Type Inference Research Articles

Verifying higher-order programs with the dijkstra monad

Modern programming languages, ranging from Haskell and ML, to JavaScript, C# and Java, all make extensive use of higher-order state. This paper advocates a new verification methodology for higher-order stateful programs, based on a new monad of predicate transformers called the Dijkstra monad. Using the Dijkstra monad has a number of benefits. First, the monad naturally yields a weakest pre-condition calculus. Second, the computed specifications are structurally simpler in several ways, e.g., single-state post-conditions are sufficient (rather than the more complex two-state post-conditions). Finally, the monad can easily be varied to handle features like exceptions and heap invariants, while retaining the same type inference algorithm. We implement the Dijkstra monad and its type inference algorithm for the F* programming language. Our most extensive case study evaluates the Dijkstra monad and its F* implementation by using it to verify JavaScript programs. Specifically, we describe a tool chain that translates programs in a subset of JavaScript decorated with assertions and loop invariants to F*. Once in F*, our type inference algorithm computes verification conditions and automatically discharges their proofs using an SMT solver. We use our tools to prove that a core model of the JavaScript runtime in F* respects various invariants and that a suite of JavaScript source programs are free of runtime errors.

Terminating constraint set satisfiability and simplification algorithms for context-dependent overloading

Abstract Algorithms for constraint set satisfiability and simplification of Haskell type class constraints are used during type inference in order to allow the inference of more accurate types and to detect ambiguity. Unfortunately, both constraint set satisfiability and simplification are in general undecidable, and the use of these algorithms may cause non-termination of type inference. This paper presents algorithms for these problems that terminate on any given input, based on the use of a criterion that is tested on each recursive step. The use of this criterion eliminates the need of imposing syntactic conditions on Haskell type class and instance declarations in order to guarantee termination of type inference in the presence of multi-parameter type classes, and allows program compilation without the need of compiler flags for lifting such restrictions. Undecidability of the problems implies the existence of instances for which the algorithm incorrectly reports unsatisfiability, but we are not aware of any practical example where this occurs.

Ambiguity and context-dependent overloading

Abstract This paper discusses ambiguity in the context of languages that support context-dependent overloading, such as Haskell. A type system for a Haskell-like programming language that supports context-dependent overloading and follow the Hindley-Milner approach of providing context-free type instantiation, allows distinct derivations of the same type for ambiguous expressions. Such expressions are usually rejected by the type inference algorithm, which is thus not complete with respect to the type system. Also, Haskell’s open world approach considers a definition of ambiguity that does not conform to the existence of two or more distinct type system derivations for the same type. The article presents an alternative approach, where the standard definition of ambiguity is followed. A type system is presented that allows only context-dependent type instantiation, enabling only one type to be derivable for each expression in a given typing context: the type of an expression can be instantiated only if required by the program context where the expression occurs. We define a notion of greatest instance type for each occurrence of an expression, which is used in the definition of a standard dictionary-passing semantics for core Haskell based on type system derivations, for which coherence is trivial. Type soundness is obtained as a result of disallowing all ambiguous expressions and all expressions involving unsatisfiability in the use of overloaded names. Following the standard definition of ambiguity, satisfiability is tested—i.e., “the world is closed” —if only if overloading is (or should have been) resolved, that is, if and only if there exist unreachable variables in the constraints on types of expressions. Nowadays, satisfiability is tested in Haskell, in the presence of multi-parameter type classes, only upon the presence of functional dependencies or an alternative mechanism that specifies conditions for closing the world, and that may happen when there exist or not unreachable type variables in constraints. The satisfiability trigger condition is then given automatically, by the existence of unreachable variables in constraints, and does not need to be specified by programmers, using an extra mechanism.

Deadlock-freedom-by-design

Over the last decade, global descriptions have been successfully employed for the verification and implementation of communicating systems, respectively as protocol specifications and choreographies. In this work, we bring these two practices together by proposing a purely-global programming model. We show a novel interpretation of asynchrony and parallelism in a global setting and develop a typing discipline that verifies choreographies against protocol specifications, based on multiparty sessions. Exploiting the nature of global descriptions, our type system defines a new class of deadlock-free concurrent systems (deadlock-freedom-by-design), provides type inference, and supports session mobility. We give a notion of Endpoint Projection (EPP) which generates correct entity code (as pi-calculus terms) from a choreography. Finally, we evaluate our approach by providing a prototype implementation for a concrete programming language and by applying it to some examples from multicore and service-oriented programming.

