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All you wanted to know about types but were afraid to ask
2014-07-08 01:02:06
Type systems are a hot topic in programming language debates. Often, however, the argument being had spend a lot of time batting about the lack of common terminology and shared information. This post hopes to reveal a swath of information about types to better form common ground for discussion of type systems of all shapes and sizes.

Type systems are a hot topic in programming language debates. Often, however, the arguments being had spend a lot of time batting about the lack of common terminology and shared information. This post hopes to reveal a swath of information about types to better form common ground for discussion of type systems of all shapes and sizes.

Corrections and extensions are very welcome.

The word "type" is easy to overload in common usage. In can refer to things as vague as elements of a domain being considered or as specific as the formal abstract definition favored by type theorists today. Furthermore, the presence or absence of types can be seen as a positive or negative trait of a language and so it's easy to see two people arguing vehemently about whether or not a language "has types" even when their definitions of "type" vary.

A wide array of common uses

In common usage, type might informally mean any mechanism whatsoever for distinguishing loose or tight classes of things. This is a very wide net to cast and below it's explored a little bit. What's important to note is not the real breadth of what "type" means but instead the importance of establishing common ground before debating about the word.

Merriam Webster says

qualities common to a number of individuals that distinguish them as an identifiable class: as (1) : the morphological, physiological, or ecological characters by which relationship between organisms may be recognized (2) : the form common to all instances of a linguistic element

and notes the synonyms "kind", "sort", and "category". In the domain of programming languages "type" is heavily overloaded between various classes of "type systems" from dynamic to static, strict to loose.

In order to discuss further it's important to consider the phase distinction in formal languages between their statics and their dynamics. The statics of a language are the properties which are inferred from the syntax of the language alone---these properties cannot reference behavior of the language under information available only at runtime. The dynamics of a language are the properties of how programs in the language evolve at runtime.

Various languages consider "types" to arise as static properties or to be available at runtime. Sometimes when types are available at runtime a language is said to have "reflection". Sometimes when types are not available at runtime a language is said to have "type erasure".

Some types are first class and available for reasoning either within the language or at least by the user. Some types are exclusively available to the compiler for accelerating code and are never made apparent to the programmer. Among those type systems which are first class you may have a burden laid upon the programmer to annotate their code with the proper types. You may also have an inference engine atop that system which eases or obviates the need to annotate programs.

Some types are purely analytic or "pluggable" in that they can constrain a program or improve our understanding of it but can never affect the semantics of the program itself. Others are intimately tied into the semantics of a language such that prior to computing all of the types of a program it's not possible to know its behavior.

Formal terminology

Among type theorists there is a more rigorous and streamlined definition of "type" which captures some (but hardly all) of the meanings above.

To invent a distinctive word, a "formal type" is a description of the statics of a language. It's assigned to a fragment of abstract syntax and it indicates the kind of value that the abstract syntax represents. For example, both 3 and 3+3 considered as abstract, static syntax might be given the same type, Natural, because they each take values as Natural numbers. For the remainder of this section, the word "formal" may be elided if the meaning is clear.

Formal types give rise to the technical notion of "type safety". A language and its type system together are type safe if the dynamics of the language respect two properties: preservation and progress. Language dynamics preserve a type if as a program "evolves" at runtime its type does not change. For instance, we might consider evaluation of the abstract syntax 3+3 as stepping like


If this evolution does not change the type then the dynamics preserve the type. A dynamics has progress if any abstract syntax either reflects a value of its type or can be reduced. In order to talk about this we thus have to give a language a notion of base values which every program attempts to compute. For instance, we can talk about the values of Natural as the natural numbers 0, 1, 2, 3, .... Thus progress entails that any program typed as a Natural either is one of those numbers or can continue to evolve.

When a language---its statics, type system, dynamics, and values together---has preservation and progress then it is type safe. What this means is that given the type of a program we know something very clear about the eventual value the program will compute.

The "unityped" argument

One nice thing about the formal types framework is that it can be applied to languages even if the language itself does not declare a choice of statics and types.

