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The contributions of our work are the following:. An instrumented, pre-packaged version of HOL Light that can be used as a reinforcement learning environment for theorem proving using our well-defined, stable Python API. Our solution comes with optimized startup capabilities for proof search, while allowing replay and strict verification of the produced proofs. Proof export and import capabilities that allow for managing large theories programmatically from the Python interface. A full-fledged, competitive automated neural theorem proving system that can automatize theorem proving in higher-order logic at tactic level directly.

A large scale reinforcement learning system that was used for training our prover.

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Comparison of neural model architectures for theorem proving purposes. Well-defined benchmarks on our HOL Light based environment to enable research and measuring progress of AI driven theorem proving in large theories. This paper is organized as follows. Although TacticToe is a great success that came with significant improvements over previous automated theorem proving systems, they do not propose an easy to use benchmark or environment for machine learning researchers.

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TacticToe does not employ deep learning nor reinforcement learning. They also provide an easy-to-use Python API for an interactive theorem prover, and they present test and training sets. While enabling automatic code extraction, it comes with a much smaller coverage of fundamental mathematics. Even including the formalization of the Feit-Thompson theorem, their benchmark comprises only theorems and lemmas, while ours features theorems and lemmas.

Besides presenting a much larger data set, we also demonstrate the feasibility of achieving state-of-the-art prover performance based on our data and environment by presenting a deep learning based theorem prover. We also report the results as theorem proving performance instead of proxy metrics. The Mizar mathematical library is probably the most comprehensive formalization effort, but its declarative style makes it hard to employ proof search, and its source code is not freely available. In contrast to our work, they use neither deep learning nor reinforcement learning.

The first use of deep neural networks for large scale theorem proving was proposed in. This work was moderately successful, finding mostly proofs for very simple theorems, especially in propositional logic. On the other hand, Metamath is not considered to be a serious contender for large scale mathematical formalization work. Here, we propose a neural prover written from scratch, relying solely on a small set of preexisting tactics and neural networks for all high level decisions. It features a few static datasets and it remains unclear how performance of machine learning models on this dataset relates to real world prover performance.

Here we describe the architecture of the evaluation and training environment. The goal of the environment is to enable artificial agents to interact with the HOL Light interactive theorem prover ITP in a replicable manner.

In order to describe our changes to HOL Light, it is helpful to establish some common terminology. The ITP provides a small number of tactics to manipulate the goal. Tactics may have tactic arguments , which can be a previously proven theorem or a list of previously proven theorems. There are also tactics that take terms as arguments, but we do not support them currently.

Applying a tactic to a goal can lead to a failure, when not all conditions are met, or is successful and produces a list of subgoals. The goal is only proven successfully, if all its subgoals are proven. In particular, if the goal is proven if the tactic application produces an empty list of subgoals. We refer to tactic applications sometimes also as proof steps.

We can think of proofs as trees, where goals are nodes and tactic applications are hyper- edges to other goals. In a successful proof, all leaves are goals with a tactic application that produced an empty list of subgoals. In order to create a stable, well-defined environment, we fix a particular version of HOL Light with a pre-selected subset of tactics and a fixed library of basic theorems, which are proved in one well-defined order.

Since it is non-trivial to find and build the exact correct set of libraries for this environment, we provide a prepackaged docker image. It can be used as a reliable black box for proof search and as reinforcement learning environment, communicated with using a simple API. We have also open sourced all the changes to the HOL Light system so that new modifications and forks are possible by third parties. The prepackaged version we provide has the following additional instrumentation, which we describe below in detail:. Logging of human-written proofs shipped with HOL Light.

Fast startup for distributed proof search. A proof checker to remove the need to trust search algorithms. We want to utilize the existing human proofs for both training and evaluation. To that effect, we have instrumented the prove method in HOL Light with extra logging code. If HOL Light is executed in proof-dump mode, each invocation of the prove function dumps the proven theorems and their proofs into files. The API provides two functions: 1 to apply tactics to goals and 2 to register theorems for future use in tactic applications. Tactic applications are completely stateless and contain the goal, the tactic to be applied, and the tactic arguments.

The poof assistant i. HOL Light in our implementation returns the outcome of the tactic application, including the list of subgoals for successful applications. The stateless tactic application interface frees us from the strict order on subgoals that HOL Light enforces in the human interface, and allows us to easily implement more advanced proof search strategies. The tactic arguments can consist of a list of theorems.

Implemented naively, this list could make the tactic application request very large and could slow down the prover. In the argument list of tactics we therefore allow theorems to be referenced by a fingerprint number. The registration of theorems is hence stateful, in contrast to tactic applications. Starting HOL Light and loading all the potentially needed libraries can take a long time - we measured it at up to 20 minutes.

This would be inhibitively long for proof search, especially in a distributed setting with thousands of workers and the startup time has to be paid for every worker. This brings the startup time of our HOL Light to mere seconds. Any bug in the implementation of a theorem prover could make its reasoning unsound, rendering the whole formalization effort futile.

For that reason, HOL Light is designed around a small trusted core of about lines of OCaml code that builds proofs from few very basic rules. The correctness of any proof found through the API thus relies on the correctness of our API implementation and the proof search itself. We thus implemented a proof checker that avoids the need for trusting the proof search and even the API. The proof checker compiles proofs into OCaml code that can be loaded in HOL Light, where they have to pass through the trusted core.

While proofs of core theorems are useful for training, we omit them in validation, since some tactics assume those theorems. Flyspeck contains most of the lemmas and theorems of the Kepler conjecture. We propose two tasks that can be measured on these benchmarks:. Predict the tactic and tactic arguments that were employed in the human proof. Prove each of the theorems in the corpora while utilizing only those theorems as tactic arguments that also humans had available. For that purpose, we provide all theorems in the three corpora in one unified list, in the order they were proven by humans.

