## Integrating decision procedures into heuristic theorem provers: A case study of linear arithmetic (1988)

Venue: | Machine Intelligence |

Citations: | 111 - 9 self |

### BibTeX

@INPROCEEDINGS{Boyer88integratingdecision,

author = {Robert S. Boyer and J Strother Moore},

title = {Integrating decision procedures into heuristic theorem provers: A case study of linear arithmetic},

booktitle = {Machine Intelligence},

year = {1988},

pages = {83--124},

publisher = {Oxford University Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

We discuss the problem of incorporating into a heuristic theorem prover a decision procedure for a fragment of the logic. An obvious goal when incorporating such a procedure is to reduce the search space explored by the heuristic component of the system, as would be achieved by eliminating from the system’s data base some explicitly stated axioms. For example, if a decision procedure for linear inequalities is added, one would hope to eliminate the explicit consideration of the transitivity axioms. However, the decision procedure must then be used in all the ways the eliminated axioms might have been. The difficulty of achieving this degree of integration is more dependent upon the complexity of the heuristic component than upon that of the decision procedure. The view of the decision procedure as a "black box " is frequently destroyed by the need pass large amounts of search strategic information back and forth between the two components. Finally, the efficiency of the decision procedure may be virtually irrelevant; the efficiency of the final system may depend most heavily on how easy it is to communicate between the two components. This paper is a case study of how we integrated a linear arithmetic procedure into a heuristic theorem prover. By linear arithmetic here we mean the decidable subset of number theory dealing with universally quantified formulas composed of the logical connectives, the identity relation, the Peano "less than " relation, the Peano addition and subtraction functions, Peano constants,

### Citations

615 |
J.S.: A Fast String Searching Algorithm
- Boyer, Moore
- 1977
(Show Context)
Citation Context ...eration between our heuristic theorem prover and its linear arithmetic procedure. The first example comes from our system's proof of the correctness of the Boyer-Moore fast string searching algorithm =-=[3]-=-. The algorithm searches for the first occurrence of a given pattern in a given text. Both the pattern and text are strings of characters over a finite alphabet. The algorithm uses an array that assoc... |

548 |
A computational logic
- Boyer, Moore
- 1979
(Show Context)
Citation Context ...nces, graphs), the user definition of new mathematical functions (e.g., prime, permutation, path), and proof by induction on well-founded relations. The logic is described precisely in Chapter III of =-=[4]-=-. Our theorem prover as it stood before we incorporated any linear arithmetic is described in Chapters V-XV of [4]. The theorem prover consists of an ad hoc collection of heuristic proof techniques. T... |

420 | Simplification by cooperating decision procedures
- Nelson, Oppen
- 1979
(Show Context)
Citation Context ...y requirements for such procedures that are not obvious when the procedures are considered in isolation. For example, much recent work on linear arithmetic procedures, e.g., that of Nelson and Oppen, =-=[13]-=-, and Shostak, [18], focuses on universally quantified formulas either with no function symbols (other than sum and difference) or with only uninterpreted function symbols. But interpreted function sy... |

104 | editors. Metafunctions: Proving Them Correct and Using Them Efficiently as - Moore, Boyer - 1981 |

97 |
Theorem proving in arithmetic without multiplication
- Cooper
- 1972
(Show Context)
Citation Context ...iplication, respectively, over the integers or rationals (according to context). Linear integer arithmetic, and thus linear Peano arithmetic, is decidable. However, integer decision procedures (e.g., =-=[8]-=-) are quite complicated compared to the many well-known decision procedures for linear inequalities over the rationals [11], [10], [1], [16], [17]. Therefore, following the tradition in program verifi... |

77 |
Deciding linear inequalities by computing loop residues
- Shostak
- 1981
(Show Context)
Citation Context ...c, is decidable. However, integer decision procedures (e.g., [8]) are quite complicated compared to the many well-known decision procedures for linear inequalities over the rationals [11], [10], [1], =-=[16]-=-, [17]. Therefore, following the tradition in program verification, we adopted a rational-based procedure, exploiting the observation that if a conjunction of inequalities is unsatisfiable over the ra... |

71 |
An efficient decision procedure for arithmetic with function symbols
- Shostak
- 1979
(Show Context)
Citation Context ...decidable. However, integer decision procedures (e.g., [8]) are quite complicated compared to the many well-known decision procedures for linear inequalities over the rationals [11], [10], [1], [16], =-=[17]-=-. Therefore, following the tradition in program verification, we adopted a rational-based procedure, exploiting the observation that if a conjunction of inequalities is unsatisfiable over the rational... |

66 |
A program verifier
- King
- 1971
(Show Context)
Citation Context ...r Peano arithmetic, is decidable. However, integer decision procedures (e.g., [8]) are quite complicated compared to the many well-known decision procedures for linear inequalities over the rationals =-=[11]-=-, [10], [1], [16], [17]. Therefore, following the tradition in program verification, we adopted a rational-based procedure, exploiting the observation that if a conjunction of inequalities is unsatisf... |

