Abstract
Apache Spark is a popular framework for writing large scale data processing applications. Our long term goal is to develop automatic tools for reasoning about Spark programs. This is challenging because Spark programs combine databaselike relational algebraic operations and aggregate operations, corresponding to (nested) loops, with User Defined Functions (UDFs). In this paper, we present a novel SMTbased technique for verifying the equivalence of Spark programs.
We model Spark as a programming language whose semantics imitates Relational Algebra queries (with aggregations) over bags (multisets) and allows for UDFs expressible in Presburger Arithmetics. We prove that the problem of checking equivalence is undecidable even for programs which use a single aggregation operator. Thus, we present sound techniques for verifying the equivalence of interesting classes of Spark programs, and show that it is complete under certain restrictions. We implemented our technique, and applied it to a few small, but intricate, test cases.
We would like to thank the reviewers for their helpful comments. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement n\(^\circ \) [321174], by Len Blavatnik and the Blavatnik Family foundation, and by the Broadcom Foundation and Tel Aviv University Authentication Initiative.
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1 Introduction
Spark [17, 29, 30] is a popular framework for writing large scale data processing applications. It is an evolution of the MapReduce paradigm, which provides an abstraction of the distributed data as bags (multisets) of items. A bag r can be accessed using higherorder operations such as map, which applies a user defined function (UDF) to all items in r; filter, which filters items in r using a given boolean UDF; and fold which aggregates items together, again using a UDF. Intuitively, map, filter and fold can be seen as extensions to the standard database operations project, select and aggregation, respectively, with arbitrary UDFs applied. Bags also support bykey, join and cartesian product operators. A language such as Scala or Python is used as Spark’s interface, allowing to embed calls to the underlying framework, as well as defining UDFs that Spark executes.
This paper shows how to harness SMT solvers to automatically reason about small subsets of Spark programs. Specifically, we are interested in developing tools that can check whether two Spark programs are equivalent and produce a witness input for the different behavior of inequivalent ones. Reasoning about the equivalence of Spark programs is challenging—not only is the problem undecidable even for programs containing a single aggregate operation, some specific intricacies arise from the fact that the input datasets are bags (rather than simple sets or individual items), and that the output might expose only a partial view of the results of UDFbased aggregations.
Our main tool for showing equivalence of Spark programs is reducing the equivalence question to the validity of a formula in Presburger arithmetic, which is a decidable theory [12, 22]. More specifically, we present a simplified model of Spark by defining \(\mathrm {SparkLite}\), a functional programming language in which UDFs are expressed over a decidable theory. We show that SMT solvers can effectively verify equivalence of and detect potential differences between Spark programs. We present different verification techniques which leverage certain semantic restrictions which, in certain cases, make the problem decidable. These restrictions can also be validated through SMT. Arguably, the most interesting aspect of our technique is that it can reason about higher order operations such as fold and foldByKey, corresponding to limited usage of loops and nested loops, respectively. The key reason for the success of our techniques is that our restrictions make it possible to automatically infer inductive hypotheses simple enough to be mechanically checked by SMT solvers, e.g., [10].
Main Results. Our main technical contributions can be summarized as follows:

We prove that verifying the equivalence of \(\mathrm {SparkLite}\) programs is undecidable even in our limited setting.

We identify several interesting restrictions of \(\mathrm {SparkLite}\) programs, and develop sound, and in certain cases complete, methods for proving program equivalence. (See Table 1, which we gradually explain in Sect. 2).

We implemented our approach on top of Z3 [10], and applied it to several interesting programs inspired by reallife Spark applications. When the implementation employs a complete method and determines that a pair of programs is not equivalent, it produces a (real) counterexample of bag elements which are witnesses for the difference between the programs. This counterexample is guaranteed to be valid for programs which have a complete verification method, and can help understand the differences between these programs.
2 Overview
For space considerations, we concentrate on presenting an informal overview through a series of simple examples, and formalize the results in [13].
Figure 1 shows two equivalent Spark programs and the formula that we use for checking their equivalence. The programs accept a bag of integer elements. They return another bag where each element is twice the value of the original element, for elements which are at least 50. The programs operate differently: P1 first multiplies, then filters, while P2 goes the other way around. \(\mathtt {map}\) and \(\mathtt {filter}\) are operations that apply a function on each element in the bag, and yield a new bag. For example, let bag R be the bag \(R=\{\!\!\{2,2,103,64\}\!\!\}\) (note that repetitions are allowed). R is an input of both P1 and P2. The \(\mathtt {map}\) operator in the first line of P1 produces a new bag, \(R'_1\), by doubling every element of R, i.e., \(R'_1=\{\!\!\{4,4,206,128\}\!\!\}\). The \(\mathtt {filter}\) operator in the second line generates bag \(R''_1\), containing the elements of \(R'_1\) which are at least 100, i.e., \(R''_1=\{\!\!\{206,128\}\!\!\}\). The second program first applies the filter operator, producing a bag \(R'_2\) of all the elements in R which are not smaller than 50, resulting in the bag \(R'_2=\{\!\!\{103,64\}\!\!\}\). P2 applies the map operator to produce bag \(R''_2\) which contains the same elements as \(R''_1\). Hence, both programs return the same value.
