Function Verification of Combinational Arithmetic Circuits

Hardware design verification is the most challenging part in overall hardware design process. It is because design size and complexity are growing very fast while the requirement for performance is ever higher. Conventional simulation-based verification method cannot keep up with the rapid increase in the design size, since it is impossible to exhaustively test all input vectors of a complex design. An important part of hardware verification is combinational arithmetic circuit verification. It draws a lot of attention because flattening the design into bit-level, known as the bit-blasting problem, hinders the efficiency of many current formal techniques. The goal of this thesis is to introduce a robust and efficient formal verification method for combinational integer arithmetic circuit based on an in-depth analysis of recent advances in computer algebra. The method proposed here solves the verification problem at bit level, while avoiding bit-blasting problem. It also avoids the expensive Groebner basis computation, typically employed by symbolic computer algebra methods. The proposed method verifies the gate-level implementation of the design by representing the design components (logic gates and arithmetic modules) by polynomials in Z2n . It then transforms the polynomial representing the output bits (called “output signature”) into a unique polynomial in input signals (called “input signature”) using gate-level information of the design. The computed input signature is then compared with the reference input signature (golden model) to determine whether the circuit behaves as anticipated. If the reference input signature is not given, our method can be used to compute (or extract) the arithmetic function of the design by computing its input signature. Additional tools, based on canonical word-level design representations (such as TED or BMD) can be used to determine the function of the computed input signature represents. We demonstrate the applicability of the proposed method to arithmetic circuit verification on a large number of designs

Formal Analysis of Arithmetic Circuits using Computer Algebra - Verification, Abstraction and Reverse Engineering

Despite a considerable progress in verification and abstraction of random and control logic, advances in formal verification of arithmetic designs have been lagging. This can be attributed mostly to the difficulty in an efficient modeling of arithmetic circuits and datapaths without resorting to computationally expensive Boolean methods, such as Binary Decision Diagrams (BDDs) and Boolean Satisfiability (SAT), that require “bit blasting”, i.e., flattening the design to a bit-level netlist. Approaches that rely on computer algebra and Satisfiability Modulo Theories (SMT) methods are either too abstract to handle the bit-level nature of arithmetic designs or require solving computationally expensive decision or satisfiability problems. The work proposed in this thesis aims at overcoming the limitations of analyzing arithmetic circuits, specifically at the post-synthesized phase. It addresses the verification, abstraction and reverse engineering problems of arithmetic circuits at an algebraic level, treating an arithmetic circuit and its specification as a properly constructed algebraic system. The proposed technique solves these problems by function extraction, i.e., by deriving arithmetic function computed by the circuit from its low-level circuit implementation using computer algebraic rewriting technique. The proposed techniques work on large integer arithmetic circuits and finite field arithmetic circuits, up to 512-bit wide containing millions of logic gates

Formal Verification of Divider and Square-root Arithmetic Circuits Using Computer Algebra Methods

A considerable progress has been made in recent years in verification of arithmetic circuits such as multipliers, fused multiply-adders, multiply-accumulate, and other components of arithmetic datapaths, both in integer and finite field domain. However, the verification of hardware dividers and square-root functions have received only a limited attention from the verification community, with a notable exception for theorem provers and other inductive, non-automated systems. Division, square root, and transcendental functions are all tied to the basic Intel architecture and proving correctness of such algorithms is of grave importance. Although belonging to the same iterative-subtract class of architectures, they widely differ from each other. IEEE floating point standard specifies square-rooting and division as basic arithmetic operation alongside the usual three basic operations. The difficulty of formally verifying hardware implementation of a divider/square-root can be attributed mostly to the modeling of its characteristic function and the high memory complexity required by standard algebraic approach. The work proposed in this thesis discusses formal verification of combinational divider and square-root circuits. Specifically, it addresses the problem of formally verifying gate-level circuits using an algebraic model. In contrast to standard verification approaches using satisfiability (SAT) or equivalence checking, the proposed method verifies whether the gate-level circuit actually performs the intended function or not, without a need for a reference design. Firstly, we present a verification methodology for a constant divider, where the divisor value is fixed to a constant integer. Albeit simpler case of verification, it provides us with the basic understanding of verification techniques and the underlying issues applicable to divider verification. Secondly, a layered verification approach is proposed for the verification of generic array dividers. Finally, the work proposed in this thesis will further analyze the divider and square-root circuits and aim to curb the memory explosion issue experienced by computer algebra based verification methods in order to successfully verify large bit-width divider-type arithmetic circuits. More specifically, a novel idea of hardware rewriting is introduced, which avoids the high memory complexity. The mentioned technique verifies a 256-bit gate-level square-root circuit with around 260,000 gates in just under 18 minutes and 127-bit gate-level divider circuit in under one minute

