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31
Invariant Generation for Psolvable Loops with Assignments
, 2008
"... We discuss interesting properties of a general technique for inferring polynomial invariants for a subfamily of imperative loops, called the Psolvable loops, with assignments only. The approach combines algorithmic combinatorics, polynomial algebra and computational logic, and it is implemented in ..."
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We discuss interesting properties of a general technique for inferring polynomial invariants for a subfamily of imperative loops, called the Psolvable loops, with assignments only. The approach combines algorithmic combinatorics, polynomial algebra and computational logic, and it is implemented in a new software package called Aligator. We present a collection of examples illustrating the power of the framework.
Providing a basin of attraction to a target region by computation of Lyapunovlike functions
 In IEEE Int. Conf. on Computational Cybernetics
, 2006
"... Abstract — In this paper, we present a method for computing a basin of attraction to a target region for nonlinear ordinary differential equations. This basin of attraction is ensured by a Lyapunovlike polynomial function that we compute using an interval based branchandrelax algorithm. This alg ..."
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Abstract — In this paper, we present a method for computing a basin of attraction to a target region for nonlinear ordinary differential equations. This basin of attraction is ensured by a Lyapunovlike polynomial function that we compute using an interval based branchandrelax algorithm. This algorithm relaxes the necessary conditions on the coefficients of the Lyapunovlike function to a system of linear interval inequalities that can then be solved exactly, and iteratively reduces the relaxation error by recursively decomposing the state space into hyperrectangles. Tests on an implementation are promising. I.
Verification Environment in Theorema
 Articles in Refereed Conference Proceedings 1. N. Popov and
"... Abstract — We present a verification environment for imperative programs (using Hoare logic) and for functional programs (using fixpoint theory) in the frame of the Theorema system (www.theorema.org). In particular, we discuss some methods for finding the invariants of loops and specifications of au ..."
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Abstract — We present a verification environment for imperative programs (using Hoare logic) and for functional programs (using fixpoint theory) in the frame of the Theorema system (www.theorema.org). In particular, we discuss some methods for finding the invariants of loops and specifications of auxiliary tail recursive functions. These methods use techniques from (polynomial) algebra and combinatorics, namely Groebner bases, variable elimination and symbolic summation (the Gosper algorithm, the technique of generating functions). The methods are demonstrated on several examples which have been treated automatically by our implementation. Index Terms — program analysis and verification, loop invariant generation, theorem proving, symbolic summation I.
A verification environment for imperative and functional programs in the theorema system
 Satellite of 2nd Balkan Conference in Informatics, 1719 November, Ohrid. Contributed talk at 2nd SouthEast European Workshop on Formal Methods (SEEFM05), ”Practical dimensions: Challenges in the business world”, Ohrid, FYR of Macedonia
, 2005
"... Abstract. We present a verification environment for imperative programs (using Hoare logic) and for functional programs (using fixpoint theory) in the frame of the Theorema system (www.theorema.org). In particular, we discuss some methods for finding the invariants of loops and of specifications of ..."
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Abstract. We present a verification environment for imperative programs (using Hoare logic) and for functional programs (using fixpoint theory) in the frame of the Theorema system (www.theorema.org). In particular, we discuss some methods for finding the invariants of loops and of specifications of auxiliary tail recursive functions. These methods use algorithms from (polynomial) algebra and combinatorics, namely Groebner bases, variable elimination and symbolic summation (the Gosper algorithm, the technique of generating functions). The techniques are demonstrated on several examples which have been treated automatically by our implementation.
Generating Polynomial Invariants with DISCOVERER and QEPCAD
"... This paper investigates how to apply the techniques on solving semialgebraic systems to invariant generation of polynomial programs. By our approach, the generated invariants represented as a semialgebraic system are more expressive than those generated with the wellestablished approaches in the ..."
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This paper investigates how to apply the techniques on solving semialgebraic systems to invariant generation of polynomial programs. By our approach, the generated invariants represented as a semialgebraic system are more expressive than those generated with the wellestablished approaches in the literature, which are normally represented as a conjunction of polynomial equations. We implement this approach with the computer algebra tools DISCOVERER and QEPCAD 1. We also explain, through the complexity analysis, why our approach is more efficient and practical than the one of [17] which directly applies firstorder quantifier elimination.