To verify that the programs are indeed equivalent, i.e., given the same inputs produce the same outputs, we encode them symbolically using formulae in firstorder logic, such that the question of equivalence boils down to proving the validity of a formula. In this example, we encode P1 as a program term: \(\phi (P1)\!=\! ite ({2*x\ge 100},{2*x},{\bot })\), and P2 as: \(\phi (P2)\!=\!2\,{*}\, ite ({x \ge 50},{x},{\bot })\), where \( ite \) denotes the ifthenelse operator and \(\bot \) is used to denote that the element has been removed. The variable symbol x can be thought of as an arbitrary element in the bag R, and the terms \(\phi (P1)\) and \(\phi (P2)\) record the effect of P1 and P2, respectively, on x. The constant symbol \(\bot \) records the deletion of an element due to not satisfying the condition checked by the \(\mathtt {filter}\) operation. The formula whose validity attests for the equivalence of P1 and P2 is \(\forall x. \phi (P1)\!=\!\phi (P2)\). It is expressible in a decidable extension of Presburger Arithmetics, which supports the special \(\bot \) symbol (see [13, Sect. 8]). Thus, its validity can be decided.
This example points out an important property of the map and filter operations, namely, their locality: they handle every element separately, with no regard to its multiplicity (the number of duplicates it has in the bag) or the presence of other elements. Thus, we can symbolically represent the effect of the program on any bag, by encoding its effect on a single arbitrary element from that bag. Interestingly, the locality property transcends to the cartesian product operator which conjoins items across bags.
Decidability. The validity of the aforementioned formula suffices to prove the equivalence of P1 and P2 due to a tacit fact: both programs operate on the same bag. Consider, however, programs \(P1'\) and \(P2'\) which receive bags \(R_1\) and \(R_2\) as inputs. \(P1'\) maps all the elements of \(R_1\) to 1 and \(P2'\) does the same for \(R_2\). Their symbolic encoding is \(\phi (P1')=(\lambda x.1)x_1\) and \(\phi (P1')=(\lambda x.1)x_2\), where \(x_1\) and \(x_2\) represent, respectively, arbitrary elements from \(R_1\) and \(R_2\). The formula \(\forall x_1,x_2.\phi (P1')=\phi (P2')\) is valid. Alas, the programs produce different results if \(R_1\) and \(R_2\) have different sizes. Interestingly, we show that unless both programs always return the empty bag, they are equivalent iff their program terms are equivalent and use the same variable symbols.^{Footnote 1} Furthermore, it is possible to decide whether a program always returns the empty bag by determining if its program term is equivalent to \(\bot \). Theorem 1 (Sect. 4.1) shows that the equivalence of \(NoAgg\) programs, i.e., ones not using aggregations, can be decided.
Usage of Inductive Reasoning. We use inductive reasoning to determine the equivalence of programs that use aggregations. Theorem 2 (presented later on) shows that equivalence in \(AggOne^{p}\), that is, of programs that use a single \(\mathtt {fold}\) operation and return a primitive value, is undecidable. Thus, we consider different classes of programs that use aggregations in limited ways.
Figure 2 contains an example of two simple equivalent \(AggOne^{p}\) programs. The programs operate over a bag of pairs (product IDs, price). The programs check if the minimal price in the bag is at least 100. The second program does this by subtracting 20 from each price in the bag and comparing the minimum to 80. P3 computes the minimal price in R using \(\mathtt {fold}\), and then returns true if it is at least 100 and false otherwise. P4 first applies \( discount \) to every element, resulting in a temporary bag \(R'\), and then computes the minimum of \(R'\). It returns true if the minimum is at least 80, and false otherwise.
The \(\mathtt {fold}\) operation combines the elements of a bag by repeatedly applying a UDF. \(\mathtt {fold}\) cannot be expressed in first order terms. Thus, we use induction to verify that two \(\mathtt {fold}\) results are equal. Roughly speaking, the induction leverages the somewhat local nature of the \(\mathtt {fold}\) operation, specifically, that it does not track how the temporarily accumulated value is obtained: Note that the elements of \(R'\) can be expressed by applying the \( discount \) function on the elements of R. Thus, intuitively, we can assume that in both programs, \(\mathtt {fold}\) iterates on the input bag R in the same order. (It is permitted to assume a particular order because the applied UDFs must be commutative for the \(\mathtt {fold}\) to be welldefined [17].^{Footnote 2}) The base of the induction hypothesis checks that the programs are equivalent when the input bags are empty, and the induction step verifies the equivalence is retained when we apply the \(\mathtt {fold}\)’s UDF on some arbitrary accumulated value and an element coming from each input bag.^{Footnote 3} In our example, when the bags are empty, both programs return true. (The \(\mathtt {fold}\) operation returns \(+\infty \).) Otherwise, we assume that after n prices checked, the minimum \(M_1\) in P3 is at least 100 iff the minimum \(M'_1\) in P4 is at least 80. The programs are equivalent if this invariant is kept after checking the next product and price (\((prod,p),(prod',p')\)) giving updated intermediate values \(M_2\) and \(M'_2\).