27 |
A new method for proving certain Presburger formulas
- Bledsoe
- 1975
(Show Context)
Citation Context ...hmetic, is decidable. However, integer decision procedures (e.g., [8]) are quite complicated compared to the many well-known decision procedures for linear inequalities over the rationals [11], [10], =-=[1]-=-, [16], [17]. Therefore, following the tradition in program verification, we adopted a rational-based procedure, exploiting the observation that if a conjunction of inequalities is unsatisfiable over ... |

26 |
Variable elimination and chaining in a resolution-based prover for inequalities
- Bledsoe, Hines
- 1980
(Show Context)
Citation Context ...n the linear procedure is very similar to the admission of universally quantified hypotheses in the formulas being proved. Thus, our work is similar in spirit to the recent work of Bledsoe and Hines, =-=[2]-=- in which arbitrary quantification is permitted. However, we make no completeness claims about our heuristics. 4 2. Background Our theorem prover deals with a quantifier free first-order logic. In add... |

22 |
A verification condition generator for FORTRAN
- Boyer, Moore
- 1981
(Show Context)
Citation Context ...mplemented include the correctness of a recursive descent parser [9], the correctness of an arithmetic simplifier now in routine use in the system [5], and the correctness of several FORTRAN programs =-=[6]. 6 3. Linea-=-r Arithmetic The theorem prover described above can easily prove by mathematical induction such simple theorems as: *1 X@LTEY �� Y@LTEZ #��# X@LTEZ *2 X-1@LTEX *3 0@LTEY #��# X@LTEX+Y, But... |

22 | Proof Checking the RSA Public Key Encryption Algorithm
- Boyer, Moore
- 1984
(Show Context)
Citation Context ...us, our original objective was achieved. Among the theorems proved by the latest version of the theorem prover are the invertibility of the Rivest, Shamir, and Adleman public key encryption algorithm =-=[7]-=-, Wilson's theorem [14], Gauss' law of 8 quadratic reciprocity and the Church-Rosser theorem [15]. All of these proofs involved a substantial amount of linear arithmetic reasoning. We now turn to our ... |

13 |
A mechanical proof of the termination of Takeuchi’s function
- Moore
- 1979
(Show Context)
Citation Context ...sor expressions in the clause. Unfortunately, irrelevant hypotheses are common in mechanically generated formulas. For example, in our system's first proof of the termination of the Takeuchi function =-=[12]-=- the proof of one lemma involved 412 cases, many of which were irrelevant. One solution to this problems is that adopted for the tail biting problem. If the linear procedure keeps track of which liter... |

4 |
Solving Problems by Formula Manipulation
- Hodes
- 1971
(Show Context)
Citation Context ...o arithmetic, is decidable. However, integer decision procedures (e.g., [8]) are quite complicated compared to the many well-known decision procedures for linear inequalities over the rationals [11], =-=[10]-=-, [1], [16], [17]. Therefore, following the tradition in program verification, we adopted a rational-based procedure, exploiting the observation that if a conjunction of inequalities is unsatisfiable ... |

3 |
An Experiment with the Boyer-Moore Theorem Prover: A Proof of the Correctness of a Simple Parser of Expressions
- Gloess
- 1980
(Show Context)
Citation Context ...niqueness of prime factorizations. All of these results are described in [4]. Other proofs discovered before the linear algorithm was implemented include the correctness of a recursive descent parser =-=[9]-=-, the correctness of an arithmetic simplifier now in routine use in the system [5], and the correctness of several FORTRAN programs [6]. 6 3. Linear Arithmetic The theorem prover described above can e... |

2 |
A Mechanical Proof of Wilson's Theorem
- Russinoff
- 1983
(Show Context)
Citation Context ...tive was achieved. Among the theorems proved by the latest version of the theorem prover are the invertibility of the Rivest, Shamir, and Adleman public key encryption algorithm [7], Wilson's theorem =-=[14]-=-, Gauss' law of 8 quadratic reciprocity and the Church-Rosser theorem [15]. All of these proofs involved a substantial amount of linear arithmetic reasoning. We now turn to our observation that theore... |

2 |
On the SUP-INF Method for Provign Presburger Formulas
- Shostak
- 1977
(Show Context)
Citation Context ...theorem prover are the invertibility of the Rivest, Shamir, and Adleman public key encryption algorithm [7], Wilson's theorem [14], Gauss' law of 8 quadratic reciprocity and the Church-Rosser theorem =-=[15]-=-. All of these proofs involved a substantial amount of linear arithmetic reasoning. We now turn to our observation that theoretical efficiency is not a good measure of the utility of a linear procedur... |