Lifting Numeric Relational Domains to Algebraic Data Types

We address the problem of verifying that the functions of a program meet their contracts, specified by pre/postconditions. We follow an approach based on constrained Horn clauses (CHCs) by which the verification problem is reduced to the problem of checking satisfiability of a set of clauses derived from the given program and contracts. We consider programs that manipulate algebraic data types (ADTs) and a class of contracts specified by catamorphisms, that is, functions defined by simple recursion schemata on the given ADTs. We show by several examples that state-of-the-art CHC satisfiability tools are not effective at solving the satisfiability problems obtained by direct translation of the contracts into CHCs. To overcome this difficulty, we propose a transformation technique that removes the ADT terms from CHCs and derives new sets of clauses that work on basic sorts only, such as integers and booleans. Thus, when using the derived CHCs there is no need for induction rules on ADTs. We prove that the transformation is sound, that is, if the derived set of CHCs is satisfiable, then so is the original set. We also prove that the transformation always terminates for the class of contracts specified by catamorphisms. Finally, we present the experimental results obtained by an implementation of our technique when verifying many non-trivial contracts for ADT manipulating programs.

Automatic generation of random values described by algebraic data types (ADTs) is often a hard task. State-of-the-art random testing tools can automatically synthesize random data generators based on ADTs definitions. In that manner, generated values comply with the structure described by ADTs, something that proves useful when testing software which expects complex inputs. However, it sometimes becomes necessary to generate structural richer ADTs values in order to test deeper software layers. In this work we propose to leverage static information found in the codebase as a manner to improve the generation process. Namely, our generators are capable of considering how programs branch on input data as well as how ADTs values are built via interfaces. We implement a tool, responsible for synthesizing generators for ADTs values while providing compiletime guarantees about their distributions. Using compile-time predictions, we provide a heuristic that tries to adjust the distribution of generators to what developers might want. We report on preliminary experiments where our approach shows encouraging results.

This work addresses the problem of verifying imperative programs that manipulate data structures, e.g., Rust programs. Data structures are usually modeled by Algebraic Data Types (ADTs) in verification conditions. Inductive invariants of such programs often require recursively defined functions (RDFs) to represent abstractions of data structures. From the logic perspective, this reduces to solving Constrained Horn Clauses (CHCs) modulo both ADT and RDF. The underlying logic with RDFs is undecidable. Thus, even verifying a candidate inductive invariant is undecidable. Similarly, IC3-based algorithms for solving CHCs lose their progress guarantee: they may not find counterexamples when the program is unsafe. We propose a novel IC3-inspired algorithm Racer for solving CHCs modulo ADT and RDF (i.e., automatically synthesizing inductive invariants, as opposed to only verifying them as is done in deductive verification). Racer ensures progress despite the undecidability of the underlying theory, and is guaranteed to terminate with a counterexample for unsafe programs. It works with a general class of RDFs over ADTs called catamorphisms. The key idea is to represent catamorphisms as both CHCs, via relationification , and RDFs, using novel abstractions . Encoding catamorphisms as CHCs allows learning inductive properties of catamorphisms, as well as preserving unsatisfiabilty of the original CHCs despite the use of RDF abstractions, whereas encoding catamorphisms as RDFs allows unfolding the recursive definition, and relying on it in solutions. Abstractions ensure that the underlying theory remains decidable. We implement our approach in Z3 and show that it works well in practice.

