Learning recursive functions from approximations

Domain theory is the mathematical theory underlying denotational semantics. This thesis presents a formalization of domain theory in the Higher Order Logic (HOL) theorem proving system along with a mechanization of proof functions and other tools to support reasoning about the denotations of functional programs. By providing a fixed point operator for functions on certain domains which have a special undefined (bottom) element, this extension of HOL supports the definition of recursive functions which are not also primitive recursive. Thus, it provides an approach to the long-standing and important problem of defining non-primitive recursive functions in the HOL system.<br /> <br />Our philosophy is that there must be a direct correspondence between elements of complete partial orders (domains) and elements of HOL types, in order to allow the reuse of higher order logic and proof infrastructure already available in the HOL system. Hence, we are able to mix domain theoretic reasoning with reasoning in the set theoretic HOL world to advantage, exploiting HOL types and tools directly. Moreover, by mixing domain and set theoretic reasoning, we are able to eliminate almost all reasoning about the bottom element of complete partial orders that makes the LCF theorem prover, which supports a first order logic of domain theory, difficult and tedious to use. A thorough comparison with LCF is provided.<br /> <br />The advantages of combining the best of the domain and set theoretic worlds in the same system are demonstrated in a larger example, showing the correctness of a unification algorithm. A major part of the proof is conducted in the set theoretic setting of higher order logic, and only at a late stage of the proof domain theory is introduced to give a recursive definition of the algorithm, which is not primitive recursive. Furthermore, a total well-founded recursive unification function can be defined easily in pure HOL by proving that the unification algorithm (defined in domain theory) always terminates; this proof is conducted by a non-trivial well-founded induction. In such applications, where non-primitive recursive HOL functions are defined via domain theory and a proof of termination, domain theory constructs only appear temporarily.

A type theoretic programming language is introduced that is based on lambda calculus with coproducts, products and inductive types, and additionally allows the definition of recursive functions in the way that is common in most functional programming languages. A formal system is presented that checks whether such a definition is structurally recursive and a soundness theorem is shown for this system. Thus all functions passing this check are ensured to terminate on all inputs. For the moment only non-mutual recursive functions are considered.

Automatic parallelization of recursive functions with rewriting rules

In this thesis, we explore techniques for the development and verification of programs in a high-level, expressive, and safe programming language. Our programs can express problems over unbounded domains and over recursive and mutable data structures. We present an implementation language flexible enough to build interesting and useful systems. We mostly maintain a core shared language for the specifications and the implementation, with only a few extensions specific to expressing the specifications. Extensions of the core shared language include imperative features with state and side effects, which help when implementing efficient systems. Our language is a subset of the Scala programming language. Once verified, programs can be compiled and executed using the existing Scala tools. We present algorithms for verifying programs written in this language. We take a layer-based approach, where we reduce, at each step, the program to an equivalent program in a simpler language. We first purify functions by transforming away mutations into explicit return types in the functions' signatures. This step rewrites all mutations of data structures into cloning operations. We then translate local state into a purely functional code, hence eliminating all traces of imperative programming. The final language is a functional subset of Scala, on which we apply verification. We integrate our pipeline of translations into Leon, a verifier for Scala. We verify the core functional language by using an algorithm already developed inside Leon. The program is encoded into equivalent first-order logic formulas over a combination of theories and recursive functions. The formulas are eventually discharged to an external SMT solver. We extend this core language and the solving algorithm with support for both infinite-precision integers and bit-vectors. The algorithm takes into account the semantics gap between the two domains, and the programmer is ultimately responsible to use the proper type to represent the data. We build a reusable interface for SMT-LIB that enables us to swap solvers transparently in order to validate the formulas emitted by Leon. We experiment with writing solvers in Scala; they could offer both a better and safer integration with the rest of the system. We evaluate the cost of using a higher-order language to implement such solvers, traditionally written in C/C++. Finally, we experiment with the system by building fully working and verified applications. We rely on the intersection of many features including higher-order functions, mutable data structures, recursive functions, and nondeterministic environment dependencies, to build concise and verified applications.