Algorithmik und Komplexität OBDD-repräsentierter Graphen

Ordered Binary Decision Diagrams (OBDDs) werden in vielen praktischen Anwendungsgebieten erfolgreich als Datenstruktur zur kompakten Repräsentation boolescher Funktionen eingesetzt. Auch sehr große Graphen werden in Bereichen wie CAD und Model Checking oft implizit durch boolesche Funktionen und OBDDs dargestellt. Diese Dissertation behandelt grundlegende graphtheoretische Probleme auf OBDD-repräsentierten Graphen und lotet die Möglichkeiten entsprechender OBDD-basierter Algorithmen aus. Zum einen werden neue Algorithmen vorgestellt und ihre Eigenschaften im Hinblick auf das Entwurfsziel sublinearer Heuristiken analysiert. Zum anderen werden Grenzen des Ansatzes durch komplexitätstheoretische Härteresultate und konkrete untere Schranken aufgezeigt

Proceedings of the 22nd Conference on Formal Methods in Computer-Aided Design – FMCAD 2022

The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing

Formal verification of pipelined microprocessors

Subject of this thesis is the formal verification of pipelined microprocessors. This includes processors with state of the art schedulers, such as the Tomasulo scheduler and speculation. In contrast to most of the literature, we verify synthesizable design at gate level. Furthermore, we prove both data consistency and liveness. We verify the proofs using the theorem proving system PVS. We verify both in-order and out-of-order machines. For verifying in-order machines, we extend the stall engine concept presented in [MP00]. We describe and implement an algorithm that does the transformation into a pipelined machine. We describe a generic machine that supports speculating on arbitraty values. We formally verify proofs for the Tomasulo scheduling algorithm with reorder buffer.Gegenstand dieser Dissertation ist die formale Verifikation von Mikroprozessoren mit Pipeline. Dies beinhaltet auch Prozessoren mit aktuellen Scheduling-Verfahren wie den Tomasulo Scheduler und spekulativer Ausfuehrung. Im Gegensatz zu weiten Teilen der bestehenden Literatur fuehren wir die Verifikation auf Gatter-Ebene durch. Des weitern beweisen wir sowohl Datenkonsistenz als auch eine obere Schranke fuer die Ausfuehrungszeit. Die Beweise werden mit dem Theorem Beweissystem PVS verifiziert. Es werden sowohl in-order Maschinen als auch out-of-order Maschinen verifiziert. Zur Verifikation der in-order Maschinen erweitern wir die Stall Engine aus [MP00]. Wir beschreiben und Implementieren ein Verfahren das die Transformation in die "pipelined machine\u27; durchfuehrt. Wir beschreiben eine generische Maschine die Spekulation auf beliebige Werte erlaubt. Wir verifizieren die Beweise fuer den Tomasulo Scheduler mit Reorder Buffer

Formale Verifikation von Mikroprozessoren mit Pipeline

Subject of this thesis is the formal verification of pipelined microprocessors. This includes processors with state of the art schedulers, such as the Tomasulo scheduler and speculation. In contrast to most of the literature, we verify synthesizable design at gate level. Furthermore, we prove both data consistency and liveness. We verify the proofs using the theorem proving system PVS. We verify both in-order and out-of-order machines. For verifying in-order machines, we extend the stall engine concept presented in [MP00]. We describe and implement an algorithm that does the transformation into a pipelined machine. We describe a generic machine that supports speculating on arbitraty values. We formally verify proofs for the Tomasulo scheduling algorithm with reorder buffer.Gegenstand dieser Dissertation ist die formale Verifikation von Mikroprozessoren mit Pipeline. Dies beinhaltet auch Prozessoren mit aktuellen Scheduling-Verfahren wie den Tomasulo Scheduler und spekulativer Ausfuehrung. Im Gegensatz zu weiten Teilen der bestehenden Literatur fuehren wir die Verifikation auf Gatter-Ebene durch. Des weitern beweisen wir sowohl Datenkonsistenz als auch eine obere Schranke fuer die Ausfuehrungszeit. Die Beweise werden mit dem Theorem Beweissystem PVS verifiziert. Es werden sowohl in-order Maschinen als auch out-of-order Maschinen verifiziert. Zur Verifikation der in-order Maschinen erweitern wir die Stall Engine aus [MP00]. Wir beschreiben und Implementieren ein Verfahren das die Transformation in die "pipelined machine'; durchfuehrt. Wir beschreiben eine generische Maschine die Spekulation auf beliebige Werte erlaubt. Wir verifizieren die Beweise fuer den Tomasulo Scheduler mit Reorder Buffer

Proceedings of the 22nd Conference on Formal Methods in Computer-Aided Design – FMCAD 2022

The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing