Nominal Unification of Higher Order Expressions with Recursive Let

A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in non-deterministic polynomial time. We also explore specializations like nominal letrec-matching for plain expressions and for DAGs and determine the complexity of corresponding unification problems.Comment: Pre-proceedings paper presented at the 26th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2016), Edinburgh, Scotland UK, 6-8 September 2016 (arXiv:1608.02534

The Undecidability of Typability in the Lambda-Pi-Calculus

The set of pure terms which are typable in the $\lambda$$\Pi$-calculus in a given context is not recursive. So there is no general type inference algorithm for the programming language Elf and, in some cases, some type information has to be mentioned by the programmer

Towards Deciding Second-order Unification Problems Using Regular Tree Automata

International audienceThe second-order unification problem is undecidable [5]. While unification procedures, like Huet's pre-unification, terminate with success on unifiable problems, they might not terminate on non-unifiable ones. There are several decidability results for unification problems with infinitely-many pre-unifiers, such as for monadic second-order problems [3]. These results are based on the regular structure of the solutions of these problems and by computing minimal unifiers. Beyond the importance of the knowledge that searching for unifiers of decidable problems always terminates, one can also use this information in order to optimize unification algorithms, such as in the case for pattern unification [10]. Nevertheless, being able to prove that the unification problem of a certain class of unification constraints is decidable is far from easy. Some results were obtained for certain syntactic restrictions on the problems (see Levy [8] for some results and references) or on the unifiers (see Schmidt-Schauß [11], Schmidt-Schauß and Schulz [12, 13] and Je˙ z [7] for some results). Infinitary unification problems, like the ones we are considering, might suggest that known tools for dealing with the infinite might be useful. One such tool is the regular tree automaton. The drawback of using regular automata for unification is, of course, their inability to deal with variables. In this paper we try to overcome this obstacle and describe an ongoing work about using regular tree automata [1] in order to decide more general second-order unification problems. The second-order unification problems we will consider are of the form λz n .x 0 t. = λz n .C(x 0 s) where C is a non-empty context [2] and x 0 does not occur in t or s. We will call such problems cyclic problems. An important result in second-order unification was obtained by Ganzinger et al. [4] and stated that second-order unification is undecidable already when there is only one second-order variable occurring twice. The unification problem they used for proving the undecidability result was an instance of the following cyclic problem. Note that we chose to use in the definition only unary second-order variables but that this restriction should not be essential. x 0 (w 1 , g(y 1 , a)) = g(y 2 , x 0 (w 2 , a)) (1) Our decidability result is obtained by posing one further restriction over cyclic problems which is based on the existence and location of variables other than the cyclic one. A sufficient condition for the decidability of second-order unification problems was given by Levy [8]. This condition states that if we can never encounter, when applying Huet's pre-unification procedure [6] to a problem, a cyclic equation, then the procedure terminates. It follows from this result that deciding second-order unification problems depends on the ability to decide cyclic problems. The rules of Huet's procedure (PUA) are given in Fig. 1. Imitation partial bindings and projection partial bindings are defined in [14] and are denoted, respectively, by PB(f, α) and PB(i, α) where α is a type, Σ a signature f ∈ Σ and i > 0

On the Limits of Second-Order Unification

Second-Order Unification is a problem that naturally arises when applying automated deduction techniques with variables denoting predicates. The problem is undecidable, but a considerable effort has been made in order to find decidable fragments, and understand the deep reasons of its complexity. Two variants of the problem, Bounded Second-Order Unification and Linear Second-Order Unification ¿where the use of bound variables in the instantiations is restricted¿, have been extensively studied in the last two decades. In this paper we summarize some decidability/undecidability/complexity results, trying to focus on those that could be more interesting for a wider audience, and involving less technical details.Peer Reviewe

Nominal Unification from a Higher-Order Perspective

Nominal Logic is a version of first-order logic with equality, name-binding, renaming via name-swapping and freshness of names. Contrarily to higher-order logic, bindable names, called atoms, and instantiable variables are considered as distinct entities. Moreover, atoms are capturable by instantiations, breaking a fundamental principle of lambda-calculus. Despite these differences, nominal unification can be seen from a higher-order perspective. From this view, we show that nominal unification can be reduced to a particular fragment of higher-order unification problems: Higher-Order Pattern Unification. This reduction proves that nominal unification can be decided in quadratic deterministic time, using the linear algorithm for Higher-Order Pattern Unification. We also prove that the translation preserves most generality of unifiers

Functions-as-Constructors Higher-Order Unification

Unification is a central operation in the construction of a range of computational logic systems based on first-order and higher-order logics. First-order unification has a number of properties that dominates the way it is incorporated within such systems. In particular, first-order unification is decidable, unary, and can be performed on untyped term structures. None of these three properties hold for full higher-order unification: unification is undecidable, unifiers can be incomparable, and term-level typing can dominate the search for unifiers. The so-called pattern subset of higher-order unification was designed to be a small extension to first-order unification that respected the basic laws governing lambda-binding (the equalities of alpha, beta, and eta-conversion) but which also satisfied those three properties. While the pattern fragment of higher-order unification has been popular in various implemented systems and in various theoretical considerations, it is too weak for a number of applications. In this paper, we define an extension of pattern unification that is motivated by some existing applications and which satisfies these three properties. The main idea behind this extension is that the arguments to a higher-order, free variable can be more than just distinct bound variables: they can also be terms constructed from (sufficient numbers of) such variables using term constructors and where no argument is a subterm of any other argument. We show that this extension to pattern unification satisfies the three properties mentioned above

Context unification is in PSPACE

Contexts are terms with one `hole', i.e. a place in which we can substitute an argument. In context unification we are given an equation over terms with variables representing contexts and ask about the satisfiability of this equation. Context unification is a natural subvariant of second-order unification, which is undecidable, and a generalization of word equations, which are decidable, at the same time. It is the unique problem between those two whose decidability is uncertain (for already almost two decades). In this paper we show that the context unification is in PSPACE. The result holds under a (usual) assumption that the first-order signature is finite. This result is obtained by an extension of the recompression technique, recently developed by the author and used in particular to obtain a new PSPACE algorithm for satisfiability of word equations, to context unification. The recompression is based on performing simple compression rules (replacing pairs of neighbouring function symbols), which are (conceptually) applied on the solution of the context equation and modifying the equation in a way so that such compression steps can be in fact performed directly on the equation, without the knowledge of the actual solution.Comment: 27 pages, submitted, small notation changes and small improvements over the previous tex

E-Unification for Second-Order Abstract Syntax

Higher-order unification (HOU) concerns unification of (extensions of) ?-calculus and can be seen as an instance of equational unification (E-unification) modulo ??-equivalence of ?-terms. We study equational unification of terms in languages with arbitrary variable binding constructions modulo arbitrary second-order equational theories. Abstract syntax with general variable binding and parametrised metavariables allows us to work with arbitrary binders without committing to ?-calculus or use inconvenient and error-prone term encodings, leading to a more flexible framework. In this paper, we introduce E-unification for second-order abstract syntax and describe a unification procedure for such problems, merging ideas from both full HOU and general E-unification. We prove that the procedure is sound and complete