Static and dynamic semantics of NoSQL languages

We present a calculus for processing semistructured data that spans differences of application area among several novel query languages, broadly categorized as "NoSQL". This calculus lets users define their own operators, capturing a wider range of data processing capabilities, whilst providing a typing precision so far typical only of primitive hard-coded operators. The type inference algorithm is based on semantic type checking, resulting in type information that is both precise, and flexible enough to handle structured and semistructured data. We illustrate the use of this calculus by encoding a large fragment of Jaql, including operations and iterators over JSON, embedded SQL expressions, and co-grouping, and show how the encoding directly yields a typing discipline for Jaql as it is, namely without the addition of any type definition or type annotation in the code.

Automating relatively complete verification of higher-order functional programs

We present an automated approach to relatively completely verifying safety (i.e., reachability) property of higher-order functional programs. Our contribution is two-fold. First, we extend the refinement type system framework employed in the recent work on (incomplete) automated higher-order verification by drawing on the classical work on relatively complete "Hoare logic like" program logic for higher-order procedural languages. Then, by adopting the recently proposed techniques for solving constraints over quantified first-order logic formulas, we develop an automated type inference method for the type system, thereby realizing an automated relatively complete verification of higher-order programs.

Вывод типов для языка Python

The article presents type inference for programs written in Python programming language. At first, type inference algorithms for parametric polymorphism that were described in the scientific literature are reviewed. The algorithms evolved from “basic” algorithm (also known as Palsberg — Schwartzbach algorithm) to Cartesian product algorithm (CPA). It was shown that CPA is both precise and efficient, however it has to be modified to be used on Python code. Afterwards, we present a new algorithm (a modification of CPA) for Python. It is shown how type inference module using the new algorithm analyses various Python language structures: constants (literals), basic Python collections, variable names, assignments, function and class definitions, attribute and index expressions. It is also shown how the algorithm deals with external (non-Python) functions using special annotations that specify output types depending on input types of the function. Afterwards, the results of work on the prototype (module that implements described type inference algorithm) are presented. The paper is concluded by an overview of possible future work directions such as generating a defect trace, i.e. description that specifies how expression got its incorrect types.

Java SAM Typed Closures: A Sound and Complete Type Inference System for Nominal Types

The last proposal for Java closures, as emerged in JSR 000335, is mainly innovative in: 1 Use of nominal types, SAM types, for closures; 2 Introduction of target types and compatibility for a contextual typing of closures; 3 Need for a type inference that reconstructs the omitted type annotations of closures and closure arguments. The paper provides a sound and complete type system, with nominal types, for such a type inference and discusses role and formalization of targeting and of compatibility in the designed inference process.

Inferences of type II extreme value distribution based on record values

It is known that if the maximum or minimum of n independent and identically distributed random variables when standardized converges as n tends to infinity to non-degenerate distribution, then it to converges to one the three extreme values distributions. Type II extreme value distribution is one of them. In this paper we will consider the record values of Type Ii extreme value distribution. Some distributional properties of the record values of this distribution will be given. Based on these properties some recurrence relations of the moments and a characterization of the Type II extreme value distribution will be presented.

Extending programming language to support object orientation in legacy systems

The contemporary software systems written in C face maintainability issues because of tight coupling. Introducing object orientation can address these problems by raising the abstraction to objects, thereby providing better programmability and understandability. However, compiling a C software with a C++ compiler is difficult because of the incompatibilities between C and C++. Some of the incompatibilities such as designated initializers are nontrivial in nature and hence are very difficult to handle by automation such as scripting or by manual efforts. Moreover, runtime support for features such as global constructors, exception handling, runtime type inference, etc. is also required in the target system. Clearly, the traditional procedural language compiler cannot provide these features. In this paper, we propose extending programming language such as C++ to support object orientation in legacy systems instead of completely redesigning them. With a case study of Linux kernel, we report major issues in providing the compile and runtime support for C++ in legacy systems, and provide a solution to these issues. Our approach paves the way for converting a large C based software into C++. The experiments demonstrate that the proposed extension saves significant manual efforts with very little change in the g++ compiler. In addition, the performance study considers other legacy systems written in C and shows that the overhead resulting from the modifications in the compiler is negligible in comparison to the functionality achieved.

Type-based safe resource deallocation for shared-memory concurrency

We propose a type system to guarantee safe resource deallocation for shared-memory concurrent programs by extending the previous type system based on fractional ownerships. Here, safe resource deallocation means that memory cells, locks, or threads are not left allocated when a program terminates. Our framework supports (1) fork/join parallelism, (2) synchronization with locks, and (3) dynamically allocated memory cells and locks. The type system is proved to be sound. We also provide a type inference algorithm for the type system and a prototype implementation of the algorithm.