Completeness of the Inductive Reasoning. In the example in Fig. 2, we use a simple form of induction by proving that two higherorder operations are equivalent iff they are equivalent on every input element and arbitrary temporarily accumulated values (\(M_1\) and \(M_1'\) in Fig. 2). Such an approach is incomplete. We now show an example for incompleteness, and a modified verification formula that is complete for a subset of \(AggOne^{p}\), called \(AggOne^{p}_{sync}\). In Fig. 3, P3 and P4 were rewritten into P5 and P6, respectively, by using \(=\) instead of \(\ge \). The rewritten programs are equivalent. We show both the “naïve” formula, similar to the formula from Fig. 2, and a revised version of it. (We explain shortly how the revised formula is obtained.) The naïve formula is not valid, since it requires that the returned values be equivalent ignoring the history of applied \(\mathtt {fold}\) operations generating the intermediate values \(M_1\) and \(M'_1\). For example, for \(M_1\!=\!60\), \(M'_1\!=\!120\), and \(p\!=\!100\), we get a spurious counterexample to equality, leading to the wrong conclusion that the programs may not be equivalent. In fact, if P5 and P6 iterate over the input bag in the same order, it is not possible that their (temporarily) accumulated values are 60 and 120 at the same time.
Luckily, we observe that, often, the \(\mathtt {fold}\) UDFs are somewhat restricted. One such natural property, is the ability to “collapse” any sequence of applications of the aggregation function f using a single application. We can leverage this property for more complete treatment of equivalence verification, if the programs collapse in synchrony; given their respective fold functions \(f_1,f_2\), initial values \(i_1,i_2\), and the symbolic representation of the program term pertaining to the folded bags \(\varphi _1, \varphi _2\), the programs collapse in synchrony if the following holds:
Note that the same input a is used to collapse both programs. In our example, \(\min (\min (+\infty ,x),y)=\min (+\infty ,a)\), and \(\min (\min (+\infty ,x20),y20)=\min (+\infty ,a20)\), for \(a=\min (x,y)\). The reader may be concerned how this closure property can be checked. Interestingly, for formulas in Presburger arithmetic, an SMT solver can decide this property.
We utilized the above closure property by observing that any pair of intermediate results can be expressed as single applications of the UDF. Surely any \(M_1\) must have been obtained by repeating applications of the form \(f_1(f_1(\cdots ))\), and similarly for \(M_1'\) with \(f_2(f_2(\cdots ))\). Therefore, in the revised formula, instead of quantifying on any \(M_1\) and \(M'_1\), we quantify over the argument a to that single application, and introduce the assumption incurred by Eq. (1). We can then write an induction hypothesis that holds iff the two fold operations return an equal result.
Handling ByKey Operations. Spark is often used to aggregate values of groups of records identified by a shared key. For example, in Fig. 4 we present two equivalent programs that given a bag of pairs of student IDs and grades, return a histogram graph of all passing grades (\({\ge }60\)), in deciles. P7 first maps each student’s grade to its decile, while making the decile the key. (The key is the first component in the pair.) Then, it computes the count of all students in a certain decile using the foldByKey operation, and filters out all nonpassing deciles (\({<}6\)) from the resulting histogram. P8 first filters out all failing grades, and then continues similarly with the histogram computation.
Verifying the equivalence of P7 and P8 is challenging because, intuitively, the bykey operation corresponds to a nested loop: It partitions the bag into “buckets” according to the key element of the bag and folds every bucket separately. Furthermore, note that the two programs fold bags which contain different keys.
Our approach to verify programs using bykey operations is based on a reduction to the problem of verifying programs using \(fold \): We rewrite the programs, so instead of applying the fold operation on one bucket at a time (as foldByKey does), we apply it on the entire bag to get the global aggregated result. We then map each key to the global aggregated result, instead of the aggregated result for the bucket. It is then possible to write an inductive hypothesis based on the rewritten program. The reduction is sound if the two compared programs partition the bag’s elements to buckets consistently: If program Q1 sends two elements to the same bucket, then Q2 must also send those two elements to the same bucket (although it does not have to be the same bucket as Q1), and vice versa. As with the property of collapsibility seen earlier, this property can also be expressed in Presburger arithmetic, and be verified using an SMT solver: for functions \(k_1\) and \(k_2\) that describe expressions for keys, we require:
Figure 4 shows the inductive hypothesis whose validity ensures the equivalence of P7 and P8, as well as the resulting instantiation of Eqs. (2) and (3). \(AggOneK^{b}\) is a sound method for verifying equivalence of pairs of programs that use single foldByKey and satisfy Eqs. (2) and (3). (See [13, Lemma 7].)^{Footnote 4}
Decidability. Table 1 characterizes the programs for which our method is applicable, together with the strength of the method.^{Footnote 5} The example programs in Fig. 1 are representative of programs that belong to the \(NoAgg\) class of programs, for which we have a decision procedure for verifying equivalence. We consider five classes of programs containing \(\mathtt {fold}\) operations. Equivalence in \(AggOne^{p}\) is undecidable, and the result is extended naturally to the special cases of \(AggOneK^{b}\), \(AggOne^{b}\) and \(AggMult^{p}\). On the other hand, \(AggOne^{p}_{sync}\) is a complete verification method. The equivalence of the programs in Figs. 2 and 3 can be verified using \(AggOne^{p}_{sync}\). Note that applying \(AggOne^{p}_{sync}\) and \(AggOneK^{b}\) require also checking the validity of Eq. (1), respectively Eqs. (2) and (3). Fortunately, these requirements are expressed in Presburger arithmetic and thus can be decided.