We present an approach to write web applications in the programming languages Haskell [8] and Curry [6]. The web applications we propose are directly based on the Hypertext Transfer Protocol (HTTP) – no additional protocol on top of HTTP is necessary. Although the main ideas are not new and already applied using other programming languages, we believe that features present in modern declarative languages – especially algebraic datatypes – serve well to represent resources on the web. 1 TRANSFER OF RESOURCE REPRESENTATIONS A resource is any piece of information named by a Uniform Resource Identifier (URI). A web application provides access to internally stored information over the Internet or accesses such information provided by other web applications. Usually, (a representation of) this information is transferred using the Hypertext Transfer Protocol (HTTP) [3] – possibly employing some protocol on top of it. Note that we do not primarily consider applications running inside a web browser like [4, 7, 9]. HTTP provides different methods to access resources from which the most important are the GET and the POST method. GET is used to retrieve some information, e.g., web browsers use GET to ask for the content of web pages. POST is used to send information to be processed by a server, e.g., entries in a web form can be sent to the server using POST. While processing the request, the server can update its internally stored information and usually sends back a result of this process. Web applications differ in their use of these two HTTP methods: RPC-style web applications are accessible through a single URI and communication is performed using the POST method. A representation of a procedure call is transferred to the server, which returns a representation of the result of this call as answer to the POST request. Note that both retrieval and update operations are performed using the POST method in RPC-style web applications and the interface of such applications is determined by the procedure calls that are understood by the server. Fielding [2] proposes another architectural style for web applications: He proposes to use the POST method only to update the internal state of the server and to use the GET method to purely retrieve information. Consequently, everything that should be available for retrieval by clients needs to be identified by an own URI. This is a big difference to RPC-style web applications that are usually accessible ?This work was partially supported by the German Research Council (DFG) grant Ha 2457/5-1. through a single URI. In the architectural style proposed by Fielding, a client sends an HTTP GET request to an URI to retrieve a representation of the resource identified by this URI corresponding to the current state of the server. To change this state, the client sends a POST request to the URI of the resource he wants to update. Fielding calls this architectural style Representational State Transfer (REST). The interface of a RESTful web application is given by the methods of HTTP and the URIs identifying the applications resources. In this work, we show how the interface given by the most important HTTP methods can be made available to Haskell or Curry programmers, hiding the details of HTTP communication.

Developing programming languages is a difficult task that requires a lot of time, effort, and expertise. Reusable programming language components make this task easier, by allowing language designers to grab off-the-shelf components for common language features. Modern functional programming languages, however, lack support for reuse of definitions, and thus language components defined using algebraic data types and pattern matching functions cannot be reused without modifying or copying existing code. To improve the situation, we introduce CS, a functional meta-language for developing reusable programming language components, that features built-in support for extensible data types and functions, as well as effects and handlers. In CS, we can define language components using algebraic data types and pattern matching functions, such that we can compose language components into larger languages and define new interpretations for existing components without modifying existing definitions.

There are a number of extended forms of algebraic data types such as type classes with existential types and generalized algebraic data types. Such extensions are highly useful but their interaction has not been studied formally so far. Here, we present a unifying framework for these extensions. We show that the combination of type classes and generalized algebraic data types allows us to express a number of interesting properties which are desired by programmers. We support type checking based on a novel constraint solver. Our results show that our system is practical and greatly extends the expressive power of languages such as Haskell and ML.

The seamless integration of relational databases and programming languages remains a major challenge. Mapping rich data types featured in general-purpose programming languages to the relational data model is one aspect of this challenge. We present a novel technique for mapping arbitrary (nonrecursive) algebraic data types to a relational data model, based on previous work on the relational representation of nested tables. Algebraic data types may be freely constructed and deconstructed in queries and may show up in the result type. The number of relational queries generated is small and statically determined by the type of the query. We have implemented the mapping in the Database Supported Haskell (DSH) library.

Algebraic data types are the natural target of pattern-matching functions. When the type is parametric then the pattern-matching functions typically support parametric polymorphism. In addition, bondi supports generic functional programming by introducing a small family of primitive, or representing data types, which can be used to define generic functions such as map, with one case for each primitive constructor. Each resulting case has a different type, so type specialisation is involved, too. During a type declaration, additional cases can be added to existing functions.

Synthesis from examples enables non-expert users to generate programs by specifying examples of their behavior. A domain-specific form of such synthesis has been recently deployed in a widely used spreadsheet software product. In this paper we contribute to foundations of such techniques and present a complete algorithm for synthesis of a class of recursive functions defined by structural recursion over a given algebraic data type definition. The functions we consider map an algebraic data type to a string; they are useful for, e.g., pretty printing and serialization of programs and data. We formalize our problem as learning deterministic sequential top-down tree-to-string transducers with a single state (1STS). The first problem we consider is learning a tree-to-string transducer from any set of input/output examples provided by the user. We show that, given a set of input/output examples, checking whether there exists a 1STS consistent with these examples is NP-complete in general. In contrast, the problem can be solved in polynomial time under a (practically useful) closure condition that each subtree of a tree in the input/output example set is also part of the input/output examples. Because coming up with relevant input/output examples may be difficult for the user while creating hard constraint problems for the synthesizer, we also study a more automated active learning scenario in which the algorithm chooses the inputs for which the user provides the outputs. Our algorithm asks a worst-case linear number of queries as a function of the size of the algebraic data type definition to determine a unique transducer. To construct our algorithms we present two new results on formal languages. First, we define a class of word equations, called sequential word equations, for which we prove that satisfiability can be solved in deterministic polynomial time. This is in contrast to the general word equations for which the best known complexity upper bound is in linear space. Second, we close a long-standing open problem about the asymptotic size of test sets for context-free languages. A test set of a language of words L is a subset T of L such that any two word homomorphisms equivalent on T are also equivalent on L. We prove that it is possible to build test sets of cubic size for context-free languages, matching for the first time the lower bound found 20 years ago.

