Arbori; the missing manuals

February 11, 2017

Number of lines of code is well known to be poor metric for virtually any purpose. One thing is indisputable though: the total volume of code is growing. Today, developers are frequently challenged with mining huge codebases. Therefore, querying parse trees is not just an academic exercise, but something of ever increasing practical value.

Blog post, though, is hardly a proper venue to introduce a new language. Here are two references to supplement it:

  1. Arbori Starter Manual by Eugene Perkov
  2. Little more detailed Semantic Analysis with Arbori

Formatting is one of the software development features that always steers controversy. To put it bluntly, everybody wants to format the code their own way. In previous SQL Dev versions the answer to that challenge was having elaborate tree-table widget spiced with several dozens of various options.

Arguably, the combined SQL and PL/SQL grammar is the most cumbersome programming language in existence, with total number of grammar productions exceeding 20000. From that perspective, the list of “Advanced Format” options offered by SQL Dev feels inadequate. To address this mismatch, the latest release introduced “Custom Format”. It offers to a developer means to specify formal conditions which SQL and PL/SQL grammatical constructs to indent,  align, pad, add extra line breaks and so on.

As usual, the best way to demonstrate a new feature is an example. Consider the following enhancement request by Stefan Poschenrieder to remove the indent of the NOT NULL ENABLE inline constraint in the following code

CREATE TABLE dim_aspect (
    dim_aspectid       NUMBER(11,0)
        NOT NULL ENABLE,
    source             VARCHAR2(50),
    hier1_desc         VARCHAR2(50),
    hierarchy1         VARCHAR2(50),
    dwh_inserteddate   DATE
);

The first step is identifying the target grammar symbol in the Code Outline panel. Here is screen snapshot illustrating it:
code outline.png
As illustrated there, positioning a cursor over the NOT keyword highlights the corresponding node in the parse tree. Therefore, the grammar symbol of interest is probably “inline_constraint“. Clicking onto the node labeled inline_constraint to double check this guess witnesses it highlighting “NOT NULL ENABLE” in the code editor. (Please be aware that clicking onto a hyperlinked looking grammar symbol has side effect of opening documentation page featuring that symbol railroad diagram definition).

As soon as the grammar symbol of interest is known, lets find it at Custom Format preferences panel. Admittedly, glaring omission there is the search widget. Yet, grammar symbols are arranged in lexicographic order, so that a reader would have no trouble spotting inline_constraint in the following formatting rule:

...
| [node) index_subpartition_clause[69,119)# 
| [node) inline_constraint & ![node) inline_constraint[14,67)
| [node) inline_ref_constraint
...

This fragment of rather convoluted disjunctive condition instructs the formatter to indent the nodes with payload “index_subpartition_clause[69,119)#“, or the nodes labeled with “inline_constraint” (but not those that also labeled as “inline_constraint[14,67)“), or the nodes labeled “inline_ref_constraint“.  Therefore, commenting out the middle conjunction

...
| [node) index_subpartition_clause[69,119)# 
--| [node) inline_constraint & ![node) inline_constraint[14,67)
| [node) inline_ref_constraint
...

is likely a solution to the problem. After making this change, a reader is advised to test what effect do modified formatting rules have on a SQL or PL/SQL code sample in the preview editor. This action is performed with the bottom-left “Run” button. The two neighbors “Import…” and “Export…” are for saving and retrieving your work. By default the modified “format.arbori” file is kept in the SQLDev product preferences directory.

Let’s reinforce this custom formatting idea with two more examples, both from Steven Feuerstein. In both cases we’ll learn some additional syntax of that formal formatting rules specification. The first example is just a bug witnessed by the following test case:

CREATE OR REPLACE PACKAGE BODY plch_pkg IS BEGIN
   FOR indx IN 1..100000 LOOP
      g_strings(    
         indx       --<--- ???
      ) := 'String ' || indx;
   END LOOP;
END;
/

The rule responsible for that unwanted indentation of the “indx” parameter is

| [node) pls_expr & [node-1) '(' & ![node) numeric_literal

It requires the parse node to be labeled with “pls_expr” but not “numeric_literal“. Also, it stipulates that the prior sibling node (formally, “node-1“) is labeled with open parenthesis. Adding the condition for the node successor not to be the closing parenthesis

| [node) pls_expr & [node-1) '(' & ![node) numeric_literal & ![node+1) ')' 

is all it takes to fix this snag.

