The `document itself`

is available locally.
Errata corrected in this version (relative to the official version held at
the JHU library) are detailed below.

The `slide deck`

from my defense is
available in PDF form.

In addition to the thesis document itself, `an addendum`

is available, containing a more fully fleshed-out
variant of the `EarthBound`

algorithm of chapter 2.

Abstract:

We investigate the design of an expressive, purely-declarative, weighted logic programming language, Dyna. Dyna is a decade-plus effort in pushing the boundaries of declarative programming and “executable mathematics;” it instantiates an unusual point in the design space, as it is both Turing-complete (unlike Datalog) and devoid of a specified execution order (unlike Prolog). That is, it is designed to be, at once, both highly expressive and rich in opportunities for automated optimization. This thesis contains two major thrusts. We first consider both the denotational (§2.1.2 and §3.1.4) and operational aspects (§2.2 to §2.5, §3.2 to §3.6, and §4) of Dyna. In particular, for operational semantics, we introduce (§2.2) and extend (through §2.5) our EarthBound solver for finite circuits; §3 considers the generalization to logic programs proper. We then turn our attention to the static analysis of this language, considering mechanisms for reasoning both about abstract notions of well-formedness of programs (§5.2) as well as more mundane concerns of realizability of programs in actual computation (§5.3 and §5.4). Along the way we endeavour to place our work in the context of the larger field of logic programming languages and present our current thoughts on future avenues of exploration.

BibTeX:

```
@PhDThesis{filardo:dyna2,
author = {Nathaniel Wesley Filardo},
institution={Johns Hopkins University},
year = {2017},
month = {10},
title = {Dyna 2: Towards a General Weighted Logic Language}
}
```

# Errata¶

The definition of idempotence in “Properties of Functions of Bags” in section 1.3 used rather than .

Footnote 19, in Algebraic Structures, in section 1.3, has been corrected to include the overlooked case of equivalent elements and .

The formula, in section 1.4, following “(recursively) considering the refutation” erroneously contained , which should have been substituted away.

In 2.2.1, “item (or, later, set) of items whose” has been corrected to “item (or, later, set of items) whose”.

It has been pointed out that the prose in 2.2.4.1 has a reading suggesting that

`FreelyManipulateM`

can directly delete a memo. The intent, as, thankfully, correctly captured by the code listing, is that`FreelyManipulateM`

uses`Flush`

to update the memo table, thereby preserving Invariant 4. As no small edit would suffice, we hope that apologetic forewarning here in the errata is sufficient.At the end of 2.3.4, the word “information” was missing.

A small typographical error has been corrected in 2.3.5.3, with no impact to the meaning of the prose.

A small grammatical error has been corrected in 2.4.3, with no impact to the meaning of the prose.

Example 16 in 2.5 has been corrected; the old prose had the right setup but contained errors in the development of the story.

Years after writing this document, I was made aware of the 1989 paper “C-expressions: a variable-free calculus for equational logic programming” by Marco Bellia and M. Eugenia Occhiuto (https://www.sciencedirect.com/science/article/pii/030439759390170X and available without a paywall at https://core.ac.uk/download/pdf/81948651.pdf) which develops a formalism very similar to that of in section 3.1 and as considered again in chapter 4. For my oversight in not finding this work in my literature survey, I can only apologize profusely.

In 3.2.1, range restriction is equivalent to -SRR, not the nonsensical -SRR.

3.2.1.2 was missing a closing parenthesis in prose.

The proof of Lemma 2 in section 3.3.4 contained a typo: “becoming equal when ” had a in the submitted version. Thankfully, the lemma body itself is stated correctly.

In 3.4.2.2, and not, as erroneously stated, .

In 3.4.2.2, a was erroneously entered as .

In figure 5 of 3.4.3.2, was missing its TeX backslash.

In 3.4.4.2, syntactic nonsense was given when the term was meant.

The last paragraph before 3.6.2.1 had an erroneous cross-reference command.

In the discussion of Abstract Unification in 4.4.1.1, a TeXnical error resulted in being rendered as .

In 4.4.3, the prose has been corrected to read “If we then remove ”, rather than proposing the removal of the just-added “”.

Theorem 3 in 5.3.2.3 contained a pair of typos: the functions and were erroneously rendered as and .

Equation 5.10 in 5.3.5 had a which should have been a .

In 5.4, should have read .

In 6.1.1, the example was of the correct general shape, but used incorrect values, which have now been corrected.

In 6.2, an erroneus “we assume that all such sets are disjoint” has been stricken and replaced with the intended “these do not include any inherited rules.”

In 6.4, it was omitted that was to be summed to obtain the result given.

In the addendum,

`Drive`

used in its loop when it should have been using