1. Introduction
To evolve and improve the type systems of mainstream objectoriented programming languages such as Java (Gosling et al., 2014), C# (2015), C++ (2011), and Scala (Odersky, 2014), which utilize class name information in defining object types and OO subtyping, a precise mathematical model of these languages is needed. A precise model of nominallytyped OOP allows accurate reasoning and analysis of these mainstream OO programming languages. Imprecise models, on the other hand, lead to inaccurate conclusions.
An object in nominallytyped OO languages is associated with its class^{1}^{1}1The term ‘type’ is overloaded. In this paper, the term has mainly two related but distinct meanings. The first meaning, mainly used by OO software developers, is a syntactic one, that directly translates to the expression ‘class, interface, or trait’ (in OO programming languages that support these constructs). In this sense, each class, interface, or trait is a type. The second meaning for ‘type,’ mainly used by mathematicians and programming languages researchers, is a semantic meaning referring to the set of instances of a corresponding class/interface/trait. In this sense, each class, interface, or trait corresponds to a type. Usually the context makes clear which sense of the two is meant, but, to emphasize, sometimes we use the term ‘class’ for the syntactic meaning. As such, unless otherwise noted the term ‘class’ in this paper should be translated in the mind of the reader to ‘class, interface or trait.’ name and the class names of its superclasses, as part of the meaning of the object. Class names, in turn, are associated with class contracts, which are usually expressed, informally, in code documentation. Class contracts are thus implicitly encoded in class names.
In nominallytyped OOP, two objects with the same structure but that have different class name information are different objects, and they have different types. The different class name information inside the two objects implies the two objects maintain different class contracts, and thus that the objects are behaviorally dissimilar. The two objects are thus considered semantically unequal. Further, in nominallytyped OO languages—where types and the subtyping relation make use of class names and of the explicitlyspecified type inheritance relation between classes—instances of two classes that are not in the inheritance hierarchy may not be replaced by each other (i.e., are not ‘assignmentcompatible’) since they may not offer the degree of behavioral substitutability intended by developers of the two classes.
Despite its clear semantic importance, class name information (henceforth, ‘nominal information’) that is embedded inside objects of many mainstream OO programming languages is not included in the most recognized denotational models of OOP that exist today. Models of OOP that lack nominal information of mainstream OO languages are structural models of OOP, not nominal ones. Examples of structurallytyped OO languages include O’Caml (Leroy et al., n.d.) (see (MacQueen, 2002)) and research languages such as Modula3 (Cardelli et al., 1989), Moby (Fisher & Reppy, 1999), Strongtalk (Bracha & Griswold, 1993), and PolyTOIL (Bruce et al., 2003). Structural models of OOP have led PL researchers to make some conclusions about OOP that contradict the intuitions of the majority of mainstream OO developers and language designers. For example, the agreement of type inheritance, at the syntactic (i.e., program code) level, and subtyping, at the semantic (i.e., program meaning) level, is a fundamental intuition of OO developers using nominallytyped OO languages. However, extant denotational models of OOP led to the inaccurate conclusion that “inheritance is not subtyping.”
Type inheritance, in classbased mainstream OO languages, is an inherently nominal notion, due to the informal association of class names with inherited class contracts. Hence the discrepancy between conclusions regarding inheritance that are based on a structural view of OOP and the intuitions of the majority of mainstream OO developers, who adopt a nominal view of OOP. This discrepancy motivated considering the inclusion of nominal information in mathematical models of OOP.
This paper presents the construction of a mathematical model of OOP, called , that includes full nominal information of mainstream OO programming languages. was first presented in (AbdelGawad, 2012) and its construction was summarized in (AbdelGawad, 2014a).
Having a model of OOP that includes nominal information of nominallytyped OOP should enable progress in the design of type systems of current and future mainstream OO languages. Some features of the type systems of these languages (e.g., generics) crucially depend on nominal information. Accurately understanding and analyzing these features, for the purposes of extending the languages or designing new languages that include them, has proven to be hard when using operational models of OOP or using structural denotational models of OOP, which lack nominal information found in nominallytyped OO languages. Having a nominal domaintheoretic model of OOP should make the analysis of features of these languages that depend on nominal information easier and more accurate. From the point of view of OO software development, having better mainstream OO languages should result in greater productivity for software developers and in them producing robust highquality software.
This paper is organized as follows. Section 2 presents a list of research related to this paper. Section 3 presents in brief the value of nominal typing to mainstream OO developers. Section 4 then starts the formal presentation of by presenting a new records domain constructor, called ‘rec,’ that is used in constructing . Section 5 presents class signatures and other related signature constructs, which are syntactic constructs used to embody the nominal information found in nominallytyped OOP. Section 6 presents the construction of , using ‘rec’ and signature constructs, then it presents a proof of the identification of inheritance and subtyping in nominallytyped OOP. Section 7 then presents in brief a comparison of to the most wellknown domaintheoretic models of OOP, namely the two structural models developed by Cardelli and by Cook. Section 8 presents the main conclusions we reached based on developing and on comparing it to other domaintheoretic models of OOP. Section 9 concludes this paper by presenting further research that can be developed based on .
