While the ease of use of its query language has certainly contributed to the phenomenal adoption of GraphQL, little has been known about fundamental computational properties of this language. To fill this gap, together with Jorge Pérez from the Universidad de Chile, we have recently embarked on a formal study of the language. The result of our work is a research paper titled “Semantics and Complexity of GraphQL” (download a pre-print of our paper). After a typical academic review process, this paper has been accepted to be presented in The Web Conference 2018. Due to the highly technical, math-heavy nature of this work, the paper may not be easily digestible for a broader, Web-developer audience. Even the summary of the paper by Adrian Colyer of “the morning paper” is still fairly technical. Therefore, in this post I aim to provide a more lightweight high-level summary of the results of our work (if you prefer a more visual summary, you may want to check out the video recording of my talk at GraphQL Europe). Hereafter, when I talk about the GraphQL language, I mean the query-specific part of the language that the GraphQL framework introduces to retrieve data from a Web server.

We have studied this language in terms of a number of typical computation-related questions that people focus on when analyzing query languages. In Theoretical Computer Science such types of questions are captured in the form of formally-defined problems and the subject of study is to achieve a mathematically-proven understanding of how difficult it is to solve any possible instance of a given problem. A typical example of such a problem in the context of database query languages is to decide for any given answer that may be in the result of a given query over a given database whether this answer is indeed in the query result. You may notice that this decision problem is related to the task of producing the query result. In fact, the difficulty of this problem (for a given query language) is one of the most commonly used formal measures to identify how computationally complex a query language is. The problem is usually called the ** evaluation problem** of a query language.

## The Evaluation Problem of GraphQL

One of our contributions in our paper has been to show that evaluation problem of the GraphQL language is “very simple”. More precisely, we show that the problem is NL-complete. It is well-known that NL-complete problems can be solved in practice by programs that use a high degree of parallelism. Hence, this gives us our first positive observation about the GraphQL language. Admittedly, this is a very abstract finding. However, to appreciate this property of the GraphQL language we may look at other well-known query languages. For instance, we may consider the Relational Algebra, which is the formal foundation of SQL. It has been shown that the complexity of the evaluation problem of the Relational Algebra is PSPACE-complete. The same has been shown for version 1.0 of the RDF query language SPARQL. Problems in this class are highly intractable. As another example, for general conjunctive queries the evaluation problem is NP-complete. NP-complete problems are commonly believed to be less complex than PSPACE-complete problems but still intractable. In contrast, there is a class called PTIME which consists of problems that are known to be tractable; that is, for each such problem there exists an algorithm whose runtime is polynomial in the size of the input. This is the case for SPARQL basic graph patterns (which comprise the conjunctive fragment of SPARQL) for which the evaluation problem is in PTIME. How does this compare to the NL-completeness of the evaluation problem of GraphQL? The good news for GraphQL fans is that NL is an even lower complexity class than PTIME.

## The Enumeration Problem of GraphQL

After having done this initial theoretical comparison of the GraphQL language to other query languages, we have looked at another, more practical computation-related problem. This problem is called the ** enumeration problem** and is concerned with how difficult it is to produce the complete result of a query. To study this problem we made our life a bit easier by considering only a particular class of GraphQL queries. We call the queries in this class “

**“. Informally, these are queries for which the process of field collection (as specified in Section 6.3.2 of the GraphQL spec) is not necessary. For these queries, and any arbitrary set of data, we have shown that the respective query results can be produced symbol-by-symbol with only constant time delay between symbols. This property is the best that one can hope for in a query language and not many query languages possess this property!**

*non-redundant queries in ground-typed normal form*At this point it has to be emphasized that focusing on non-redundant queries in ground-typed normal form is not a limitation of our work. In contrast! As we also show in our paper, every arbitrary GraphQL query can be rewritten into a non-redundant, ground-typed query that is guaranteed to produce the same result. For the necessary rewriting rules refer to Proposition 3.9 in our paper.

