Proof assistants, such as Isabelle HOL, offer tools to facilitate inductive theorem proving. Isabelle experts know how to use these tools effectively; however they did not have a systematic way to encode their expertise. To address this problem, we present our domain-specific language, LiFtEr. LiFtEr allows experienced Isabelle users to encode their induction heuristics in a style independent of any problem domain. LiFtEr's interpreter mechanically checks if a given application of induction tool matches the heuristics, thus transferring experienced users' expertise to new Isabelle users.
The most well-known approach for ITP, called waterfall @cite_18 , was invented for a first-order logic, which cannot handle induction without jeopardizing the soundness by introducing axioms. Jiang al followed this approach and ran multiple waterfalls @cite_0 to automate ITP in HOL light @cite_5 . However, when deciding induction variables, they naively picked the first free variable with recursive type and left the selection of appropriate induction variables as future work.
@cite_0 @cite_5 @cite_18
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1. Summary In Part I, four ostensibly different theoretical models of induction are presented, in which the problem dealt with is the extrapolation of a very long sequence of symbols—presumably containing all of the information to be used in the induction. Almost all, if not all problems in induction can be put in this form. Some strong heuristic arguments have been obtained for the equivalence of the last three models. One of these models is equivalent to a Bayes formulation, in which a priori probabilities are assigned to sequences of symbols on the basis of the lengths of inputs to a universal Turing machine that are required to produce the sequence of interest as output. Though it seems likely, it is not certain whether the first of the four models is equivalent to the other three. Few rigorous results are presented. Informal investigations are made of the properties of these models. There are discussions of their consistency and meaningfulness, of their degree of independence of the exact nature of the Turing machine used, and of the accuracy of their predictions in comparison to those of other induction methods. In Part II these models are applied to the solution of three problems—prediction of the Bernoulli sequence, extrapolation of a certain kind of Markov chain, and the use of phrase structure grammars for induction. Though some approximations are used, the first of these problems is treated most rigorously. The result is Laplace's rule of succession. The solution to the second problem uses less certain approximations, but the properties of the solution that are discussed, are fairly independent of these approximations. The third application, using phrase structure grammars, is least exact of the three. First a formal solution is presented. Though it appears to have certain deficiencies, it is hoped that presentation of this admittedly inadequate model will suggest acceptable improvements in it. This formal solution is then applied in an approximate way to the determination of the “optimum” phrase structure grammar for a given set of strings. The results that are obtained are plausible, but subject to the uncertainties of the approximation used. Horn clause logic programming can be extended to include abduction with integrity constraints. In the resulting extension of logic programming, negation by failure can be simulated by making negative conditions abducible and by imposing appropriate denials and disjunctions as integrity constraints. This gives an alternative semantics for negation by failure, which generalises the stable model semantics of negation by failure. The abductive extension of logic programming extends negation by failure in three ways: (1) computation can be perfonned in alternative minimal models, (2) positive as well as negative conditions can be made abducible, and (3) other integrity constraints can also be accommodated. * This paper was written while the first author was at Imperial College. 235 Introduction The tenn "abduction" was introduced by the philosopher Charles Peirce [1931] to refer to a particular kind of hypothetical reasoning. In the simplest case, it has the fonn: From A and A fB infer B as a possible "explanation" of A. Abduction has been given prominence in Charniak and McDennot's [1985] "Introduction to Artificial Intelligence", where it has been applied to expert systems and story comprehension. Independently, several authors have developed deductive techniques to drive the generation of abductive hypotheses. Cox and Pietrzykowski [1986] construct hypotheses from the "dead ends" of linear resolution proofs. Finger and Genesereth [1985] generate "deductive solutions to design problems" using the "residue" left behind in resolution proofs. Poole, Goebel and Aleliunas [1987] also use linear resolution to generate hypotheses. All impose the restriction that hypotheses should be consistent with the "knowledge base". Abduction is a fonn of non-monotonic reasoning, because hypotheses which are consistent with one state of a knowledge base may become inconSistent when new knowledge is added. Poole [1988] argues that abduction is preferable to noh-monotonic logics for default reasoning. In this view, defaults are hypotheses fonnulated within classical logic rather than conclusions derived withln some fonn of non-monotonic logic. The similarity between abduction and default reasoning was also pointed out in [Kowalski, 1979]. In this paper we show how abduction can be integrated with logic programming, and we concentrate on the use of abduction to generalise negation by failure. Conditional Answers Compared with Abduction In the simplest case, a logic program consists of a set of Horn Clauses, which are used backward to_reduce goals to sub goals. The initial goal is solved when there are no subgollls left; Unifying first-order logic and probability is a long-standing goal of AI, and in recent years many representations combining aspects of the two have been proposed. However, inference in them is generally still at the level of propositional logic, creating all ground atoms and formulas and applying standard probabilistic inference methods to the resulting network. Ideally, inference should be lifted as in first-order logic, handling whole sets of indistinguishable objects together, in time independent of their cardinality. Poole (2003) and (2005, 2006) developed a lifted version of the variable elimination algorithm, but it is extremely complex, generally does not scale to realistic domains, and has only been applied to very small artificial problems. In this paper we propose the first lifted version of a scalable probabilistic inference algorithm, belief propagation (loopy or not). Our approach is based on first constructing a lifted network, where each node represents a set of ground atoms that all pass the same messages during belief propagation. We then run belief propagation on this network. We prove the correctness and optimality of our algorithm. Experiments show that it can greatly reduce the cost of inference. We present three approaches for unsupervised grammar induction that are sensitive to data complexity and apply them to Klein and Manning's Dependency Model with Valence. The first, Baby Steps, bootstraps itself via iterated learning of increasingly longer sentences and requires no initialization. This method substantially exceeds Klein and Manning's published scores and achieves 39.4 accuracy on Section 23 (all sentences) of the Wall Street Journal corpus. The second, Less is More, uses a low-complexity subset of the available data: sentences up to length 15. Focusing on fewer but simpler examples trades off quantity against ambiguity; it attains 44.1 accuracy, using the standard linguistically-informed prior and batch training, beating state-of-the-art. Leapfrog, our third heuristic, combines Less is More with Baby Steps by mixing their models of shorter sentences, then rapidly ramping up exposure to the full training set, driving up accuracy to 45.0 . These trends generalize to the Brown corpus; awareness of data complexity may improve other parsing models and unsupervised algorithms.