The Head Condition and Polymorphic Recursion

We consider prescriptive type systems for logic programs (as in Godel or Mercury). In such systems, the typing is static, but it guarantees an operational property: if a program is well-typed, then all derivations starting in a well-typed query are again well-typed. This property has been called subject reduction. We show that this property can also be phrased as a property of the proof-theoretic semantics of logic programs, thus abstracting from the usual operational (top-down) semantics. This proof-theoretic view leads us to questioning a condition which is usually considered necessary for subject reduction, namely the head condition. It states that the head of each clause must have a type which is a variant (and not a proper instance) of the declared type. We provide a more general condition, thus reestablishing a certain symmetry between heads and body atoms. The condition ensures that in a derivation, the types of two unified terms are themselves unifiable. We discuss possible implications of this result. We also discuss the relationship between the head condition and polymorphic recursion, a concept known in functional programming.

As typed logic programming becomes more mainstream, system building tools like partial deduction systems will need to be mapped from untyped languages to typed ones. It is important, however, when mapping techniques across that the new techniques should exploit the type system as much as possible. In this paper, we show how norms, which play a crucial role in termination analysis, can be generated from the prescribed types of a logic program. Interestingly, the types highlight restrictions of earlier norms and suggest how these norms can be extended to obtain some very general and powerful notions of norm which can be used to measure any term in an almost arbitrary way. We see our work on norm derivation as a contribution to the termination analysis of typed logic programs which, in particular, forms an essential part of offline partial deduction systems.

We define an inference system for modular strictness analysis of functional programs by extending a conjunctive strictness logic with polymorphic and conditional properties. This extended set of properties is used to define a syntax-directed, polymorphic strictness analysis based on polymorphic recursion whose soundness is established via a translation from the polymorphic system into the conjunctive system. From the polymorphic analysis, an inference algorithm based on constraint resolution is derived and shown complete for variant of the polymorphic analysis. The algorithm deduces at the same time a property and a set of hypotheses on the free variables of an expression which makes it suitable for analysis of program with module structure.

We introduce a new semantics for logic programming languages. It generalises the traditional Herbrand universe semantics, and specialises the semantics of logical relations, as used in analysing parametricity in functional and imperative programming languages. We outline a typed logic programming language, give it this semantics, and show how it supports structured development of logic programs as advocated by Sterling et al. In particular, it gives semantics for some dynamic aspects of logic programs.

A new declarative programming paradigm, constraint functional logic programming (CFLP) is presented, which unifies systematically all features of constraint, functional and logic programming. CFLP extends constraint functional programming (CFP), which supports equation solving over a functional program, to general constraint programming permitting the use of constraints for programming as well as for posing queries. It also generalises constraint logic programming (CLP), by admitting user-defined functions as a purely functional subsystem of a CLP language. This paper presents the motivation for designing CFLP and its semantics. Programming examples are given in the concrete CFLP language Falcon.KeywordsLogic ProgramLogic ProgrammingConstraint ProgrammingConstraint SystemFunctional ProgrammingThese 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.

Languages that integrate functional and logic programming with a complete operational semantics are based on narrowing, a unification-based goal-solving mechanism which subsumes the reduction principle of functional languages and the resolution principle of logic languages. In this article, we present a partial evaluation scheme for functional logic languages based on an automatic unfolding algorithm which builds narrowing trees. The method is formalized within the theoretical framework established by Lloyd and Shepherdson for the partial deduction of logic programs, which we have generalized for dealing with functional computations. A generic specialization algorithm is proposed which does not depend on the eager or lazy nature of the narrower being used. To the best of our knowledge, this is the first generic algorithm for the specialization of functional logic programs. We also discuss the relation to work on partial evaluation in functional programming, term-rewriting systems, and logic programming. Finally, we present some experimental results with an implementation of the algorithm which show in practice that the narrowing-driven partial evaluator effectively combines the propagation of partial data structures (by means of logical variables and unification) with better opportunities for optimization (thanks to the functional dimension).

In this paper we show how we can use size and groundness analyses lifted to regular and (polymorphic) Hindley/Milner typed programs to determine more accurate termination of (type correct) programs. Type information for programs may be either inferred automatically or declared by the programmer. The analysis of the typed logic programs is able to completely reuse a framework for termination analysis of untyped logic programs by using abstract compilation of the type abstraction. We show that our typed termination analysis is uniformly more accurate than untyped termination analysis for regularly typed programs, and demonstrate how it is able to prove termination of programs which the untyped analysis can not.

In functional logic programs, rules are applicable independently of textual order, i.e., any rule can potentially be used to evaluate an expression. This is similar to logic languages and contrary to functional languages, e.g., Haskell enforces a strict sequential interpretation of rules. However, in some situations it is convenient to express alternatives by means of compact default rules. Although default rules are often used in functional programs, the non-deterministic nature of functional logic programs does not allow to directly transfer this concept from functional to functional logic languages in a meaningful way. In this paper we propose a new concept of default rules for Curry that supports a programming style similar to functional programming while preserving the core properties of functional logic programming, i.e., completeness, non-determinism, and logic-oriented uses of functions. We discuss the basic concept and sketch an initial implementation of it which exploits advanced features of functional logic languages.

In functional logic programs, rules are applicable independently of textual order, i.e., any rule can potentially be used to evaluate an expression. This is similar to logic languages and contrary to functional languages, e.g., Haskell enforces a strict sequential interpretation of rules. However, in some situations it is convenient to express alternatives by means of compact default rules. Although default rules are often used in functional programs, the non-deterministic nature of functional logic programs does not allow to directly transfer this concept from functional to functional logic languages in a meaningful way. In this paper, we propose a new concept of default rules for Curry that supports a programming style similar to functional programming while preserving the core properties of functional logic programming, i.e., completeness, non-determinism, and logic-oriented use of functions. We discuss the basic concept and propose an implementation which exploits advanced features of functional logic languages.

The basics of logic programming, as embodied by Prolog, are generally well-known in the programming language community. However, more advanced techniques, such as tabling, answer subsumption and probabilistic logic programming fail to attract the attention of a larger audience. The cause for the community's seemingly limited interest lies with the presentation of these features: the literature frequently focuses on implementations and examples that do little to aid the understanding of non-experts in the field. The key point is that many of these advanced logic programming features can be characterised in more generally known, more accessible terms. In my research I try to reconcile these advanced concepts from logic programming (Tabling, Answer subsumption and probabilistic programming) with concepts from functional programming (effects, monads and applicative functors).

We introduce a typed functional programming language LPL (acronym for Light linear Programming Language) in which all valid programs run in polynomial time, and which is complete for polynomial time functions. LPL is based on lambda-calculus, with constructors for algebraic data-types, pattern matching and recursive definitions, and thus allows for a natural programming style. The validity of LPL programs is checked through typing and a syntactic criterion on recursive definitions. The higher order type system is designed from the ideas of Light linear logic: stratification, to control recursive calls, and weak exponential connectives §, !, to control duplication of arguments.

We demonstrate the pragmatic value of the principal typing property, a property distinct from ML's principal type property, by studying a type system with principal typings. The type system is based on rank 2 intersection types and is closely related to ML. Its principal typing property provides elegant support for separate compilation, including smartest recompilation and incremental type inference. Moreover, it motivates a new rule for typing recursive definitions that can type some interesting examples of polymorphic recursion.

Several formal systems, such as resolution and minimal model semantics, provide a framework for logic programming. In this article, we will survey the use ofstructural proof theoryas an alternative foundation. Researchers have been using this foundation for the past 35 years to elevate logic programming from its roots in first-order classical logic into higher-order versions of intuitionistic and linear logic. These more expressive logic programming languages allow for capturing stateful computations and rich forms of abstractions, including higher-order programming, modularity, and abstract data types. Term-level bindings are another kind of abstraction, and these are given an elegant and direct treatment within both proof theory and these extended logic programming languages. Logic programming has also inspired new results in proof theory, such as those involving polarity and focused proofs. These recent results provide a high-level means for presenting the differences between forward-chaining and backward-chaining style inferences. Anchoring logic programming in proof theory has also helped identify its connections and differences with functional programming, deductive databases, and model checking.

We study the complexity of computable and Σ$_{\rm 1}^{\rm 0}$ inductive definitions of sets of natural numbers. For we example, we show how to assign natural indices to monotone Σ$_{\rm 1}^{\rm 0}$-definitions and we use these to calculate the complexity of the set of all indices of monotone Σ$_{\rm 1}^{\rm 0}$-definitions which are computable. We also examine the complexity of new type of inductive definition which we call weakly finitary monotone inductive definitions. Applications are given in proof theory and in logic programming.

Teleo-Reactive (TR) robotic agent programs comprise sequences of guarded action rules clustered into named parameterised procedures. Their ancestry goes back to the first cognitive robot, Shakey. Like Shakey, a TR programmed robotic agent has a deductive Belief Store comprising constantly changing predicate logic percept facts, and knowledge facts and rules for querying the percepts. In this paper we introduce TR programming using a simple example expressed in the teleo-reactive programming language TeleoR, which is a syntactic extension of QuLog, a typed logic programming language used for the agent’s Belief Store. We give a formal definition of the regression property that rules of TeleoR procedures should satisfy, and an informal operational semantics of the evaluation of a TeleoR procedure call. We then formally express key features of the evaluation in LTL. Finally we show how this LTL formalisation can be used to prove that a procedure’s rules satisfy the regression property by proving it holds for one rule of the example TeleoR program. The proof requires us: to formally link a TeleoR agent’s percept beliefs with sensed configurations of the external environment; to link the agent’s robotic device action intentions with actual robot actions; to specify the eventual physical effects of the robot’s actions on the environment state.