Limitations. We restrict ourselves to programs using map, filter, cartesian product, fold, and foldByKey where UDFs are defined in Presburger Arithmetic. We forbid self products—it is possible, but technically cumbersome, to extend our work to support selfproducts. However, supporting operators such as union and subtract can be tricky because of the bag semantics. Presburger arithmetic can be implemented with solvers such as Cooper’s algorithm [9]. For simplicity we use Z3 which does not support full Presburger arithmetic, but supports the fragment of Presburger arithmetic used in this paper. Z3 also supports uninterpreted functions, which are useful to prove equivalence of other classes of Spark programs, but this is beyond the scope of this paper.
3 The \(\mathrm {SparkLite}\) Language
In this section, we describe \(\mathrm {SparkLite}\), a simple functional programming language based on the operations provided by Spark [29].
Preliminaries. We denote a (possibly empty) sequence of elements coming from a set X by \(\overline{X}\). An ifthenelse expression \( ite (p,e,e')\) denotes an expression that evaluates to e if p holds and to \(e'\) otherwise. A bag m over a domain X is a multiset, i.e., a set which allows for repetitions, with elements taken from X. We denote the multiplicity of an element x in bag m by m(x), where for any x, either \(0<m(x)\) or m(x) is undefined. We write \(x \in m\) as a shorthand for \(0<m(x)\). We write \(\{\!\!\{x;n(x) \mid x \in X \wedge \phi (x)\}\!\!\}\) to denote a bag with elements from X satisfying some property \(\phi \) with multiplicity n(x), and omit the conjunct \(x \in X\) if X is clear from context. We denote the size (number of elements) of a bag m by m and the empty bag by \(\{\!\!\{\}\!\!\}\). We denote the ith component of a tuple x by \(p_i(x)\), and extend \(p_i(\cdot )\) to bags containing tuples in the natural way.
SparkLite. The syntax of \(\mathrm {SparkLite}\) is defined in Fig. 5. \(\mathrm {SparkLite}\) supports two primitive types: integers (\(\mathtt {Int}\)) and booleans (\(\mathtt {Boolean}\)). On top of this, the user can define record types \(\varvec{\tau }\), which are tuples of primitive types, and Bags:^{Footnote 6} \( Bag _{\varvec{\tau }}\) is (the type of) bags containing elements of type \(\varvec{\tau }\). We refer to primitive types and records as basic types, and, by abuse of notation, range over them using \(\varvec{\tau }\). We use e to denote a basic expression containing only basic types, written in Presburger arithmetics extended to include tuples in a straightforward way. (See [13, Sect. 8].) We range over variables using \(\varvec{v}\) and \(\varvec{r}\) for variables of basic types and Bag, respectively.
A program \( \varvec{P}(\overline{\varvec{r} \mathbin {\mathtt {:}} Bag _{\varvec{\tau }}}, \overline{\varvec{v} \mathbin {\mathtt {:}} \varvec{\tau }})\,\, \mathbin {\mathtt {=}}\,\, \overline{ Fdef } \, \overline{ PFdef }\, E \,\, \eta \) is comprised of a header and a body, which are separated by the \(\mathtt {=}\) sign. The header contains the name of the program (\(\varvec{P}\)) and a sequence of the names and types of its input formal parameters, which may be Bags (\(\overline{\varvec{r}}\)) or records or primitive types (\(\overline{\varvec{v}}\)). The body of the program is comprised of two sequences of function declarations (\(\overline{ Fdef }\) and \(\overline{ PFdef }\)), variable declarations (E), and the program’s main expression (\(\eta \)). \(\overline{ Fdef }\) binds function names \(\varvec{f}\) with firstorder lambda expressions, i.e., to a function which takes as input a sequence of arguments of basic types and returns a value of a basic type. \(\overline{ PFdef }\) associates function names \(\varvec{F}\) with a restricted form of secondorder lambda expressions, which we refer to as parametric functions. As in the Kappa Calculus [15], a parametric function \(\varvec{F}\) receives a sequence of basic expressions and returns a first order function. Parametric functions can be instantiated to form an unbounded number of functions from a single pattern. For example, \( \mathbin {\mathtt {def}}\,\, \mathtt {addC} \mathbin {\mathtt {=}} \lambda x:\mathtt {Int}.\,\lambda y:\mathtt {Int}.~ x + y:\mathtt {Int} \) can create any first order function which adds a constant to its argument, e.g., \(\mathtt {addC(1)}=\lambda x:\mathtt {Int}.\,1+x:\mathtt {Int}\) and \(\mathtt {addC(2)}=\lambda x:\mathtt {Int}.\,2+x:\mathtt {Int}\).
The program declares variables with a sequence of let expressions which bind general expressions to variables. A general expression is either a basic expression (e), a bag expression (\(\mu \)), or an aggregate expression (\(\mathtt {fold}\mathtt {(}\varvec{e},f\mathtt {)}\mathtt {(}\mu \mathtt {)}\) or \(\mathtt {foldByKey}\mathtt {(}\varvec{e},f\mathtt {)}\mathtt {(}\mu \mathtt {)}\)). The expression \(\mathtt {cartesian}^{}(\mu , \mu ')\) returns the cartesian product of \(\mu \) and \(\mu '\). \(\mathtt {map}\mathtt {(}f\mathtt {)}\mathtt {(\mu )}\) produces a \(Bag \) by applying the unary UDF f to every element x of \(\mu \). \(\mathtt {filter}\mathtt {(}f\mathtt {)}\mathtt {(\mu )}\) evaluates to a copy of \(\mu \), except that all elements in \(\mu \) which do not satisfy f are removed. The aggregate expression \(\mathtt {fold}\mathtt {(}\varvec{e},f\mathtt {)}\mathtt {(}\mu \mathtt {)}\) accumulates the results obtained by iteratively applying the binary UDF f to every element x in a \(Bag \) \(\mu \) in some arbitrary order together with the accumulated result obtained so far, which is initialized to the initial element \(\varvec{e}\). If \(\mu \) is empty, then \(\mathtt {fold}\mathtt {(}\varvec{e},f\mathtt {)}\mathtt {(}\mu \mathtt {)}=\varvec{e}\). The \(\mathtt {foldByKey}\mathtt {(}\varvec{e},f\mathtt {)}\) operation applied on a \(Bag \) \(\mu \) of record type \(K\times V\) produces a \(Bag \) of pairs, where every key \(k\in K\) which appears in \(\mu \) is associated with the result obtained by applying \(\mathtt {fold}\mathtt {(}\varvec{e},f\mathtt {)}\) to the \(Bag \) containing all the values associated with k in \(\mu \).
