Higher Order Unification via Explicit Substitutions

AbstractHigher order unification is equational unification for βη-conversion. But it is not first order equational unification, as substitution has to avoid capture. Thus, the methods for equational unification (such as narrowing) built upon grafting (i.e., substitution without renaming) cannot be used for higher order unification, which needs specific algorithms. Our goal in this paper is to reduce higher order unification to first order equational unification in a suitable theory. This is achieved by replacing substitution by grafting, but this replacement is not straightforward as it raises two major problems. First, some unification problems have solutions with grafting but no solution with substitution. Then equational unification algorithms rest upon the fact that grafting and reduction commute. But grafting and βη-reduction do not commute in λ-calculus and reducing an equation may change the set of its solutions. This difficulty comes from the interaction between the substitutions initiated by βη-reduction and the ones initiated by the unification process. Two kinds of variables are involved: those of βη-conversion and those of unification. So, we need to set up a calculus which distinguishes between these two kinds of variables and such that reduction and grafting commute. For this purpose, the application of a substitution of a reduction variable to a unification one must be delayed until this variable is instantiated. Such a separation and delay are provided by a calculus of explicit substitutions. Unification in such a calculus can be performed by well-known algorithms such as narrowing, but we present a specialised algorithm for greater efficiency. At last we show how to relate unification in λ-calculus and in a calculus with explicit substitutions. Thus, we come up with a new higher order unification algorithm which eliminates some burdens of the previous algorithms, in particular the functional handling of scopes. Huet's algorithm can be seen as a specific strategy for our algorithm, since each of its steps can be decomposed into elementary ones, leading to a more atomic description of the unification process. Also, solved forms in λ-calculus can easily be computed from solved forms in λσ-calculus

Normalisation for higher-order calculi with explicit substitutions

Explicit substitutions (ES) were introduced as a bridge between the theory of rewrite systems with binders and substitution, such as the λ-calculus, and their implementation. In a seminal paper Melliès observed that the dynamical properties of a rewrite system and its ES-based implementation may not coincide: he showed that a strongly normalising term (i.e. one which does not admit infinite derivations) in the λ-calculus may lose this status in its ES-based implementation. This paper studies normalisation for the latter systems in the general setting of higher-order rewriting: Based on recent work extending the theory of needed strategies to non-orthogonal rewrite systems we show that needed strategies normalise in the ES-based implementation of any orthogonal pattern higher-order rewrite system.Facultad de Informátic

Normalisation for higher-order calculi with explicit substitutions

Explicit substitutions (ES) were introduced as a bridge between the theory of rewrite systems with binders and substitution, such as the λ-calculus, and their implementation. In a seminal paper Melliès observed that the dynamical properties of a rewrite system and its ES-based implementation may not coincide: he showed that a strongly normalising term (i.e. one which does not admit infinite derivations) in the λ-calculus may lose this status in its ES-based implementation. This paper studies normalisation for the latter systems in the general setting of higher-order rewriting: Based on recent work extending the theory of needed strategies to non-orthogonal rewrite systems we show that needed strategies normalise in the ES-based implementation of any orthogonal pattern higher-order rewrite system.Facultad de Informátic

Higher Order Unification via Explicit Substitutions

Projet COQ, Projet PARA,Higher order unification is equational unification for $\beta\eta$-conversion. But it is not first order equational unification, as substitution has to avoid capture. Thus the methods for equational unification (such as narrowing) built upon grafting (i.e. substitution without renaming), cannot be used for higher order unification, which needs specific algorithms. Our goal in this paper is to reduce higher order unification to first order equational unification in a suitable theory. This is achived by replacing substitution by grafting, but this replacement is not straightforward as it raises two major problems. First, some unification problems have solutions with grafting but no solution with substitution. Then equational unification algorithms rest upon the fact that grafting and reduction commute. But grafting and $\beta\eta$-reduction do not commute in $\lambda$-calculus and reducing an equation may change the set of its solutions. This difficulty comes from the interaction between the substitutions initiated by $\beta \eta$-reduction and the ones initiated by the unification process. The difference is at the variable level. Two kinds of variables are involved: those of $\beta\eta$-conversion and those of unification. So, we need to set up a calculus which distinguishes these two kinds of variables and such that reduction and grafting commute. For that, the application of a substitution of a reduction variable to a unification one must be delayed until this variable is instantiated. Such a separation and a delay are provided by a calculus of explicit substitutions. Unification in such a calculus can be performed by well-known algorithms such as narrowing, but we present a specialized algorithm for a greater efficiency. At last we show how to relate unification in $\lambda$-calculus and in a calculus with explicit substitutions. Thus we come up with a new higher order unification algorithm which eliminates some burdens of the previous algorithms, in particular the functional handling of scopes. Huet's algorithm, can be seen as a specific strategy for our algorithm, since each of its step is decomposed in elementary ones, giving a more atomic description of the unification process. Also, solved forms in $\lambda$-calculus can easily be computed from solved forms in \ls-calculus