For instance, we can examine a dynamically typed language in a trivial sense by giving every syntactically correct program the same type which reads something like "either and integer or a string or an exception or a dictionary of integers or a dictionary of strings or an array of integers or and array of dictionaries of integers or..." eventually encompassing every kind of value allowable within the language. Often this large union of types is called Any.

Under a type system like this any language is automatically type safe! This should not be thought of as a strength of the language but instead it should suggest a crucial weakness to the notion of type safety itself: if the types are too boring themselves then preservation and progress don't mean much.

This leads to a desire to give type systems which are refinements of Any. Sometimes compilers of dynamic languages will automatically attempt to infer these more refined types as this can lead to improved performance. Sometimes first-class type systems allow for giving both vague and refined types together within a program leading to hybrid type systems where the degree of "interestingness" or "safety" of the given types can vary throughout a program.

It's also sometimes seen as derogatory when a language is considered to only have Any as its type. It certainly means a lack of formal type analysis available statically and it suggests that language features exist which prevent this static analysis from being possible. However, the lack of a refined type system is different from the impossibility of finding such a system.

Pluggable types and general static analysis

Languages which do not fix their formal type system---or, even more weakly, languages where the semantics are not dependent on the types---have the option to be analyzed under many different kinds of statics. These are sometimes called "pluggable types".

In this circumstance the UX of typing may feel very different. Instead of being something performed at each compilation, static analyses might be performed optionally by a third-party tool. The analysis may be partial and multiple tools might be combined together in order to build a complete static report.

Languages of formal types

Advanced forms of formal types can involve their own language for merely describing the types themselves. This often involves the notion of "type constructors" such as List which are applied to other types like Integer to form complex types like List Integer, i.e. lists of integers. Usually these languages are described as "type languages" while the language being described by the types is the "term language". So, [1,2,3] is a value in the term language which has the formal type List Integer which is written in the type language.

As the complexity of the type system itself grows it may begin to resemble formal language all on its own and it then becomes desirable for the "types to have types". This second level of typing is sometimes called "kinds". The third level is sometimes called "sorts".

It's can be desirable for the type languages at each higher level to be simpler and simpler so that there isn't an infinite stack of typing levels to handle. This provides a tradeoff on the expressiveness of the type language. It's also often desirable for these languages to not be Turing Complete as it's important to be able to say two types are equivalent (and thus their programs take the same values). Turing Completeness would cause equivalence to be undecidable in general and mean that type checking could sometimes fail to terminate.

On the other hand, some languages have embraced unification of the types and terms. In other words, the very language which the types describe is indentical to the language of the types. This produces an infinite hierarchy of types and all the complexities involved therein. Typically these languages have "dependent types" which allow information to bleed between the layers of types. It's also typical that these languages restrict Turing Completeness at every layer for all the reasons discussed previously.

As a final note, the complexity of type languge is usually related to the "interestingness" of types. Very boring type languages will have only very weak notions of "type safety" while very complex type languages will usually include both boring notions of type safety and extremely interesting ones. At the most complex end, dependently typed languages typically have types which can express arbitrary mathematical theorems and the programs which inhabit these types an be seen as proofs of validity of the theorem.

More practically, complex type languages can capture information about complex abstraction patterns being used in the term language. Typically, a term language with a weak type language will trade off simplicity and speed of analysis for loss of abstraction power. Since complex type languages are at the very edge of research today this is perhaps the most key tradeoff in the design of the statics of a language.

Dynamic types

Outside of the realm of "formal types" explored previously there are many kinds of type systems available dynamically in languages. For instance, many languages express notions of reflection which allows a language to get ahold of the type of a value at runtime. For instance, in Javascript we can write

> typeof(3)

to reflect on the type of a value like 3.

The existence of reflection implies that the runtime of a language maintains fragments of some type language attached to each value. In formal terms these are sometimes called "tags" to differentiate from formal types, but in dynamic language communities they are referred to simply as types.


As dynamic types are available at runtime if type errors occur they only occur when the program actually runs into an inconsistency of types. Oftentimes this occurs in dynamic languages as explicit or implicit contract checking where assertions about arguments are made to ensure that their values are appropriate.

