W. Bledsoe. A new method for proving certain Presburger formulas. In Proc. of the 4 th Joint Conf. on Artificial Intelligence, pages 15--21, 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a user settable option. Certification of proofs run with caching turned o# will provide evidence that the mechanism itself does not interfere with validity. However, other uses of the because rule are more fundamental. The SupInf tactic, which implements Bledsoe s Sup Inf decision procedure[5, 14] for linear arithmetic, invokes because if the decision procedure says yes. We do not believe the rule itself is objectionable, but if Nuprl proofs are to be believed, the uses of because must be accounted for. This is currently an active project in the Nuprl group at Cornell. Acknowledgments ....

W. Bledsoe. A new method for proving certain Presburger formulas. In Proc. of the 4 th Joint Conf. on Artificial Intelligence, pages 15--21, 1975.

....because rule may account for 15 or more of the rules in a proof. When the Auto tactic discharges a well formedness goal by appealing to the proof cache, it simply completes the proof by invoking the because rule. Similarly, the SupInf tactic, which implements Bledsoe s Sup Inf decision procedure[5, 13] for linear arithmetic, invokes because if the decision procedure says yes. We do not believe the rule itself is objectionable, but if Nuprl proofs are to be believed, the uses of because must be accounted for. This is currently an active project in the Nuprl group at Cornell. Acknowledgments ....

W. Bledsoe. A new method for proving certain Presburger formulas. In Proc. of the 4 th Joint Conf. on Articial Intelligence, pages 15-21, 1975.

....algorithms for linear integer constraints based, directly or indirectly, on projection. The relationship between this approach and transitivity based methods is quite close, since transitive closure can be thought of as a cumulative form of projection with redundancy elimination. The SUPINF method [2, 20] is complete over the real numbers, but the class of integer constraints on which it is complete is not clearly defined. The Omega test [15, 16] and the algorithm of [10] adapt Fourier s projection algorithm for real numbers to integers. The former computes a disjunction of constraints, whereas ....

Bledsoe, W.W.: A new method for proving certain Presburger formulas. Proc. of the 4 th Joint Conf. on Artificial Intelligence. (1975) 15--21.

.... (fflfi 1 ) ffl fi2 ff1 ) fi 3 ff1 ) fflfi 1 ) ffl fi 2 ff1 ) fi 3 ff1 ) fi 1 ff1 fi 2 ) fi 3 fi1 fi2 ) In the second branch, we have: fi3 ff1 ) fflfi1 ) ff 1 fi3 ) fi 1 ffl) The simplification carried out is basically a normalisation via SUP INF procedure [Ble75] which attempts to remove variables, ffl and ff 1 , leaving behind predicates exclusively in terms of ffi i g i21 : 3 . Only the predicate condition for the first branch can be successfully simplified, but not the second branch. This means that the first branch can be eliminated by supplying a ....

W.W. Bledsoe. A new method for proving certain Presburger formulae. In Proc of ICJAI, pages 15--21, 1975.

....inequalities expressing the conditions for a successful a mating task could be back propagated to derive initial conditions for the plan variables. To perform the constraint propagation symbolically, Brooks built a constraint manipulation system [14] based on the SUP Gamma INF method of Bledsoe [7], whereby the extreme values of an unknown variable were recursively expressed in terms of subvariables exteme values. This method of constraint propagation was also employed in TWAIN [95] to parameterize the task skeletons selected for the interwoven subproblems in an assembly task. 1.4.4 ....

Bledsoe, W.W. (1975), "A new method for proving certain Presburger formulas", IJCAI at Tibilsi, Georgia, U.S.S.R.

....many well known decision procedures for linear inequalities over the rationals [ Therefore, following the tradition in program verification, we adopted a rational based procedure. 1 Shostak [1977] attributes this result to Oppen. It is this tradition of work in the rationals [ Bledsoe, 1975; Shostak, 1977; 1979; Boyer and Moore, 1988 ] that we question in this paper. Here we report on an experimental comparison of decision procedures on 10000 randomly generated formulae, in order to explain this tradition to some extent. However, the results are very surprising broadly, they show ....