2. Related Research
is a domaintheoretic model of nominallytyped OOP. Dana Scott invented and developed—with others including Gordon Plotkin—the fields of domain theory and denotational semantics (e.g., see Scott, 1976; Stoy, 1977; Smyth & Plotkin, 1982; Scott, 1983; Gunter & Scott, 1990; Gierz et al., 2003; Cartwright et al., 2016). The development of denotational semantics has been motivated by researching the semantics of functional programming languages such as Lisp (McCarthy, 1963, 1996) and ML (Gordon et al., 1978; Milner et al., 1997).
Research on the semantics of OOP has taken place subsequently. Cardelli built the first widely known denotational model of OOP (Cardelli, 1984, 1988a). Cardelli’s work was pioneering, and naturally, given the research on modeling functional programming extant at that time, the model Cardelli constructed was a structural denotational model of OOP that lacked nominal information.^{2}^{2}2Significantly, Cardelli in fact also hinted at looking for investigating nominal typing (on page 2 of (Cardelli, 1988b)). Cardelli’s hint, unfortunately, went largely ignored for years. Cook and his colleagues built on Cardelli’s work to separate the notions of inheritance and subtyping (Cook, 1989; Cook & Palsberg, 1989; Cook et al., 1990). Later, other researchers (such as (Bruce, 2002) and (Simons, 2002)) promoted Cardelli and Cook’s structural view of OOP, and promoted conclusions based on this view.
Martin Abadi, with Luca Cardelli, later presented operational models of OOP (Abadi & Cardelli, 1994, 1996). These models also had a structural view of OOP. Operational models with a nominal view of OOP got later developed however. In their seminal work, Atsushi Igarashi, Benjamin Pierce, and Philip Wadler presented Featherweight Java (FJ) (Igarashi et al., 2001) as an operational model of a nominallytyped OO language. Even though not the first operational model of nominallytyped OOP (for example, see (Drossopoulou et al., 1999), (Nipkow & Von Oheimb, 1998) and (Flatt et al., 1998, 1999)), FJ is the most widelyknown operational model of (a tiny core subset of) a nominallytyped OO language, namely Java.^{3}^{3}3It is worthy to mention that —as a more foundational domaintheoretic model of nominallytyped OO languages (including Java)—provides a denotational justification for the inclusion of nominal information in Featherweight Java.
Other research that is similar to one presented here, but that had different research interests and goals, is that of Reus and Streicher (Reus, 2002; Reus & Streicher, 2002; Reus, 2003). In (Reus, 2003), an untyped denotational model of classbased OOP is developed. Type information is largely ignored in this work (object methods and fields have no type signatures) and some nominal information is included with objects only to analyze OO dynamic dispatch. The model of (Reus, 2003) was developed to analyze mutation and imperative features of OO languages and for developing specifications of OO software and the verification of its properties. Analyzing the differences between structurallytyped and nominallytyped OO type systems was not a goal of Reus and Streicher’s research, and in their work the identification of inheritance and subtyping was, again (as in FJ), assumed rather than proven as a consequence of nominality and nominal typing.
3. The Value of NominalTyping in OOP
In this section we briefly present the value of nominaltyping and nominalsubtyping to OO software developers and OO language designers. More details on the value of nominaltyping and nominalsubtyping can be found in (AbdelGawad, 2016b).
As hinted to in the Introduction (Section 1), the main semantic value of nominaltyping to mainstream OOP lies in the association of type (i.e., class/interface/trait) names with behavioral contracts that are part of the public interface of objects, making typing and subtyping in nominallytyped OO languages closer to semantic typing and semantic subtyping than structuraltyping and structuralsubtyping are. Designing their software based on having public behavioral contracts allows OO developers to design robust software (Bloch, 2008).
The semantic value of nominal type information leads nominallytyped and structurallytyped OO languages to have different views of type names, where type names in nominallytyped OOP have fixed meanings (tied to the public contracts) while in structurallytyped OOP (in agreement with the tradition in functional programming) type names are viewed as mere ‘shortcuts for type expressions’ that can thus change their meanings, e.g., upon inheritance. This difference in viewing type names leads OO developers using structurallytyped OO languages to face problems—such as spurious subtyping, missing subsumption, and spurious binary methods (see (AbdelGawad, 2016b))—that are not found in nominallytyped OO languages.
Further, the identification of type inheritance with OO subtyping (‘inheritance is subtyping’) resulting from nominaltyping (which we prove in this paper) enables nominallytyped OO languages to present OO developers with a simple conceptual model during the OO software design process.
Finally, due to the ubiquity of the need for objects in OOP to be “autognostic”
(selfaware, i.e. recursive, see (Cook, 2009)) and given that recursive
data values can be typed using recursive types (MacQueen et al., 1986), the ease by which recursive
types can be expressed in nominallytyped OO languages is a decided benefit
nominaltyping offers to OO software developers and designers (Pierce, 2002). More details on the benefits of nominaltyping can be found in (AbdelGawad, 2016b).
Without further ado, we now start the presentation of as a model of OOP that includes full nominal information found in many mainstream OO languages.
4. ‘Rec’ (), A New Records Domain Constructor