We denote the meaning of a \(\mathrm {SparkLite}\) program P by \(\llbracket P \rrbracket \), which receives input environments \(\rho _0\), assigning values to P’s formal variables, to either bags or basic types. (See [13, Sect. 7].)
Remarks. We assume that the signature of UDFs given to either map, filter, fold or foldByKey match the type of the \(Bag \) on which they are applied. Also, to ensure that the meaning of \(\mathtt {fold}\mathtt {(}\varvec{e},f\mathtt {)}\mathtt {(}\varvec{r}\mathtt {)}\) and \(\mathtt {foldByKey}\mathtt {(}\varvec{e},f\mathtt {)}\mathtt {(}\varvec{r}\mathtt {)}\) is well defined, i.e., we require, as Spark does [17], that f be commutative on its second argument: \(\forall x,y_1,y_2. f(f(x,y_1),y_2)=f(f(x,y_2),y_1)\).
4 Verifying Equivalence of \(\mathrm {SparkLite}\) Programs
Programs \(P_1\) and \(P_2\) are comparable if they receive the same sequence of formal input parameters, and produce the same output type. They are equivalent if, in addition, for any input environment \(\rho _0\), it holds that \(\llbracket P_1 \rrbracket (\rho _0) = \llbracket P_2 \rrbracket (\rho _0)\). We assume that we only check the equivalence of comparable programs. Also, without loss of generality, we define programs without let expressions; as variables are never reassigned, these can always be eliminated by substituting every variable by its definition. We can now state our result regarding decidability of \(NoAgg\) programs, defined as programs without aggregate terms. (cf. [13, Sect. 9].)
Theorem 1
The equivalence of programs in the \(NoAgg\) class is decidable.
Unsurprisingly, however, equivalence in the general case is undecidable. The reduction in [13, Theorem 2] from the halting problem for 2counter machines shows that verifying equivalence of \(AggOne^{p}\) programs, is an undecidable problem.
Theorem 2
The problem of deciding whether two arbitrary \(AggOne^{p}\) \(\mathrm {SparkLite}\) programs are equivalent is undecidable.
4.1 Program Terms
The first step of our technique is the construction of program terms: Given a program P with main expression \(\eta \), we generate a program term \(\phi (P)\) which, roughly speaking, reflects the effect of the program on arbitrary elements taken from its input bags. It is obtained by applying the translation function \(\phi \), shown in Fig. 6, on P’s main expression. \(\phi \) recursively traverses the expression and generates a logical term over the vocabulary of builtin operations and UDFs defined in P. The base case of the recursion is input bag variables r, which \(\phi \) replaces with fresh variables \(\mathbf {x}_{r}\). We refer to these variables as representative variables. Translation of a \(\mathrm {SparkLite}\) operation on Bags produces a term corresponding to the application of its UDF on a single Bag element, which is a new bag expression: A \(\mathtt {map}(f)(\mu )\) operation is translated into the expression received by applying the lambda expression that corresponds to f, on the program term of \(\mu \). A \(\mathtt {filter}(f)(\mu )\) operation is translated to an \( ite \) expression which returns the program term of \(\mu \) on the then branch and \(\bot \) on the else branch. The \(\mathtt {cartesian}^{}(\mu ,\mu ')\) operation is translated to a pair of program terms pertaining to its arguments. Note that in the absence of aggregate operations, \(\phi (\cdot )\) is a firstorder term and thus can be used directly in formulas.
Aggregate operations require iterating over all the elements of \(\mu \). Therefore, it is clear that the translation of \(\mathtt {fold}\) cannot be masqueraded as a firstorder term. For \(\mathtt {fold}(e,f)(\mu )\) we are using a special operator \([{\phi (\mu )}]_{i,f}\), where \(\phi (\mu )\) is the term pertaining to the bag being folded, i is the initial value, and f is the fold function. We refer to \([{\phi (\mu )}]_{i,f}\) as an aggregate term.
RepVarSet. For an expression \(\mu \) consisting only of input bags and input parameters of basic types, \( RepVarSet (\mu )\) denotes the set of all representative variables corresponding to the input bags appearing in \(\mu \). We can thus similarly define \( RepVarSet (P)\) for the main expression of P. \(\mathsf {FV}(P)\) denotes the entire set of free variables (both representative and nonbag inputs) in the program term of P.
