## Automating elementary number-theoretic proofs using Gröbner bases

Citations: | 4 - 0 self |

### BibTeX

@MISC{Harrison_automatingelementary,

author = {John Harrison},

title = {Automating elementary number-theoretic proofs using Gröbner bases},

year = {}

}

### OpenURL

### Abstract

Abstract. We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain number-theoretic predicates such as ‘divisible by’, ‘congruent ’ and ‘coprime’; one notable example in this class is the Chinese Remainder Theorem (for a specific number of moduli). The method is based on a reduction to ideal membership assertions that are then solved using Gröbner bases. As well as illustrating the usefulness of the procedure on examples, and considering some extensions, we prove a limited form of completeness for properties that hold in all rings. 1

### Citations

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(Show Context)
Citation Context ...prime’, abbreviates ∃x y. sx + ty = 1. Over the integers, coprime(m, n) holds precisely if m and n have no common factors besides ±1. This equivalence is proved in many elementary number theory texts =-=[2, 8]-=-. We attempt to explain any algebraic terminology as it is used, but a reader may find it helpful to refer to an algebra textbook such as [21] for more on rings, polynomials and ideals. It is worth no... |

350 |
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(Show Context)
Citation Context ...alence is proved in many elementary number theory texts [2, 8]. We attempt to explain any algebraic terminology as it is used, but a reader may find it helpful to refer to an algebra textbook such as =-=[21]-=- for more on rings, polynomials and ideals. It is worth noting that we tend to blur the distinction between three distinct notions of ‘polynomial’: (i) a first-order formula in the language of rings, ... |

219 |
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Citation Context ...ere is a polynomial expression for the witness e in terms of the other variables. To prove the ideal membership goal, the most natural and straightforward technique is to apply Buchberger’s algorithm =-=[5]-=- to find a Gröbner basis for the ideal, and then show that y − x reduces to 0 w.r.t. this basis. A suitably instrumented version of the algorithm can actually produce the explicit cofactor polynomials... |

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(Show Context)
Citation Context ...sting subtasks of larger formal proofs. Some algorithms implement decision procedures for theories or logical fragments known to be decidable, such as Cooper’s algorithm [7] for Presburger arithmetic =-=[17]-=-. Others are more heuristic in nature, e.g. automated induction proofs employing conjecture generalization [4], though many of these can be understood in a general framework of proof planning [6]. Her... |

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Citation Context ...aling with relatively uninteresting subtasks of larger formal proofs. Some algorithms implement decision procedures for theories or logical fragments known to be decidable, such as Cooper’s algorithm =-=[7]-=- for Presburger arithmetic [17]. Others are more heuristic in nature, e.g. automated induction proofs employing conjecture generalization [4], though many of these can be understood in a general frame... |

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Citation Context ...tic [17]. Others are more heuristic in nature, e.g. automated induction proofs employing conjecture generalization [4], though many of these can be understood in a general framework of proof planning =-=[6]-=-. Here we present a new algorithm for a useful class of elementary numbertheoretic properties. We will introduce and motivate the procedure by focusing on the integers Z, though we will see later that... |

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Citation Context ...nd in particular we obtain the witnesses w = u, z = v. 6 Implementation We have implemented a simple prototype of the routine, containing fewer than 100 lines of code, in the HOL Light theorem prover =-=[9]-=-; in version 2.20, it is included in the standard release. The implemented version does not yet use the extension to multiple equations using auxiliary variables, and some of the initial normalization... |

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Citation Context ... implementation. Allowing a richer quantifier structure soon leads to undecidability, even in the linear case; for example multiplication can be defined in terms of divisibility, successor and 1 only =-=[18]-=-, so even that theory is undecidable. In contrast, we allow more or less unrestricted use of multiplication, which in principle leads to undecidability. But the approach of seeking properties true in ... |

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(Show Context)
Citation Context ...problem above takes in the degenerate case (n = 0 and k = 0) of proving that a Diophantine equation has no solutions over the integers: ∀x. �m i=1 ei(x) = 0 ⇒ ⊥. Since this is known to be undecidable =-=[16]-=- while ideal membership over the integers is decidable [1] it follows that our test based on ideal membership cannot be both sound and complete. And indeed, it is not hard to find examples of incomple... |