For the purpose of constructing , we introduce a new domain constructor. In addition to including nominal information of mainstream OOP, models records as tagged finite functions rather than infinite functions, as another improvement over extant domaintheoretic models of OOP (particularly that of Cardelli and other models built directly on top of it, such as Cook’s.)
Due to the finiteness of the shape of an object (the shape of an object is the set of names/labels of its fields and methods), and due to the flatness of the domain of labels when labels are formulated as members of a computational domain, modeling objects in motivates defining a new domain constructor that is similar to but somewhat different from conventional functional domain constructors. This domain constructor, , called ‘rec,’ constructs tagged finite functions, which we call record functions. Record functions are explicitly finite mathematical objects.
A domain , constructed using , is the domain of record functions modeling records with labels from a flat domain of labels to an arbitrary domain of values. Below we present the records domain constructor, , then we discuss its mathematical properties. The definition of makes use of standard definitions of basic domain theory (See, for example, (Cartwright et al., 2016). A summary of domain theory notions used to construct is presented in (AbdelGawad, 2014b) and in Appendix A of (AbdelGawad, 2012, 2013a).)
4.1. Record Functions
A record can be viewed as a finite mapping from a set of labels (as member names) to fields or methods. Thus, we model records using explicitly finite record functions. A record function is a finite function paired with a tag representing the input domain of the function. The tag of a record function modeling a record represents the set of labels of the record. In agreement with the definition of shapes of objects, we similarly call the set of labels of a record the shape of the record. The tag of a record function thus tells the shape of the record.
4.2. Definition of
Let be the flat domain containing all record labels plus an extra improper bottom label, , that makes be a domain. (All computational domains must have a bottom element.) Let be an arbitrary domain, with approximation ordering and bottom element . Domain contains the values that members of records are mapped to.
Let denote the subdomain relation (see Definition 6.2 in (Cartwright et al., 2016).) If we let range over arbitrary finite subdomains of (all subdomains contain ), then we define the domain as the domain of record functions from to , where the universe, , of domain is defined by the equation
(1) 
with sets defined as
(2) 
and where is a function that maps the shape corresponding to a domain to a unique tag in a countable set of tags (whose exact format does not need to be specified), and where is the standard domain of strict continuous functions from into . Tags are needed in record functions to ensure that the records domain constructor is a continuous, in fact computable, domain constructor.
To illustrate, using a record is modeled by a record function . It should be noted that allows constructing the (unique) record function
that models the empty record (one with an empty set of labels, for which .)
The approximation ordering, , over elements of is defined as follows. The bottom element approximates all elements of the domain . Nonbottom elements and in with unequal tags are unrelated to one another. On the other hand, elements and with the same tag are ordered by their embedded functions (which must be elements of the same domain.) Formally, for two nonbottom record functions in that are defined over the same , where , if
and
where and are elements in , then we define
Having defined the records domain constructor , we now discuss its mathematical properties.
Theorem 4.1.
Given a flat countable domain of labels and an arbitrary domain , is a domain.
Proof.
See Appendix B. ∎
Because in the construction of we use to construct domains as least fixed points of functions over domains, where the constructed domains need to be subdomains of Scott’s universal domain, , we need to ascertain that has the domaintheoretic properties needed for it to be used inside these functions. We thus need to prove that is a continuous function over its input domain , i.e., that, as a function over domains, is monotonic with respect to the subdomain relation, , and that preserves least upper bounds of domains under that relation.
Theorem 4.2.
Domain constructor is a continuous function over flat domains and arbitrary domains .
Proof.
See Appendix B. ∎
5. Class Signatures
In this section we present formal definitions for class signatures and related constructs. Class signatures and other signature constructs are syntactic constructs that capture nominal information found in objects of mainstream OO software. Embedding class signature closures (formally defined below) in objects of makes them nominal objects, thereby making objects more precise models of objects in mainstream OO languages such as Java, C#, C++, and Scala.
Class signatures formalize the notion of object interfaces. A class signature corresponding to a class in nominallytyped mainstream OOP is a concrete expression the interface of the class, i.e., of how instances of the class should be viewed and interacted with by other objects (“the outside world”).^{4}^{4}4Object interfaces are also discussed in (AbdelGawad, 2016b), (AbdelGawad, 2013b) and Ch. 2 of (AbdelGawad, 2013a).
To capture nominal information of nominallytyped mainstream OOP, we define three syntactic signature constructs: (1) class signatures, (2) class signature environments, and (3) class signature closures. Additionally, fields and methods, respectively, have (4) field signatures and (5) method signatures.
5.1. Class Signatures
If is the set of all class names, and is the set of all member (i.e., field and method) names, we define a set that includes all class signatures by the equation
(3) 
where and are the crossproduct and finitesequences set constructors, respectively, is the set of field signatures, and is the set of method signatures.
The equation for expresses that a class signature corresponding to a certain class is composed of four components:

The class name (also used as a signature name for the class signature),

A finite sequence of names of immediate supersignatures of the signature, i.e., of signatures corresponding to immediate superclasses of the class,

A finite sequence of field signatures corresponding to class fields, and

A finite sequence of method signatures corresponding to class methods.
The use of signature names (members of ) inside signatures characterizes class signatures as nominal constructs, where two signatures with different names but that are otherwise equal are different signatures.
The second component of a signature, a (possibly empty) sequence of signature names (i.e., a member of ), is the immediate supersignature names component of the class signature. Having names of immediate supersignatures of a class signature explicitly included as a component of the class signature is an essential and critical feature in the modeling of nominal subtyping in nominallytyped OOP. Explicitly specifying the supersignatures of a class signature identifies the nominal structure of the class hierarchy immediately above the named class. This also agrees with the inheritance of the contract associated with class names, which is a crucial semantic component of what is intended to be inherited in nominallytyped mainstream OOP.
The equation for field signatures expresses that a field signature is a pair of a field name (a member of ) and a class signature name. Similarly, the equation for method signatures expresses that a method signature is a triple of a method name, a sequence of class signature names (for the method parameters), and a signature name (for the method result).
Not all members of set are class signatures. To agree with our intuitions about describing the interfaces of classes and their instances, a member of is a class signature if its supersignature names component, its field signatures component and its method signatures component (i.e., the second, third and fourth components of ) have no duplicate signature names, field names, and method names, respectively (For simplicity, method overloading is not modeled in our model of OOP.) It should be noted, however, that field names and method names are in separate name spaces and thus we allow a field and a method to have the same name.
Information in class signatures is derived from the text of classes of OO programs. Given that interfaces of objects are the basis for defining types in OO type systems, class signatures are the formal basis for nominallytyped OO type systems, so as to confirm that objects are used consistently and properly within a program ((AbdelGawad, 2016b), Ch. 2 of (AbdelGawad, 2013a), and (AbdelGawad, 2013b), give more details on types and typing in OOP.)
5.2. Signature Environments
A signature environment is a finite set of class signatures that has unique class names, where each signature name is associated with exactly one class signature in the environment. (Accordingly, function application notation can be used to refer to particular class signatures in a signature environment. If is a signature name guaranteed to be the name of some class signature in a signature environment , we use function application notation, , to refer to this particular class signature.) In addition to requiring the uniqueness of signature names, a finite set of class signatures needs to satisfy certain consistency conditions to function as a signature environment. A signature environment specifies two relations between signature names: an immediate supersignature relation and a directreference (adjacency) relation (The first relation is a subset of the second.) These two relations can be represented as directed graphs. The consistency conditions on a signature environment constrain these two relations and their corresponding graphs.
As such, a finite set of class signatures is a signature environment if and only if (i) A class signature, with the right signature name, belongs to for each signature reference in each class signature of , (ii) The graph for the supersignatures relation for is an acyclic graph (This constraint forces any signature environment to have at least one class signature that has no supersignatures, i.e., its second component is the empty sequence), and (iii) The set of field signatures and method signatures of each class signature in is a superset of the set of field signatures and method signatures of each supersignature named by the supersignatures component of .
In agreement with inheritance in mainstream OO languages, the last condition makes class signatures in signature environments reflect the explicit inheritance information in classbased OOP, by requiring a class signature to only extend (i.e., add to) the set of members supported by an explicitlyspecified supersignature. Requiring the members of a class signature to be a superset of the members of all of its supersignatures means that exact matching of member signatures is required. This requirement thus enforces an invariant subtyping rule for field and method signatures, mimicking the rule used in mainstream OO languages (such as Java and C#) before the addition of generics. This condition can be relaxed but we do not do so in this paper. More details are available in (AbdelGawad, 2012).
5.3. Signature Closures
Inside a class signature, class names can be viewed as “pointers” that refer to other class signatures. Without bindings of class names to corresponding class signatures, a single class signature that has name references to other class signatures is not a closed entity on its own. This motivates the notion of a signature closure. A closure of a class signature is a set of class signatures (a signature environment, in particular) that offers bindings to class names referred to in all elements of the set, such that the whole set has no “dangling pointers” in its references to other class signatures (i.e., is referentiallyclosed) and has no redundant class signatures relative to some main class signature in the set (called the root class signature of the closure.) A signature closure thus “closes” the root class signature by providing bindings for all class names referenced, directly or indirectly, in the signature. This motivates the following formal definition of signature closures.
A signature closure is a pair of a signature name and a signature environment. A pair of a signature name and a signature environment is a signature closure if and only if there exists a class signature in with signature name and if the directreference (adjacency) relation corresponding to is referentiallyclosed relative to , and if this relation is the smallest such relation. Class signature is then called the root class signature of . Relative to the root class signature, a signature environment is minimal, i.e., contains no unnecessary class signatures. This minimality condition ensures that all class signatures in the signature environment of a signature closure are accessible via paths in the adjacency graph of the signature environment starting from (the node in the graph corresponding to) the root signature name, i.e., that the signature environment has no redundant class signatures unnecessary for the root class signature.
Similar to a single class signature, when viewed as a “closed class signature” a signature closure has a name: namely, that of its root class signature; has member signatures: namely, field and method signatures of its root class signature; has a fields shape and a methods shape: namely, those of its root class signature; and it has immediate supersignature names: namely, those of its root class signature. A signature closure, not just a class signature, is the full formal expression of the notion of object interfaces. Each class in a classbased OOP program has a corresponding class signature and a corresponding class signature closure. The nominal information in a class signature closure is an invariant of all instances of the class (including the behavioral contracts associated with class names.)
5.4. Relations on Signatures
For class signatures and , we define where is an equivalence relation on sequences that ignores the order (and repetitions) of elements of a sequence. For two field signatures and , Similarly, for two method signatures and , (Here, sequence equality, not sequence equivalence, is used. For method parameter signature names, order and repetitions do matter.)
Two signature environments are equal if and only if they are equal as sets. Two signature closures are equal if and only if they are equal as pairs. Equal signature closures have the same root class signature name and equal signature environments.
Finally, a relation between signature environments that is needed when we discuss inheritance is the extension relation on signature environments. A signature environment extends a signature environment (written ) if binds the names defined in to exactly the same class signatures as does. Viewed as sets, is a superset of . Thus,
5.5. Subsigning and Inheritance
The supersignatures component of class signatures defines an ordering relation between signature closures. We call this relation between signature closures subsigning. The subsigning relation between class signature closures models the inheritance relation between classes in classbased OOP.
A signature closure is an immediate subsignature () of a signature closure if the signature environment (i.e., the second component) of is an extension () of the signature environment of and the signature name of is a member of the supersignature names component of the root class signature of , i.e.,
The subsigning relation, , between signature closures is the reflexive transitive closure of the immediate subsigning relation (). To illustrate the definitions given in this section, Appendix A presents a few examples of signature constructs, and presents examples of signature closures that are in the subsigning relation.
The inclusion of class contracts in deciding the subsigning relation makes the subsigning relation a more accurate reflection of a true “isa” (substitutability) relationship than the structural subtyping relation used in structurallytyped OOP. This makes subsigning capture the fact that subtyping in nominallytyped OOP is more semantically accurate than structural subtyping, as mentioned earlier, and as is explained in more detail in (AbdelGawad, 2016b).
6. : A Model of Nominal OOP
Using the records domain constructor () presented in Section 4 and signature constructs presented in Section 5, in this section we now present the construction of as a more precise model of nominallytyped mainstream OOP.
The construction of proceeds in two steps. First, the solution of a simple recursive domain equation defines a preliminary domain of raw objects, where an object in contains (1) a signature closure that encodes nominal information of nominallytyped OOP, and contains bindings for object members in two separate records: (2) a record for fields of the object, and (3) a record for methods of the object.
A simple recursive definition of objects with signature information does not force signature information embedded in objects to conform with their member bindings. Accordingly, in the second step of the construction of , invalid objects in the constructed preliminary domain of objects are “filtered out” producing a domain of proper objects that model nominal objects of mainstream OO software. Invalid objects are ones where the signature information is inconsistent with member bindings in the member records. The filtering of the preliminary domain is done by defining a projection function on the preliminary domain .
We call the model having the preliminary domain defined by the domain equation ‘’. Our target model, , is the one containing the image domain resulting from applying the filtering function on the preliminary domain of .
6.1. Construction of
The domain equation defining , and thence , uses two flat domains and . Domain is the flat domain of labels, and domain is the flat domain of signature closures (Section 5).
The domain equation that describes is
(4) 
where the main domain defined by the equation, , is the domain of raw objects, is the strict product domain constructor, and is the records domain constructor (Section 4). Equation (4) states that every raw object (i.e., every element in ) is a triple of:

A signature closure (i.e., a member of ),

A fields record (i.e., a member of ), and

A methods record (i.e., a member of where is the strict continuous functions domain constructor, and is the finitesequences domain constructor.)
Domain of is the solution of Equation (4). Applying the iterative leastfixed point (LFP) construction method from domain theory (Cartwright et al., 2016), the construction of proceeds in iterations, driven by the structure of the righthand side (RHS) of Equation (4). The RHS of the equation is viewed as a continuous function over domains (given the continuity of all used domain constructors, and that constructor composition preserves continuity.) Details of the iterative construction of are presented in (AbdelGawad, 2012).
The second step in constructing is the definition of a projection/filtering function, filter, to map domain of to the domain of valid objects modeling objects of nominallytyped OOP. For this, first, we define an object in to be valid as follows.
Definition 6.1.
An object in is valid if it is the bottom object , or if it is a nonbottom object such that

The fields shape and the methods shape of are exactly the same as (i.e., equal to) the shape of and the shape of , respectively,

Nonbottom valid objects bound to field names in have signature closures that subsign the signature closures for corresponding fields in , and

Nonbottom functions bound to method names in conform to corresponding method signatures in , where by conformance the functions are required to

take in sequences of valid objects whose embedded signature closures subsign (componentwise) the corresponding sequences of method parameter signature closures in , prepended with itself (for the implicit parameter self/this), and

return valid objects with signature closures that subsign the corresponding return value signature closures specified in the method signatures in .

As a direct translation of Definition 6.1, the function filter mapping into ( is a proper subdomain of ) is defined using the following three recursive function definitions, presented using lazy functional language pseudocode.