....that same theory sometimes goes by the name Bledsoe real arithmetic. Kreisel and Krivine [ 1967 ] these are Hodes procedure for Presburger rational arithmetic [ 1971 ] and Cooper s procedure for Presburger integer arithmetic [ 1972 ] There is also the Sup Inf family of procedures due to Bledsoe [ 1975 ] and latterly improved by Shostak [ 1977; 1979 ] The rational based procedures are an attempt to overcome the complexity of integer based procedures. It can be easily seen that there are formulae true in rational arithmetic and false in natural arithmetic and vice versa. For instance, 9x:2x = ....

W. W. Bledsoe. A new method for proving certain Presburger formulas. In Proceedings of the 4th International Joint Conference on Artificial Intelligence, Tbilisi, Georgia, U. S. S. R., 3--8 September 1975.

....in disjunctive normal form. Cooper s algorithm does not require formulas to be transformed into disjunctive normal form and may be better for formulas that would be expensive to put into disjunctive normal form (although our methods for handling negation address this as well) The SUP INF method [Ble75, Sho77] is a semi decision procedure. It sometimes detects solutions when only real solutions exist and it cannot be used for symbolic quantified variable elimination. H.P. Williams [Wil76] describes an extension of Fourier elimination to integers. His scheme leads to a much more explosive growth ....

W. W. Bledsoe. A new method for proving certain presburger formulas. In Advance Papers, 4th Int. Joint Conference on Artif. Intell., Tibilisi, Georgia, U.S.S.R, 1975.

....inequality reasoning. Arith cannot handle arbitrary integer linear programming problems. These come up frequently when one type checks types parameterized by integers. For this reason, we are implementing the Sup Inf method for linear programming problems over the integers and the rationals [10]. We hope to verify this implementation using our work on reflection. See section 3.4 for details. 3.3.5 Rewriting Term rewriting is a very useful technique for theorem proving. Some systems base all their reasoning on rewriting [14] Nuprl has a term rewriting package using conversions [25] ....

W. W. Bledsoe. A new method for proving certain Presburger formulas. In 4th International Joint Conference on Artificial Intelligence, pages 15--21, Tiblsi, 1975.

....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 verification, we adopted a rational based procedure, exploiting the observation that if a conjunction of inequalities is unsatisfiable over the rationals it is unsatisfiable over the integers. Such a procedure is sound but incomplete. For ....

W. W. Bledsoe. A New Method for Proving Certain Presburger Formulas. Advance Papers, Fourth Int. Joint Conf. on Art. Intell., Tbilisi, Georgia, U.S.S.R., September, 1975, pp. 15-20.

....is classical, as Howe s work illustrates [19] It also allows the interpretation of recursive mathematics that all functions are given by Turing machines or Lisp programs. It also allows an Intuitionistic interpretation. One way to describe this style is to relate it to the work of Bishop [3] who showed that real, complex, and abstract analysis could be formalized in this neutral way. 1.4 Outline In section 1 we present the basic ideas from Nuprl needed for this article. Surprisingly little is required, and we claim that this basic material is mostly as readable as the mathematical ....

....[1] Stuart F. Allen. A non type theoretic semantics for type theoretic language. PhD thesis, Cornell University, 1987. 2] Y. Bertot, G. Kahn, and L. Th ery. Proof by pointing. In Theoretical Aspects of Computer Software, Lecture Notes in Computer Science, volume 789, pages 141 160, 1994. [3] E. Bishop. Foundations of Constructive Analysis. McGraw Hill, NY, 1967. 4] P. Borras, D. Cl ement, T. Despeyroux, J. Incerpi, G. Kahn, B. Lang, and V. Pascual. Centaur: the system. In Software Engineering Notes, volume 13(5) Third Symposium on Software Development Environments, 1988. 5] N. ....

[Article contains additional citation context not shown here]

W. W. Bledsoe. A new method for proving certain Presburger formulas. Fourth Intl. Joint Conf. on A.I., Tblisi, USSR, September 1975.

