## Extensions to a Generalization Critic for Inductive Proof (1996)

Venue: | 13th Conference on Automated Deduction |

Citations: | 12 - 7 self |

### BibTeX

@INPROCEEDINGS{Ireland96extensionsto,

author = {Andrew Ireland and Alan Bundy},

title = {Extensions to a Generalization Critic for Inductive Proof},

booktitle = {13th Conference on Automated Deduction},

year = {1996},

pages = {47--61},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

In earlier papers a critic for automatically generalizing conjectures in the context of failed inductive proofs was presented. The critic exploits the partial success of the search control heuristic known as rippling. Through empirical testing a natural generalization and extension of the basic critic emerged. Here we describe our extended generalization critic together with some promising experimental results. 1 Introduction A major obstacle to the automation of proof by mathematical induction is the need for generalization. A generalization is underpinned by the cut-rule of inference. In a goal-directed framework, therefore, a generalization introduces an infinite branching point into the search space. It is known [13] that the cut-elimination theorem does not hold for inductive theories. Consequently heuristics for controlling generalization play an important role in the automation of inductive proof. There are a number of different kinds of generalization. In this paper we present...

### Citations

548 |
A Computational Logic
- Boyer, Moore
- 1979
(Show Context)
Citation Context ...onstruction. 8 basic critic are presented in table I. All the examples require accumulator generalization and therefore cannot be proved automatically by other inductive theorem provers such as nqthm =-=[2]-=-. The relative performance of the basic and extended critics on the example conjectures is recorded in table II. The lemmata used in motivating the generalizations are presented in table III while the... |

272 | The use of explicit plans to guide inductive proofs
- Bundy
- 1988
(Show Context)
Citation Context ...ation can be identified and secondly, how the construction of the required generalized conjecture can be automated. 2 Background 2.1 Proof Methods and Critics We build upon the notion of a proof plan =-=[3]-=- and tactic-based theorem proving [7]. While a tactic encodes the low-level structure of a family of proofs a proof plan expressions the highlevel structure. In terms of automated deduction, a proof p... |

177 |
An overview of Prolog
- Miller, Nadathur
- 1988
(Show Context)
Citation Context ...during the search for generalizations. Our extended critic has been implemented and integrated within the CL AM proof planner [5]. The implementation makes use of the higher-order features of -Prolog =-=[15]-=-. Below we document the testing of our implementation. 7.1 Experimental Results The results presented in [11] for the basic critic were replicated by the extended critic. The extended critic, however,... |

163 | Rippling: A heuristic for guiding inductive proofs
- Bundy, Stevens, et al.
- 1993
(Show Context)
Citation Context ... term tree positions. Note that a sideways ripple is only performed if progress is made towards a sink. In general, a wave-rule may combine all three forms. For a complete description of rippling see =-=[1, 4]-=-. 2.3 A Critic for Discovering Generalizations In terms of the ripple method, the need for an accumulator generalization can be explained in terms of the failure of a sideways ripple due to the absenc... |

133 |
Edinburgh LCF - A mechanised logic of computation
- Gordon, Milner, et al.
- 1979
(Show Context)
Citation Context ... how the construction of the required generalized conjecture can be automated. 2 Background 2.1 Proof Methods and Critics We build upon the notion of a proof plan [3] and tactic-based theorem proving =-=[7]-=-. While a tactic encodes the low-level structure of a family of proofs a proof plan expressions the highlevel structure. In terms of automated deduction, a proof plan guides the search for a proof. Th... |

130 |
The oyster-clam system
- Bundy, Harmelen, et al.
- 1990
(Show Context)
Citation Context ...ion that certain accumulator occurrences are more constrained than others during the search for generalizations. Our extended critic has been implemented and integrated within the CL AM proof planner =-=[5]-=-. The implementation makes use of the higher-order features of -Prolog [15]. Below we document the testing of our implementation. 7.1 Experimental Results The results presented in [11] for the basic c... |

59 |
Programming: the derivation of algorithms
- Kaldewaij
- 1990
(Show Context)
Citation Context ...plied by the user. A common strategy for discovering invariants is to start with a desired post-condition from which the invariant is derived by a process of weakening. The notion of a tail invariant =-=[12]-=- represents one such way of deriving an invariant. The search for a tail invariant is appropriate when the desired post-condition takes the form: r = f(X; Y) 9 That is, as the number of definitions an... |