Given our positive finding regarding the constant time delay, we can also infer the following property of the GraphQL language: The time required to produce the complete query result (for any non-redundant query in ground-typed normal form) depends linearly on the size of this result. This is another highly desirable property of a query language!

## The Issue of Exponential Result Size Blow-up in GraphQL

The only caveat regarding the latter finding is that there are cases in which the size of the result of a GraphQL query depends exponentially on the size of the query. For a simple example, consider querying data with a GraphQL schema whose query type has a field that allows us to retrieve data about a person, say Alice; this data has a scalar field name with value Alice and another field knows with an array of two data objects about two other persons; each of these data objects also has a knows field referring back to Alice. Then, for a query of the form

start { knows { knows { ... { knows { name } } ... } } }

in which 2xN knows fields are nested, the result contains the value Alice 2^N times! In other words, increasing the size of the query linearly (adding two more levels of nesting in each step) will cause the size of the query result to increase exponentially. This is not just a theoretical issue. In the introduction of our paper we describe an experiment in which we have observed the issue when querying the public GraphQL API of Github. As a sidenote, and in all fairness to the GraphQL language, such an exponential result size blow-up is possible in many query languages (all that it needs is the possibility of expressing conjunctive queries).

While we are the first to formally quantify the possible extend of the result size blow-up of GraphQL, the basic issue of potentially huge results has been recognized before in the GraphQL community. Existing approaches to address this issue are to restrict queries i) based on their nesting depth, ii) based on a calculation of the ** maximum** possible number of result nodes at each level of the nesting depth, or ii) based on some cost estimation or complexity estimation for which users have to provide various cost factors for different elements of GraphQL schemas. Notice that all these approaches use heuristics and none of them takes into account the actual data being queried. As a consequence, these approaches fall short in providing a robust solution for the issue as they can fail in both directions: discarding requests for which the response may be of a manageable size, and allowing requests for which producing the complete response is too resource intensive.

## Our Solution to the Result Size Issue

Instead of relying on data-agnostic heuristics and estimates, we introduce an approach which is accurate. This approach comes in the form of an algorithm that returns the * exact size* of the result of any given non-redundant GraphQL query in ground-typed normal form (see above) over any possible data set (without actually producing this query result). We use this algorithm to prove that the result size can be computed in an amount of time that depends only linearly on the product of the query size times the data set size. Hence, our algorithm achieves this complexity bound. The idea of the algorithm is to recursively consider every subquery and add up the sizes of results of the subquery for every part of the queried data for which the subquery has to be evaluated. During the recursive process, data objects are labeled with the result sizes of subqueries evaluated at them. These labels are then used to avoid the repeated calculation of subquery result sizes.

As an example, consider some simple example data that may be illustrated as the following graph of objects (where the node named *r* in the graph corresponds to the query object that would have to be specified in a GraphQL schema for this data).

Moreover, consider the following GraphQL query.

query { start { advisor { univ { name } } friend { univ { name } } } }

Apparently, the result of this query over our example data would be the following JSON object.

start: { advisor: { univ: { name: UCh } } friend: { univ: { name: UCh } } }

Now, instead of immediately starting to produce this result, suppose we first want to calculate its size. We measure the size of GraphQL query results in terms of symbols, where every field name counts as a symbol, and so does every scalar value and every special character (such as colons and curly braces). For instance, the aforementioned result of our example query happens to be of size 26. However, we want to calculate this size without already having produced the query result. To this end, we may first look at the root field start of the query object, and we observe that the number of result symbols that would be produced from this field is 4+*x* where the number 4 covers the symbols in the first and the last line of the example result and *x* is the sum of the result symbols obtained from going to the data object *u* and evaluating each of the two subqueries, *q1* = advisor{univ{name}} and *q2* = friend{univ{name}}. To make the following explanation more concise, let us denote the size of these two subquery results by writing *size(q1,u)* and *size(q2,u)*, respectively. Thus, we know that the size of the complete query result is equivalent to