....(e.g. convert x = 1 into 1 x 1) Once we have projected away y and z, we then compute the gist of the red equations with respect to the black equations. 3. 4 Related Work Several authors have explored methods for using integer programming methods to decide subclasses of Presburger formulas [Ble75, Sho77, JM87] The work of [Ble75, Sho77] cannot handle nested, alternating quantifiers. The work described in [JM87] can only handle constraints of the form v v 0 c (for variables v and v 0 and constant c) These limitations prevent this use of these techniques for the types of ....

....1) Once we have projected away y and z, we then compute the gist of the red equations with respect to the black equations. 3. 4 Related Work Several authors have explored methods for using integer programming methods to decide subclasses of Presburger formulas [Ble75, Sho77, JM87] The work of [Ble75, Sho77] cannot handle nested, alternating quantifiers. The work described in [JM87] can only handle constraints of the form v v 0 c (for variables v and v 0 and constant c) These limitations prevent this use of these techniques for the types of dependence analysis problems we need to ....

W. W. Bledsoe. A new method for proving certain presburger formulas. In Advance Papers, 4th Int. Joint Conference on Artif. Intell., Tibilisi, Georgia, U.S.S.R, 1975.

....(e.g. convert x = 1 into 1 x 1) Once we have projected away y and z, we then compute the gist of the red equations with respect to the black equations. 3. 4 Related Work Several authors have explored methods for using integer programming methods to decide subclasses of Presburger formulas [Ble75, Sho77, JM87] The work of [Ble75, Sho77] cannot handle nested, alternating quantifiers. The work described in [JM87] can only handle constraints of the form v v 0 c (for variables v and v 0 and constant c) These limitations prevent the use of these techniques for the types of dependence ....

....1) Once we have projected away y and z, we then compute the gist of the red equations with respect to the black equations. 3. 4 Related Work Several authors have explored methods for using integer programming methods to decide subclasses of Presburger formulas [Ble75, Sho77, JM87] The work of [Ble75, Sho77] cannot handle nested, alternating quantifiers. The work described in [JM87] can only handle constraints of the form v v 0 c (for variables v and v 0 and constant c) These limitations prevent the use of these techniques for the types of dependence analysis problems we need to ....

W. W. Bledsoe. A new method for proving certain presburger formulas. In Advance Papers, 4th Int. Joint Conference on Artif. Intell., Tibilisi, Georgia, U.S.S.R, 1975.

....in disjunctive normal form. Cooper s algorithm does not require formulas to be transformed into disjunctive normal form and may be better for formulas that would be expensive to put into disjunctive normal form (although our methods for handling negation address this as well) The SUP INF method [Ble75, Sho77] is a semi decision procedure. It sometimes detects solutions when only real solutions exist and it cannot be used for symbolic quantified variable elimination. H.P. Williams [Wil76] describes an extension of Fourier elimination to integers. His scheme leads to a much more explosive growth ....

W. W. Bledsoe. A new method for proving certain presburger formulas. In Advance Papers, 4th Int. Joint Conference on Artif. Intell., Tibilisi, Georgia, U.S.S.R, 1975.

.... conditions which can be encoded under the following form ( p 1 (Y, X 1 ) p n (Y, Xn ) a(X 1 [ Xn ) where Y is a list variable, X 1 , X n are vectors of natural number variables, a(X 1 [ X n ) is a linear arithmetic formula, and p 1 , p n are predicates recursively defined [2, 4]. An example taken from [4] of such a form is min(Y,X 1 ) max(Y,X 2 ) 8K,L (LX 1 0 K ) L X 2 K) where min(Y,X 1 ) and max(Y,X 2 ) mean that X 1 and X 2 are the minimum and maximum elements of the list Y. Boyer and Moore have proposed to integrate a decision procedure for linear arithmetic ....

Bledsoe, W.W. (1975). "A New Method for Proving certain Presburger Formulas", Proc. 4th Intl. Joint Conf. on Artificial Intelligence, Tbilissi, pp. 15-20.