fun filter(o:):
match o with ((nm,se), fr, mr)
if (sfshp(se(nm)) != recshp(fr))
(smshp(se(nm)) != recshp(mr))
return // nonmatching shapes
else // lazily construct closest valid object to o
match se(nm), fr, mr with
(_, _, [(, )  i=1,,m ],
[(, , )  j=1,,n]),
(frtag, {  i=1,,m}),
(mrtag, {  j=1,,n})let si = se_clos(se, )
let misj = map(se_clos(se), [nm::])
// nm is prepended to to handle ‘this’
let mosj = se_clos(se, )
return ((nm,se),
(frtag, { filterobjsig(si,)  i=1,,m}),
(mrtag, { filtermethsig(misj, mosj, )
 j=1,,n}))
fun filterobjsig(ss:, o:):
match o with (s, _, _)
if (s ss)
return filter(o) // closest valid object to o
else
return // no subsigning
fun filtermethsig(in_s:, out_s:, m:):
return (let vos = map2(filterobjsig, in_s, )
in filterobjsig(out_s, m(vos)))
In the definition of filter, functions sfshp and smshp compute field and method shapes of signatures, while function recshp computes shapes of records. Function se_clos(se,nm) computes a signature closure corresponding to signature name nm whose first component is nm and whose second component is the minimal subset of signature environment se that makes se_clos(se,nm) a signature closure. To handle this/self a “curried” version of se_clos is passed to the map function. Additionally, domain is the domain of nonempty sequences of signature closures (nonempty because methods are always passed in the object this/self), and domains and are auxiliary domains of raw methods and methods, respectively. The function map2 is the twodimensional version of map (i.e., takes a binary function and two input lists as its arguments.)
In words, the definition of the filtering function filter states that the function takes an object of and returns a corresponding valid object of . If the object is invalid because of nonmatching shapes in the signature closure of and its member records, filter returns the bottom object (in domain , is the closest valid object to an invalid object with nonequal shapes in its signature and records.) Otherwise, has matching signature and record shapes but may have objects bound to its fields, or taken in or returned by its methods, whose signature closure does not subsign the corresponding signature closures in the signature closure of . In this case, filter lazily constructs and returns the closest valid object in domain to , where all nonbottom fields and nonbottom methods of are guaranteed (via functions filterobjsig and filtermethsig, respectively) to have signature closures that subsign the corresponding signature closures in the signature closure of .
Function filterobjsig checks if its input object o has a signature closure s that subsigns a required declared signature closure ss. If s is not a subsignature of ss, filterobjsig returns . If it is, the function calls filter on o, thereby returning the closest valid object to o.
For methods, when filtermethsig is applied to a method m it returns a valid method that when applied to the same input as m, returns the closest valid object to the output object of m that subsigns the declared output signature closure out_s corresponding to the sequence of valid objects closest (componentwise) to that (again, componentwise) subsigns the declared sequence of input signature closures in_s prepended with the signature closure of the object enclosing m (to properly filter the first argument object in , which is the value for this/self.)
Having defined the filtering function filter, the proof that domain , as defined by filter, is a welldefined computable subdomain of is presented in Appendix B.
6.2. Class Types
As constructed, is a nominal model of OOP, because objects of domain of include signatures specifying the associated class contracts maintained by the objects (including inherited contracts.) This nominal information encoded in signatures provides a framework for naturally partitioning the domain of objects into sets defining class types, where a type is a set of similar objects.
First, we define exact class types. The exact class type corresponding to a class C is the set of all objects tagged with the signature closure for C.^{5}^{5}5In Java, for example, objects in the exact type for a class C are precisely those for which the getClass() method returns the class object for C. Next, it should be noted that a cardinal principle of nominallytyped mainstream OOP is that objects from subclasses of a class C conform to the contract of class C and can be used in place of objects constructed using class C (i.e., in place of objects in the exact class type of C.) Hence, the natural type associated with class C, called the class type corresponding to or designated by C, consists of the objects in class C plus the objects in all subclasses of class C. In nominallytyped OO languages, the class type designated by class C is not the exact class type for C but the union of all exact types corresponding to classes that subclass (i.e., inherit from) class C, including class C itself.
Motivated by this discussion, we define class types in as interpretations of signature closures. For a signature closure , its interpretation is a subdomain of domain , having the same underlying approximation ordering of domain and whose universe is defined by the equation
(5) 
In other words, the class type designated by a class is the interpretation of the signature closure corresponding to the class, which, in turn, is the set of all objects in domain of with a signature closure that subsigns , or the bottom object . Given that subsigning in models OO inheritance, the definition of class types is in full agreement with intuitions of mainstream OO developers.
Having defined class types, it should be noted that a class type is always a nonempty domain (i.e., always has some nonbottom object) because the object
(where is the fields shape of and is the methods shape of ) is always a valid constructed object (i.e., is an object of domain of that passes filtering to domain of .) This object is a member of by Equation (5). The nonemptiness of class types is used in the proof of the identification of inheritance and subtyping.
6.3. Inheritance is Subtyping
After we constructed , and after we defined class types in agreement with intuitions of mainstream OO developers, we can now easily see what it means for nominallytyped OO type systems to completely identify inheritance and subtyping. We express this statement formally as follows: Two signature closures corresponding to two classes are in the subsigning relation if and only if the class types denoted by the two signature closures are in the subset relation (i.e., the two classes are in the inheritance relation if and only if the corresponding class types are in the nominal subtyping relation.) We prove the correspondence between inheritance and subtyping in the following theorem.
