On subgoal induction

published as cite EWD:EWD576pubComputer Scienc

The Use of Proof Planning for Cooperative Theorem Proving

AbstractWe describebarnacle: a co-operative interface to theclaminductive theorem proving system. For the foreseeable future, there will be theorems which cannot be proved completely automatically, so the ability to allow human intervention is desirable; for this intervention to be productive the problem of orienting the user in the proof attempt must be overcome. There are many semi-automatic theorem provers: we call our style of theorem provingco-operative, in that the skills of both human and automaton are used each to their best advantage, and used together may find a proof where other methods fail. The co-operative nature of thebarnacleinterface is made possible by the proof planning technique underpinningclam. Our claim is that proof planning makes new kinds of user interaction possible.Proof planning is a technique for guiding the search for a proof in automatic theorem proving. Common patterns of reasoning in proofs are identified and represented computationally as proof plans, which can then be used to guide the search for proofs of new conjectures. We have harnessed the explanatory power of proof planning to enable the user to understand where the automatic prover got to and why it is stuck. A user can analyse the failed proof in terms ofclam's specification language, and hence override the prover to force or prevent the application of a tactic, or discover a proof patch. This patch might be to apply further rules or tactics to bridge the gap between the effects of previous tactics and the preconditions needed by a currently inapplicable tactic

Abduction in Well-Founded Semantics and Generalized Stable Models

Abductive logic programming offers a formalism to declaratively express and solve problems in areas such as diagnosis, planning, belief revision and hypothetical reasoning. Tabled logic programming offers a computational mechanism that provides a level of declarativity superior to that of Prolog, and which has supported successful applications in fields such as parsing, program analysis, and model checking. In this paper we show how to use tabled logic programming to evaluate queries to abductive frameworks with integrity constraints when these frameworks contain both default and explicit negation. The result is the ability to compute abduction over well-founded semantics with explicit negation and answer sets. Our approach consists of a transformation and an evaluation method. The transformation adjoins to each objective literal $O$ in a program, an objective literal $not(O)$ along with rules that ensure that $not(O)$ will be true if and only if $O$ is false. We call the resulting program a {\em dual} program. The evaluation method, \wfsmeth, then operates on the dual program. \wfsmeth{} is sound and complete for evaluating queries to abductive frameworks whose entailment method is based on either the well-founded semantics with explicit negation, or on answer sets. Further, \wfsmeth{} is asymptotically as efficient as any known method for either class of problems. In addition, when abduction is not desired, \wfsmeth{} operating on a dual program provides a novel tabling method for evaluating queries to ground extended programs whose complexity and termination properties are similar to those of the best tabling methods for the well-founded semantics. A publicly available meta-interpreter has been developed for \wfsmeth{} using the XSB system.Comment: 48 pages; To appear in Theory and Practice in Logic Programmin

Capturing Hiproofs in HOL Light

Hierarchical proof trees (hiproofs for short) add structure to ordinary proof trees, by allowing portions of trees to be hierarchically nested. The additional structure can be used to abstract away from details, or to label particular portions to explain their purpose. In this paper we present two complementary methods for capturing hiproofs in HOL Light, along with a tool to produce web-based visualisations. The first method uses tactic recording, by modifying tactics to record their arguments and construct a hierarchical tree; this allows a tactic proof script to be modified. The second method uses proof recording, which extends the HOL Light kernel to record hierachical proof trees alongside theorems. This method is less invasive, but requires care to manage the size of the recorded objects. We have implemented both methods, resulting in two systems: Tactician and HipCam