56 | The Use of Planning Critics in Mechanizing Inductive Proofs
- Ireland
- 1992
(Show Context)
Citation Context ...serve the failure of a proof attempt. Such interventions typically result in the user generalizing their conjecture or supplying additional lemmata to the prover. Through the notion of a proof critic =-=[10]-=- we have attempted to automate this process. Critics provide the proof planning framework with an exception handling mechanism which enables the partial success of a proof plan to be exploited in sear... |

34 |
Krivine, Elements of mathematical logic
- Kreisel, L
- 1967
(Show Context)
Citation Context ...tion. A generalization is underpinned by the cut-rule of inference. In a goal-directed framework, therefore, a generalization introduces an infinite branching point into the search space. It is known =-=[13]-=- that the cut-elimination theorem does not hold for inductive theories. Consequently heuristics for controlling generalization play an important role in the automation of inductive proof. There are a ... |

28 | Difference Unification
- Basin, Walsh
- 1992
(Show Context)
Citation Context ... term tree positions. Note that a sideways ripple is only performed if progress is made towards a sink. In general, a wave-rule may combine all three forms. For a complete description of rippling see =-=[1, 4]-=-. 2.3 A Critic for Discovering Generalizations In terms of the ripple method, the need for an accumulator generalization can be explained in terms of the failure of a sideways ripple due to the absenc... |

28 | Mechanical proofs about computer programs
- Good
- 1985
(Show Context)
Citation Context ...rifying an imperative program. This is reflected in the fact that some of the major contributions in this area rely to a large extent upon user interaction, e.g. in the gypsy verification environment =-=[6]-=- all loop invariants are supplied by the user. A common strategy for discovering invariants is to start with a desired post-condition from which the invariant is derived by a process of weakening. The... |

20 |
Using Middle-Out reasoning to Guide Inductive Theorem Proving
- Hesketh
- 1991
(Show Context)
Citation Context ...tended critic have automatically generated (3), the required generalization of (2). A proof of (3) can be constructed by CL AM completely automatically. 8 Related Work Jane Hesketh in her thesis work =-=[9]-=- also tackled the problem of accumulator generalization in the context of proof planning. Her approach, however, did not deal with multiple accumulators. By introducing the primary and secondary class... |

17 | Prototyping a parallel vision system in standard ML
- Michaelson, Scaife
- 1995
(Show Context)
Citation Context ...= (x y (reduce x z)) Figure 1: Example list processing functions This conjecture was provided 2 by an independent research group working on the development parallel systems from functional prototypes =-=[14]-=-. Their development process involves formal proof. Currently proofs are constructed by hand and represent a time consuming hurdle to the research project. Having failed to prove conjecture (1) by hand... |

14 | Using middle-out reasoning to control the synthesis of tail-recursive programs
- Hesketh, Bundy, et al.
- 1992
(Show Context)
Citation Context ...ch broader than ours in that she unified a number of different kinds of generalization. Moreover, she was also able to synthesize tail-recursive functions given equivalent naive recursive definitions =-=[8]-=-. 9 Future Work Our results for the extended critic have been promising. More testing is planned. We believe that our technique is not restricted to reasoning about functional programs. This will be r... |

5 |
An automatic generalization method for the inductive proof of replicated and parallel architectures
- Pierre
- 1995
(Show Context)
Citation Context ...ic invariants. 9.2 Hardware Verification We also believe that our technique is applicable in the context of hardware verification. For instance, we believe that it subsumes the procedure described in =-=[16]-=- for generalizing hardware specifications. 9.3 User Interaction The critic mechanism was motivated by a desire to build an automatic theorem prover which was more robust than the conventional provers.... |

3 |
Productive use of failure in inductive proof. Research Paper 716
- Ireland, Bundy
- 1995
(Show Context)
Citation Context ...l success of a proof plan to be exploited in search for a proof. The mechanism works by allowing proof patches to be associated with different patterns of precondition failure. We previously reported =-=[11]-=- various ways of patching of inductive proofs based upon the partial success of the ripple method described below. 2.2 A Method for Guiding Inductive Proof In the context of mathematical induction the... |