An abstract interpretation algorithm for residuating logic programs

Residuation is an operational mechanism for the integration of functions into logic programming languages. The residuation principle delays the evaluation of functions during the unification process until the arguments are sufficiently instantiated. This has the advantage that the deterministic nature of functions is preserved but the disadvantage of incompleteness: if the variables in a delayed function call are not instantiated by the logic program, this function can never be evaluated and some answers which are logical consequences of the program are lost. In order to detect such situations at compile time, we present an abstract interpretation algorithm for this kind of programs. The algorithm approximates the possible residuations and instantiation states of variables during program execution. If the algorithm computes an empty residuation set for a goal, then it is ensured that the concrete execution of the goal does not end with a nonempty set of residuations which cannot be evaluated due to insufficient instantiation of argument variables

Analysis of nonlinear constraints in CLP(R)

Solving nonlinear constraints over real numbers is a complex problem. Hence constraint logic programming languages like CLP(R) or Prolog III solve only linear constraints and delay nonlinear constraints until they become linear. This efficient implementation method has the disadvantage that sometimes computed answers are unsatisfiable or infinite loops occur due to the unsatisfiability of delayed nonlinear constraints. These problems could be solved by using a more powerful constraint solver which can deal with nonlinear constraints like in RISC-CLP(Real). Since such powerful constraint solvers are not very efficient, we propose a compromise between these two extremes. We characterize a class of CLP(R) programs for which all delayed nonlinear constraints become linear at run time. Programs belonging to this class can be safely executed with the efficient CLP(R) method while the remaining programs need a more powerful constraint solver

Snapshot generation in a constructive object-oriented modeling language

CooML is an object-oriented modeling language where specifications are theories in a constructive logic designed to handle incomplete information. In this logic we view snapshots as a formal counterpart of object populations, which are associated with specifications via the constructive interpretation of logical connectives. In this paper, we introduce the \u201csnapshot semantics\u201d of CooML and we describe a snapshot generation (SG) algorithm, which can be applied to validate specifications in the spirit of OCL-like constraints over UML models. Differently from the latter and from the standard BHK semantics, the logic allows us to exploit a notion of partial validation that is appropriate to encodings characterised by incomplete information. SG is akin to model generation in answer set programming. We show that the algorithm is sound and complete so that its successful termination implies consistency of the system

Programming with narrowing: A tutorial

AbstractNarrowing is a computation implemented by some declarative programming languages. Research in the last decade has produced significant results on the theory and foundation of narrowing, but little has been published on the use of narrowing in programming. This paper introduces narrowing from a programmer’s viewpoint; shows, by means of examples, when, why and how to use narrowing in a program; and discusses the impact of narrowing on software development activities such as design and maintenance. The examples are coded in the programming language Curry, which provides narrowing as a first class feature

Records for logic programming

CFT is a new constraint system providing records as logical data structure for constraint (logic) programming. It can be seen as a generalization of the rational tree system employed in Prolog II, where finer-grained constraints are used, and where subtrees are identified by keywords rather than by position. CFT is defined by a first-order structure consisting of so-called feature trees. Feature trees generalize the ordinary trees corresponding to first-order terms by having their edges labeled with field names called features. The mathematical semantics given by the feature tree structure is complemented with a logical semantics given by five axiom schemes, which we conjecture to comprise a complete axiomatization of the feature tree structure. We present a decision method for CFT, which decides entailment and disentailment between possibly existentially quantified constraints. Since CFT satisfies the independence property, our decision method can also be employed for checking the satisfiability of conjunctions of positive and negative constraints. This includes quantified negative constraints such as forall yforall z(xneq f(y,z)). The paper also presents an idealized abstract machine processing negative and positive constraints incrementally. We argue that an optimized version of the machine can decide satisfiability and entailment in quasi-linear time

Relational Programming in miniKanren: Techniques, Applications, and Implementations

Thesis (Ph.D.) - Indiana University, Computer Sciences, 2009The promise of logic programming is that programs can be written relationally, without distinguishing between input and output arguments. Relational programs are remarkably flexible—for example, a relational type-inferencer also performs type checking and type inhabitation, while a relational theorem prover generates theorems as well as proofs and can even be used as a simple proof assistant. Unfortunately, writing relational programs is difficult, and requires many interesting and unusual tools and techniques. For example, a relational interpreter for a subset of Scheme might use nominal unification to support variable binding and scope, Constraint Logic Programming over Finite Domains (CLP(FD)) to implement relational arithmetic, and tabling to improve termination behavior. In this dissertation I present miniKanren, a family of languages specifically designed for relational programming, and which supports a variety of relational idioms and techniques. I show how miniKanren can be used to write interesting relational programs, including an extremely flexible lean tableau theorem prover and a novel constraint-free binary arithmetic system with strong termination guarantees. I also present interesting and practical techniques used to implement miniKanren, including a nominal unifier that uses triangular rather than idempotent substitutions and a novel “walk”-based algorithm for variable lookup in triangular substitutions. The result of this research is a family of languages that supports a variety of relational idioms and techniques, making it feasible and useful to write interesting programs as relations