Logical Reduction of Metarules

International audienceMany forms of inductive logic programming (ILP) use metarules, second-order Horn clauses, to define the structure of learnable programs and thus the hypothesis space. Deciding which metarules to use for a given learning task is a major open problem and is a trade-off between efficiency and expressivity: the hypothesis space grows given more metarules, so we wish to use fewer metarules, but if we use too few metarules then we lose expressivity. In this paper, we study whether fragments of metarules can be logically reduced to minimal finite subsets. We consider two traditional forms of logical reduction: subsumption and entailment. We also consider a new reduction technique called derivation reduction, which is based on SLD-resolution. We compute reduced sets of metarules for fragments relevant to ILP and theoretically show whether these reduced sets are reductions for more general infinite fragments. We experimentally compare learning with reduced sets of metarules on three domains: Michalski trains, string transformations, and game rules. In general, derivation reduced sets of metarules outperform subsumption and entailment reduced sets, both in terms of predictive accuracies and learning times

Learning programs by learning from failures

We describe an inductive logic programming (ILP) approach called learning from failures. In this approach, an ILP system (the learner) decomposes the learning problem into three separate stages: generate, test, and constrain. In the generate stage, the learner generates a hypothesis (a logic program) that satisfies a set of hypothesis constraints (constraints on the syntactic form of hypotheses). In the test stage, the learner tests the hypothesis against training examples. A hypothesis fails when it does not entail all the positive examples or entails a negative example. If a hypothesis fails, then, in the constrain stage, the learner learns constraints from the failed hypothesis to prune the hypothesis space, i.e. to constrain subsequent hypothesis generation. For instance, if a hypothesis is too general (entails a negative example), the constraints prune generalisations of the hypothesis. If a hypothesis is too specific (does not entail all the positive examples), the constraints prune specialisations of the hypothesis. This loop repeats until either (i) the learner finds a hypothesis that entails all the positive and none of the negative examples, or (ii) there are no more hypotheses to test. We introduce Popper, an ILP system that implements this approach by combining answer set programming and Prolog. Popper supports infinite problem domains, reasoning about lists and numbers, learning textually minimal programs, and learning recursive programs. Our experimental results on three domains (toy game problems, robot strategies, and list transformations) show that (i) constraints drastically improve learning performance, and (ii) Popper can outperform existing ILP systems, both in terms of predictive accuracies and learning times.Comment: Accepted for the machine learning journa

Making sense of sensory input

This paper attempts to answer a central question in unsupervised learning: what does it mean to "make sense" of a sensory sequence? In our formalization, making sense involves constructing a symbolic causal theory that both explains the sensory sequence and also satisfies a set of unity conditions. The unity conditions insist that the constituents of the causal theory -- objects, properties, and laws -- must be integrated into a coherent whole. On our account, making sense of sensory input is a type of program synthesis, but it is unsupervised program synthesis. Our second contribution is a computer implementation, the Apperception Engine, that was designed to satisfy the above requirements. Our system is able to produce interpretable human-readable causal theories from very small amounts of data, because of the strong inductive bias provided by the unity conditions. A causal theory produced by our system is able to predict future sensor readings, as well as retrodict earlier readings, and impute (fill in the blanks of) missing sensory readings, in any combination. We tested the engine in a diverse variety of domains, including cellular automata, rhythms and simple nursery tunes, multi-modal binding problems, occlusion tasks, and sequence induction intelligence tests. In each domain, we test our engine's ability to predict future sensor values, retrodict earlier sensor values, and impute missing sensory data. The engine performs well in all these domains, significantly out-performing neural net baselines. We note in particular that in the sequence induction intelligence tests, our system achieved human-level performance. This is notable because our system is not a bespoke system designed specifically to solve intelligence tests, but a general-purpose system that was designed to make sense of any sensory sequence

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

Constructive approaches to Program Induction

Search is a key technique in artificial intelligence, machine learning and Program Induction. No matter how efficient a search procedure, there exist spaces that are too large to search effectively and they include the search space of programs. In this dissertation we show that in the context of logic-program induction (Inductive Logic Programming, or ILP) it is not necessary to search for a correct program, because if one exists, there also exists a unique object that is the most general correct program, and that can be constructed directly, without a search, in polynomial time and from a polynomial number of examples. The existence of this unique object, that we term the Top Program because of its maximal generality, does not so much solve the problem of searching a large program search space, as it completely sidesteps it, thus improving the efficiency of the learning task by orders of magnitude commensurate with the complexity of a program space search. The existence of a unique Top Program and the ability to construct it given finite resources relies on the imposition, on the language of hypotheses, from which programs are constructed, of a strong inductive bias with relevance to the learning task. In common practice, in machine learning, Program Induction and ILP, such relevant inductive bias is selected, or created, manually, by the human user of a learning system, with intuition or knowledge of the problem domain, and in the form of various kinds of program templates. In this dissertation we show that by abandoning the reliance on such extra-logical devices as program templates, and instead defining inductive bias exclusively as First- and Higher-Order Logic formulae, it is possible to learn inductive bias itself from examples, automatically, and efficiently, by Higher-Order Top Program construction. In Chapter 4 we describe the Top Program in the context of the Meta-Interpretive Learning approach to ILP (MIL) and describe an algorithm for its construction, the Top Program Construction algorithm (TPC). We prove the efficiency and accuracy of TPC and describe its implementation in a new MIL system called Louise. We support theoretical results with experiments comparing Louise to the state-of-the-art, search-based MIL system, Metagol, and find that Louise improves Metagol’s efficiency and accuracy. In Chapter 5 we re-frame MIL as specialisation of metarules, Second-Order clauses used as inductive bias in MIL, and prove that problem-specific metarules can be derived by specialisation of maximally general metarules, by MIL. We describe a sub-system of Louise, called TOIL, that learns new metarules by MIL and demonstrate empirically that the metarules learned by TOIL match those selected manually, while maintaining the accuracy and efficiency of learning. iOpen Acces