In the second example

CREATE PACKAGE fmt AS
    PROCEDURE create_checklist (
        user_id_in       IN INTEGER,
        question_id_in   IN INTEGER
    );
    --<-- wanted extra line break 
    PROCEDURE remove_checklist (
        checklist_id_in   IN INTEGER
    );

    PROCEDURE remove_checklist (
        user_id_in       IN INTEGER,
        question_id_in   IN INTEGER
    );

END fmt;

the formatted code have only single line breaks after each package member. The formatting rule responsible for this functionality is “extraLines“. It formally a disjunction of the two rules “sql_stmts” and “significant_statements“. In the later rule we see the following conjunctive condition:

| [node) basic_decl_item & [node+20 < node)

The syntax of first part is familiar: it requires the node to be labeled as “basic_decl_item“.

The second part is little more cumbersome. The “[ node” and “node )” refer to the node beginning and ending position, correspondingly. Each node in the parse tree recognizes a sequence of tokens. In our example nodes labeled as “basic_decl_item” recognizes a sequence of tokens beginning with the PROCEDURE keyword and ending with semicolon. The beginning position is the offset of the PROCEDURE keyword, while the ending position is the offset of the semicolon. The condition  “[node+20 < node)” requires more than 20 tokens between the node beginning and ending position, or informally the package member to be of “sufficient length”.

Please note that package data members are recognized as “basic_decl_item” as well. Therefore, the proper fix depends on if a user wants to inset double line breaks after data members or not. The length condition was a naive attempt on distinguishing data and procedures, but more elaborate condition, distinguishing data and procedures is possible too.

Let’s walk trough yet another example demonstrating how flexible it is. Consider the following example adapted from ToadWorld ER:

Can we vertically align THEN values of CASE statement ? Current Format:

SELECT CASE
         WHEN a.col5 = b.col5 THEN 'COL5'
         WHEN a.col55555555555555 = b.col55 THEN 'COL55555'
         WHEN a.col555555 = b.col55555 THEN 'COL555555'
       END AS VAL
FROM   table1;

Desired Format:

SELECT CASE
         WHEN a.col5 = b.col5                THEN 'COL5'
         WHEN a.col55555555555555 = b.col55  THEN 'COL55555'
         WHEN a.col555555   = b.col5555      THEN 'COL555555'
       END AS VAL
FROM   table1;

For the reference, the default formatting in SQLDeveloper looks less than desirable:

SELECT
        CASE WHEN
            a.col5 = b.col5
        THEN
            'COL5'
        WHEN
            a.col55555555555555 = b.col55
        THEN
            'COL55555'
        WHEN
            a.col555555 = b.col55555
        THEN
            'COL555555'
        END
    AS val
FROM
    table1

However, to satisfy this request it requires exactly 4 changes of the formatting program.

First, before taking on alignments, we should fix unwanted indentation. How about not indenting conditions a.col5 = b.col5 and some such? By examining the parse tree, we notice that the code fragment a.col5 = b.col5 is formally a condition. Therefore, it must be that the line

| [node) condition  &  [node-1) 'WHEN'

of the formatting program that is responsible for the effect. Comment (or delete it), and rerun the formatter to confirm this hypothesis:

SELECT
        CASE WHEN a.col5 = b.col5 THEN
            'COL5'
        WHEN a.col55555555555555 = b.col55 THEN
            'COL55555'
        WHEN a.col555555 = b.col55555 THEN
            'COL555555'
        END
    AS val
FROM
    table1;

What about not indenting expressions, like string literal ‘COL5’? Just comment out the following rule:

--| [node) expr  &  [node-1) 'THEN'

The effect of this change, however, is somewhat discouraging:

SELECT
        CASE WHEN a.col5 = b.col5 THEN 'COL5' WHEN a.col55555555555555 = b.col5 THEN 'COL55555' WHEN a.col555555 = b.col5 THEN 'COL555555' WHEN a.col55555555
 = b.col5 THEN 'COL5555555' END
    AS val
FROM
    table1;

