## Complex quantifier elimination in HOL (2001)

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Venue: | TPHOLs 2001: Supplemental Proceedings |

Citations: | 10 - 2 self |

### BibTeX

@INPROCEEDINGS{Harrison01complexquantifier,

author = {John Harrison},

title = {Complex quantifier elimination in HOL},

booktitle = {TPHOLs 2001: Supplemental Proceedings},

year = {2001},

pages = {159--174}

}

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### Abstract

Abstract. Building on a simple construction of the complex numbers and a proof of the Fundamental Theorem of Algebra, we implement, as a HOL derived inference rule, a decision method for the first order algebraic theory of C based on quantifier elimination. Although capable of solving some mildly interesting problems, we also implement a more efficient semidecision procedure for the universal fragment based on Gröbner bases. This is applied to examples including the automatic proof of some simple geometry theorems. The general and universal procedures present an interesting contrast in that the latter can exploit the finding-checking separation to achieve greater efficiency, though this feature is only partly exploited in the present implementation. 1

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Linear programming and extensions
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(Show Context)
Citation Context ...he qi constants. As is pointed out in [22], similar certificates can be produced by algorithms that are more efficient on large problems, such as the standard simplex algorithm for linear programming =-=[6]-=-. We consider the ability to separate proof and checking in this way to be fundamentally important in implementing certain (semi)decision procedures both soundly and reasonably efficiently in a fully-... |

219 |
Ein Algorithmus zum auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal
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Citation Context ...eed to be able to prove negative results of the form ‘there is no common solution to this conjunction of equations’. It turns out that the standard Gröbner basis algorithm [27] invented by Buchberger =-=[3]-=- allows us to do this easily. The Gröbner basis algorithm has a strong resemblance to Knuth-Bendix completion [17], which it predated by several years. The basic idea is to order the monomials ax k1 1... |

103 | On the Rapid Computation of Various Polylogarithmic Constants
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(Show Context)
Citation Context .../\ (y pow 2 = Cx (&3)) ==> ((x * y) pow 2 = Cx (&6)) The following takes 10.5 seconds. It is a genuine ‘practical’ example in that, when verifying in HOL the polylogarithmic series for π presented in =-=[1]-=- we wanted to prove that x 2 + √ 2x + 1 has no real roots. The following theorem presents a simple route to that result since as it is universal its restriction to R also holds, and clearly x 4 + 1 ha... |

92 | Theorem Proving with the Real Numbers
- Harrison
- 1998
(Show Context)
Citation Context ...nly comes at the cost of increasing the complexity of the polynomials in othersvariables. However, compared with the more complicated and difficult quantifier elimination procedure for R described in =-=[12]-=-, the efficiency of the present procedure is not calamitously bad. This is to be expected, as the decidability of R appears to be an inherently more difficult problem [7]. To start with a trivial exam... |

85 |
Mechanical Geometry Theorem Proving
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- 1988
(Show Context)
Citation Context ... proof reconstruction could be done much more efficiently if the resulting coefficient polynomials were reduced. Wu’s method has been impressively applied to a huge range of geometry problems by Chou =-=[4]-=-, and we have taken some examples and coordinate translations from there. The use of Gröbner bases in geometry theorem proving has been investigated by several researchers, notably Kapur [15]. In comp... |

59 | Metatheory and reflection in theorem proving: a survey and critique
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- 1995
(Show Context)
Citation Context ...enting certain (semi)decision procedures both soundly and reasonably efficiently in a fully-expansive context, often much more practical and flexible than the use of so-called ‘reflection principles’ =-=[10]-=-. In order to verify the final polynomial identity in HOL, we implemented simple routines for maintaining polynomials in canonical sum-of-monomials form ordered according to the monomial ordering used... |

24 | Efficiency in a fully-expansive theorem prover
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- 1993
(Show Context)
Citation Context ...he finding-checking separation for efficiency. In fact, it’s quite striking that the first reasonably efficient fully-expansive decision procedure for linear arithmetic over N, implemented by Boulton =-=[2]-=- used exactly the same kind of certificate to separate search from inference-based proof reconstruction, the only differences being that the pi were linear polynomials and the qi constants. As is poin... |

15 |
A refutational approach to geometry theorem proving
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- 1988
(Show Context)
Citation Context ...heoretical device when proving the full Hilbert Nullstellensatz from a weak special case (see below). Seidenberg [25] also used it when presenting his quantifier elimination procedure for R and Kapur =-=[15]-=- appears to have been the first to use it in a practical computer implementation. Although there are general methods that can eliminate a block of existential quantifiers even when the polynomials inc... |