We present RADA, a portable, scalable tool for reasoning about formulas containing algebraic data types using catamorphism (fold) functions. It can work as a back-end for reasoning about recursive programs that manipulate algebraic types. RADA operates by successively unrolling catamorphisms and uses either CVC4 and Z3 as reasoning engines. We have used RADA for reasoning about functional implementations of complex data structures and to reason about guard applications that determine whether XML messages should be allowed to cross network security domains. Promising experimental results demonstrate that RADA can be used in several practical contexts.

Metamorphic relations (MRs) describe the invariant relationships between program inputs and outputs. By checking for violations of MRs, faults in programs can be detected. Identifying MRs manually is a tedious and error-prone task. In this paper, we propose AutoMR, a novel method for systematically inferring and cleansing MRs. AutoMR can discover various types of equality and inequality MRs through a search method (particle swarm optimization). It also employs matrix singular-value decomposition and constraint solving techniques to remove the redundant MRs in the search results. Our experiments on 37 numerical programs from two popular open source packages show that AutoMR can effectively infer a set of accurate and succinct MRs and outperform the state-of-the-art method. Furthermore, we show that the discovered MRs have high fault detection ability in mutation testing and differential testing.

Search Based Software Engineering techniques are emerging as important tools for software maintenance. Foremost among these is Genetic Improvement, which has historically applied the stochastic techniques of Genetic Programming to optimize pre-existing program code. Previous work in this area has not generally preserved program semantics and this article describes an alternative to the traditional mutation operators used, employing deterministic proof search in the sequent calculus to yield semantics-preserving transformations on algebraic data types. Two case studies are described, both of which are applicable to the recently-introduced ‘grow and graft’ technique of Genetic Improvement: the first extends the expressiveness of the ‘grafting’ phase and the second transforms the representation of a list data type to yield an asymptotic efficiency improvement.

Generating good random values described by algebraic data types is often quite intricate. State-of-the-art tools for synthesizing random generators serve the valuable purpose of helping with this task, while providing different levels of invariants imposed over the generated values. However, they are often not built for composability nor extensibility, a useful feature when the shape of our random data needs to be adapted while testing different properties or sub-systems. In this work, we develop an extensible framework for deriving compositional generators, which can be easily combined in different ways in order to fit developers' demands using a simple type-level description language. Our framework relies on familiar ideas from the a la Carte technique for writing composable interpreters in Haskell. In particular, we adapt this technique with the machinery required in the scope of random generation, showing how concepts like generation frequency or terminal constructions can also be expressed in the same type-level fashion. We provide an implementation of our ideas, and evaluate its performance using real-world examples.

Datalog is a carefully restricted logic programming language. What makes Datalog attractive is its declarative fixpoint semantics: Datalog queries consist of simple Horn clauses, yet Datalog solvers efficiently compute all derivable tuples even for recursive queries. However, as we argue in this paper, Datalog is ill-suited as a programming language and Datalog programs are hard to write and maintain. We propose a “new” frontend for Datalog: functional programming with sets called functional IncA. While programmers write recursive functions over algebraic data types and sets, we transparently translate all code to Datalog relations. However, we retain Datalog’s strengths: Functions that generate sets can encode arbitrary relations and mutually recursive functions have fixpoint semantics. We also ensure that the generated Datalog program terminates whenever the original functional program terminates, so that we can apply off-the-shelve bottom-up Datalog solvers. We demonstrate the versatility and ease of use of functional IncA by implementing a type checker, a program transformation, an interpreter of the untyped lambda calculus, two data-flow analyses, and clone detection of Java bytecode.

We propose an extension of the join-calculus with pattern matching on algebraic data types. Our initial motivation is twofold: to provide an intuitive semantics of the interaction between concurrency and pattern matching; to define a practical compilation scheme from extended join-definitions into ordinary ones plus ML pattern matching. To assess the correctness of our compilation scheme, we develop a theory of the applied join-calculus, a calculus with value-passing and value matching.KeywordsPattern MatchTopological OrderMessage ContentClosed ProcessReaction RuleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.