Here we need to look onto parse tree yet again to identify what grammar symbol would indent the entire WHEN-THEN fragments. The symbol in question is the  searched_case_expression#, so that the missing indentation rule alternative is just

| [node) searched_case_expression#

With this amendments the sample code formats to:

SELECT
        CASE
            WHEN a.col5 = b.col5 THEN 'COL5'
            WHEN a.col55555555555555 = b.col55 THEN 'COL55555'
            WHEN a.col555555 = b.col5555 THEN 'COL555555'
        END
    AS val
FROM
    table1;

Please note that even though there is a set options for CASE expressions (which, again, is a concession to traditional format design), flipping those options won’t produce format as clean as we just did.

So far we made 3 custom formatting program changes . Now we are ready to take on alignments. Let’s first investigate how current  formatting rules align the equalities,  because reverse engineering similar functionality is the easiest way to master new skills. Here is the code:

/**
 * All alignments (paddings) ...
**/
paddedIdsInScope: (
-- types
  [id) identifier & [id+1) datatype & [scope) relational_properties
| [id) decl_id & ([id+1) prm_spec_unconstrained_type | [id+1) 'IN' | [id+1) 'OUT' ) & [scope) fml_part
| [id) decl_id & ([id+1) constrained_type | [id+1) object_d_rhs) & [scope) adt_definition
| [id) decl_id & ([id+1) constrained_type | [id+1) object_d_rhs) & [scope) decl_list
-- =>
| [id) sim_expr & [id+1) '=' & [id+1+1) '>' & [scope) paren_expr_list
-- :=
| [id) name & [id+1) ':' & [id+1+1) '=' & [scope) seq_of_stmts
-- =
| [id) column & [id+1) '=' & [id+1+1) expr & [scope) where_clause
| [id) column & [id+1) '=' & [id+1+1) expr & [scope) on_using_condition
) & scope < id ->

The first highlighted condition specify that we are interested in columns, followed by equality, followed by some expression. They would be chosen together with the containing where_clause. Therefore, if we choose columns within the CASE scope

| [id) column & [id+1) '=' & [id+1+1) expr & [scope) case_expression

then it would align the equalities (but not THEN expressions):

SELECT
        CASE
            WHEN a.col5                = b.col5 THEN 'COL5'
            WHEN a.col55555555555555   = b.col55 THEN 'COL55555'
            WHEN a.col555555           = b.col55555 THEN 'COL555555'
        END
    AS val
FROM
    table1;

The condition we are after is a slight variation of thereof:

| [id) condition & [id+1) 'THEN' & [id+1+1) expr & [scope) case_expression

which produces:

SELECT
        CASE
            WHEN a.col5 = b.col5               THEN 'COL5'
            WHEN a.col55555555555555 = b.col55 THEN 'COL55555'
            WHEN a.col555555 = b.col55555      THEN 'COL555555'
        END
    AS val
FROM
    table1;

Querying Parse Trees

October 12, 2016

In previous blog entry we have glimpsed into new SQL Developer 4.2 feature — parse trees. Again, this is something that development tools don’t readily expose with the most likely reason being the lack of foresight what users can do with this information. Blunt answer proposed here is: “query it”.

In general, querying  is a formal and meaningful way to ask questions about something. In context of parsing the structure of the objects which we ask those questions about is well defined. In other words, the input for a query is a parse tree. Then, what is the structure of query answer?

There are at least two possibilities. The most natural is perhaps to have the output of the query to have the same structure as the input, that is the tree. However, I yet to see a compelling query language over trees, so perhaps choosing something more familiar to database developer – a relation (table) – would be a better choice?

A [query] language which takes an input in one form (a tree), but produces the results in another form (a relation/table) might seem counter intuitive, yet we’ll demonstrate that it is practical. The first objection might be that such query language, which input is a structure of one kind, but the output is something else is “not closed”. This is the matter of wrong appearance. Trees are binary relations; therefore, it is just that our proposed query language makes no effort to keep the query structures confined to this narrow kind of relations.

As we are going to talk in depth about this new query language, it would become awkward without naming it. So let’s give it a name — Arbori. (The name credit goes to Brian Jeffries from SQL Dev team). With this prelude, let’s give a taste of Arbori query.

