## Integrating reasoning about ordinal arithmetic into ACL2 (2004)

Venue: | In Formal Methods in Computer-Aided Design: 5th International Conference – FMCAD-2004, LNCS |

Citations: | 7 - 5 self |

### BibTeX

@INPROCEEDINGS{Manolios04integratingreasoning,

author = {Panagiotis Manolios and Daron Vroon},

title = {Integrating reasoning about ordinal arithmetic into ACL2},

booktitle = {In Formal Methods in Computer-Aided Design: 5th International Conference – FMCAD-2004, LNCS},

year = {2004},

pages = {82--97},

publisher = {Springer–Verlag}

}

### OpenURL

### Abstract

Abstract. Termination poses one of the main challenges for mechanically verifying infinite state systems. In this paper, we develop a powerful and extensible framework based on the ordinals for reasoning about termination in a general purpose programming language. We have incorporated our work into the ACL2 theorem proving system, thereby greatly extending its ability to automatically reason about termination. The resulting technology has been adopted into the newly released ACL2 version 2.8. We discuss the creation of this technology and present two case studies illustrating its effectiveness. 1

### Citations

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Term rewriting and all that
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(Show Context)
Citation Context ...focused on various restricted instances of the termination problem. For example, much of the current research on termination is aimed at providing termination proofs for Term Rewriting Systems (TRSs) =-=[2, 1, 8]-=-. Most of the remaining research is focused on developing algorithms and heuristics for the automatic generation of appropriate wellfounded measure functions [14, 19, 7, 6, 5]. Since termination is an... |

544 |
The Logical and
- Moore
- 1997
(Show Context)
Citation Context ...ng proved, provided the instance has a smaller measure according to the chosen measure function. The ACL2 theorem prover is an example of the so-called Boyer-Moore school of inductive theorem proving =-=[3, 4]-=-. It is an integrated system of ad hoc proof techniques that include simplification, generalization, induction and many other techniques. Simplification is, however, the key technique and includes the... |

280 |
Computer-aided reasoning: an approach
- Kaufmann, Manolios, et al.
- 2000
(Show Context)
Citation Context ...g languages. To that end, we develop a powerful and extensible framework —based on our previous work on ordinal arithmetic [15, 16]— for reasoning about termination in the ACL2 theorem proving system =-=[11, 12]-=-. Our choice of ACL2 for this project was based on two criteria. Since termination is unsolvable, we wanted a system with theorem proving support and in which termination plays a key role. ACL2 meets ... |

226 | Termination of term rewriting using dependency pairs
- Arts, Giesl
- 2000
(Show Context)
Citation Context ...focused on various restricted instances of the termination problem. For example, much of the current research on termination is aimed at providing termination proofs for Term Rewriting Systems (TRSs) =-=[2, 1, 8]-=-. Most of the remaining research is focused on developing algorithms and heuristics for the automatic generation of appropriate wellfounded measure functions [14, 19, 7, 6, 5]. Since termination is an... |

185 | The size-change principle for program termination
- Lee, Jones, et al.
- 2001
(Show Context)
Citation Context ...or Term Rewriting Systems (TRSs) [2, 1, 8]. Most of the remaining research is focused on developing algorithms and heuristics for the automatic generation of appropriate wellfounded measure functions =-=[14, 19, 7, 6, 5]-=-. Since termination is an undecidable problem, this research focuses on solving decidable fragments and is generally presented in terms of toy languages that lack the full functionality of programming... |

120 |
A complete method for the synthesis of linear ranking functions
- Podelski, Rybalchenko
- 2004
(Show Context)
Citation Context ...or Term Rewriting Systems (TRSs) [2, 1, 8]. Most of the remaining research is focused on developing algorithms and heuristics for the automatic generation of appropriate wellfounded measure functions =-=[14, 19, 7, 6, 5]-=-. Since termination is an undecidable problem, this research focuses on solving decidable fragments and is generally presented in terms of toy languages that lack the full functionality of programming... |

68 | A.: Automating the dependency pair method
- Hirokawa, Middeldorp
- 2005
(Show Context)
Citation Context ...focused on various restricted instances of the termination problem. For example, much of the current research on termination is aimed at providing termination proofs for Term Rewriting Systems (TRSs) =-=[2, 1, 8]-=-. Most of the remaining research is focused on developing algorithms and heuristics for the automatic generation of appropriate wellfounded measure functions [14, 19, 7, 6, 5]. Since termination is an... |