Theorem 6.1.
For two signature closures and denoting class types and , we have
(6) 
Proof.
Based on Equation (5), and the nonemptiness of class types, the proof of this theorem is simple.
Case: The (only if) direction:
If , by applying the definition of (i.e., Equation (5)) all elements of belong to (the variable in Equation (5) is instantiated to , and is a common member in all class types.) Thus, .
Case: The (if) direction:
By the nonemptiness of there exists a nonbottom object of with signature closure . If , then . By Equation (5) all nonbottom members of must have a signature closure that subsigns . When applied to we thus have . ∎
We should notice in the proof above that it is the nominality of objects of (i.e., the embedding of signature closures into objects) that makes being a superset of imply that has as one of its supersignatures, and vice versa. The simplicity of the proof is a clear indication of the naturalness of the definitions for class signatures and class types.
7. Compared to Structural Models of OOP
Having presented , in this section we briefly compare to the most wellknown structural domaintheoretic models of OOP, namely the model of Cardelli, which we call , and that of Cook, which we call .
Comparing to and reveals that includes full class name information while and totally ignore this information, based on the different views of type names adopted by each of the models. Objects in and are viewed as mere (plain) records, while in they are viewed as records that maintain contracts, which are referred to via nominal information, with nominal information being part of the identity of objects.
, and also have different views of types, type inheritance and subtyping, where behavioral contracts (via type name information) are part of the identity of types in , and thus are respected in type inheritance and subtyping, but contracts are ignored in and . In addition, and model recursive types, while does not. This leads (due to nominality) and (due to lack of recursive types) to identify type inheritance with OO subtyping while breaks that identification.
More details on the differences and similarities between , and can be found in (AbdelGawad, 2016a).
8. Conclusions
Based on realizing the semantic value of nominaltyping, in this paper we presented as a model of OOP that includes nominal information found in nominallytyped mainstream OO software. The inclusion of nominal information as part of the identity of objects and class types in led us to readily prove that type inheritance, at the syntactic level, and subtyping, at the semantic level, completely agree in nominallytyped OOP. A comparison of to structural models of OOP revealed nominal and structural models of OOP have different views on fundamental notions of OOP. It is necessary, we thus believe, to include nominal information in any accurate model of nominallytyped mainstream OOP. By its inclusion of nominal information, offers a chance to understand and advance OOP and current OO languages based on a firmer semantic foundation.
9. Future Work
One immediate possible future work that can be built on top of research presented in this paper is to define a minimal nominallytyped OO language, e.g., in the spirit of FJ (Igarashi et al., 2001), then, in a standard straightforward manner, give the denotational semantics of program constructs of this language in . The type safety of this language can then be proven using the given denotational semantics.
Generics add to the expressiveness of type systems of nominallytyped OO programming languages (Bank et al., 1996; Bracha et al., 1998; Cartwright & Steele, 1998; Langer, 2015; Bloch, 2008; Gosling et al., 2014; CSh, 2015; Sca, 2014). Another possible future work that can be built on top of is to produce a denotational model of generic nominallytyped OOP. Such a model may provide a chance for a better analysis of features of generics in nominallytyped mainstream OO languages and thus provide a chance for suggesting improvements and extensions to the type systems of these languages.
Acknowledgments
The authors are thankful to Benjamin Pierce for the feedback he offered on motivating and presenting .
Appendix A Class Signature Examples
To illustrate the definitions of signature constructs given in Section 5, in this appendix we present a few examples of signature constructs. Assuming the following OO class definitions (in Javalike pseudocode),
we define the corresponding class signatures
(Object, [], [], [(equals, [Object], Boolean)]),
(Boolean, [Object], …), and
(Pair, [Object], [(first, Object), (second, Object)],
[(equals, [Object], Boolean), (swap, [], Pair)])
and, hence, define signature environments {, },
and
{, , }, and
the signature closures (Object, ),
and = (Pair, ).
We can immediately see, using the definition of extension and the definitions of immediate subsigning and subsigning in Section 5, that , , and . The last conclusion expresses the fact that class Pair inherits from class Object, and the second to last conclusion expresses that class Pair is an immediate subclass of class Object (The reader is encouraged to find other similar conclusions based on the definitions of classes Object, Boolean and Pair given above.)
Appendix B Proofs
In this appendix we present proofs of main theorems in this paper, pertaining to the properties of the records domain constructor , and to the filtering of to . These proofs ascertain the welldefinedness of and of the filtering, and thus their appropriateness for being used in constructing .
b.1. The Domain of Record Functions has an Effective Presentation
It is straightforward to confirm that constructs a domain. To prove that constructs domains given an arbitrary domain and a domain (with a fixed interpretation as a flat domain of labels), we build an effective presentation of the finite elements of , assuming an effective presentation of the finite elements of and . We prove that these finite elements form a finitary basis of the records domain. Since has a fixed interpretation, domain constructor can be considered as being parametrized only by domain .
Given an effective presentation of where , we define, for all , the finite sequences
where , and
(7) 
The size , of , is the number of ones in the binary expansion of , and thus with equality only when is one less than a power of 2. only when , and in this case (the empty label sequence)^{6}^{6}6The definition of is patterned after a similar construction presented in Dana Scott’s “Data Types as Lattices” (Scott, 1976). Unlike the case in Scott’s construction, here, in the LHS of Equation (7), is doubled—i.e., the binary expansion of is “shifted left” by one position—to guarantee , and thus guarantee that is never an element of .. It is easy to confirm that there is a onetoone correspondence between the set of natural numbers and the set of distinct finite label sequences .