34 |
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(Show Context)
Citation Context ...prime’, abbreviates ∃x y. sx + ty = 1. Over the integers, coprime(m, n) holds precisely if m and n have no common factors besides ±1. This equivalence is proved in many elementary number theory texts =-=[2, 8]-=-. We attempt to explain any algebraic terminology as it is used, but a reader may find it helpful to refer to an algebra textbook such as [21] for more on rings, polynomials and ideals. It is worth no... |

22 |
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Citation Context ... as the top node, such that for every node B in the tree, there is a clause in the axiom set that can be instantiated so its conclusion is B and its antecedent atoms are the nodes below B in the tree =-=[10]-=-. This special canonical proof format for deductions from Horn clauses allows us to deduce some interesting consequences. We start with a theorem due to Simmons [19, 12, 21]: Theorem 1. Let p1(x), . .... |

22 |
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Citation Context ...e only over the class of rings in general. There are established results for decidability of universal linear formulas in the language of Presburger arithmetic including divisibility by non-constants =-=[3, 15]-=-, though we are not aware of any actual implementation. Allowing a richer quantifier structure soon leads to undecidability, even in the linear case; for example multiplication can be defined in terms... |

19 | Ideal Membership in Polynomial Rings over the Integers
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(Show Context)
Citation Context ...) of proving that a Diophantine equation has no solutions over the integers: ∀x. �m i=1 ei(x) = 0 ⇒ ⊥. Since this is known to be undecidable [16] while ideal membership over the integers is decidable =-=[1]-=- it follows that our test based on ideal membership cannot be both sound and complete. And indeed, it is not hard to find examples of incompleteness, where the existential assertion holds over Z but t... |

17 |
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Citation Context ...lution proofs [14]. In order to state these special properties of Horn clause theories, it is more convenient to consider first-order logic without special treatment of equality. By a standard result =-=[13]-=-, a formula is valid in first-order logic with equality iff it is a general first-order consequence of the set of equivalence and congruence properties of equality for the language at issue. In partic... |

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6 |
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(Show Context)
Citation Context ...e the normal Buchberger algorithm for polynomial ideals over Q, implemented in HOL via int_ideal_cofactors. Properly speaking, we should use a version of Buchberger’s algorithm tailored to the ring Z =-=[11]-=-. For example, consider proving just x+y = 0∧x−y = 0 ⇒ x = 0. This does not hold in all rings (e.g. set x = y = 1 in the integers modulo 2). The Gröbner basis algorithm over the rationals, however, wo... |

5 |
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(Show Context)
Citation Context ... are the nodes below B in the tree [10]. This special canonical proof format for deductions from Horn clauses allows us to deduce some interesting consequences. We start with a theorem due to Simmons =-=[19, 12, 21]-=-: Theorem 1. Let p1(x), . . . , pr(x) and p(x) be polynomials with integer coefficients over the variables x = x1, . . . , xl. Then the following holds in all commutative rings with 1: ∀x1, . . . , xl... |

5 |
The solution of a decision problem for several classes of rings
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(Show Context)
Citation Context ... are the nodes below B in the tree [10]. This special canonical proof format for deductions from Horn clauses allows us to deduce some interesting consequences. We start with a theorem due to Simmons =-=[19, 12, 21]-=-: Theorem 1. Let p1(x), . . . , pr(x) and p(x) be polynomials with integer coefficients over the variables x = x1, . . . , xl. Then the following holds in all commutative rings with 1: ∀x1, . . . , xl... |

3 |
Semantical completeness theorems in logic and algebra
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(Show Context)
Citation Context ..., and so the special results we will note for Horn clause theories are not directly applicable, though analogous results can be derived for general theories by considering canonical resolution proofs =-=[14]-=-. In order to state these special properties of Horn clause theories, it is more convenient to consider first-order logic without special treatment of equality. By a standard result [13], a formula is... |

1 |
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- 1974
(Show Context)
Citation Context ...e only over the class of rings in general. There are established results for decidability of universal linear formulas in the language of Presburger arithmetic including divisibility by non-constants =-=[3, 15]-=-, though we are not aware of any actual implementation. Allowing a richer quantifier structure soon leads to undecidability, even in the linear case; for example multiplication can be defined in terms... |

1 | Annotated English version by [20 - Warsaw - 1930 |