it would follow that the Mitchell order is less than κ ++ . It is shown that the Mitchell order is at most the Mahlo rank if the Mitchell ord...

A simple model of genetic algorithm convergence is presented. An "ideal" iteration may be specified, in terms of a random variable for the weight distribution. The behavior of an "actual" iteration may in some cases be sufficiently close to the ideal behavior that useful quantitative information is obtained. In any case, a useful framework for anal...

In an earlier paper the author defined N1L configurations, and stated a conjecture concerning them which would lead to an improvement by a constant factor to the sphere-packing bound for linear double error correcting codes. Here a computer search is presented, in an effort to gather evidence on the conjecture.

Some theorems are proved about configurations in binary linear codes. A conjecture is made about configurations in linear double error correcting codes which would lead to an upper bound better by an factor than the sphere packing bound.

A construction of limits and colimits in the category of topological structures over a first order language is given. The construction readily specializes to the full subcategory of models of an equational theory.

The principle of “collecting the universe” justifies axioms asserting the existence of large cardinals which can be “built up” by iterating Mahlo’s operation. In a previous paper [the author, Math. Log. Q. 39, 79–95 (1993; Zbl 0803.03030)] “schemes” were used to define iterations of length up to κ + . This paper gives a method for defining iteratio...

Using the methods used to obtain O(log N) routings of permutations, basic algorithms on the star graph which are faster than those previously reported in the literature are obtained. In particular, an O(log2N) sort and an O(log N) Fourier transform are presented. A simulation of meshes is also given.

The star-connected cycles (SCC) graph was recently proposed as an attractive interconnection network for parallel processing, using a star graph to connect cycles of nodes. This paper presents an analytical solution for the problem of the average distance of the SCC graph. We divide the cost of a route in the SCC graph into three components, and sh...

The star-connected cycles (SCC) graph was recently proposed as an attractive interconnection network for parallel processing, using a star graph to connect cycles of nodes. The paper presents an analytical solution for the problem of the average distance of the SCC graph. They divide the cost of a route in the SCC graph into three components, and s...

Star connected cycles are shown to be an undirected Cayley graph, and the graph automorphism group is determined. A routing algorithm is given, which finds an optimal path in polynomial time. The diameter, and tight bounds on the average distance, are computed. Keywords: Distributed computing, interconnection network, routing, star connected cycles...

Binary tree structures have been very useful in solving divide-and-conquer type of problems. Embedding binary trees into another network-the host network-helps in designing solutions for the host network using the known solutions on binary trees. Embedding arbitrary binary trees into networks, in particular into the hypercube, has been addressed in...

An enumeration of star graphs is given which has many useful properties. For example an arbitrary prefix or suffix is connected; indeed the diameter is O(n). As a consequence, there is an O(n) interval broadcast algorithm. Prefixes which have t(n-1)! vertices for some t are especially well-behaved. The topology of, embeddings in, and algorithms for...

The question of simulating a completely healthy hypercube with a degraded one (one with some faulty processors) has been considered by several authors. We consider the question for the star-graph interconnection network. With suitable assumptions on the fault probability, there is, with high probability, a bounded distance embedding of K<sub>n</sub...

The problem of whether an assignment in the hypercube H_n, where the distance from the source to the destination is bounded, can be routed with minimum distance and bounded congestion is considered. It is shown that this is so, if assignments of given “type” can be so routed. Of particular interest is whether for distance 3 and fixed type...

Higher types can readily be added to set theory, Bernays-Morse set theory being an example. A type for each ordinal is added in [2]. Adding higher types to set theory provides a neat solution to the problem of how to handle higher type categories. We give the basic definitions, and prove cocompleteness of some higher type categories. MSC: 14A15.

Adding higher types to set theory differs from adding inaccessible cardinals, in that higher type arguments apply to all sets rather than just ordinary ones. Levy's reflection axiom is justified, by considering the principle that we can pretend that the universe is a set, together with methods of Gaifman [8]. We reprove some results of Gaifman, and...

The basic properties of generic oracles are reviewed, and proofs given that they separate and and are weakly incompressible. A new notion of generic oracle, called t-generic, is defined. It is shown that t-generic oracles do not exist, and consequently a nondeterministic oracle machine which for any oracle X accepts the tautologies relativized to X...

For each value of the parameters A,n,d, a linear program exists whose integer solutions correspond to codes. The Plotkin bound gives a necessary and sufficient condition on n/d for feasibility. Some further simple remarks on the tableau of the linear program can be made; it can also be modified to consider only linear codes. For A divisible by 4, c...

A periodic sorting network consists of a sequence of identical blocks. In this paper, the periodic balanced sorting network, which consists of log n blocks, is introduced. Each block, called a balanced merging block, merges elements on the even input lines with those on the odd input lines. The periodic balanced sorting network sorts n items in O (...

The partial derivatives of the universal anomaly with respect to the orbital parameters have been derived, with either t or r as the independent variables. The theory is applied to an orbit determination example using three right ascensions and declinations with a moving observer. The case of the minimization of transit time for orbit transfer is a...

Neural networks "compute" though not in the way that traditional computers do. It is necessary to accept their weaknesses to use their strengths. We discuss some of the assumptions and constraints that govern operation of neural nets, describe one particular ...

Neural networks "compute" though not in the way that traditional computers do. It is necessary to accept their weaknesses to use their strengths. We discuss some of the assumptions and constraints that govern operation of neural nets, describe one particular ...

New relationships between questions in combinatorial number theory are shown. The Erdos-Turan conjecture is shown to be equivalent to the finite problem; the Erdos-Turan conjecture for N is shown to follow from that for Z; and the additive number theory of the integers and that of the finite bit strings under bitwise mod 2 addition are contrasted....

An Armstrong relation for a set of functional dependencies (FDs) is a relation that satisfies each FD implied by the set but no FD that is not implied by it. The structure and size (number of tuples) of Armstrong relatsons are investigated. Upper and lower bounds on the size of minimal-sized Armstrong relations are derived, and upper and lower boun...

This paper introduces a new sorting network, called the balanced sorting network, that sorts n items in O([lgn]2) time using (n/2)(lgn)2 comparators. Although these bounds are comparable to many existing sorting networks, the balanced sorting network possess some distinct advantages. In particular, its structure is highly regular consisting of a se...

Equations f@@@@ &equil; g@@@@ between polynomial time computable functions can be represented by sets of propositional formulas. If f@@@@ &equil; g@@@@ is provable in certain arithmetic systems, then polynomial length proofs of the representing formulas exist in certain propositional systems. Two cases of this phenomenon and a general theory are gi...