Example 1
Consider the main expression \( \eta = {\mathtt {filter}}\mathtt {(}{geq(100)}\mathtt {)(}{{\mathtt {map}}\mathtt {(}{double}\mathtt {)(}{R}\mathtt {)}}\mathtt {)} \) of the program \(P1''\) obtained by inlining the let expressions in program P1 (see Sect. 2), defining the doubling function as \(double=\lambda x.2*x\), and instantiating the parametric function \(geq=\lambda y.\,\lambda x. x \ge y\) to act as the condition of the filter. The program term of \(P1''\) is \(\phi (P1'')=ite(2 * \mathbf {x}_{R} \ge 100, 2 * \mathbf {x}_{R}, \bot )\). Intuitively, we can learn how \(P1''\) affects every element of, e.g., input Bag \(\{\!\!\{2,2,103,64\}\!\!\}\), by treating \(\phi (P1'')\) as a “function” of \(\mathbf {x}_{R}\) and “applying” it to 2, 2, 103, and 64. It is easy to see that \(\mathsf {FV}(P1'')= RepVarSet (P1'')=\{\mathbf {x}_{R}\}\). Consider now instead \(P5'\) also obtained by inlining of the let expressions in P5. In this case, \(\phi (P5'')=[{\mathbf {x}_{R}}]_{+\infty ,\lambda A,(x,y). ite(A<y,A,y)}=100\).
4.2 Verifying Equivalence of \(\mathrm {SparkLite}\) Programs with Aggregation
In this section, we discuss the generation of inductive hypotheses for programs with aggregations. We focus on the \(AggOne^{p}\) and \(AggOne^{p}_{sync}\) methods (recall Table 1), applicable on programs with a single fold operation. For space reasons, we relegate to [13, Sects. 13 and 14] the discussion of the other methods: \(AggOne^{b}\), \(AggMult^{p}\) and \(AggOneK^{b}\), which are all sound techniques generalizing \(AggOne^{p}\).
We note that in the presence of \(\mathtt {fold}\) operations, The resulting terms are no longer legal terms in first order logic, and thus, we cannot use them directly in formulae. Instead, we extract out of them a set of formulae whose validity, intuitively, amounts to the establishment of an inductive invariant regarding the effect of \(\mathtt {fold}\) operations.
Verifying Equivalence of \(AggOne^{p}\) Programs. Arguably, the simplest class of programs with aggregations is the class of programs that return a primitive expression that depends on the result of the aggregation operation. Technically, a pair of \(\mathrm {SparkLite}\) programs is in class \(AggOne^{p}\) if each program P in the pair belongs to \({AggOne^{p}}\), i.e., there is a an expression g in Presburger Arithmetic with a single free variable x such that the program term of P is of the form \(g[[{\phi (\mu )}]_{i,f}/x]\), where \(\mu \) is a bag expression that does not include \(\mathtt {fold}\) or \(\mathtt {foldByKey}\) operations; that is, if \(\phi (P)\) can be obtained by substituting x in g with the aggregate term pertaining to the application of a \(\mathtt {fold}\) operation on \(\mu \). In the following, we refer to g as P’s top expression. By abuse of notation, we use the functional notation g(t) as a shorthand for g[t / x], the expression obtained by substituting the term t with g’s free variable. Similarly, given an expression e with two free variables x and y, we write \(e(t_1,t_2)\) as a shorthand for \(e[t_1/x,t_2/y]\).
Lemma 1 formalizes the sound method that we used in Sect. 2 to show that P3 and P4 (see Fig. 2) are equivalent.
Lemma 1
(Sound Method for Verifying Equivalence of \(Agg^1\) Programs). Let \(P_1\) and \(P_2\) be \(AggOne^{p}\) programs such that \(\mathsf {FV}(P_1) = \mathsf {FV}(P_2)\). Assume that \(\phi (P_1)=g_1([{\phi (\mu _1)}]_{{i}_1,f_1})\) and \(\phi (P_2)=g_2([{\phi (\mu _2)}]_{{i}_2,f_2})\), where \(f_1=\lambda x,y.\,e_1\) and \(f_2=\lambda x,y.\,e_2\). \(P_1\) and \(P_2\) are equivalent if the following conditions hold:
Intuitively, Eqs. (5) and (6) formalize the concept of inductive reasoning described in Sect. 2 for the base of the induction and the induction step, respectively. Equation (4) requires that the free variables of the folded bag expressions use the same representative variables. It ensures that the two \(\texttt {fold}\) operations iterate over bags of the same size. Note that we do not require that the bag folded by the two programs be equivalent. However, in Eq. (6) we still use the fact that corresponding elements in the two folded bags can be produced by instantiating the program terms \(e_{1,2}\) with corresponding elements from the input bags.
Complete Verification Techniques for Subclasses of \(AggOne^{p}\mathbf{.}\) Lemma 1 provides a sound, but incomplete, verification technique. This means that there are cases in which a pair of equivalent programs does not satisfy one or more of the requirements of Lemma 1. Luckily, some of these cases can be identified and subsequently have their equivalence verified using other methods. As a simple example, in [13] we show that the equivalence of \(\mathrm {SparkLite}\) programs whose \(\mathtt {fold}\) operations return a constant value can be reduce to the (decidable) problem of verifying equivalence of \(NoAgg\) programs. We now describe the \(AggOne^{p}_{sync}\) verification method.