Reim & ReImInfer

Reference immutability ensures that a reference is not used to modify the referenced object, and enables the safe sharing of object structures. A pure method does not cause side-effects on the objects that existed in the pre-state of the method execution. Checking and inference of reference immutability and method purity enables a variety of program analyses and optimizations. We present ReIm, a type system for reference immutability, and ReImInfer, a corresponding type inference analysis. The type system is concise and context-sensitive. The type inference analysis is precise and scalable, and requires no manual annotations. In addition, we present a novel application of the reference immutability type system: method purity inference. To support our theoretical results, we implemented the type system and the type inference analysis for Java. We include a type checker to verify the correctness of the inference result. Empirical results on Java applications and libraries of up to 348kLOC show that our approach achieves both scalability and precision.

Multivariate amortized resource analysis

We study the problem of automatically analyzing the worst-case resource usage of procedures with several arguments. Existing automatic analyses based on amortization or sized types bound the resource usage or result size of such a procedure by a sum of unary functions of the sizes of the arguments. In this article we generalize this to arbitrary multivariate polynomial functions thus allowing bounds of the form mn which had to be grossly overestimated by m 2 + n 2 before. Our framework even encompasses bounds like ∑ i,j≤ n m i m j where the m i are the sizes of the entries of a list of length n . This allows us for the first time to derive useful resource bounds for operations on matrices that are represented as lists of lists and to considerably improve bounds on other superlinear operations on lists such as longest common subsequence and removal of duplicates from lists of lists. Furthermore, resource bounds are now closed under composition which improves accuracy of the analysis of composed programs when some or all of the components exhibit superlinear resource or size behavior. The analysis is based on a novel multivariate amortized resource analysis. We present it in form of a type system for a simple first-order functional language with lists and trees, prove soundness, and describe automatic type inference based on linear programming. We have experimentally validated the automatic analysis on a wide range of examples from functional programming with lists and trees. The obtained bounds were compared with actual resource consumption. All bounds were asymptotically tight, and the constants were close or even identical to the optimal ones.

An error-tolerant type system for variational lambda calculus

Conditional compilation and software product line technologies make it possible to generate a huge number of different programs from a single software project. Typing each of these programs individually is usually impossible due to the sheer number of possible variants. Our previous work has addressed this problem with a type system for variational lambda calculus (VLC), an extension of lambda calculus with basic constructs for introducing and organizing variation. Although our type inference algorithm is more efficient than the brute-force strategy of inferring the types of each variant individually, it is less robust since type inference will fail for the entire variational expression if any one variant contains a type error. In this work, we extend our type system to operate on VLC expressions containing type errors. This extension directly supports locating ill-typed variants and the incremental development of variational programs. It also has many subtle implications for the unification of variational types. We show that our extended type system possesses a principal typing property and that the underlying unification problem is unitary. Our unification algorithm computes partial unifiers that lead to result types that (1) contain errors in as few variants as possible and (2) are most general. Finally, we perform an empirical evaluation to determine the overhead of this extension compared to our previous work, to demonstrate the improvements over the brute-force approach, and to explore the effects of various error distributions on the inference process.

A type system for counting instances of software components

We identify an abstract language for component software based on process algebra. Besides the usual operators for sequential, alternative and parallel composition, it has primitives for instantiating components and for deleting instances of components. We define an operational semantics for our language and give a type system in which types express quantitative information on the components involved in the execution of the expressions of the language. Included in this information is for each component the maximum number of instances that are simultaneously active during the execution of the expression. The type system is compositional by the novel use of ‘deficit types’. The type inference algorithm runs in time quadratic in the size of the input. We consider extensions of the language with loops and tail recursion, and with a scope mechanism. We illustrate the approach with some examples, one on UML diagram refinement and one on counting objects on the free store in C++.

The undecidability of type related problems in the type-free style System F with finitely stratified polymorphic types

We consider here a number of variations on System F that are predicative second-order systems whose terms are intermediate between the Curry style and the Church style. As in the Church style, the terms we deal with here contain the information on where universal quantifier elimination and introduction in the type inference process must take place. However, they omit the information on what types are involved in the rules, which is similar to Curry forms. This can be viewed as a version of the partial type reconstruction problem considered by Boehm and Pfenning in which type erasure is done in a systematic way. In this paper we prove the undecidability of the type checking, type inference, and typability problems for the system. This demonstrates that the reason for undecidability is not the absence of the information where the second-order rules should be applied but the actual shape of the polymorphic types to be used in the derivation. Moreover, the proof works for the predicative version of the system with finitely stratified polymorphic types. The result includes bounds on the Leivant levels of types used in the instances leading to undecidability.

