Ambiguity and Object-Role Modeling (ORM)— Part 3

In a previous article I explained why ORM2, the second incarnation of Niam/Object-Role Modeling, is inherently ambiguous when we consider Join Subset Constraints and a formalism which allows for ambiguity.

In this article I will describe another way to show that Object-Role Modeling may be inherently ambiguous by virtue of its formalisation under a theory of finite-model theory and when analysed under Ehrenfeucht Fraisse (EF) Games of a second order, which naturally sees sentences of Object-Role Modeling under higher-order logic.

I have not found a paper that describes why Dr Halpin, the ostensible inventor of Object-Role Modeling, chose to map an homomorphism from a theory under finite model theory (KL or Knowledge Language) and sentences of Niam (the precursor to Object-Role Modeling) as an ostensible isomorphism under Dr Halpin’s PhD thesis; but it is safe to say that Halpin perceived that if sentences of Niam were viewed as sentences under finite model theory then they would not fall under higher-order logic and therefore not fall under Gödel's Incompleteness Theorems. The idea of Dr Halpin’s thesis being that sentences of Niam, or diagrams in the case of Niam, would in some way be unambiguous. Object-Role Modeling is oft quoted as unambiguous and we explore the voracity of that claim here.

For those suitably familiar with model theory, the history of formal logic and Ehrenfeucht Fraisse (EF) Games, that thesis has its challenges. The reason is that it presupposes that the interpreter of a Niam/ORM diagram actively chooses the same interpretation as the creator of a Niam/ORM diagram every time and is not a free operator capable of independently choosing any interpretation that fits sentences of the theory.

I have found no journal submitted paper that has acknowledged that Niam/ORM diagrams can naturally be interpreted under higher-order logic. In many ways that reflects a resistance to the reality of EF Games and to some degree discounts the independent intelligence of other sentient beings. My feelings are that we should at least acknowledge higher-order interpretations of Object-Role Modeling if we are to have any credibility in the logic community.

The formalism (syntax) of Object-Role Modeling has no negation, which makes it laborious to recreate Godel’s incompleteness theorems using ORM itself. It’s not impossible, but laborious under a standard interpretation. Of course, Godel’s incompleteness theorems are about non-standard interpretations of a theory of higher-order logic and it took Russell and Whitehead 10 years to write Principia Mathematica over which Godel’s theorems were written, and another 20 years until Godel found the weaknesses in Principia Mathematica.

For our purposes, let us at least here acknowledge that there are higher-order interpretations of Object-Role Modeling.

Higher-Order Interpretations of Object-Role Modeling

The picture below is of a man reaching for a Part in a Bin in a Warehouse. The associated YouTube video (if you click on the picture) demonstrates the creation of the model and how the Fact Type Reading (Part is in Bin in Warehouse) is formulated in the Boston Object-Role Modeling software.

“Part is in Bin in Warehouse” is known as a Fact Type, and “Part ‘123’ is in Bin ‘1’ in Warehouse ‘Syndey’” is known as a Fact.

Object-Role Modeling allows you to create sample populations of Facts within the diagram itself within what is known as a Fact Table, as in the example below:

A sample population within a Fact Table in ORM

Extending our diagram by objectifying the Fact Type, we allow our model to store a quantity of each type of part stored in a bin in a warehouse:

Object-Role Modeling was originally synthesised as a language, Niam, ostensibly isomorphic with a theory of finite model theory called KL (Knowledge Language) itself synthesised under Finite Model Theory (FMT). Theorems, sentences, or diagrams of ORM (at least under the Niam interpretation) have an ostensible 1:1 mapping to theorems of KL, as below:

An ORM diagram with corresponding theorems of Knowledge Language (KL)

The ORM diagram is on the left, the theorems of KL are on the right.

NB Halpin’s thesis would not represent Objectified/Nested Fact Types in the manner of the KL theorems above, but in a form as below. Irrespectively, the duplicator of an EF Game gets to choose and interpretation and where populations of objectified fact types have identity in their own right, as under nominalism. Please refer to Halpin’s thesis, pages 80–81 for more information:

Extract from Dr Halpin’s PhD Thesis — On ‘Nesting’

The idea of Dr Halpin’s thesis was to suppose that because KL is a theory that can be interpreted under formal logic and model theory, then ORM to must also be considered as an equitable formalism.

