Universal Instance Research Articles

THE REVERSE MATHEMATICS OF THE THIN SET AND ERDŐS–MOSER THEOREMS

AbstractThe thin set theorem for n-tuples and k colors ( $\operatorname {\mathrm {\sf {TS}}}^n_k$ ) states that every k-coloring of $[\mathbb {N}]^n$ admits an infinite set of integers H such that $[H]^n$ avoids at least one color. In this paper, we study the combinatorial weakness of the thin set theorem in reverse mathematics by proving neither $\operatorname {\mathrm {\sf {TS}}}^n_k$ , nor the free set theorem ( $\operatorname {\mathrm {\sf {FS}}}^n$ ) imply the Erdős–Moser theorem ( $\operatorname {\mathrm {\sf {EM}}}$ ) whenever k is sufficiently large (answering a question of Patey and giving a partial result towards a question of Cholak Giusto, Hirst and Jockusch). Given a problem $\mathsf {P}$ , a computable instance of $\mathsf {P}$ is universal iff its solution computes a solution of any other computable $\mathsf {P}$ -instance. It has been established that most of Ramsey-type problems do not have a universal instance, but the case of Erdős–Moser theorem remained open so far. We prove that Erdős–Moser theorem does not admit a universal instance (answering a question of Patey).

On a generalization of “Eight Blocks to Madness” puzzle

We consider a puzzle such that a set of colored cubes is given as an instance. Each cube has unit length on each edge and its surface is colored so that what we call the Surface Color Condition is satisfied. Given a palette of six colors, the condition requires that each face should have exactly one color and all faces should have different colors from each other. The puzzle asks to compose a 2×2×2 cube that satisfies the Surface Color Condition from eight suitable cubes in the instance. Note that cubes and solutions have 30 varieties respectively. In this paper, we give answers to three problems on the puzzle: (i) For every subset of the 30 solutions, is there an instance that has the subset exactly as its solution set? (ii) Create a maximum sized infeasible instance (i.e., one having no solution). (iii) Create a minimum sized universal instance (i.e., one having all 30 solutions). We solve the problems with the help of a computer search. We show that the answer to (i) is no. For (ii) and (iii), we show examples of the required instances, where their sizes are 23 and 12, respectively. The answer to (ii) solves one of the open problems that were raised in Berkove et al. (2008).

Degrees bounding principles and universal instances in reverse mathematics

A Turing degree dbounds a principle P of reverse mathematics if every computable instance of P has a d-computable solution. P admits a universal instance if there exists a computable instance such that every solution bounds P. We prove that the stable version of the ascending descending sequence principle (SADS) as well as the stable version of the thin set theorem for pairs (STS(2)) does not admit a bound of low2 degree. Therefore no principle between Ramsey's theorem for pairs (RT22) and SADS or STS(2) admits a universal instance. We construct a low2 degree bounding the Erdős–Moser theorem (EM), thereby showing that the previous argument does not hold for EM. Finally, we prove that the only Δ20 degree bounding a stable version of the rainbow Ramsey theorem for pairs (SRRT22) is 0′. Hence no principle between the stable Ramsey theorem for pairs (SRT22) and SRRT22 admits a universal instance. In particular the stable version of the Erdős–Moser theorem does not admit one. It remains unknown whether EM admits a universal instance.

On Concrete Universals : A Modern Treatment Using Category Theory

Today it would be considered bad metaphysics to think that among all the concrete instances of a property there could be a universal instance so that all instances had the property by virtue of participating in that concrete universal. Yet there is a mathematical theory, category theory, dating from the mid-20th century that shows how to precisely model concrete universals within the Platonic Heaven of mathematics. This paper, written for the philosophical logician, develops this category-theoretic treatment of concrete universals along with a new concept to abstractly model the functions of a brain.

On Concrete Universals: A Modern Treatment Using Category Theory

Today it would be considered "bad Platonic metaphysics" to think that among all the concrete instances of a property there could be a universal instance so that all instances had the property by virtue of participating in that concrete universal. Yet there is a mathematical theory, category theory, dating from the mid-20th century that shows how to precisely model concrete universals within the "Platonic Heaven" of mathematics. This paper, written for the philosophical logician, develops this category-theoretic treatment of concrete universals along with a new concept to abstractly model the functions of a brain.

An algorithmic view of pseudochaos

The controversial concept of quantum chaos – the dynamical chaos in bounded mesoscopic quantum systems – is presented as the most important and universal instance of a new generic dynamical phenomenon: pseudochaos. The latter characterizes the irregular behaviour of dynamical systems with discrete energy and/or frequency spectrum, which include classical systems with discrete phase space. The question of randomness is addressed in terms of the algorithmic theory of dynamical systems, using the Arnold cat map for illustration. The relationship between the algorithmic theory and ergodic theory is discussed.