11 | Verifying the accuracy of polynomial approximations in HOL
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- 1997
(Show Context)
Citation Context ...tation of univariate polynomial functions as lists of coefficients (without gaps, starting at the constant term). In fact, we simply re-used the existing HOL Light theory of real polynomial functions =-=[11]-=-. Essentially all the proofs were simply taken over with trivial changes, except for some with no complex analog (e.g. those concerned with ordering or, like Sturm’s theorem, specific to R), or where ... |

8 |
Tarski’s elimination theory for real closed fields
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(Show Context)
Citation Context ...procedure for R described in [12], the efficiency of the present procedure is not calamitously bad. This is to be expected, as the decidability of R appears to be an inherently more difficult problem =-=[7]-=-. To start with a trivial example 1 , which is solved in 4.6 seconds. 2 |- !x y. (x pow 2 = Cx (&2)) /\ (y pow 2 = Cx (&3)) ==> ((x * y) pow 2 = Cx (&6)) The following takes 10.5 seconds. It is a genu... |

7 | Automated geometric reasoning: Dixon resultants, Gröbner bases, and characteristic sets
- Kapur
(Show Context)
Citation Context ...able results. For an excellent comparative survey of the application to geometry theorem proving of the characteristic set method, Gröbner bases and yet another technique called Dixon resultants, see =-=[16]-=-. There seems no entirely satisfactory theoretical explanation for just why so many geometrical statements turn out to be true if the natural algebraic formulation is extended to the complex numbers, ... |

6 | Gröbner bases in type theory
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- 1998
(Show Context)
Citation Context ...f Buchberger’s algorithm with a fully-expansive proof reconstruction, there have actually been two formal verifications of a version of Buchberger’s algorithm by Théry [26] and by Coquand and Persson =-=[5]-=-. Although both verifying algorithms and reconstructing proofs have their strengths and weaknesses, we think the latter approach is quite promising as an approach to actually using the method as a dec... |

3 |
Uber die Darstellung der ganzen Zahlen als Summen von nten Potenzen ganzer Zahlen
- Hurwitz
- 1908
(Show Context)
Citation Context .... The former is expected from the context, while the latter should be obvious given that all other terms in the sum treat the variables symmetrically. We confess that we consulted the original source =-=[14]-=- before making these trivial observations. The Gröbner-based method is usually far more efficient on purely universal formulas than the general procedure outlined above, and by more intelligent simpli... |

2 |
Grundlagen der Geometrie,” Teubner, 1899. English translation by Townsend, Open Court, 1902; reprinted 1950. The second German edition was translated by Unger; this was revised and enlarged by Bernays, Open Court
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- 1971
(Show Context)
Citation Context ... would be interesting to try to relate the coordinate definitions to more traditional definitions in the style of Euclid’s Elements, and to the modern axiomatic treatment of geometry given by Hilbert =-=[13]-=-. 4 Seehttp://www.cs.kun.nl/gi/projects/fta/xindex.html. 5 Seehttp://www.cl.cam.ac.uk/ mn200/research.sAcknowledgements Thanks to Freek Wiedijk for interesting me in the formalization of the complex n... |

1 |
volume 123 of Graduate Texts in Mathematics
- Ebbinghaus
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(Show Context)
Citation Context ...presentation of polynomial functions does not impose canonicality). |- !p. ˜(?a l. ˜(a = Cx(&0)) /\ ALL (\b. b = Cx(&0)) l /\ (p = CONS a l)) ==> ?z. poly p z = Cx(&0) We formalize a proof taken from =-=[8]-=-, an inductive refinement [19, 9] of the classic ‘minimum modulus’ proof à la Argand. The crucial analytical component is the assertion that a continuous complex function attains its minimum modulus i... |

1 |
On the fundamental theorem of algebra
- Estermann
- 1956
(Show Context)
Citation Context ...nctions does not impose canonicality). |- !p. ˜(?a l. ˜(a = Cx(&0)) /\ ALL (\b. b = Cx(&0)) l /\ (p = CONS a l)) ==> ?z. poly p z = Cx(&0) We formalize a proof taken from [8], an inductive refinement =-=[19, 9]-=- of the classic ‘minimum modulus’ proof à la Argand. The crucial analytical component is the assertion that a continuous complex function attains its minimum modulus in a closed disc in the complex pl... |