Code quality metrics, or code guidelines can be viewed as queries or constraints over a code base. For example, Trivadis coding guideline #33 reads: “Always close locally opened cursors“. Reformulating it as a query, it would sound like this: “Find all cursors which are not closed in the enclosing procedure“.

When querying parse trees it is always insightful to focus on a concrete code sample and figure out what parse tree nodes we are after.

CREATE PACKAGE BODY "Trivadis code guideline #33" AS
--bad
    PROCEDURE not_close_cursor (
        out_count   OUT INTEGER
    ) AS
        CURSOR c1 IS
            SELECT COUNT(*) FROM all_users;

    BEGIN
        out_count := 0;
        OPEN c1;
        FETCH c1 INTO out_count;
    END not_close_cursor;

-- Good

    PROCEDURE close_cursor (
        out_count   OUT INTEGER
    ) AS
        CURSOR c1 IS
        CURSOR c1 IS
            SELECT COUNT(*) FROM all_users;

    BEGIN
        out_count := 0;
        OPEN c1;
        FETCH c1 INTO out_count;
        CLOSE c1;
    END close_cursor;

END;

The parse tree for this code is

QueryingParseTrees1.png

The cursor declaration is a node with payload “cursor_d” (1). It has two children, the node labeled with keyword “CURSOR”, followed by the identifier “C1”.

Next, we identify the parse node where the cursor is closed (2). It is a node labeled as “close_statement” and we have to make sure that its second child matches the name of the cursor that we have identified on previous step.

Finally, the cursor declaration and closing nodes that we just have identified have to be in the same scope, that is there has to be an ancestor node corresponding to the procedure that encloses both (3).

This is all the thought that is needed. The further development is just the formal translation of those 3 insights into formal Arbori query.

However,  a proposition of mastering new query language is something that was hard to convince SQL Dev feature decision makers. This is why Arbori query is hidden and could only be enabled by an auxiliary command:

set hidden param arboriEditor = true;

Executing this command results in appearance of an additional button on the Code Outline panel:

QueryingParseTrees2.png

Clicking it invokes Arbori dockable query panel. For our purposes it is convenient to organize all three panels side-by-side:

QueryingParseTrees3.png

If a reader feels intimidated by a perspective of learning yet another query language, then this perception is without merit. In fact, anybody proficient in SQL should have no problem learning it. In SQL terms, each Arbori query accesses a single input object – the parse tree, which allows to greatly simplify query syntax: there is no need for the FROM clause. Also the attributes restriction is implicit from the predicates (or is achieved by other means), so that there is no SELECT clause. The entire query is reduced to a single WHERE clause.

Unlike SQL, each query is named. Here we started writing a query appropriately named “leaking_cursors”. There is no substance in query itself yet (it is invalid Arbori syntax).

The major Arbori query components are unary predicates. Basically, we want to say that the node payload is such and such. Before we expressed our interest finding a nodes labeled “cursor_d”. This is how it is formally written in Arbori syntax:

QueryingParseTrees4.png

We need to introduce an attribute name — “cursor_decl” (1) — and specify that a parse node named “cursor_decl” is required to have a payload “cursor_d” (2). If this syntax feels alien to the reader, written in SQL this would look something like this:

select cursor_decl
from parse_tree
where payload(cursor_decl).includes('cursor_d')

Admittedly, coupling square and round parenthesis in Arbori might be too much for some readers. Please don’t hesitate to propose a better syntax!

Executing this simple query produces the list of cursor declarations (3). Changing list selection updates node references on the parse tree and, as I described in the previous article, tree node selection is consistent with selection in the original PL/SQL or SQL code editor. The numbers in the odd brackets/parenthesis [13,15) mean semi-open interval/segment of integers: at position 13 of the input PL/SQL sample code there is a token “CURSOR”, while at the next position 14 there is a token “C1”.

This is some progress towards desired query, so let’s find all the nodes labeled “subprg_body”. We know how to do that already, so we just add one more condition. More importantly, we need a condition which requires procedure node to be an ancestor of the cursor declaration node. This is formally specified as a binary ordering relation enclosed_proc < cursor_decl:

QueryingParseTrees5.png

Informally, ancestor date of birth precedes that of descendant. A reader familiar with concept of nested sets  would find this notation quite intuitive, for example the node [49,51) “cursor_decl” is nested inside the [41,80) “enclosed_proc” node.