....procedures for a given theory may vary depending on their degree of completeness (i.e. which formulas they can decide) and their complexity, which are traded off against each other. Two decision procedures for Presburger arithmetic are available 3 . The first is based on the Sup Inf method [Ble75] which efficiently decides a subset of the theory; the other is an implementation of Cooper s algorithm [Coo72] which is a decision procedure for the entire theory. The Sup Inf method is complete for rational quantifier free Presburger arithmetic, and can be extended to handle uninterpreted ....

W.W. Bledsoe. A new method for proving certain Presburger formulas. In Proc. of the 4 th International Joint Conference on Artificial Intelligence, pages 15--21, September 1975.

....depended on basic properties of order and equivalence relations, such as symmetry, antisymmetry, reflexivity, irreflexivity, transitivity and linearity. Arithmetic Reasoner I implemented a new inference rule for solving linear inequalities over the integers, based on Bledsoe s sup inf algorithm [Ble75] The chief enhancement I made was to take full advantage of the linear arithmetic properties of non linear arithmetic functions and non arithmetic functions that have integer values. Type Checker All type checking in the Nuprl system is done by proof. Enhancements I made to the type checking ....

....inequalities over the rationals, but integer linear programming is NP complete. In practice in theorem proving, simple adaptations of methods over the rationals have worked well for the integers. I chose to implement in Nuprl a tactic that uses the Sup Inf method for solving integer inequalities [Ble75] The basic algorithm considers a conjunction of inequalities 0 e 1 : 0 e p where the e i are linear expressions over the rationals in variables x 1 : x n and determines whether or not there exists an assignment of values to the x j that satisfies the conjunction. The algorithm works ....

W. W. Bledsoe. A new method for proving certain Presburger formulas. In 4th International Joint Conference on Artificial Intelligence, pages 15--21, Tiblsi, 1975.

....much experience with adapting components written by others. It has led us to understand what we called the open black box problem. In [26] we describe a four year effort to encorporate linear arithmetic decision procedures into Nqthm. We studied and implemented procedures by Hodes [27] Bledsoe [28], Shostak [29] and Nelson and Oppen [30] The primary benefit promised by the inclusion of a decision procedure in a general purpose setting is to relieve the general purpose device of the need to explore the search space attributable to the decidable fragment. Our study indicates that in order ....

W. W. Bledsoe, "A New Method for Proving Certain Presburger Formulas", Advance Papers, Fourth Int. Joint Conf. on Art. Intell., Tbilisi, Georgia, U.S.S.R., September 1975, pp. 15-20.

....of the standard limit theorems from analysis (Bledsoe, Boyer, and Henneman 1972) using the limit heuristic and the proofs of many difficult theorems in analysis (Ballantyne and Bledsoe 1977) using nonstandard analysis. PROVER, augmented with special rules to handle integer linear inequalities (Bledsoe 1975), was the proof engine for a very large program verification project (Good 1985; Good, London, and Bledsoe 1975) In Ballantyne and Bledsoe (1982) PROVER was used to study how mathematicians create examples and counterexamples to conjectures and how they are used to guide proofs. The program ....

....the proofs of many difficult theorems in analysis (Ballantyne and Bledsoe 1977) using nonstandard analysis. PROVER, augmented with special rules to handle integer linear inequalities (Bledsoe 1975) was the proof engine for a very large program verification project (Good 1985; Good, London, and Bledsoe 1975). In Ballantyne and Bledsoe (1982) PROVER was used to study how mathematicians create examples and counterexamples to conjectures and how they are used to guide proofs. The program tried to mimic a mathematician at a board drawing pictures of a problem. It was successful at discovering minimal ....

[Article contains additional citation context not shown here]

Bledsoe, W. W. 1975. A New Method for Proving Certain Presburger Formulas. In Proceedings of the Fourth International Joint Conference on Artificial Intelligence, 15-21. Menlo Park, Calif.: International Joint Conferences on Artificial Intelligence.

No context found.

W W Bledsoe. A new method for proving certain Presburger formulae. In Proc of ICJAI, pages 15--21, 1975.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC