Nominal Abstraction

Recursive relational specifications are commonly used to describe the computational structure of formal systems. Recent research in proof theory has identified two features that facilitate direct, logic-based reasoning about such descriptions: the interpretation of atomic judgments through recursive definitions and an encoding of binding constructs via generic judgments. However, logics encompassing these two features do not currently allow for the definition of relations that embody dynamic aspects related to binding, a capability needed in many reasoning tasks. We propose a new relation between terms called nominal abstraction as a means for overcoming this deficiency. We incorporate nominal abstraction into a rich logic also including definitions, generic quantification, induction, and co-induction that we then prove to be consistent. We present examples to show that this logic can provide elegant treatments of binding contexts that appear in many proofs, such as those establishing properties of typing calculi and of arbitrarily cascading substitutions that play a role in reducibility arguments.Comment: To appear in the Journal of Information and Computatio

Constraint Handling Rules with Binders, Patterns and Generic Quantification

Constraint Handling Rules provide descriptions for constraint solvers. However, they fall short when those constraints specify some binding structure, like higher-rank types in a constraint-based type inference algorithm. In this paper, the term syntax of constraints is replaced by $\lambda$-tree syntax, in which binding is explicit; and a new $\nabla$ generic quantifier is introduced, which is used to create new fresh constants.Comment: Paper presented at the 33nd International Conference on Logic Programming (ICLP 2017), Melbourne, Australia, August 28 to September 1, 2017 16 pages, LaTeX, no PDF figure

End-to-End Differentiable Proving

We introduce neural networks for end-to-end differentiable proving of queries to knowledge bases by operating on dense vector representations of symbols. These neural networks are constructed recursively by taking inspiration from the backward chaining algorithm as used in Prolog. Specifically, we replace symbolic unification with a differentiable computation on vector representations of symbols using a radial basis function kernel, thereby combining symbolic reasoning with learning subsymbolic vector representations. By using gradient descent, the resulting neural network can be trained to infer facts from a given incomplete knowledge base. It learns to (i) place representations of similar symbols in close proximity in a vector space, (ii) make use of such similarities to prove queries, (iii) induce logical rules, and (iv) use provided and induced logical rules for multi-hop reasoning. We demonstrate that this architecture outperforms ComplEx, a state-of-the-art neural link prediction model, on three out of four benchmark knowledge bases while at the same time inducing interpretable function-free first-order logic rules.Comment: NIPS 2017 camera-ready, NIPS 201

An Implementation of the Language Lambda Prolog Organized around Higher-Order Pattern Unification

This thesis concerns the implementation of Lambda Prolog, a higher-order logic programming language that supports the lambda-tree syntax approach to representing and manipulating formal syntactic objects. Lambda Prolog achieves its functionality by extending a Prolog-like language by using typed lambda terms as data structures that it then manipulates via higher-order unification and some new program-level abstraction mechanisms. These additional features raise new implementation questions that must be adequately addressed for Lambda Prolog to be an effective programming tool. We consider these questions here, providing eventually a virtual machine and compilation based realization. A key idea is the orientation of the computation model of Lambda Prolog around a restricted version of higher-order unification with nice algorithmic properties and appearing to encompass most interesting applications. Our virtual machine embeds a treatment of this form of unification within the structure of the Warren Abstract Machine that is used in traditional Prolog implementations. Along the way, we treat various auxiliary issues such as the low-level representation of lambda terms, the implementation of reduction on such terms and the optimized processing of types in computation. We also develop an actual implementation of Lambda Prolog called Teyjus Version 2. A characteristic of this system is that it realizes an emulator for the virtual machine in the C language a compiler in the OCaml language. We present a treatment of the software issues that arise from this kind of mixing of languages within one system and we discuss issues relevant to the portability of our virtual machine emulator across arbitrary architectures. Finally, we assess the the efficacy of our various design ideas through experiments carried out using the system

Free Monads, Intrinsic Scoping, and Higher-Order Preunification

Type checking algorithms and theorem provers rely on unification algorithms. In presence of type families or higher-order logic, higher-order (pre)unification (HOU) is required. Many HOU algorithms are expressed in terms of $\lambda$-calculus and require encodings, such as higher-order abstract syntax, which are sometimes not comfortable to work with for language implementors. To facilitate implementations of languages, proof assistants, and theorem provers, we propose a novel approach based on the second-order abstract syntax of Fiore, data types \`a la carte of Swierstra, and intrinsic scoping of Bird and Patterson. With our approach, an object language is generated freely from a given bifunctor. Then, given an evaluation function and making a few reasonable assumptions on it, we derive a higher-order preunification procedure on terms in the object language. More precisely, we apply a variant of $E$-unification for second-order syntax. Finally, we briefly demonstrate an application of this technique to implement type checking (with type inference) for Martin-L\"of Type Theory, a dependent type theory

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

Nominal Logic Programming

Nominal logic is an extension of first-order logic which provides a simple foundation for formalizing and reasoning about abstract syntax modulo consistent renaming of bound names (that is, alpha-equivalence). This article investigates logic programming based on nominal logic. We describe some typical nominal logic programs, and develop the model-theoretic, proof-theoretic, and operational semantics of such programs. Besides being of interest for ensuring the correct behavior of implementations, these results provide a rigorous foundation for techniques for analysis and reasoning about nominal logic programs, as we illustrate via examples.Comment: 46 pages; 19 page appendix; 13 figures. Revised journal submission as of July 23, 200