Linearity Analysis for Automatic Differentiation
"... Linearity analysis determines which variables depend on which other variables and whether the dependence is linear or nonlinear. One of the many applications of this analysis is determining whether a loop involves only linear loopcarried dependences and therefore the adjoint of the loop may be rev ..."
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Linearity analysis determines which variables depend on which other variables and whether the dependence is linear or nonlinear. One of the many applications of this analysis is determining whether a loop involves only linear loopcarried dependences and therefore the adjoint of the loop may be reversed and fused with the computation of the original function. This paper specifies the dataflow equations that compute linearity analysis. In addition, the paper describes using linearity analysis with array dependence analysis to determine whether a loopcarried dependence is linear or nonlinear.
ChangeOfBases Abstractions for NonLinear Systems.
, 2012
"... We present abstraction techniques that transform a given nonlinear dynamical system into a linear system or an algebraic system described by polynomials of bounded degree, such that, invariant properties of the resulting abstraction can be used to infer invariants for the original system. The abstr ..."
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We present abstraction techniques that transform a given nonlinear dynamical system into a linear system or an algebraic system described by polynomials of bounded degree, such that, invariant properties of the resulting abstraction can be used to infer invariants for the original system. The abstraction techniques rely on a changeofbasis transformation that associates each state variable of the abstract system with a function involving the state variables of the original system. We present conditions under which a given change of basis transformation for a nonlinear system can define an abstraction. Furthermore, the techniques developed here apply to continuous systems defined by Ordinary Differential Equations (ODEs), discrete systems defined by transition systems and hybrid systems that combine continuous as well as discrete subsystems. The techniques presented here allow us to discover, given a nonlinear system, if a change of bases transformation involving degreebounded polynomials yielding an algebraic abstraction exists. If so, our technique yields the resulting abstract system, as well. This approach is further extended to search for a change of bases transformation that abstracts a given nonlinear system into a system of linear differential inclusions. Our techniques enable the use of analysis techniques for linear systems to infer invariants for nonlinear systems. We present preliminary evidence of the practical feasibility of our ideas using a prototype implementation. 1
Probabilistic Program Analysis with Martingales
"... We present techniques for the analysis of infinite state probabilistic programs to synthesize probabilistic invariants and prove almostsure termination. Our analysis is based on the notion of (super) martingales from probability theory. First, we define the concept of (super) martingales for loop ..."
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We present techniques for the analysis of infinite state probabilistic programs to synthesize probabilistic invariants and prove almostsure termination. Our analysis is based on the notion of (super) martingales from probability theory. First, we define the concept of (super) martingales for loops in probabilistic programs. Next, we present the use of concentration of measure inequalities to bound the values of martingales with high probability. This directly allows us to infer probabilistic bounds on assertions involving the program variables. Next, we present the notion of a super martingale ranking function (SMRF) to prove almost sure termination of probabilistic programs. Finally, we extend constraintbased techniques to synthesize martingales and supermartingale ranking functions for probabilistic programs. We present some applications of our approach to reason about invariance and termination of small but complex probabilistic programs.
Experimental Program Verification in the
"... Abstract. We describe practical experiments of program verification in the frame of the Theorema system (www.theorema.org). This includes both functional programs (using fixpoint theory), as well as imperative programs (using Hoare logic). By comparing different approaches we are trying to find gene ..."
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Abstract. We describe practical experiments of program verification in the frame of the Theorema system (www.theorema.org). This includes both functional programs (using fixpoint theory), as well as imperative programs (using Hoare logic). By comparing different approaches we are trying to find general schemes which are useful for practical work. The Theorema system offers facilities for working with higherorder predicate logic formulae (including various general and domainoriented provers) and also for defining and testing algorithms both in functional and in imperative styles. We generate verification conditions as naturalstyle predicate logic formulae, which can be then proven by Theorema, by issuing naturalstyle proofs which are human–readable.
Symbolic Computation and Program Verification. Proving Partial Correctness and Synthesizing Optimal Algorithms ∗
"... We present methods for checking the partial correctness of, respectively to optimize, imperative programs, using polynomial algebra methods, namely resultant computation and quantifier elimination (QE) by cylindrical algebraic decomposition (CAD). The results are very promising but also show that th ..."
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We present methods for checking the partial correctness of, respectively to optimize, imperative programs, using polynomial algebra methods, namely resultant computation and quantifier elimination (QE) by cylindrical algebraic decomposition (CAD). The results are very promising but also show that there is room for improvement of algebraic algorithms. 1