My dissertation "Mathematics of Domains" is available below.

Our review paper linked from partialmetric.org: Michael Bukatin, Ralph Kopperman, Steve Matthews, and Homeira Pajoohesh. Partial Metric Spaces.

Our Tbilisi paper: Michael Bukatin and Steve Matthews. Linear Models of Computation and Program Learning. GCAI 2015,

**Our new paper**: Michael Bukatin and Jon Anthony. Dataflow
Matrix Machines and V-values: a Bridge between Programs and Neural Nets.
In "K + K = 120" Festschrift, 2017.

**Analytic Properties of Approximation-Based Models of
Computations: a Tutorial** is available as
tutorial.ps.gz. This tutorial explains
key ideas of our work on partial and relaxed metrics and co-continuous
valuations and measures on approximation domains via a simple example
of interval numbers.

Michael A. Bukatin, Svetlana Yu. Shorina.
On a Smyth Conjecture.
*Topology Proceedings*, ** 24** (1999), 57-70.
Free PDF on the site of Topology Proceedings

This electronic version of 06/19/00 is the same as the text published in the proceedings of 14th Summer Conference on General Topology and its Applications.

This paper is a follow-up of the following paper:

Michael A. Bukatin, Svetlana Yu. Shorina.
*Relaxed Metrics, Maximal Points, and Negative Information.*

A preliminary draft is available as neginfo.ps.gz. The comments and critique are welcome. Currently available is Version 0.6 of September 24, 1998.

This paper was presented at
MFPS 14
conference
(see also)
and is a follow-up of our earlier paper on
**relationships between measures and
generalized distances on domains:**

Michael A. Bukatin, Svetlana Yu. Shorina.
Partial Metrics and Co-continuous Valuations.
In M. Nivat, ed., Foundations of Software
Science and Computation Structures, *Lecture Notes in Computer Science*,
** 1378**, 125-139, Springer, 1998.

This electronic version of 01/08/98 is the same as the text published in LNCS 1378 --- the proceedings of the FoSSaCS, a part of ETAPS'98.

An extended preliminary version (Version 0.5 of October 14, 1997) is available as ccval_draft.ps.gz.

The paper introduces a new notion of co-continuity for valuations and shows how to build continuous partial and relaxed metrics from co-continuous valuations on domains.

Michael A. Bukatin, Joshua S. Scott.
Towards Computing Distances between Programs via Scott Domains.
In S. Adian, A. Nerode, eds., Logical Foundations of
Computer Science, *Lecture Notes in Computer Science*,
** 1234**, 33-43, Springer, 1997.

This electronic version of 03/07/97 is the same as the text published in LNCS 1234 --- the proceedings of the 4th International Symposium on Logical Foundations of Computer Science (LFCS'97).

We show that a computable distance between two equal partially
defined objects, *x=y*, must be non-zero under very weak
assumptions, thus making **partial metrics** by
Steve Matthews
and our own **relaxed metrics** the preferable
way to define semantically meaningful distances between
programs.

My co-author Joshua Scott received his undergraduate degree in Math from Brandeis (Spring 1996) and now studies at Northeastern. The key results were obtained during our joint studies on July 4, 1996 holidays.

Some extra details and an inductive,
rather than deductive, angle of presentation
can be found at
Towards Computing Distances between Programs via Domains
(version of 12/05/96).
It has a scary subtitle:
*a Symmetric Continuous Generalized Metric for
Scott Topology on Continuous Scott
Domains with Countable Bases*.
The inductive presentation means that
the results are presented in the sequence
and order they were discovered, and guiding motivations
at each step are emphasized, not hidden.
The deductive presentation (as in the published version
of our paper) means that the results are arranged into
an elegant logical scheme, from the general versions
of definitions to partial cases, but the traces of how this
scheme was obtained are largely erased.

The previous version of this paper (version 3.0 of 10/06/96) is somewhat longer, but it does not contain the references to the related papers by S. G. Matthews and by S. J. O'Neill and discussion of the differences between our construction and the one by O'Neill, as we did not know that these results are related. It was presented on 10/09/96.

I discussed my latest efforts in this direction during my talk Relaxed Metrics on Domains, Measures and Invariance given at Third Northeastern Conference on Topological Methods in Programming Language Semantics.

The following three papers study domains for denotational semantics as sets of theories of logical calculi. They are reviewed in my dissertation proposal.

Subdomains for Algebraic Information Systems solves the question "which subsets of domains should be considered subdomains?".

Information Systems and Retractions: an Elementary Treatment studies finitary retractions.

Information Systems and Complete Lattices is an abstract of my talk on the intuition behind the use of non-reflexive logics for description of non-algebraic domains (see my dissertation proposal for details).

This material was published as:

Michael A. Bukatin. Logic of Fixed Points and Scott Topology. Topology Proceedings, 26, 2002, 433-468.

My dissertation proposal is a review of the field and a position paper. It is fairly controversial, because it is trying to make a case that things are pretty bad in the programming language theory and practice, and because it is also trying to explain why they are so bad. Comments and flames are welcome at this e-mail address. To retrieve it click on the title:

Continuous Functions as Models for Programs: Mathematics and Applications.

The current version of the complete dissertation is available below.

- Part 1. Introduction.
- Part 2. Logic of Fixed Points for Domains.
- Part 3. Elements of Analysis on Domains.

Final version (January 6, 2002), 135 pages.

Dissertation draft (gzipped PostScript, 245K): Mathematics of Domains (PDF, 571K): Mathematics-of-Domains.pdf

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