Behzad Akbarpour

MetiTarski: An Automatic Theorem Prover for Real-Valued Special Functions

Many theorems involving special functions such as ln, exp and sin can be proved automatically by MetiTarski: a resolution theorem prover modified to call a decision procedure for the theory of real closed fields. Special functions are approximated by upper and lower bounds, which are typically rational functions derived from Taylor or continued fraction expansions. The decision procedure simplifies clauses by deleting literals that are inconsistent with other algebraic facts. MetiTarski simplifies arithmetic expressions by conversion to a recursive representation, followed by flattening of nested quotients. Applications include verifying hybrid and control systems.

Formal Reasoning about Expectation Properties for Continuous Random Variables

Expectation (average) properties of continuous random variables are widely used to judge performance characteristics in engineering and physical sciences. This paper presents an infrastructure that can be used to formally reason about expectation properties of most of the continuous random variables in a theorem prover. Starting from the relatively complex higher-order-logic definition of expectation, based on Lebesgue integration, we formally verify key expectation properties that allow us to reason about expectation of a continuous random variable in terms of simple arithmetic operations. In order to illustrate the practical effectiveness and utilization of our approach, we also present the formal verification of expectation properties of the commonly used continuous random variables: Uniform, Triangular and Exponential.

MetiTarski: An Automatic Prover for the Elementary Functions

Many inequalities involving the functions ln, exp, sin, cos, etc., can be proved automatically by MetiTarski: a resolution theorem prover (Metis) modified to call a decision procedure (QEPCAD) for the theory of real closed fields. The decision procedure simplifies clauses by deleting literals that are inconsistent with other algebraic facts, while deleting as redundant clauses that follow algebraically from other clauses. MetiTarski includes special code to simplify arithmetic expressions.

Formal verification of analog designs using MetiTarski

MetiTarski, an automatic theorem prover for inequalities on real-valued elementary functions, can be used to verify properties of analog circuits. First, a closed form solution to the model of the circuit is obtained. We present two techniques for obtaining the closed form solution. One is based on piecewise linear modeling and the inverse Laplace transform. The other is based on small-signal analysis and transfer function theory. Second, the properties of interest are turned into a set of inequalities involving analytic functions, which are proved automatically using MetiTarski. We verify properties concerning oscillation and the change in gain due to component tolerances.

Formalization of fixed-point arithmetic in HOL

This paper addresses the formalization in higher-order logic of fixed-point arithmetic. We encoded the fixed-point number system and specified the different quantization modes in fixed-point arithmetic such as the directed and even quantization modes. We also considered the formalization of exceptions detection and their handling like overflow and invalid operation. An error analysis is then performed to check the correctness of the quantized result after carrying out basic arithmetic operations, such as addition, subtraction, multiplication and division against their mathematical counterparts. Finally, we showed by an example how this formalization can be used to enable the verification of the transition from floating-point to fixed-point algorithmic level in the signal processing design flow.

Applications of MetiTarski in the Verification of Control and Hybrid Systems

MetiTarski, an automatic proof procedure for inequalities on elementary functions, can be used to verify control and hybrid systems. We perform a stability analysis of control systems using Nichols plots, presenting an inverted pendulum and a magnetic disk drive reader system. Given a hybrid systems specified by a system of differential equations, we use Maple to obtain a problem involving the exponential and trigonometric functions, which MetiTarski can prove automatically.

Extending a resolution prover for inequalities on elementary functions

Experiments show that many inequalities involving exponentials and logarithms can be proved automatically by combining a resolution theorem prover with a decision procedure for the theory of real closed fields (RCF). The method should be applicable to any functions for which polynomial upper and lower bounds are known. Most bounds only hold for specific argument ranges, but resolution can automatically perform the necessary case analyses. The system consists of a superposition prover (Metis) combined with John Harrisons RCF solver and a small amount of code to simplify literals with respect to the RCF theory.

Formal verification of analog circuits in the presence of noise and process variation

We model and verify analog designs in the presence of noise and process variation using an automated theorem prover, MetiTarski. Due to the statistical nature of noise, we propose to use stochastic differential equations (SDE) to model the designs. We find a closed form solution for the SDEs, then integrate the device variation due to the 0.18¿m fabrication process and verify properties using MetiTarski. We illustrate the proposed approach on an inverting Op-Amp Integrator and a Band-Gap reference bias circuit.

Verifying a Synthesized Implementation of IEEE-754 Floating-Point Exponential Function using HOL

Deep datapath and algorithm complexity have made the verification of floating-point units a very hard task. Most simulation and reachability analysis verification tools fail to verify a circuit with a deep datapath like most industrial floating-point units. Theorem proving, however, offers a better solution to handle such verification. In this paper, we have hierarchically formalized and verified a hardware implementation of the IEEE-754 table-driven floating-point exponential function algorithm using the higher-order logic (HOL) theorem prover. The high ability of abstraction in the HOL verification system allows its use for the verification task over the whole design path of the circuit, starting from gate-level implementation of the circuit up to a high-level mathematical specification.

Formalization of Cadence SPW Fixed-Point Arithmetic in HOL

This paper addresses the formalization in higher-order logic of fixed-point arithmetic based on the SPW (Signal Processing WorkSystem) tool. We encoded the fixed-point number system and specified the different rounding modes in fixed-point arithmetic such as the directed and even rounding modes.We also considered the formalization of exceptions detection and their handling like overflow and invalid operation. An error analysis is then performed to check the correctness of the rounding and to verify the basic arithmetic operations, addition, subtraction, multiplication and division against their mathematical counterparts. Finally, we showed by an example how this formalization can be used to enable the verification of the transition from the floating-point to fixed-point algorithmic levels in the design flow of signal processors.