Predicate Anti-unification in (Constraint) Logic Programming

The concept of anti-unification refers to the process of determining the most specific generalization (msg) of two or more input program objects. In the domain of logic programming, anti-unification has primarily been investigated for computing msgs of tree-like program structures such as terms, atoms, and goals (the latter typically seen as ordered sequences). In this work, we study the anti-unification of whole predicate definitions. We provide a definition of a predicate generalization that allows to characterize the problem of finding the most specific generalization of two predicates as a (computationally hard) search problem. The complexity stems from the fact that a correspondence needs to be constructed between (1) some of the arguments of each of the predicates, (2) some of the clauses in each of the predicate's definitions, and (3) some of the body atoms in each pair of associated clauses. We propose a working algorithm that simultaneously computes these correspondences in a greedy manner. While our algorithm does not necessarily compute the most specific generalization, we conjecture that it allows to compute, in general, a sufficiently good generalization in an acceptable time

Inductive learning in Shared Neural Multi-Spaces

The learning of rules from examples is of continuing interest to machine learning since it allows generalization from fewer training ex- amples. Inductive Logic Programming (ILP) generates hypothetical rules (clauses) from a knowledge base augmented with (positive and negative) examples. A successful hypothesis entails all positive examples and does not entail any negative example. The Shared Neural Multi-Space (Shared NeMuS) structure encodes first order expressions in a graph suitable for ILP-style learning. This paper explores the NeMuS structure and its re- lationship with the Herbrand Base of a knowledge-base to generate hy- potheses inductively. It is demonstrated that inductive learning driven by the knowledge-base structure can be implementated successfully in the Amao cognitive agent framework, including the learning of recursive hypotheses

Matching Logic

This paper presents matching logic, a first-order logic (FOL) variant for specifying and reasoning about structure by means of patterns and pattern matching. Its sentences, the patterns, are constructed using variables, symbols, connectives and quantifiers, but no difference is made between function and predicate symbols. In models, a pattern evaluates into a power-set domain (the set of values that match it), in contrast to FOL where functions and predicates map into a regular domain. Matching logic uniformly generalizes several logical frameworks important for program analysis, such as: propositional logic, algebraic specification, FOL with equality, modal logic, and separation logic. Patterns can specify separation requirements at any level in any program configuration, not only in the heaps or stores, without any special logical constructs for that: the very nature of pattern matching is that if two structures are matched as part of a pattern, then they can only be spatially separated. Like FOL, matching logic can also be translated into pure predicate logic with equality, at the same time admitting its own sound and complete proof system. A practical aspect of matching logic is that FOL reasoning with equality remains sound, so off-the-shelf provers and SMT solvers can be used for matching logic reasoning. Matching logic is particularly well-suited for reasoning about programs in programming languages that have an operational semantics, but it is not limited to this

Proceedings of the Workshop on the lambda-Prolog Programming Language

The expressiveness of logic programs can be greatly increased over first-order Horn clauses through a stronger emphasis on logical connectives and by admitting various forms of higher-order quantification. The logic of hereditary Harrop formulas and the notion of uniform proof have been developed to provide a foundation for more expressive logic programming languages. The λ-Prolog language is actively being developed on top of these foundational considerations. The rich logical foundations of λ-Prolog provides it with declarative approaches to modular programming, hypothetical reasoning, higher-order programming, polymorphic typing, and meta-programming. These aspects of λ-Prolog have made it valuable as a higher-level language for the specification and implementation of programs in numerous areas, including natural language, automated reasoning, program transformation, and databases

A new algebraic structure in the standard model of particle physics

We introduce a new formulation of the real-spectral-triple formalism in non-commutative geometry (NCG): we explain its mathematical advantages and its success in capturing the structure of the standard model of particle physics. The idea, in brief, is to represent $A$ (the algebra of differential forms on some possibly-noncommutative space) on $H$ (the Hilbert space of spinors on that space), and to reinterpret this representation as a simple super-algebra $B=A\oplus H$ with even part $A$ and odd part $H$. $B$ is the fundamental object in our approach: we show that (nearly) all of the basic axioms and assumptions of the traditional real-spectral-triple formalism of NCG are elegantly recovered from the simple requirement that $B$ should be a differential graded $\ast$-algebra (or "$\ast$-DGA"). Moreover, this requirement also yields other, new, geometrical constraints. When we apply our formalism to the NCG traditionally used to describe the standard model of particle physics, we find that these new constraints are physically meaningful and phenomenologically correct. In particular, these new constraints provide a novel interpretation of electroweak symmetry breaking that is geometric rather than dynamical. This formalism is more restrictive than effective field theory, and so explains more about the observed structure of the standard model, and offers more guidance about physics beyond the standard model.Comment: 30 pages, no figures, matches JHEP versio

The New Normal: We Cannot Eliminate Cuts in Coinductive Calculi, But We Can Explore Them

In sequent calculi, cut elimination is a property that guarantees that any provable formula can be proven analytically. For example, Gentzen's classical and intuitionistic calculi LK and LJ enjoy cut elimination. The property is less studied in coinductive extensions of sequent calculi. In this paper, we use coinductive Horn clause theories to show that cut is not eliminable in a coinductive extension of LJ, a system we call CLJ. We derive two further practical results from this study. We show that CoLP by Gupta et al. gives rise to cut-free proofs in CLJ with fixpoint terms, and we formulate and implement a novel method of coinductive theory exploration that provides several heuristics for discovery of cut formulae in CLJ.Comment: Paper presented at the 36th International Conference on Logic Programming (ICLP 2019), University Of Calabria, Rende (CS), Italy, September 2020, 16 page