4 + *size(q1,u)* + *size(q2,u)*

What remains is to calculate the sizes of the two subquery results, which we can do in a recursive fashion using a similar reasoning as before: First, let’s focus on *size(q1,u)*. By looking at the data object *u*, we observe that the root field advisor of subquery *q1* is present in *u*. Therefore, we know that the subquery result will contain 4 result symbols from the advisor field plus the symbols obtained from going to data object *v* and evaluating the subquery *q3* = univ{name}. Hence, we have that

*size(q1,u)* = 4 + *size(q3,v)*

and we need to enter the recursion again to calculate *size(q3,v)*. In this case, we have that

*size(q3,v)* = 4 + *size(q4,w)*

with subquery *q4* = name, and another recursion step is needed for *size(q4,w)*. This step now brings us to a base case of the recursion because, by looking at data object *w*, we see that the result of subquery *q4* is name:UCh and, thus, contains exactly 3 symbols. At this point, our algorithm adds a label to data object *w* to indicate that the result size of subquery *q4* at this object is 3. Next, the recursion comes back to calculating the value of *size(q3,v)*, for which it now becomes clear that this value is 7. So, data object *v* is labeled to indicate that the result of subquery *q3* at this object is 7. Returning now to the calculation of *size(q1,u)*, it becomes clear that *size(q1,u)*=11 and the data object *u* is labeled accordingly.

Being back at the initial step of the recursion now, we still have to calculate the value of *size(q2,u)*. Observe that *q2* contains the subquery *q3* that we have seen before and it also has to be evaluated at the same data object as before (that is, object *v*). Therefore, to calculate

*size(q1,u)* = 4 + *size(q3,v)*

it is now possible to obtain the value of *size(q3,v)* by reading the corresponding label at object *v* instead of going into the recursion again for *size(q3,v)*. The remaining steps of the calculation should not be difficult to guess at this point.

As can be observed from this example, labeling the data objects during the execution of our algorithm enables the algorithm to avoid repeating the same calculations and, thus, to achieve the polynomial runtime (even in cases in which actually producing the query result would have an exponential runtime). For a pseudo-code representation of the complete algorithm refer to our paper. We also have a prototypical JavaScript implementation of the algorithm.

Given this algorithm, we propose that GraphQL servers should perform the following steps for any given query:

- First, compute the size of the response, which can be done efficiently by using our algorithm.
- If the size is too big, reject query.
- Else, inform the size to the client, and
- Produce and send the response byte by byte.

## Conceptual Contributions

Before wrapping up, I would like to also highlight some of the conceptual contributions that we have made in addition to all the aforementioned technical contributions. As a foundation of our work, we have developed a formally more precise definition of the GraphQL language. The main reason why this was necessary is that the semantics of GraphQL queries—i.e., the definition of what the expected result of any given query is—is given in the GraphQL specification by means of a recursive program specified by pseudo code. This recursion is based on an operation to resolve any field in a query. Surprisingly, this operation is not fully specified and, instead, simply assumes access to an “*internal function […] for determining the […] value of [the] field.*” While the lack of a more precise definition of this internal function is likely to be intentional (to allow for implementations of GraphQL on top of arbitrary database back-ends), it makes a systematic analysis of the GraphQL language unworkable. As a basis for providing a well-defined formalization of the language, we had to introduce a formal model that defines the notion of a so-called ** GraphQL graph**. This notion is an artifact of realizing that GraphQL enables users to query an underlying data set in terms of a virtual, graph-like view of this data set. The form of this view depends on the GraphQL schema being used, and the resolver functions that implement the schema can be understood to establish the view. Our notion of a GraphQL graph is an abstraction to conceptualize such a view independent of its relationship to the underlying data set. As a sidenote for readers who are familiar with the notion of Property Graphs as supported by popular graph database systems like Neo4J and by the Apache Tinkerpop graph processing framework, a GraphQL graph is very similar to a Property Graph (with a few additional, GraphQL-specific features). The technical details of all these definitions can be found in our paper.

## Thanks

As a final remark, I want to thank the GraphQL community (in particular, the initial creators of the original GraphQL spec) for the excellent documentation of GraphQL which has made our work of formalizing the language a pleasant journey.