We introduce a new theorem prover for classical higher-order logic named auto2. The prover is designed to make use of human-specified heuristics when searching for proofs. The core algorithm is a best-first search through the space of propositions derivable from the initial assumptions, where new propositions are added by user-defined functions called proof steps. We implemented the prover in Isabelle HOL, and applied it to several formalization projects in mathematics and computer science, demonstrating the high level of automation it can provide in a variety of possible proof tasks.
A similar blackboard'' approach is used for heuristic theorem proving by @cite_7 , where the focus is on proving real inequalities. The portion of our system concerning inequalities is not as sophisticated as what is implemented there. Instead, our work can be viewed as applying a similar technique to all forms of reasoning.
@cite_7
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We describe a general method for verifying inequalities between real-valued expressions, especially the kinds of straightforward inferences that arise in interactive theorem proving. In contrast to approaches that aim to be complete with respect to a particular language or class of formulas, our method establishes claims that require heterogeneous forms of reasoning, relying on a Nelson-Oppen-style architecture in which special-purpose modules collaborate and share information. The framework is thus modular and extensible. A prototype implementation shows that the method is promising, complementing techniques that are used by contemporary interactive provers. In this paper we prove lower bounds on randomized multiparty communication complexity, mainly in the message-passing model, where messages are sent player-to-player. Some of our results apply to the blackboard model, where each message is written on a blackboard for all players to see. We introduce a new technique for proving such bounds, called symmetrization, which is natural, intuitive, and often easy to use. For example, for the problem where each of @math players gets a bit-vector of length @math , and the goal is to compute the coordinatewise XOR of these vectors, we prove a tight lower bound of @math in the blackboard model. For the same problem with AND instead of XOR, we prove a lower bound of roughly @math in the message-passing model (assuming @math ) and @math in the blackboard model. We also prove lower bounds for bitwise majority, for a graph-connectivity problem, and for other problems; the technique seems applicable to a wide range of other problems as well. A... Motivated by settings in which predictive models may be required to be non-discriminatory with respect to certain attributes (such as race), but even collecting the sensitive attribute may be forbidden or restricted, we initiate the study of fair learning under the constraint of differential privacy. We design two learning algorithms that simultaneously promise differential privacy and equalized odds, a 'fairness' condition that corresponds to equalizing false positive and negative rates across protected groups. Our first algorithm is a private implementation of the equalized odds post-processing approach of [, 2016]. This algorithm is appealingly simple, but must be able to use protected group membership explicitly at test time, which can be viewed as a form of 'disparate treatment'. Our second algorithm is a differentially private version of the oracle-efficient in-processing approach of [, 2018] that can be used to find the optimal fair classifier, given access to a subroutine that can solve the original (not necessarily fair) learning problem. This algorithm is more complex but need not have access to protected group membership at test time. We identify new tradeoffs between fairness, accuracy, and privacy that emerge only when requiring all three properties, and show that these tradeoffs can be milder if group membership may be used at test time. We conclude with a brief experimental evaluation. We applied two state-of-the-art machine learning techniques to the problem of selecting a good heuristic in a first-order theorem prover. Our aim was to demonstrate that sufficient information is available from simple feature measurements of a conjecture and axioms to determine a good choice of heuristic, and that the choice process can be automatically learned. Selecting from a set of 5 heuristics, the learned results are better than any single heuristic. The same results are also comparable to the prover's own heuristic selection method, which has access to 82 heuristics including the 5 used by our method, and which required additional human expertise to guide its design. One version of our system is able to decline proof attempts. This achieves a significant reduction in total time required, while at the same time causing only a moderate reduction in the number of theorems proved. To our knowledge no earlier system has had this capability.