Well-structured contracts can ensure that violations of program invariants are quickly detected as the program is exercised at runtime. They also have the full expressiveness of the language available to them for structuring the details of a contract.

It's often interesting to compare contracts and formal dependent types as they appear to have similar expressiveness. Contracts operate dynamically during program operation and check decidable properties of the values of a running program. Dependent types express static invariants which all possible runnings of a program will prove. Complex contracts can be easily added to a program though no matter how many times you run a program you can never be certain that your contracts are always upheld. Complex dependent types are very challenging to introduce to a program as they impose structure across the entirety of the program eventually ensuring that all possible runtimes respect the properties of the types.

Reflection and formal type safety

Languages with reflection are often thought to be formally type unsafe as you can trivially write programs which reflect on type information and use it to coerce, in other words to change their own type and break preservation. As discussed before, though, formal type safety must be considered relative to a choice of formal type system and we can always type everything as Any and get it back.

Ultimately, however, languages must usually trade off between simple type languages and the power of reflection. Unbounded reflection usually requires very complex type system to have interesting type safety. Bounded (a.k.a. "type safe") reflection is possible in moderately powerful formally typed languages, though, and most uses of reflection can be performed in bounded systems.

Types as user interfaces

When a language fixes a formal or informal type system it will often use that type system to provide feedback to the programmer or tester about errors in the program.

For instance, type information (again, formally, tag information) available at runtime in dynamically typed languages with reflection is often used to make meaningful error messages when a program throws an exception or gets stuck. This improves the programmer user experience by revealing information available to the runtime system or interpreter.

The most notorious user experience with types arises in compiled, statically typed languages where the compiler will refuse to proceed if it cannot complete a typecheck. In other words, the compiler examines the syntax of the entire program and ensures that the types are consistent throughout. When inconsistencies are found it rejects the entire program and reports the inconsistencies.

Usually, the former user experience is the hallmark of the general class of "dynamically typed languages" while the latter user experience is associated with "statically typed languages". In fact, this user experience distinction is often considered the primary feature distinguishing the two.

Generally, this is a good heuristic, but it's important to note where it breaks down.

For instance, some dynamically typed languages with pluggable type systems might be able to provide ahead-of-time warnings about possible type failures (see Erlang's Dialyzer or Clojure's typed-clojure). It's also very possible to write a compiler for a statically typed language which compiles programs with type inconsistencies and merely ignores them until they actually cause problems at runtime (see GHC Haskell's -fdefer-type-errors flag).

Generally it's an open research question as to what the best user interfaces for information afforded by either formal or informal types are. This research is well-developed for dynamic types but often more anemic for static types.

Types and programming performance

Ultimately, there is usually a desire for empirical evidence that some kind of type system (or lack thereof) improves coding capacity by some measure or another. These empirical studies are often valuable, but also often lacking in the kind of internal or extenal validity needed to draw convincing conclusions.

Lacking potent empirical evidence, performance arguments often come down to stylistic values or feel which relegates the question again to one of user experience and psychology.

Testing versus typing

One argument for style and performance is the difference between testing and typing. This is common enough to deserve special consideration.

Program tests are capable of providing evidence for achievement of a much wider set of goals than the guarantees of types can provide. In professional programming it's ubiquitous to design program testing regimes which ensure that deliverables achieve the business value they were designed to while it's almost inconceivable to believe that even the very most sophisticated type language could express such a demand.

Type systems are faster to run and provide comprehensive guarantees (so long as the system exhibits type safety) on program behavior. It is impossible to write tests which can efficiently enumerate all possible program states to ensure that no edge cases remain, though any expressive type system can succinctly eliminate edge cases by simply making invalid programs inexpressible.

Ultimately, there is very little reason to believe that testing would subsume all of typing or visa versa, however the middle ground between the two use cases is sometimes contested. Extensive unit test suites often test program properties which could be encoded in types and sufficiently abstract program components can be completely guaranteed of their value via expressive typing. In this middle ground there is unlikely to be a silver bullet---specific program performance needs should be analyzed to determine whether the tradeoffs of static or dynamic analysis work best.