A Lambda-Free Higher-Order Recursive Path Order

International audienceWe generalize the recursive path order (RPO) to higher-order terms without λ-abstraction. This new order fully coincides with the standard RPO on first-order terms also in the presence of currying, distinguishing it from previous work. It has many useful properties, including well-foundedness, transitivity, stability under substitution, and the subterm property. It appears promising as the basis of a higher-order superposition calculus

Denotation by Transformation: Towards Obtaining a Denotational Semantics by Transformation to Point-free Style

It has often been observed that a point-free style of programming provides a more abstract view on programs. We aim to use the gain in abstraction to obtain a denotational semantics for functional logic languages in a straightforward way. Here we propose a set of basic operations based on which arbitrary functional logic programs can be transformed to point-free programs. The resulting programs are strict but, nevertheless, the semantics of the original program is preserved. There is a one-to-one mapping from the primitives introduced by the transformation to operations in relation algebra. This mapping can be extended to obtain a relation algebraic model for the whole program. This yields a denotational semantics which is on one hand closely related to point-free functional logic programs and on the other hand connects to the well developed field of algebraic logic including automatic proving

Variable elimination for building interpreters

In this paper, we build an interpreter by reusing host language functions instead of recoding mechanisms of function application that are already available in the host language (the language which is used to build the interpreter). In order to transform user-defined functions into host language functions we use combinatory logic : lambda-abstractions are transformed into a composition of combinators. We provide a mechanically checked proof that this step is correct for the call-by-value strategy with imperative features.Comment: 33 page

Deciding Confluence and Normal Form Properties of Ground Term Rewrite Systems Efficiently

It is known that the first-order theory of rewriting is decidable for ground term rewrite systems, but the general technique uses tree automata and often takes exponential time. For many properties, including confluence (CR), uniqueness of normal forms with respect to reductions (UNR) and with respect to conversions (UNC), polynomial time decision procedures are known for ground term rewrite systems. However, this is not the case for the normal form property (NFP). In this work, we present a cubic time algorithm for NFP, an almost cubic time algorithm for UNR, and an almost linear time algorithm for UNC, improving previous bounds. We also present a cubic time algorithm for CR

Extending SMT Solvers to Higher-Order Logic

International audienceSMT solvers have throughout the years been able to cope with increasingly expressive formulas, from ground logics to full first-order logic (FOL). In contrast, the extension of SMT solvers to higher-order logic (HOL) is mostly un-explored. We propose a pragmatic extension for SMT solvers to support HOL reasoning natively without compromising performance on FOL reasoning, thus leveraging the extensive research and implementation efforts dedicated to efficient SMT solving. We show how to generalize data structures and the ground decision procedure to support partial applications and extensionality, as well as how to reconcile quantifier instantiation techniques with higher-order variables. We also discuss a separate approach for redesigning an HOL SMT solver from the ground up via new data structures and algorithms. We apply our pragmatic extension to the CVC4 SMT solver and discuss a redesign of the veriT SMT solver. Our evaluation shows they are competitive with state-of-the-art HOL provers and often outperform the traditional encoding into FOL

A Transfinite Knuth-Bendix Order for Lambda-Free Higher-Order Terms

International audienceWe generalize the Knuth-Bendix order (KBO) to higher-order terms without λ-abstraction. The restriction of this new order to first-order terms coincides with the traditional KBO. The order has many useful properties, including transitivity, the subterm property, compatibility with contexts (monotonicity), stability under substitution, and well-foundedness. Transfinite weights and argument coefficients can also be supported. The order appears promising as the basis of a higher-order superposition calculus