Collapsing towers of interpreters

Given a tower of interpreters, i.e., a sequence of multiple interpreters interpreting one another as input programs, we aim to collapse this tower into a compiler that removes all interpretive overhead and runs in a single pass. In the real world, a use case might be Python code executed by an x86 runtime, on a CPU emulated in a JavaScript VM, running on an ARM CPU. Collapsing such a tower can not only exponentially improve runtime performance, but also enable the use of base-language tools for interpreted programs, e.g., for analysis and verification. In this paper, we lay the foundations in an idealized but realistic setting. We present a multi-level lambda calculus that features staging constructs and stage polymorphism: based on runtime parameters, an evaluator either executes source code (thereby acting as an interpreter) or generates code (thereby acting as a compiler). We identify stage polymorphism, a programming model from the domain of high-performance program generators, as the key mechanism to make such interpreters compose in a collapsible way. We present Pink, a meta-circular Lisp-like evaluator on top of this calculus, and demonstrate that we can collapse arbitrarily many levels of self-interpretation, including levels with semantic modifications. We discuss several examples: compiling regular expressions through an interpreter to base code, building program transformers from modi ed interpreters, and others. We develop these ideas further to include reflection and reification, culminating in Purple, a reflective language inspired by Brown, Blond, and Black, which realizes a conceptually infinite tower, where every aspect of the semantics can change dynamically. Addressing an open challenge, we show how user programs can be compiled and recompiled under user-modified semantics.Parts of this research were supported by ERC grant 321217, NSF awards 1553471 and 1564207, and DOE award DE-SC0018050

Learning programs with magic values

A magic value in a program is a constant symbol that is essential for the execution of the program but has no clear explanation for its choice. Learning programs with magic values is difficult for existing program synthesis approaches. To overcome this limitation, we introduce an inductive logic programming approach to efficiently learn programs with magic values. Our experiments on diverse domains, including program synthesis, drug design, and game playing, show that our approach can (i) outperform existing approaches in terms of predictive accuracies and learning times, (ii) learn magic values from infinite domains, such as the value of pi, and (iii) scale to domains with millions of constant symbols

Inductive logic programming at 30: a new introduction

Inductive logic programming (ILP) is a form of machine learning. The goal of ILP is to induce a hypothesis (a set of logical rules) that generalises training examples. As ILP turns 30, we provide a new introduction to the field. We introduce the necessary logical notation and the main learning settings; describe the building blocks of an ILP system; compare several systems on several dimensions; describe four systems (Aleph, TILDE, ASPAL, and Metagol); highlight key application areas; and, finally, summarise current limitations and directions for future research.Comment: Paper under revie

Logic Programs as Declarative and Procedural Bias in Inductive Logic Programming

Machine Learning is necessary for the development of Artificial Intelligence, as pointed out by Turing in his 1950 article ``Computing Machinery and Intelligence''. It is in the same article that Turing suggested the use of computational logic and background knowledge for learning. This thesis follows a logic-based machine learning approach called Inductive Logic Programming (ILP), which is advantageous over other machine learning approaches in terms of relational learning and utilising background knowledge. ILP uses logic programs as a uniform representation for hypothesis, background knowledge and examples, but its declarative bias is usually encoded using metalogical statements. This thesis advocates the use of logic programs to represent declarative and procedural bias, which results in a framework of single-language representation. We show in this thesis that using a logic program called the top theory as declarative bias leads to a sound and complete multi-clause learning system MC-TopLog. It overcomes the entailment-incompleteness of Progol, thus outperforms Progol in terms of predictive accuracies on learning grammars and strategies for playing Nim game. MC-TopLog has been applied to two real-world applications funded by Syngenta, which is an agriculture company. A higher-order extension on top theories results in meta-interpreters, which allow the introduction of new predicate symbols. Thus the resulting ILP system Metagol can do predicate invention, which is an intrinsically higher-order logic operation. Metagol also leverages the procedural semantic of Prolog to encode procedural bias, so that it can outperform both its ASP version and ILP systems without an equivalent procedural bias in terms of efficiency and accuracy. This is demonstrated by the experiments on learning Regular, Context-free and Natural grammars. Metagol is also applied to non-grammar learning tasks involving recursion and predicate invention, such as learning a definition of staircases and robot strategy learning. Both MC-TopLog and Metagol are based on a $\top$-directed framework, which is different from other multi-clause learning systems based on Inverse Entailment, such as CF-Induction, XHAIL and IMPARO. Compared to another $\top$-directed multi-clause learning system TAL, Metagol allows the explicit form of higher-order assumption to be encoded in the form of meta-rules.Open Acces