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.

FACILE is a language which symmetrically integrates concurrent and functional programming. It is a typed and statically scoped language. The language supports both function and process abstractions. Recursive functions may be defined and used within processes, and processes can be dynamically created during expression evaluation. Inter-process communication is by hand-shaking on typed channels. Typed channels, functions and process scripts are first-class values.In this paper, we present the “core” syntax of FACILE. We then describe an abstract machine C-FAM which executes concurrent processes evaluating functional style expressions. The operational definition of FACILE is presented by giving compilation rules from FACILE to C-FAM instructions, and execution rules for the abstract machine. An overview of the FACILE environment is also discussed.KeywordsFunctional ProgrammingConcurrent ProgrammingAbstract MachineFunctional LanguageAbstract Data TypeThese 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.

This paper was inspired by [FBW 94]. An arbitrary upper bound on the size of some program for the target function suffices for the learning of some program for this function. In [FBW 94] it was discovered that if “learning” is understood as “identification in the limit,” then in some programming languages it is possible to learn a program of size not exceeding the bound, while in some other programming languages this is not possible.We have studied three other learning types, namely, “finite identification,” “co-learning” and “confidence-learning.” These three types are very different. Co-learning with the considered additional information in the form “an arbitrary upper bound for the size of the minimal program” allows the learning of the class of all recursive functions. “Finite identification” does not allow this. “Confidence-learning” is strong enough to learn the class of all recursive functions even without the additional information. However, the results of our paper show exactly the opposite rating for the capabilities of learning programs not exceeding the size given by the bound.For finite identification it is still possible to identify small programs with additional information in some programming languages but not in all of them. For co-learning it is not possible in any programming language. These results contrast to the result in [FKS 94] showing that an arbitrary class of recursive functions is co-learnable if and only if it is identifiable in the limit. Finally, for confidence-learning it is in general not possible to identify small programs with additional information.KeywordsProgramming LanguageTarget FunctionRecursive FunctionInductive InferenceIdentification TypeThese 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.

WITH REFINED LINEARITY TEST: AN EXAMPLE OF RECURSIVE NETWORK STRUCTURE IN FP2, A FUNCTIONAL PARALLEL PROGRAMMING LANGUAGE Ricardo Caferra and Philippe Jorrand LIFIA Laboratoire d 'Informatique Fondamentale et d'Intelligence Artificielle IMAG GRENOBLE FRANCE Unification of first order terms is an essential primitive not only in resolution-like theorem provers but also in most AI languages. A rigorous approach to the design of a unification process with a high degree of internal parallelism has been recently proposed in [I ]. Starting with a trivial unification algorithm described by a recursive function in FP2 (a Functional Parallel Programming language developped at LIFIA), that process is obtained through successive transformational steps:

Corecursive definitions are usually only meaningful in functional languages with lazy evaluation semantics, because their domain and range may contain cyclic data graphs. By inspection of the call stack, it is possible in a strict evaluation environment to detect cycles in a computation, and thus transform finite input graphs infinite time. This paper presents a virtual machine with suitable cycle handling primitives and operational semantics to implement strict evaluation of corecursive functions. We discuss the impact on calling conventions and definition constructs, and demonstrate the relevance of the introduced features by application to the domain of infinite precision decimal arithmetics.KeywordsVirtual MachineOutput ParameterRecursive FunctionFunctional LanguageCycle DetectionThese 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.