Deterministic parallelism via liquid effects

Shared memory multithreading is a popular approach to parallel programming, but also fiendishly hard to get right. We present Liquid Effects , a type-and-effect system based on refinement types which allows for fine-grained, low-level, shared memory multi-threading while statically guaranteeing that a program is deterministic. Liquid Effects records the effect of an expression as a for- mula in first-order logic, making our type-and-effect system highly expressive. Further, effects like Read and Write are recorded in Liquid Effects as ordinary uninterpreted predicates, leaving the effect system open to extension by the user. By building our system as an extension to an existing dependent refinement type system, our system gains precise value- and branch-sensitive reasoning about effects. Finally, our system exploits the Liquid Types refinement type inference technique to automatically infer refinement types and effects. We have implemented our type-and-effect checking techniques in CSOLVE, a refinement type inference system for C programs. We demonstrate how CSOLVE uses Liquid Effects to prove the determinism of a variety of benchmarks.

Fast and precise hybrid type inference for JavaScript

JavaScript performance is often bound by its dynamically typed nature. Compilers do not have access to static type information, making generation of efficient, type-specialized machine code difficult. We seek to solve this problem by inferring types. In this paper we present a hybrid type inference algorithm for JavaScript based on points-to analysis. Our algorithm is fast , in that it pays for itself in the optimizations it enables. Our algorithm is also precise , generating information that closely reflects the program's actual behavior even when analyzing polymorphic code, by augmenting static analysis with run-time type barriers. We showcase an implementation for Mozilla Firefox's JavaScript engine, demonstrating both performance gains and viability. Through integration with the just-in-time (JIT) compiler in Firefox, we have improved performance on major benchmarks and JavaScript-heavy websites by up to 50%. Inference-enabled compilation is the default compilation mode as of Firefox 9.

CCured

In this paper we propose a scheme that combines type inference and run-time checking to make existing C programs type safe. We describe the CCured type system, which extends that of C by separating pointer types according to their usage. This type sysiem allows both pointers whose usage can be verified statically to be type safe, and pointers whose safety must be checked at run time. We prove a type soundness result and then we present a surprisingly simple type inference algoriihm that is able lo infer the appropriate pointer kinds for existing C programs. Our experience wiih the CCured sysiem shows that the inference is very effective for many C programs, as it is able to infer that most or all of the pointers are statically verifiable to be type safe. The remaining pointers are instrumented wiih efficient run-time checks to ensure that they are used safely. The resulting performance loss due to run-time checks is 0-150%, which is several times better than comparable approaches thai use only dynamic checking. Using CCured we have discovered programming bugs in established C programs such as several SPECINT95 benchmarks.

A Bi-Directional Refinement Algorithm for the Calculus of (Co)Inductive Constructions

The paper describes the refinement algorithm for the Calculus of (Co)Inductive Constructions (CIC) implemented in the interactive theorem prover Matita. The refinement algorithm is in charge of giving a meaning to the terms, types and proof terms directly written by the user or generated by using tactics, decision procedures or general automation. The terms are written in an "external syntax" meant to be user friendly that allows omission of information, untyped binders and a certain liberal use of user defined sub-typing. The refiner modifies the terms to obtain related well typed terms in the internal syntax understood by the kernel of the ITP. In particular, it acts as a type inference algorithm when all the binders are untyped. The proposed algorithm is bi-directional: given a term in external syntax and a type expected for the term, it propagates as much typing information as possible towards the leaves of the term. Traditional mono-directional algorithms, instead, proceed in a bottom-up way by inferring the type of a sub-term and comparing (unifying) it with the type expected by its context only at the end. We propose some novel bi-directional rules for CIC that are particularly effective. Among the benefits of bi-directionality we have better error message reporting and better inference of dependent types. Moreover, thanks to bi-directionality, the coercion system for sub-typing is more effective and type inference generates simpler unification problems that are more likely to be solved by the inherently incomplete higher order unification algorithms implemented. Finally we introduce in the external syntax the notion of vector of placeholders that enables to omit at once an arbitrary number of arguments. Vectors of placeholders allow a trivial implementation of implicit arguments and greatly simplify the implementation of primitive and simple tactics.