61 | Practical Methods for Proving Program Termination
- Colón, Sipma
- 2002
(Show Context)
Citation Context ...or Term Rewriting Systems (TRSs) [2, 1, 8]. Most of the remaining research is focused on developing algorithms and heuristics for the automatic generation of appropriate wellfounded measure functions =-=[14, 19, 7, 6, 5]-=-. Since termination is an undecidable problem, this research focuses on solving decidable fragments and is generally presented in terms of toy languages that lack the full functionality of programming... |

54 |
Synthesis of linear ranking functions
- Colón, Sipma
- 2001
(Show Context)
Citation Context |

15 |
An early program proof by Alan Turing
- Morris, Jones
- 1984
(Show Context)
Citation Context ...ic to a unique ordinal. In this sense, ordinals are the most general setting for termination arguments. This is why Turing says that for proving termination, “it is natural to give an ordinal number” =-=[18]-=-. 3.1 Ordinal Arithmetic Given a well-ordered structure, 〈A, <A〉, we denote the unique ordinal that is isomorphic to this structure as Ord(A, <A). Ordinal addition is defined as follows. Given two ord... |

13 | O.: A heuristic for the automatic generation of ranking functions
- Dams, Gerth, et al.
(Show Context)
Citation Context |

11 | Algorithms for Ordinal Arithmetic
- Manolios, Vroon
- 2003
(Show Context)
Citation Context ...ning about the termination of arbitrary programs written in actual programming languages. To that end, we develop a powerful and extensible framework —based on our previous work on ordinal arithmetic =-=[15, 16]-=-— for reasoning about termination in the ACL2 theorem proving system [11, 12]. Our choice of ACL2 for this project was based on two criteria. Since termination is unsolvable, we wanted a system with t... |

9 | Ordinal arithmetic in acl2
- Manolios, Vroon
- 2003
(Show Context)
Citation Context ...ning about the termination of arbitrary programs written in actual programming languages. To that end, we develop a powerful and extensible framework —based on our previous work on ordinal arithmetic =-=[15, 16]-=-— for reasoning about termination in the ACL2 theorem proving system [11, 12]. Our choice of ACL2 for this project was based on two criteria. Since termination is unsolvable, we wanted a system with t... |

9 | Multiset Relations: a Tool for Proving Termination
- Ruiz–Reina, Alonso, et al.
- 2000
(Show Context)
Citation Context ... study illustrates how other users have used our ordinal arithmetic library to mechanically prove complex termination arguments. 6.1 Legacy Books: Multiset Case Study ACL2’s multiset ordering library =-=[20]-=- makes significant use of the ordinals. A multiset is a set in which items can appear more than once. For example, {1, 3, 2, 2, 4} is a multiset over the natural numbers which contains two 2’s. Given ... |

9 | Efficient simulation of formal processor models
- Wilding, Greve, et al.
(Show Context)
Citation Context ...”) subset of Common Lisp. We assume basic knowledge of Common Lisp syntax. Because it is a programming language, ACL2 is executable, and execution can reach speeds comparable to programs written in C =-=[22]-=-. The logic of ACL2 is a first-order predicate calculus with equality, recursive function definitions, and mathematical induction. The primitive built-in functions are axiomatized. For example, one ax... |

6 |
Proving theorems about pure lisp functions
- Boyer, Moore
- 1975
(Show Context)
Citation Context ...ng proved, provided the instance has a smaller measure according to the chosen measure function. The ACL2 theorem prover is an example of the so-called Boyer-Moore school of inductive theorem proving =-=[3, 4]-=-. It is an integrated system of ad hoc proof techniques that include simplification, generalization, induction and many other techniques. Simplification is, however, the key technique and includes the... |

4 |
Proof of Dixon’s lemma using the ACL2 theorem prover via an explicit ordinal mapping
- Sustik
- 2003
(Show Context)
Citation Context ... termination arguments, not on automatically proving termination for a decidable fragment of the termination problem. As an example of this generality, our work has been used to prove Dickson’s Lemma =-=[21]-=-, which plays a crucial role in proving the termination of Buchberger’s algorithm for finding Gröbner bases of polynomial ideals (see Section 6.2). Our work can also be used to reason about reactive s... |

2 | A Formal Proof of Dickson’s Lemma in ACL2
- Martín–Mateos, Alonso, et al.
- 2003
(Show Context)
Citation Context ...f of the termination of Buchberger’s algorithm for finding a Gröbner basis of a polynomial ideal, and is therefore an important step toward the larger goal of formalizing results from algebra in ACL2 =-=[17]-=-. Sustik made essential use of the ordinals and our library, as his proof depends heavily on the ordinals and could not have been proved in older versions of ACL2 without essentially building up a the... |