Given an effective presentation of the finite elements of , an effective presentation of the finite elements of , the domain of (nonstrict) sequences of length () of elements of , is
where, for ,
is the onetoone tupling function (also called the Cantor tupling function), and
is the onetoone Cantor pairing function.
Now, let
where, again, is the number of ones in the binary expansion of , and
The sequence of the finite elements of can then be presented as , and for ,
Given the decidability of the consistency () and lub () relations for finite elements of , the presentation of the finite elements of is effective, since, for record functions and as defined in Section 4.2, under the approximation ordering defined by Equation 4.2, the consistency relation
(8) 
is decidable (given the finiteness of records), and the lub relation
(9) 
is recursive (handling or in the definitions of and is obvious. All record functions are consistent with , and the lub of a record function and is .)
Lemma B.1 ( constructs domains).
Under , elements of form a finitary basis of .
Proof.
Lemma B.1 actually proves that is a computable function that maps a pair of a flat domain and a domain to the corresponding record domain. The presumption is that no effective presentation is necessary for the flat domain because distinct indices for elements of will simply mean distinct labels . If is a flat countably infinite domain (which implies it has an effective presentation) and is an arbitrary domain, then the lemma asserts that is a domain with an effective presentation that is constructible from the effective presentations for and .
b.2. Domain Constructor is Continuous
Lemma B.2 ( is monotonic).
For domains and , and a flat domain of labels ,
Proof.
First, we prove that is monotonic with respect to the subset relation on the universe of its input, i.e., that . Then, given that the approximation ordering on (as a subdomain of ) is the restriction of the approximation ordering on , we prove that the elements of (as members of ) form a domain under the approximation ordering of , and thus that is a subdomain of .
Since , then . For arbitrary where , we thus have
Thus, . Accordingly, for sets (the elements of with tag ) and (the elements of with tag ), as defined in Equation 2 of Section 4.2, we have . Thus,
Thus,
(10) 
Next, since is a subdomain of when restricted to elements of , we know: (i) the approximation relation on is the approximation relation on restricted to ; (ii) consistent pairs of are consistent pairs in ; and (iii) lubs, in , of consistent pairs of elements of are also their lubs in . Thus, for , , and
Hence, according to the definition of the approximation, consistency and lub relations for (Equations (4.2), (8) and (9)), the lub, in , of a consistent pair of records is also their lub in . That is, respectively, for , we have
(11) 
(12) 
and
(13) 
From equations (10), (11), (12), (13), and the fact that is the bottom element of both and , we can conclude using Definition 6.2 in (Cartwright et al., 2016) that
∎
In addition to being monotonic, continuity of a domain constructor asserts that the lub of domains it constructs using a chain of input domains is the domain it constructs using the lub of the chain of input domains (i.e., that, for , the lub of a chain of input domains gets mapped by to the lub, say domain , of the chain of output domains .)
Lemma B.3 ( preserves lubs.).
For a chain of domains , if , , and , then .
Proof.
Let be the lub of the chain of domains (’s form a chain by the monotonicity of .) Domain is thus the union of domains , i.e., .
Domain is equal to because each element in ( is a record function) is an element of a domain for some . Given is a subset of , will also appear in .
Similarly, a record function in is an element of a domain for some , because every finite subset of has to appear in one (given that is a chain of domains.) Thus, by the definition of , is also a member of .
This proves that . ∎
b.3. Filtering is a Finitary Projection
In this section we prove that function filter, as defined in Section 6.1, is indeed a finitary projection, and thus that the domain of valid objects (Definition 6.1 in Section 6.1) defined by the filtering function is a subdomain of Scott’s universal domain , and thus is indeed a domain.
To do so, we first prove a number of auxiliary propositions regarding domain .
Proposition B.1.
In domain , higherranked objects do not approximate lowerranked ones, i.e., implies
Proof.
By strong induction on rank of objects. ∎
To prove that filter defines a projection, in the sequel we use the inductivelydefined predicate valid (as defined by Definition 6.1 in Section 6.1) that applies to objects of . Note that, in addition to , objects with empty field and method records provide base cases for the definition of valid.
Lemma B.4 (filter returns the closest valid object that approximates its input object).
For an object of , filter() valid(filter()) ( valid() filter())
Proof.
By strong induction on rank of objects, noting that, for the base case, filter(o) diverges (i.e., “returns” ) for the rank 0 input object , and if an object of rank 1 is invalid then filter(o) also returns (no distinct objects of rank 1 approximate each other.) Proposition B.1 is used for the inductive case.∎
Theorem B.1.
filter is a finitary projection.
Proof.
We prove that filter is a finitary projection, on four steps.

filter is a retraction: filter(filter(o)) = filter(o)
Proof.
Obvious from definition of filter, and that, by Lemma B.4, function filter returns a valid object (i.e., valid(filter(o))). ∎

filter approximates identity: filter(o) o
Proof.
By Lemma B.4. ∎

filter is a continuous function
Proof.
Direct, from the continuity of functions used to define filter (such as recshp, map, se_clos, etc.), and noting the closure of continuous functions under composition and lambda abstraction. ∎

filter is finitary
Proof.
The condition in point 2 of Theorem 8.5 in (Cartwright et al., 2016), namely
can be rewritten for the filtering function filter as
(14) Objects of domain are in onetoone correspondence with principal ideals over their finitary basis. The filtering function filter returns, as its output, the closest valid object to its input object (The object returned is a welldefined object, and it is a fixed point of the filtering function.) Thus, given that objects correspond to strong ideals in the finitary basis of , they correspond to downwardclosed sets. Condition (14) is thus true for all objects in .∎
Based on the definition of finitary projections, function filter is thus a finitary projection.∎
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