The g o a l is a provable statement, i. The t a c t i c is the ID of one of a preselected small set of tactics currently consisting of 41 tactics that led to a successful proof. The a r g l i s t is the list of theorems that were passed to a tactic application as arguments. Additionally, there is a special argument signifying that the argument list was empty.

The n e g a r g l i s t is an optional list of non-arguments that is not actually necessary for any proof. They are collected during proof search in our reinforcement learning pipeline, and the list is empty for all the examples generated from the human proof logs. Before training and evaluation, we have split the top level theorems into three subsets: training, validation and test set in a ratio. More expressive logics, such as Higher-order logics , allow the convenient expression of a wider range of problems than first order logic, but theorem proving for these logics is less well developed.

It has the sources of many of the systems mentioned above. From Wikipedia, the free encyclopedia. This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. April Learn how and when to remove this template message. Symbolic computation Computer-aided proof Automated reasoning Formal verification Logic programming Proof checking Model checking Proof complexity Computer algebra system Program analysis computer science General Problem Solver Metamath language for formalized mathematics.

Verlag Louis Neuert. Breslau: Wilhelm Kobner. Archived from the original PDF on Retrieved Principia Mathematica 1st ed. Cambridge University Press. Principia Mathematica 2nd ed. Warszawa: 92— Ki Springer : 2— Retrieved 2 September McCune Journal of Automated Reasoning. The New York Times. Retrieved 15 July The automation of proof by mathematical induction. Journal of Symbolic Computation. Springer, Berlin, Heidelberg, Categories : Formal methods Automated theorem proving. Hidden categories: Articles needing additional references from April All articles needing additional references.

Namespaces Article Talk. Views Read Edit View history. There are three key components in a PBE system. We leverage a divide-and-conquer-based deductive search paradigm to inductively reduce the problem of synthesizing a program expression of a certain kind that satisfies a given specification into sub-problems that refer to sub-expressions or sub-specifications. We leverage features of the program structure as well of the outputs generated by the program on test inputs.

We leverage active-learning techniques based on clustering inputs and synthesizing multiple programs. Each of these three PBE components leverage both formal methods and heuristics. We make the case for synthesizing these heuristics from training data using appropriate machine learning methods.

This not only leads to better heuristics, but also enables easier development, maintenance, and even personalization of a PBE system. He is the inventor of the Flash Fill feature in Microsoft Excel used by hundreds of millions of people. Machine learning systems are often inscrutable black boxes: they are effective predictors but lack human-understandable explanations. I will describe a new paradigm for influence-directed explanations that address this problem.

Influence-directed explanations shed light on the inner workings of black-box machine learning systems by identifying components that causally influence system behavior and by providing human-understandable interpretation to the concepts represented by these components. I will describe instances of this paradigm that are model-agnostic and instances that are specific to deep neural networks.

Influence-directed explanations serve as a foundation for reasoning about fairness and privacy of machine learning models. Our initial exploration suggests that formal methods for analysis of probabilistic systems have an important role to play in efficient generation of influence-directed explanations. Joint work with colleagues at Carnegie Mellon University.

He is Director of the Accountable Systems Lab. His research focuses on enabling real-world complex systems to be accountable for their behavior, especially as they pertain to privacy, fairness, and security. His work has helped create foundations and tools for accountable big data systems and cryptographic protocols.

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Specific examples include a privacy compliance tool chain deployed at Microsoft. Datta obtained Ph. I will briefly survey recent and expected developments in AI and their implications. Beyond these, one must expect that AI capabilities will eventually exceed those of humans across a range of real-world-decision making scenarios.


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Should this be a cause for concern, as Elon Musk, Stephen Hawking, and others have suggested? And, if so, what can we do about it? While some in the mainstream AI community dismiss the issue, I will argue instead that a fundamental reorientation of the field is required.

Instead of building systems that optimize arbitrary objectives, we need to learn how to build systems that will, in fact, be beneficial for us. I will show that it is useful to imbue systems with explicit uncertainty concerning the true objectives of the humans they are designed to help, as well as the ability to learn more about those objectives from observation of human behaviour.

His research covers a wide range of topics in artificial intelligence, with an emphasis on the long-term future of artificial intelligence and its relation to humanity. He has developed a new global seismic monitoring system for the nuclear-test-ban treaty and is currently working to ban lethal autonomous weapons. Deep learning has led to rapid progress being made in the field of machine learning and artificial intelligence, leading to dramatically improved solutions of many challenging problems such as image understanding, speech recognition, and automatic game playing.

Despite these remarkable successes, researchers have observed some intriguing and troubling aspects of the behaviour of these models. A case in point is the presence of adversarial examples which make learning based systems fail in unexpected ways.

Such behaviour and the difficultly of interpreting the behaviour of neural networks is a serious hindrance in the deployment of these models for safety-critical applications. In this talk, I will review the challenges in developing models that are robust and explainable and discuss the opportunities for collaboration between the formal methods and machine learning communities. Pushmeet is a principal scientist and team leader at DeepMind. His research revolves around Intelligent Systems and Computational Sciences.

In terms of application domains, he is interested in goal-directed conversation agents, machine learning systems for healthcare, and 3D reconstruction and tracking for augmented and virtual reality. Artificial Intelligence will improve productivity across a broad range of applications and across a broad range of industries. The benefit to humanity will be substantial but there are important lessons to learn and steps to take.

This talk will address some of the issues to be considered and explain why software and community are key to success with AI. Twitter: AlisonBLowndes. Deep neural networks have achieved impressive experimental results in image classification, but can surprisingly be unstable with respect to adversarial perturbations, that is, minimal changes to the input image that cause the network to misclassify it.