Automated Generation of Non-Linear Loop Invariants Utilizing Hypergeometric Sequences

Analyzing and reasoning about safety properties of software systems becomes an especially challenging task for programs with complex flow and, in particular, with loops or recursion. For such programs one needs additional information, for example in the form of loop invariants, expressing properties to hold at intermediate program points. In this paper we study program loops with non-trivial arithmetic, implementing addition and multiplication among numeric program variables. We present a new approach for automatically generating all polynomial invariants of a class of such programs. Our approach turns programs into linear ordinary recurrence equations and computes closed form solutions of these equations. These closed forms express the most precise inductive property, and hence invariant. We apply Gr\"obner basis computation to obtain a basis of the polynomial invariant ideal, yielding thus a finite representation of all polynomial invariants. Our work significantly extends the class of so-called P-solvable loops by handling multiplication with the loop counter variable. We implemented our method in the Mathematica package Aligator and showcase the practical use of our approach.Comment: A revised version of this paper is published in the proceedings of ISSAC 201

Recurrence extraction for functional programs through call-by-push-value

The main way of analyzing the complexity of a program is that of extracting and solving a recurrence that expresses its running time in terms of the size of its input. We develop a method that automatically extracts such recurrences from the syntax of higher-order recursive functional programs. The resulting recurrences, which are programs in a call-by-name language with recursion, explicitly compute the running time in terms of the size of the input. In order to achieve this in a uniform way that covers both call-by-name and call-by-value evaluation strategies, we use Call-by-Push-Value (CBPV) as an intermediate language. Finally, we use domain theory to develop a denotational cost semantics for the resulting recurrences.Comment: POPL 202

Generating rate equations for complex enzyme systems by a computer-assisted systematic method

<p>Abstract</p> <p>Background</p> <p>While the theory of enzyme kinetics is fundamental to analyzing and simulating biochemical systems, the derivation of rate equations for complex mechanisms for enzyme-catalyzed reactions is cumbersome and error prone. Therefore, a number of algorithms and related computer programs have been developed to assist in such derivations. Yet although a number of algorithms, programs, and software packages are reported in the literature, one or more significant limitation is associated with each of these tools. Furthermore, none is freely available for download and use by the community.</p> <p>Results</p> <p>We have implemented an algorithm based on the schematic method of King and Altman (KA) that employs the topological theory of linear graphs for systematic generation of valid reaction patterns in a GUI-based stand-alone computer program called <it>KAPattern</it>. The underlying algorithm allows for the assumption steady-state, rapid equilibrium-binding, and/or irreversibility for individual steps in catalytic mechanisms. The program can automatically generate MathML and MATLAB output files that users can easily incorporate into simulation programs.</p> <p>Conclusion</p> <p>A computer program, called <it>KAPattern</it>, for generating rate equations for complex enzyme system is a freely available and can be accessed at <url>http://www.biocoda.org</url>.</p

Analyzing Behavioural Scenarios over Tabular Specifications Using Model Checking

Tabular notations, in particular SCR specifications, have proved to be a useful means for formally describing complex requirements. The SCR method offers a powerful family of analysis tools, known as the SCR Toolset, but its availability is restricted by the Naval Research Laboratory of the USA. This toolset applies different kinds of analysis considering the whole set of behaviours associated with a requirements specification. In this paper we present a tool for describing and analyzing SCR requirements descriptions, that complements the SCR Toolset in two aspects. First, its use is not limited by any institution, and resorts to a standard model checking tool for analysis; and second, it allows to concentrate the analysis to particular sets of behaviours (subsets of the whole specifications), that correspond to particular scenarios explicitly mentioned in the specification. We take an operational notation that allows the engineer to describe behavioural "scenarios" by means of programs, and provide a translation into Promela to perform the analysis via Spin, an efficient off-the-shelf model checker freely available. In addition, we apply the SCR method to a Pacemaker system and we use its tabular specification as a running example of this article.Comment: In Proceedings LAFM 2013, arXiv:1401.056