The next step, is finding the nodes corresponding to cursor closures. This is easy:

[cursor_close_stmt) close_stmt 

We can also add a requirement for “cursor_close_stmt” to be nested inside “enclosed_proc”. However, we also need to express new requirement:

  • The nodes “cursor_decl” and “cursor_close_stmt” refer to the same cursor name

This warrants additional Arbori syntax, and slight tactical maneuver. First, let’s forget about “cursor_decl” and “cursor_close_stmt” and shift our attention onto their children. Specifically, we are interested in their children which convey the cursor name, that is the ones labeled “identifier” (among other things). Let’s call these modified attributes “cursor_decl_id” and “cursor_close_stmt_id”. Since we have conditions onto their parents, it is handy to just have a convenient way to refer to them. That Arbori syntax is a caret after an attribute name. For example

[cursor_close_stmt_id^) close_stmt 

postulates that a parent node of “cursor_close_stmt_id” is labeled “close_stmt”. There is one more additional syntax: specifically we want “cursor_decl_id” and “cursor_close_stmt_id” to refer to the same identifier:

?cursor_close_stmt_id = ?cursor_decl_id

Now we have all the ingredients to finalize our query. Realizing, however, that we found not quite yet what we were after, lets rename it into “well_behaved_cursors”:

QueryingParseTrees6.png

We are just one step away from finishing the task. One more query is wanted, named appropriately as “all_cursors”, which is the variant of “well_behaved_cursors” that ignores cursor closing statements. Then, we need to subtract “well_behaved_cursors” from “all_cursors”.

Apart from exotic syntax, little of what I have just described would raise brows of a reader with database background. Query gradually evolving for performing more and more complex tasks is something database developers and DBAs routinely do. Querying the code is quite easy!

Reshaping Parse Trees

September 22, 2016

Introduction

Parsing and internal parse structures — trees — are not something that development tools readily expose. It is assumed that most users are comfortable with numerous features that depend on parsing, such as formatting, code block collapsing/expanding, code advisors and so on, but aren’t really interested learning the formal code structure. In SQL Developer 4.2 we have attempted to encourage users to experiment with parse trees.

The location, where the new feature is introduced, is somewhat obscure — it is the former 4.2 “Outline” panel, which is was accessible via “Quick Outline” context menu item. Minor name change reflects the amended functionality

reshapingparsetrees1

Perhaps some explanation of what you see at the left panel is warranted. Oracle PL/SQL language is ADA derivative, which formal grammar can be found here. In the above example, the top most node labeled subprg_spec has been recognized as matching the first production for the rule

subprg_spec	: PROCEDURE_ IDENTIFIER .fml_part.
		| FUNCTION_  designator .fml_part. RETURN_ ty_mk
		;

Consequently, it has 3 children labeled PROCEDURE, TEST (which is identifier) and , fml_part.

For some obscure reason, there is no trace of this genuine formal grammar in oracle PL/SQL documentation, and “railroad” syntax diagrams are different. The good news, however, is that oracle documentation SQL grammar matches almost verbatim to the parse trees that SQL Developer exposes.

Playing with parse trees

Selecting a node on the tree highlights the recognized portion of the code in the editorreshapingparsetrees2

Some nodes, such as those labeled with fml_part in the example above, are collapsed. This is because parse trees for even moderately long code become overwhelming. Expanding tree branch manually is quite laborious; this is why there are 2 more effective ways to accomplish it.

The first alternative is to click “expand branch” context menu item
reshapingparsetrees3

Even though the root node is not collapsed, this context menu item is still there, and this action will expand all the collapsed descendant branches.

Selectively collapsing/expanding tree branches

The second alternative is little more sophisticated. Suppose, in our working example you would like to have all the nodes with payload fml_part expanded. Then, you click onto the second button on the toolbar (it is marked with plus symbol and “Collapse nodes with payload” tooltip).

reshapingparsetrees4

If you check the fml_part menu item, then all the nodes with this payload become expanded.