A shortcut fusion approach to accumulations

Reversible functional languages have been proposed that use patterns symmetrically for matching and building data: A pattern used on the left-hand side of a function rule takes apart a data structure, and a pattern used on the right-hand side of a rule builds a data structure. When calling a function in reverse, the meaning of patterns are reversed, so patterns on the right-hand side take apart data and patterns on the left-hand side build data. If using a pattern to build data creates a node with exactly one reference, a node that is taken apart must by symmetry also have exactly one reference. This implies linearity: A node that is built or taken apart has exactly one incoming reference. A recursive function can take apart one data structure while building two or more new copies, allowing multiple uses of values, but the new copies will also have one reference each. Linearity makes garbage collection simple: Whenever a node is taken apart by pattern matching, it is freed. The cost is that multiple uses of a value require deep copies. The reason for the linearity restriction is the symmetry between constructing and deconstructing nodes: Since constructing a node creates exactly one reference to this node, the inverse can only be applied if the reference count is exactly one. We can overcome this limitation if constructing a node can return a node with multiple references. We achieve this through maximal sharing: If a newly constructed node is identical to an already existing node, we return a pointer to the existing node (increasing its reference count) instead of allocating a new node with reference count one. This allows multiple references to nodes while retaining the symmetry of pattern matching and construction, and it allows values to be used multiple times without making deep copies. To avoid searching the entire heap for an identical node, we use hash-consing to restrict the search to a small segment of the heap. We estimate how large this segment needs to be to give a low probability of allocation failure with acceptable utilisation and support this estimate with experiments. We sketch how a functional program can be translated to use the memory manager, and we test the memory manager both with artificial test programs and a hand-compiled functional program.

Dependent types provide a strong foundation for specifying and verifying rich properties of programs through type-checking. The earliest implementations combined dependency, which allows types to mention program variables; with type-level computation, which facilitates expressive specifications that compute with recursive functions over types. While many recent applications of dependent types omit the latter facility, we argue in this paper that it deserves more attention, even when implemented without dependency.In particular, the ability to use functional programs as specifications enables statically-typed metaprogramming: programs write programs, and static type-checking guarantees that the generating process never produces invalid code. Since our focus is on generic validity properties rather than full correctness verification, it is possible to engineer type inference systems that are very effective in narrow domains. As a demonstration, we present Ur, a programming language designed to facilitate metaprogramming with first-class records and names. On top of Ur, we implement Ur/Web, a special standard library that enables the development of modern Web applications. Ad-hoc code generation is already in wide use in the popular Web application frameworks, and we show how that generation may be tamed using types, without forcing metaprogram authors to write proofs or forcing metaprogram users to write any fancy types.

This paper proposes FUnCAL, a functional redesign of the graph transformation language UnCAL. A large amount of graph-structured data are widely used, including biological database, XML with IDREFs, WWW, and UML diagrams in software engineering. UnCAL is a language designed for graph transformations, i.e., extracting a subpart of a graph data and converting it to a suitable form, as what XQuery does for XMLs. A distinguished feature of UnCAL is its semantics that respects bisimulation on graphs; this enables us to reason about UnCAL graph transformations as recursive functions, which is useful for reasoning as well as optimization. However, there is still a gap to apply the program-manipulation techniques studied in the programming language literature directly to UnCAL programs, due to some special features in UnCAL, especially markers. In this paper, following the observation that markers can be emulated by tuples and λ-abstractions, we transform UnCAL programs to a restricted class of usual (thus, marker-free) functional ones. By this translation, we can reason, analyze or optimize UnCAL programs as usual functional programs. Moreover, we introduce a type system for showing that a small modification to the usual lazy semantics is enough to run well-typed functional programs as finite-graph transformations in a terminating way.

Verifying relational properties of functional programs by first-order refinement

We describe techniques for synthesis and verification of recursive functional programs over unbounded domains. Our techniques build on top of an algorithm for satisfiability modulo recursive functions, a framework for deductive synthesis, and complete synthesis procedures for algebraic data types. We present new counterexample-guided algorithms for constructing verified programs. We have implemented these algorithms in an integrated environment for interactive verification and synthesis from relational specifications. Our system was able to synthesize a number of useful recursive functions that manipulate unbounded numbers and data structures.