In Sect. 2 we showed that although programs P5 and P6 do not satisfy the requirements of Lemma 1, we can verify their equivalence using a more specialized verification technique, \(AggOne^{p}_{sync}\). We now present a more detailed discussion of \(AggOne^{p}_{sync}\). We recall that the three main properties of pairs of programs that \(AggOne^{p}_{sync}\) applies to are (1) both belong to \(AggOne^{p}\); (2) the folds in both programs can be collapsed; and (3) the process of collapsing the folds can be done in synchrony.
The collapsing property states that any value produced by consecutive applications of the \(\mathtt {fold}\) UDF can be obtained by a single application. For example, if the UDF is \(sum\!=\!\lambda x,y.\,x+y\) and the initial value is 0, then the result obtained by applying sum consecutively on any two elements a and b can also be obtained by applying sum once on \(a+b\). Also, recall that the bag being folded contains elements which are obtained via a sequence of \(\mathtt {map}\), \(\mathtt {filter}\) and \(\mathtt {cartesian}\) operations applied to elements taken out of the input bags. Synchronized collapsing occurs when given the same input elements to two consecutive applications of the \(\mathtt {fold}\) UDF, it is possible to collapse them both using the same input element.
Thus, synchronized collapsing is a semantic property of \(\mathtt {fold}\) UDFs, aggregated terms, and initial values of a pair of programs that belong to \(AggOne^{p}_{sync}\). In the following, we denote by \(\mathsf {FV}_{\mathsf {r}}(P)\) and \(\mathsf {FV}_{\mathsf {b}}(P)\) the subsets of \(\mathsf {FV}(P)\) comprised of bag, respectively, nonbag, input formal parameters.
Definition 1
(The \(AggOne^{p}_{sync}\) Class). Let \(P_1\) and \(P_2\) be \(AggOne^{p}\) programs such that \(\mathsf {FV}(P_1) = \mathsf {FV}(P_2)\). Assume that \(\phi (P_1)=g_1([{\phi (\mu _1)}]_{{i}_1,f_1})\) and \(\phi (P_2)=g_2([{\phi (\mu _2)}]_{{i}_2,f_2})\), where \(f_1=\lambda x,y.\,e_1\) and \(f_2=\lambda x,y.\,e_2\). We say that \(P_1\) and \(P_2\) belong together to \(AggOne^{p}_{sync}\), denoted by \(\langle P_1,P_2 \rangle \in AggOne^{p}_{sync}\), if the following conditions hold:
Note that in Eq. (8), all applications of the \(\mathtt {fold}\) UDF functions agree on the values of the nonbag input formal parameters used to “generate” the accumulated elements. Also note that checking if \(\langle P_1,P_2 \rangle \in AggOne^{p}_{sync}\) involves determining the validity of an additional decidable formula, namely Eq. (8). Theorem 3 shows that verifying the equivalence of a pair of programs in \(AggOne^{p}_{sync}\) effectively reduces to checking a single application of the \(\mathtt {fold}\) UDFs.
Theorem 3
(Equivalence in \(AggOne^{p}_{sync}\) is Decidable ). Let \(P_1\) and \(P_2\) be \(AggOne^{p}\) programs as in Lemma 1, such that \(\langle P_1,P_2 \rangle \in AggOne^{p}_{sync}\). \(P_1\) and \(P_2\) are equivalent if and only if the following holds:
5 Prototype Implementation
We developed a prototype implementation verifying the equivalence of Spark programs. The tool is written in Python 2.7 and uses the Z3 Python interface to prove formulas. We ran our experiments on a 64bit Windows host with a quad core 3.40 GHz Intel Core i76700U processor, with 32 GB memory. The tool accepts pairs of Spark program written using the Python interface, determines the class of \(\mathrm {SparkLite}\) program they belong to, and verifies their equivalence using the appropriate method.
A total of 23 testcases of both equivalent and nonequivalent instances were tested, including all the examples from this paper. In Fig. 7, we highlight test cases inspired by real Spark uses taken from [17, 28] and online resources (e.g., opensource Spark clients), and belong to one of the defined \(\mathrm {SparkLite}\) classes. The full list of tested programs appears in [13, Sect. 15]. They include join optimizations, different aggregations, and various UDFs. For each instance, the tool either verifies that the given programs are equivalent, or produces a counterexample, that is, an input for which the programs produce different outputs. Each example was analyzed in less than 0.5 s. It is also interesting to note that most examples with a primitive aggregation output are verified using \(AggOne^{p}_{sync}\) and not \(AggOne^{p}\), indicating that the \(AggOne^{p}_{sync}\) class is not esoteric, but wide enough to cover useful programs. Our tool was able to prove the equivalence of all equivalent programs, and find counterexamples for inequivalent ones, with the exception of \(P15''\) and \(P16''\) which belong to \(AggOne^{p}\). While it is immediate that these programs are equivalent (we note the intermediate fold results in both programs are the same, and apply the same transformation on the fold result), our tool was not able to show the equivalence. This is because the \(AggOne^{p}_{sync}\) technique is not applicable to this particular example, as count is not a collapsible fold function, and the \(AggOne^{p}\) technique is effective only when the equivalence claim is inductive, which is not the case here.
6 Related Work and Conclusion
The problem considered (i.e., determining equivalence of expressions accessing a dataset) is a classic topic in database theory. Query containment and equivalence were first studied in seminal work by Chandra and Merlin [2]. This work was extended in numerous papers, e.g., [18] for queries with inequalities and [4] for acyclic queries. Of most relevance to this paper are the extensions to queries evaluated under bag and bagset semantics [3], and to aggregate queries, e.g., [7, 8, 14]. The latter papers consider specific aggregate functions, such as min, count, sum and average, or aggregate functions defined by operations over abelian monoids. In comparison, we do not restrict UDFs to monoids, and provide a different characterization for decidability.