The myth of man's fall and the basic rule for psychoanalytic therapy discovered by sigmund freud: Free association—a program against hubris

Siirala M. The of Man's Fall and the Basic Rule for Psychoanalytic Therapy Discovered by Sigmund Freud: Free Association-A Program against Hubris. Int Forum Psychoanal 1997;6:33-39. Stockholm, ISSN 0803-706X. The title contains the thesis of the paper. Hubris is here: the godlike position of knowing good and evid. The mythical story about man's self-estrangement, as it is told in the biblical canon about three thousand years ago, depicts more a rise than a fall, as it entails a seizure of the throne of the one and absolute “Thou”. On good grounds that myth can be regarded as a particularly universal instance of man's self- and existence interpretation. In any case it seems obvious that the basic rule, as it implies giving voting right to everything announcing itself in man, even and especially to all that is defying his “already-knowing”, his open or latent judgements, is of crucial value on his way towards an authentic wholeness as a person. Consequences for the conception of not only therapeutic knowledge but knowledge in general, are discussed. How the myth of man's fall throws essential light upon deep depression, especially as met in psychoanalytic therapy, is demonstrated by means of two case histories.

Query abbreviation in the Entity-Relationship data model

This paper examines the idea of context inference, which is the ability of a query processor to infer the appropriate relationships between the classes mentioned in a query. Context inference has been the basis of Universal Instance (UI) data models. We show that context inference can be applied to the Entity-Relationship model, resulting in a compact, high-level language without the undesirable restrictions associated with UI databases. We also show how the traditional notions of inheritance and path expressions can be considered to be special cases of context inference.

An extended universal instance data model

The universal instance (UI) data model is extended to include the semantic notion of generalization as well as specialization. It is shown how this extension gives the model significantly more modelling capability and overcomes the well-known limitations of other UI models. In particular, the extended model allows both multiply defined attributes and multiple independent relationships. Consequently, the extended UI model has the modelling capability of ER-diagrams, together with the capability to handle nonnavigational (i.e. attribute-based) queries. The semantics of both the standard and extended UI models are also studied, and it is shown that UI models have an unexpectedly rich semantics. Finally, some comparisons with object-oriented models are made. The extended UI model is a purely value-oriented model, but it appears to parallel object-oriented models closely. Some consequences of this similarity are discussed.

An Extension of "Representative Instances and γ-Acyclic Relational Schemes"

Let R be a γ-acyclic relational scheme, and let F be the set of functional dependencies (FD's) embodied in R. Given an existence constrained database r over R, it was shown in [1] that it is possible to connect tuples from different relations in r and construct a universal instance L, possibly containing null values δ, such that the total projection of L onto R yields exactly the set r. Moreover, conditions were given which guarantee that this L would satisfy the functional dependency with nulls (NFD) counterparts of FD's, in F. The purpose of this note is to generalize the latter result and show that under the same conditions, L actually satisfies NFD counterparts of FD's in the closure F+ of F.

Construction of Universal Instances for Loop-Free Network Databases Using a Join-Like Operation

In this paper, we give a polynomial-time method to construct effectively the unique universal instance, using as few nulls as possible, from any loop-free network database, via a minimal information extension of natural join. Our results can be seen as concretely and quickly implementing the universal relation view for databases which are not pairwise consistent.

Representative instances and γ-acyclic relational schemes

In this paper, we study under what conditions will a pairwise inconsistent relational database ≪R,r≫ have a universal/representative instance L. If R is γ-acyclic and r satisfies all existence constraints, then it is possible to construct a universal instance L, using unmarked nulls, whose total projections onto R yield exactly the relations in r. We show that L would actually be a representative instance under a set of functional dependencies if R satisfies the additional mild condition: for any functional dependency X → A where A is a single attribute, whenever XA is contained in two relation schemes R and R' of R, it follows that R ∩R' is a relation scheme of R, having X as one of its keys.

A characterization of globally consistent databases and their correct access paths

The representative instance is proposed as a representation of the data stored in a database whose relations are not the projections of a universal instance. Database schemes are characterized for which local consistency implies global consistency. (Local consistency means that each relation satisfies its own functional dependencies; global consistency means that the representative instance satisfies all the functional dependencies.) A method of efficiently computing projections of the representative instance is given, provided that local consistency implies global consistency. Throughout, it is assumed that a cover of the functional dependencies is embodied in the database scheme in the form of keys.