Effects of in-chain and off-chain substitutions on spin fluctuations in the spin-Peierls compound CuGeO_3

The effect of in-chain and off-chain substitutions on 1D spin fluctuations in the spin-Peierls compound CuGeO_3 has been studied using Raman scattering in order to understand the interplay between defect induced states, enhanced spin-spin correlations and the ground state of low dimensional systems. In-chain and off-chain substitutions quench the spin-Peierls state and induce 3D antiferromagnetic order at T\leq 5 K. Consequently a suppression of a 1D gap-induced mode as well as a constant intensity of a spinon continuum are observed at low temperatures. A 3D two-magnon density of states now gradually extends to higher temperatures T\leq 60K compared with pure CuGeO_3. This effect is more pronounced in the case of off-chain substitutions (Si) for which a N\'eel state occurs over a larger substitution range, starting at very low concentrations. Besides, additional low energy excitations are induced. These effects, i.e. the shift of a dimensional crossover to higher temperatures are due to an enhancement of the spin-spin correlations induced by a small amount of substitutions. The results are compared with recent Monte Carlo studies on substituted spin ladders, pointing to a similar instability of coupled, dimerized spin chains and spin ladders upon substitution.Comment: 14 pages, 6 eps figures, to be published in PR

A Theory of Explicit Substitutions with Safe and Full Composition

Many different systems with explicit substitutions have been proposed to implement a large class of higher-order languages. Motivations and challenges that guided the development of such calculi in functional frameworks are surveyed in the first part of this paper. Then, very simple technology in named variable-style notation is used to establish a theory of explicit substitutions for the lambda-calculus which enjoys a whole set of useful properties such as full composition, simulation of one-step beta-reduction, preservation of beta-strong normalisation, strong normalisation of typed terms and confluence on metaterms. Normalisation of related calculi is also discussed.Comment: 29 pages Special Issue: Selected Papers of the Conference "International Colloquium on Automata, Languages and Programming 2008" edited by Giuseppe Castagna and Igor Walukiewic

Normalisation for higher-order calculi with explicit substitutions

Explicit substitutions (ES) were introduced as a bridge between the theory of rewrite systems with binders and substitution, such as the λ-calculus, and their implementation. In a seminal paper Melliès observed that the dynamical properties of a rewrite system and its ES-based implementation may not coincide: he showed that a strongly normalising term (i.e. one which does not admit infinite derivations) in the λ-calculus may lose this status in its ES-based implementation. This paper studies normalisation for the latter systems in the general setting of higher-order rewriting: Based on recent work extending the theory of needed strategies to non-orthogonal rewrite systems we show that needed strategies normalise in the ES-based implementation of any orthogonal pattern higher-order rewrite system.Facultad de Informátic

Higher-Order Horn Clauses

A generalization of Horn clauses to a higher-order logic is described and examined as a basis for logic programming. In qualitative terms, these higher-order Horn clauses are obtained from the first-order ones by replacing first-order terms with simply typed λ-terms and by permitting quantification over all occurrences of function symbols and some occurrences of predicate symbols. Several proof-theoretic results concerning these extended clauses are presented. One result shows that although the substitutions for predicate variables can be quite complex in general, the substitutions necessary in the context of higher-order Horn clauses are tightly constrained. This observation is used to show that these higher-order formulas can specify computations in a fashion similar to first-order Horn clauses. A complete theorem proving procedure is also described for the extension. This procedure is obtained by interweaving higher-order unification with backchaining and goal reductions, and constitutes a higher-order generalization of SLD-resolution. These results have a practical realization in the higher-order logic programming language called λProlog

A Case Study on Logical Relations using Contextual Types

Proofs by logical relations play a key role to establish rich properties such as normalization or contextual equivalence. They are also challenging to mechanize. In this paper, we describe the completeness proof of algorithmic equality for simply typed lambda-terms by Crary where we reason about logically equivalent terms in the proof environment Beluga. There are three key aspects we rely upon: 1) we encode lambda-terms together with their operational semantics and algorithmic equality using higher-order abstract syntax 2) we directly encode the corresponding logical equivalence of well-typed lambda-terms using recursive types and higher-order functions 3) we exploit Beluga's support for contexts and the equational theory of simultaneous substitutions. This leads to a direct and compact mechanization, demonstrating Beluga's strength at formalizing logical relations proofs.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759

LeoPARD --- A Generic Platform for the Implementation of Higher-Order Reasoners

LeoPARD supports the implementation of knowledge representation and reasoning tools for higher-order logic(s). It combines a sophisticated data structure layer (polymorphically typed {\lambda}-calculus with nameless spine notation, explicit substitutions, and perfect term sharing) with an ambitious multi-agent blackboard architecture (supporting prover parallelism at the term, clause, and search level). Further features of LeoPARD include a parser for all TPTP dialects, a command line interpreter, and generic means for the integration of external reasoners.Comment: 6 pages, to appear in the proceedings of CICM'2015 conferenc