In the field of verification and programming languages, several works address properties of relational algebra operators. Most notably, Cosette [6], is a fully automated prover for SQL equivalences, which provides a proof or a counterexample to equivalence by utilizing both a theorem prover and a solver. The approach supports standard SQL features as well as predetermined aggregation functions such as count, sum, and average. On the other hand, by addressing Spark programs, our approach focuses on custom UDFs for selects, projections, and aggregation. Similarly, Spec# [20] has a fixed set of comprehensions such as sum, count, min and max, fitted into templates with both filters and expression terms akin to map, which are encoded into the SMT solver using specialized axioms, e.g. the distribution of plus over min/max. Our techniques, on the other hand, extract automatically properties of comprehensions to define suitable verification conditions for equivalence. El Ghazi and Taghdiri [11] took the SMT solver approach to verify relational constraints in Alloy [16], in order to be able to provide proofs, and not just counterexamples. There is, however, no guarantee on completeness, or the ability of the solver to provide a proof. It differs from this work, which carefully defines criteria for decidability and soundness, even in the expense of expressivity. Loncaric et al. [21] utilize a smallmodel property of sets to verify synthesized data structures which is similar to the one we leverage in the \(NoAgg\) method. We extend this property to bags and aggregate operations. Smith and Albarghouthi [25] presented an algorithm for synthesizing Spark programs by analyzing user examples fitted into higherorder sketches. They use SMTs to verify commutativity of the \(fold \) UDFs. Chen et al. [5], studied the decidability of the latter problem. We use SMT to verify program equality assuming that the \(fold \) UDFs are commutative. In this sense, our approaches are complementary.
There are also generic frameworks for verifying functional programs, such as F* [27] and Liquid Types [23, 24]. These prove program safety via type checking, which also utilizes SMT to check validity of implications. Both approaches require additional manual effort to verify programs like the ones we explore: in Liquid Types, there is no notion of equivalence, so a suitable summary must be given that holds for both programs. In F*, equivalence can be expressed via assertions, but verifying assertions in F* is incomplete with respect to inductive data types, such as lists. Appropriate invariants must be provided manually, essentially the same ones that are constructed automatically in this paper. Another approach to verifying functional programs is applied by Leon [1, 26], whose engine is based on decision procedures for the quantifierfree theory of algebraic data types with different fold functions, which allow handling recursive functions with firstorder constraints. However, the approach relies on finite unrolling of the recursive calls, thus it cannot verify the equivalence of two programs when the equivalence property is not inductive by itself. In contrast, our approach is successful because of the novel specialized treatment of synchronous collapsible UDFs.
Dafny [19] supports functional programming, inductive data types, higherorder functions, and also provides some automatic induction. Dafny can automatically verify our \(NoAgg\) test cases. However, applying it to certain \(AggOne^{p}\) programs required supplying auxiliary lemmas. For example, verifying the equivalence of P15 and P16 required the use of a lemma asserting that multiplying the sum of elements in a bag by three produce the same result as summing the bag obtained by multiplying every element by three. Essentially, the lemma establishes equivalence relations between subprograms, and gives rise to a possible heuristic extension of our tool by searching for relations between subprograms.
Conclusion. The main conceptual contribution of this paper is that the problem of checking program equivalence of \(\mathrm {SparkLite}\) programs, which reflect an interesting subset of Spark programs, can be addressed via a reduction to the validity of formulas in a decidable fragment of firstorder logic. We believe the foundations laid in this paper will lead to the development of tools that handle formal verification and optimization of more classes of programs written in Spark and similar frameworks, e.g., ones with nested aggregations and unions.
Notes
 1.
Recall that intuitively, these variables pertain to arbitrary elements in the input bags. In our example, \(\phi (P1')\) uses variable \(x_1\) and \(\phi (P2')\) uses \(x_2\).
 2.
We note that our results do not require UDFs to be associative, however, Spark does.
 3.
Note that \(AggOne^{p}\) programs can fold bags produced by a sequence of filter, map, and cartesian product operations. Our approach is applicable to such programs because if the program terms of two folded bags use the same variable symbols, then any selection of elements from the input bags produces an element in the bag being folded in one program iff it produces an element in the bag that the other program folds. (See Lemma 1).
 4.
Our approach is not sound if Eqs. (2) and (3) do not hold. To illustrate such a case, consider a hypothetical a case in which \(P7'\) computes the histogram by deciles, \(P8'\) by percentiles, and then both programs map all the elements to a constant, ignoring the aggregated value. \(P7'\) produces at most 10 elements (one per decile), while \(P8'\) produces at most 100, so they are clearly inequivalent.
 5.
Due to space considerations, we do not discuss equivalence of programs from mixed syntactic classes with comparable output types. In essence, there is a reduction from these instances such that one of the methods presented here will be applicable.
 6.
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Grossman, S., Cohen, S., Itzhaky, S., Rinetzky, N., Sagiv, M. (2017). Verifying Equivalence of Spark Programs. In: Majumdar, R., Kunčak, V. (eds) Computer Aided Verification. CAV 2017. Lecture Notes in Computer Science(), vol 10427. Springer, Cham. https://doi.org/10.1007/9783319633909_15
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