A sharp eyed reader might have noticed that, by default, select is expanded, while seq_of_stmts is collapsed. This is because code outline is working not only for PL/SQL editor but also SQL Worksheet.  In SQL worksheet queries — select statements — are the main laborers, this is why their structure is expanded. However, if you work mostly in PL/SQL editor, then those select statements are less interesting. For example, a select query which is a part of a cursor declaration can be collapsed, unless a user decides to investigate it. As for seq_of_stmts, again, PL/SQL package bodies are frequently very large, so having collapsed sequence of statements — the bulk of procedure or function — is handy.

Selectively hiding the nodes

Another way to bring a parse tree to manageable size is hiding the nodes. In general, if node disappears, something must be done about this node children. Naturally, orphans become attached to node’s parent. For example, deleting node B, reattaches grand children C and D to node A.

reshapingparsetrees5

Which nodes are not so much interesting? Consider a typical SQL query

reshapingparsetrees6

A chain of tree nodes marked with select_list faithfully reflects SQL grammar, but a user might want to see more concise picture.  The 3rd (“Hide nodes with payload”) button on toolbar serves this purpose

reshapingparsetrees7

Unchecking the select_list menu item (it is unchecked by default) reshapes the tree like this

reshapingparsetrees8

Conclusion

There is only so much that tree reshaping can do. For large trees it is laborious to search for interesting information. In the next installment we’ll describe an approach which is more satisfying for a reader with database background — querying.

SQL Developer 4.2 early adopter release is out, but SQL performance analysis improvements somehow slipped from the list of additional features. Here is detailed walkthrough of three amendments:
– Cancelling Long Running Queries (while extracting partial statistics)
– Object Hyperlinks
– Hints

Two years ago we have ventured into mathematical foundation of relational theory. From algebraic geometry perspective relations were viewed as Finite Varieties. In the followup  we were able to describe functional dependencies via explicit analytic formulas and provide intuitive interpretation of the classic result from database dependency theory — Heath’s theorem. Recent arXiv preprint provides a little more  coherent exposition of those ideas together with some excursion into Quantum Theory.

Here we complement the aforementioned arXiv article with demonstration that your average off-the-shelf Computer Algebra System is actually a [rudimentary] RDBMS. The CAS system of choice is CoCoA.

The starting point is being able to exhibit a system of polynomial equations constraining a relation defined as a set of set of tuples.

Given a binary relation with attributes x and y

[x y]
 1 1
 2 1
 3 2

we execute the following series of CoCoA commands. First, we need to specify the polynomial ring:

Use XY ::= QQ[x,y];

Then, list the tuples:

Points := mat([[1, 1], [2, 1], [3, 2]]);

Finally, specify the ideal, and print it out:

I := IdealOfPoints(XY, Points);
I;

The output:

ideal(y^2 -3*y +2, x*y -x -3*y +3, x^2 -3*x -2*y +4)

Next, for the second relation:

[x z]
 1 1
 2 1
 3 1
 3 2

we execute the series of commands

Use XZ ::= QQ[x,z];
Points := mat([[1, 1], [2, 1], [3, 1],[3, 2]]);
J := IdealOfPoints(XZ, Points);
J;

which outputs

ideal(z^2 -3*z +2, x*z -x -3*z +3, x^3 -6*x^2 +11*x -6)

What is the natural join of the two relations? It is the sum I+J. However, we must switch to polynomial ring which contains both XY and XZ:

Use XYZ ::= QQ[x,y,z];
I:=ideal(y^2 -3*y +2, x*y -x -3*y +3, x^2 -3*x -2*y +4);
J:=ideal(z^2 -3*z +2, x*z -x -3*z +3, x^3 -6*x^2 +11*x -6);

After redefining verbatim both ideals in the larger ring (can it be accomplished easier?), we calculate their “join”:

I+J;

which outputs

ideal(y^2 -3*y +2, x*y -x -3*y +3, x^2 -3*x -2*y +4, 
z^2 -3*z +2, x*z -x -3*z +3, x^3 -6*x^2 +11*x -6)

This is, again an ideal of points, which is evident with the command

RationalSolve(GBasis(I+J));
[[1, 1, 1], [2, 1, 1], [3, 2, 1], [3, 2, 2]]

Here we leveraged GBasis function as a way to convert an ideal into list (because RationalSolve accepts a list, not an ideal).

