Ruben Gamboa

The LDL system prototype

The logic data language (LDL) system provides a declarative logic-based language and integrates relational database and logic programming technologies so as to support advanced data and knowledge-based applications. A comprehensive overview of the system and a description of LDL language and the compilation techniques employed to translate LDL queries into target query execution plans on the stored data are presented. The architecture and runtime environment of the system and the optimization techniques employed in order to improve the performance and assure the safety of the compiled queries are given. The experience gained so far with the system and application areas where the LDL approach appears to be particularly effective are discussed. >

Nonstandard Analysis in ACL2

ACL2 refers to a mathematical logic based on applicative Common Lisp, as well as to an automated theorem prover for this logic. The numeric system of ACL2 reflects that of Common Lisp, including the rational and complex-rational numbers and excluding the real and complex irrationals. In conjunction with the arithmetic completion axioms, this numeric type system makes it possible to prove the nonexistence of specific irrational numbers, such as √2. This paper describes ACL2(r), a version of ACL2 with support for the real and complex numbers. The modifications are based on nonstandard analysis, which interacts better with the discrete flavor of ACL2 than does traditional analysis.

Curve-based representation of moving object trajectories

In recent years, many emerging database applications deal with continuously moving data objects - each data object moves continuously and frequently reports its current location, moving direction, and speed to the database server. A database server for these applications keeps track of the trajectories of individual moving objects and processes queries referring to the past or future trajectories. Related techniques view a moving object trajectory as a sequence of connected line segments. However, most natural moving objects, such as airplanes, vessels, and vehicles, draw a smooth trajectory with no angles. This paper presents our curve-based trajectory representation models. The presented results show that the curve-based models provide much more accurate trajectories than the line-based models when we have the same amount of data (same number of reported points). In other words, the curve-based models require a smaller amount of data while providing the same accuracy in trajectory representation.

Mechanical Verification of a Square Root Algorithm Using Taylor's Theorem

The IBM Power4 processor uses Chebyshev polynomials to calculate square root. We formally verified the correctness of this algorithm using the ACL2(r) theorem prover. The proof requires the analysis on the approximation error of Chebyshev polynomials. This is done by proving Taylors theorem, and then analyzing the Chebyshev polynomial using Taylor polynomials. Taylors theorem is proven by way of non-standard analysis, as implemented in ACL2(r). Since a Taylor polynomial has less accuracy than the Chebyshev polynomial of the same degree, we used hundreds of Taylor polynomial generated by ACL2(r) to evaluate the error of a Chebyshev polynomial.

Mechanically verifying the correctness of the Fast Fourier Transform in ACL2

In [10], Misra introduced the powerlist data structure, which is well suited to express recursive, data-parallel algorithms. In particular, Misra showed how powerlists could be used to give simple descriptions to very complex algorithms, such as the Fast Fourier Transform (FFT). Such simplicity in presentation facilitates reasoning about the resulting algorithms, and in fact Misra was able to give a stunningly simple proof of the correctness of the FFT. In this paper, we show how this proof can be verified using ACL2. This strengthens Misras case that powerlists provide a suitable framework in which to define and reason about parallel algorithms, particularly using mechanical tools to aid in reasoning.

Automatic differentiation in ACL2

In this paper, we describe recent improvements to the theory of differentiation that is formalized in ACL2(r). First, we show how the normal rules for the differentiation of composite functions can be introduced in ACL2(r). More important, we show how the application of these rules can be largely automated, so that ACL2(r) can automatically define the derivative of a function that is built from functions whose derivatives are already known. Second, we show a formalization in ACL2(r) of the derivatives of familiar functions from calculus, such as the exponential, logarithmic, power, and trigonometric functions. These results serve as the starting point for the automatic differentiation tool described above. Third, we describe how users can add new functions and their derivatives, to improve the capabilities of the automatic differentiator. In particular, we show how to introduce the derivative of the hyperbolic trigonometric functions. Finally, we give some brief highlights concerning the implementation details of the automatic differentiator.

Extending dynamic constraint detection with disjunctive constraints

The languages of current dynamic constraint detection techniques are often specified by fixed grammars of universal properties. These properties may not be sufficient to express more subtle facts that describe the essential behavior of a given program. In an effort to make the dynamically recovered specification more expressive and program-specific we propose the state space partitioning technique as a solution which effectively adds program-specific disjunctive properties to the language of dynamic constraint detection. In this paper we present ContExt, a prototype implementation of the state space partitioning technique which relies on Daikon for dynamic constraint inference tasks. In order to evaluate recovered specifications produced by ContExt, we develop a methodology which allows us to measure quantitatively how well a particular recovered specification approximates the essential specification of a programs behavior. The proposed methodology is then used to comparatively evaluate the specifications recovered by ContExt and Daikon on two examples.

Extending Dynamic Constraint Detection with Polymorphic Analysis

The general technique for dynamically detecting likely invariants, as implemented by Daikon, lacks specific object- oriented support for polymorphism. Daikon examines only the declared type of a variable which prohibits it from examination of the runtime variables in the presence of polymorphism. The approach presented in this paper extends the technique to consider the runtime type of a polymorphic variable, which may have different declared and runtime types. The runtime behavior of a polymorphic variable is captured by polymorphic constraints which have the form of an implication with the name of the runtime class in the antecedent. We demonstrate the improved accuracy of the dynamically detected specification on the Money example from the JUnit testing framework tutorial. Polymorphic constraints are shown to reveal the specification of the runtime behavior of the example.

Theory Extension in ACL2(r)

ACL2(r) is a modified version of the theorem prover ACL2 that adds support for the irrational numbers using nonstandard analysis. It has been used to prove basic theorems of analysis, as well as the correctness of the implementation of transcendental functions in hardware. This paper presents the logical foundations of ACL2(r). These foundations are also used to justify significant enhancements to ACL2(r).

A Formalization of Powerlist Algebra in ACL2

In Misra (ACM Trans Program Lang Syst 16(6):1737–1767, 1994), Misra introduced the powerlist data structure, which is well suited to express recursive, data-parallel algorithms. Moreover, Misra and other researchers have shown how powerlists can be used to prove the correctness of several algorithms. This success has encouraged some researchers to pursue automated proofs of theorems about powerlists (Kapur 1997; Kapur and Subramaniam 1995, Form Methods Syst Des 13(2):127–158, 1998). In this paper, we show how ACL2 can be used to verify theorems about powerlists. We depart from previous approaches in two significant ways. First, the powerlists we use are not the regular structures defined by Misra; that is, we do not require powerlists to be balanced trees. As we will see, this complicates some of the proofs, but on the other hand it allows us to state theorems that are otherwise beyond the language of powerlists. Second, we wish to prove the correctness of powerlist algorithms as much as possible within the logic of powerlists. Previous approaches have relied on intermediate lemmas which are unproven (indeed unstated) within the powerlist logic. However, we believe these lemmas must be formalized if the final theorems are to be used as a foundation for subsequent work, e.g., in the verification of system libraries. In our experience, some of these unproven lemmas presented the biggest obstacle to finding an automated proof. We illustrate our approach with two case studies involving Batcher sorting and prefix sums.