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The Damas-Milner Calculus is the typed -calculus underlying the type system for ML and several other strongly typed polymorphic functional languages such as Miranda 1 and Haskell. Mycroft has extended its problematic monomorphic typing rule for recursive definitions with a polymorphic typing rule. He proved the resulting type system, which we call the Milner-Mycroft Calculus, sound with respect to Milner's semantics, and showed that it preserves the principal typing property of the Damas-Milner Calculus. The extension is of practical significance in typed logic programming languages and, more generally, in any language with (mutually) recursive definitions. In this paper we show that the type inference problem for the Milner-Mycroft Calculus is log-space equivalent to semi-unification, the problem of solving subsumption inequations between first-order terms. This result has been proved independently by Kfoury, Tiuryn, and Urzyczyn. In connection with the recently establish...

Interpretation of Logic Programs Roberto Barbuti , Roberto Giacobazzi , Giorgio Levi Dipartimento di Informatica Universit`a di Pisa Corso Italia 40, 56125 Pisa fbarbuti,giaco,levig@di.unipi.it in ACM Transactions on Programming Languages and Systems Vol 15, January 1993 Abstract The theory of abstract interpretation provides a formal framework to develop advanced dataflow analysis tools. The idea is to define a non-standard semantics which is able to compute, in finite time, an approximated model of the program. In this paper we define an abstract interpretation framework based on a fixpoint approach to the semantics. This leads to the definition, by means of a suitable set of operators, of an abstract fixpoint characterization of a -model associated with the program. Thus, we obtain a specializable abstract framework for bottom-up abstract interpretations of definite logic programs. The specialization of the framework is shown on two examples, namely ground dependence analysis and depth-k analysis.

An informal tutorial is presented for program synthesis, with an emphasis on deductive methods. According to this approach, to construct a program meeting a given specification, we prove the existence of an object meeting the specified conditions. The proof is restricted to be sufficiently constructive, in the sense that, in establishing the existence of the desired output, the proof is forced to indicate a computational method for finding it. That method becomes the basis for a program that can be extracted from the proof. The exposition is based on the deductive-tableau system, a theorem-proving framework particularly suitable for program synthesis. The system includes a nonclausal resolution rule, facilities for reasoning about equality, and a well-founded induction rule. INTRODUCTION This is an introduction to program synthesis, the derivation of a program to meet a given specification. It focuses on the deductive approach, in which the derivation task is regarded as a problem of ...

We present SLDNFA, an extension of SLDNF-resolution for abductive reasoning on abductive logic programs. SLDNFA solves the floundering abduction problem: non-ground abductive atoms can be selected. SLDNFA provides also a partial solution for the floundering negation problem. Different abductive answers can be derived from an SLDNFA-refutation; these answers provide different compromises between generality and comprehensibility. Two extensions of SLDNFA are proposed which satisfy stronger completeness results. The soundness of SLDNFA and its extensions is proven. Their completeness for minimal solutions with respect to implication, cardinality and set inclusion is investigated. The formalisation of SLDNFA presented here is an update of an older version presented in [13] and does not rely on skolemisation of abductive atoms. 1

The Davis-Putnam-Logemann-Loveland procedure (DPLL) was introduced in the early

Abstract. FDPLL is a directly lifted version of the well-known Davis-Putnam-Logeman-Loveland (DPLL) procedure. While DPLL is based on a splitting rule for case analysis wrt. ground and complementary literals, FDPLL uses a lifted splitting rule, i.e. the case analysis is made wrt. non-ground and complementary literals now. The motivation for this lifting is to bring together successful first-order techniques like unification and subsumption to the propositionally successful DPLL procedure. At the heart of the method is a new technique to represent first-order interpretations, where a literal specifies truth values for all its ground instances, unless there is a more specific literal specifying opposite truth values. Based on this idea, the FDPLL calculus is developed and proven as strongly complete. 1

In the last ten years declaration-free programming languages with a polymorphic typing discipline (ML, B) have been developed to approximate the flexibility and conciseness of dynamically typed languages (LISP, SETL) while retaining the safety and execution efficiency of conventional statically typed languages (Algol68, Pascal). These polymorphic languages can be type checked at compile time, yet allow functions whose arguments range over a variety of types. We investigate several polymorphic type systems, the most powerful of which, termed Milner-Mycroft Calculus, extends the so-called let-polymorphism found in, e.g., ML with a polymorphic typing rule for recursive definitions. We show that semi-unification, the problem of solving inequalities over firstorder terms, characterizes type checking in the Milner-Mycroft Calculus to polynomial time, even in the restricted case where nested definitions are disallowed. This permits us to extend some infeasibility results for related combinato...

Abstract. We show how declarative diagnosis techniques can be extended to cope with verification of operational properties, such as computed answers, and of abstract properties, such as types and groundness dependencies. The extension is achieved by using a simple semantic framework, based on abstract interpretation. The resulting technique (abstract diagnosis) leads to elegant bottom-up and top-down verification methods, which do not require to determine the symptoms in advance, and which are effective in the case of abstract properties described by finite domains.

Schwartz et al. described an optimization to implement built-in abstract types such as sets and maps with efficient data structures. Their transformation rests on the discovery of finite universal sets, called bases, to be used for avoiding data replication and for creating aggregate data structures that implement associative access by simpler cursor or pointer access. The SETL implementation used global analysis similar to classical dataflow for typings and for set inclusion and membership relationships to determine bases. However, the optimized data structures selected by this optmization did not include a primitive linked list or array, and all optimized data structures retained some degree of hashing. Hence, this heuristic approach did not guarantee a uniform improvement in performance over the use of default representations. The analysis was complicated by SETL's imperative style, weak typing, and low level control structures. The implemented optimizer was large (about 20,000 line...