The thesis holds, for Niam at least, if we consider a side-by-side comparison as sufficient enough to map an isomorphism and if we only consider Object-Role Modeling under finite model theory and a lower-order logic.

The trouble with saying that Object-Role Modeling is necessarily unambiguous is that that position negates, or totally ignores, that Object-Role Modeling can also (by virtue of its mapping to KL) be considered under higher-order logic when analysed under Ehrenfeucht Fraisse Games extended to higher-order logic.

If we look at our ORM diagram again we define the ‘Stock’ Fact Type under KL, and also the sample population:

Theorems of Knowledge Language (KL)

x,y and z in our population, S, correspond to part ‘123’, bin ‘1’ and warehouse ‘Sydney’ in our ORM diagram, which I provide below for ease of reading:

Sample populations in an Object-Role Model

Our relationship between x, y and z in ‘Stock’, in the set of KL theorems above, is what is called a relation; Stock maps a relation with x, y and z as terms. Halpin defines ‘relations’ as different from tuples or functions, but for our purposes we choose an interpretation where they are functions.

Our higher order interpretation comes, under EF Games and a higher order interpretation of ORM, when we have output of the Stock function as a term in the second function which maps stock items to the quantity stored. X and y in our secondary function then corresponds to the population of a function, ‘S’ for ‘Stock, and y which ultimately equates to a value.

A function of a function under non-standard interpretation of ORM/KL

While the standard interpretation of ORM, in a somewhat convoluted way, would not have this interpretation, analysing Object-Role Modeling diagrams under EF Games extended to higher-order logic, sees Fact Types with Roles connected to Objectified Fact Types as functions with functional outputs as terms.

NB An interpretation of Object-Role Modeling under Nominalism would also see populations under the Stock Fact Type as individuals and as terms to functions linked to the Objectified ‘Stock’ Fact Type. i.e. Under nominalism one views Fact Types joining Objectified Fact Types as functions accepting as terms the output of other functions as individuals.

Under EF Games extended to higher-order logic Object-Role Modeling naturally has a higher-order interpretation. Indeed, for those that are not familiar with Dr Halpin’s PhD thesis and a standard interpretation of ORM, a higher-order interpretation of ORM is arguably the most natural interpretation of ORM.

Ambiguity and Object-Role Modeling

For serious researchers, and for those curiously investigative, this is trivially done under non-standard analysis of Object-Role Modeling. Below are theorems of Russell/Whiteheads theory formalised under Principia Mathematica (PM)(top) and their corresponding Object-Role Model/Niam diagram (below) where we substitute natural language words for symbols of PM themselves. The theorems of PM may manifest as sample populations in Fact Tables. The sample below is PM’s synthesis of “1 + 1 = 2” under PM.

1 + 1 = 2 under Principia Mathematica, and under non-standard analysis of Object-Role Modeling

NB Halpin’s thesis presupposes that it is not possible to substitute alternate symbols to things such as the textual name of Entity Types. Gödel put paid to this by emphasising that interpretation is outside the role of the creator of a sentence; EF Games adopts the same.

The same theorem can be expressed as a sample population of a Fact Type over a single Entity Type where the Entity Type, Symbol, has values constrained to the symbols of Principia Mathematica.

A theorem of Principia Mathematica expressed as an Objec-Role Model

Without proof, we propose here that using such techniques all the theorems of Principia Mathematica can be faithfully recreated in Object-Role Modeling such that Gödel's incompleteness theorems may apply to Object-Role Modeling under a non-standard interpretation of ORM. Higher-order interpretations of ORM are naturally derived under Ehrenfeucht Fraisse Games over Object-Role Modeling by the duplicator in an EF Game.

This is to say that claims of Object-Role Modeling being unambiguous seem readily to fall short under a more detailed appreciation of the history of formal logic, model theory and logic as analysed under games.

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