Next, projection is an elimination ideal. Once again, it is zero-dimensional ideal of points, so that a typical database user would like to list the tuples:

RationalSolve(GBasis(elim(z, I+J)));
[[1, 1], [2, 1], [3, 2]]

Finally, let’s hint what are counterparts of the rest of RA operations. The set union is the intersection of an ideals. The set difference is colon ideal. The only operation which doesn’t have an obvious analog in polynomial algebra is the least challenging one — renaming.

Recently, Litak & Mikulas & Hidders published the extended version of their earlier work. However, in a typical mathematical tradition following the famous saying “when fine building is revealed to the public eye, the scaffolding should be removed”, reading it requires some detective work. In particular, lets investigate the motivation behind their axiom system:

x ^ (y v z) = (x ^ (z v (R00 ^ y))) v (x ^ (y v (R00 ^ z))). % AxRH1

(R00 ^ (x ^ (y v z))) v (y ^ z) =
= ((R00 ^ (x ^ y)) v z) ^ ((R00 ^ (x ^ z)) v y). % AxRH2

(x ^ y) v (x ^ z) = x ^ ( ((x v z) ^ y) v ((x v y) ^ z) ). % AxRL1

Let’s start with well known lattice inequality

x v (y ^ z) > (x v y) ^ (x v z).

Here we abused notation a little and assume the order symbol > to be reflexive relation. If the LHS of inequality is greater (or equal) than the RHS, then perhaps one can hope to substitute y and z with greater values at RHS, or with lower values at LHS so that inequality becomes an identity? This is the technique employed by Sergey Polin when refuting McKenzie’s conjecture. Specifically, he introduced a non-decreasing chain of values

y1 = y
z1 = z

y2 = y v (x ^ z1).
z2 = z v (x ^ y1).

y3 = y v (x ^ z2).
z3 = z v (x ^ y2).
...

together with weakened distributivity law

x v (y ^ z) = (x v yN) ^ (x v zN).

Unfortunately, literal Polin-style weakened distributivity law fails in relational lattices for n=1,…,6, at least. However, Polin’s technique can be naturally generalized when an algebra offers constant(s). The axiom AxRH1 is weakened distributivity law with y elevated to y v (R00 ^ z) and z elevated to y v (R00 ^ z).

In a similar venue, the axiom AxRL1 is Polin-style weakening applied to LHS of distributivity law. (Litak et.al. attribute the axiom to Padmanabhan work.)

Genesis of AxRH2 is probably various earlier discovered conditional distributivity identities involving the header relation R00. For professional logician converting implication into equality is a walk in the park.

How about the dual distributivity law of union over join? The stricter conditional distributivity of union over join established in an earlier paper hints that the axiom might be more complicated. One might have to lower/elevate both sides of inequality. After some trial permutations of variables and the header constant R00, it revealed the following identity:

(x v (y ^ (x v z))) ^ (x v (z ^ (x v y))) = x v ((y v (x ^ R00)) ^ (z v (x ^ R00))).

This can be translated into relational calculus expression

(( exists p1 exists p2 exists p6(x(p1,p3,p5,p7) | (y(p2,p3,p6,p7)& exists p1 exists p3 exists p4 exists p6(x(p1,p3,p5,p7) | z(p4,p5,p6,p7))))& exists p1 exists p4 exists p6(x(p1,p3,p5,p7) | (z(p4,p5,p6,p7)& exists p1 exists p5 exists p2 exists p6(x(p1,p3,p5,p7) | y(p2,p3,p6,p7)))))
<->
exists p1(x(p1,p3,p5,p7) | ( exists p2 exists p6 exists p1 exists p5(y(p2,p3,p6,p7) | (x(p1,p3,p5,p7)&$F))& exists p4 exists p6 exists p1 exists p3(z(p4,p5,p6,p7) | (x(p1,p3,p5,p7)&$F))))).

which validity is easily verified with Prover9.

This axiom doesn’t seem to be independent from the {AxRH1,AxRH2,AxRL1} with Mace4 models lookup up to cardinality 26 at least, although Prover9 fails to confirm it after 3 hours running time. (This is not entirely surprising, with union over join distributivity law being often enforced by join over union).