Doctor of Philosophy

dissertationFormal verification of hardware designs has become an essential component of the overall system design flow. The designs are generally modeled as finite state machines, on which property and equivalence checking problems are solved for verification. Reachability analysis forms the core of these techniques. However, increasing size and complexity of the circuits causes the state explosion problem. Abstraction is the key to tackling the scalability challenges. This dissertation presents new techniques for word-level abstraction with applications in sequential design verification. By bundling together k bit-level state-variables into one word-level constraint expression, the state-space is construed as solutions (variety) to a set of polynomial constraints (ideal), modeled over the finite (Galois) field of 2^k elements. Subsequently, techniques from algebraic geometry -- notably, Groebner basis theory and technology -- are researched to perform reachability analysis and verification of sequential circuits. This approach adds a "word-level dimension" to state-space abstraction and verification to make the process more efficient. While algebraic geometry provides powerful abstraction and reasoning capabilities, the algorithms exhibit high computational complexity. In the dissertation, we show that by analyzing the constraints, it is possible to obtain more insights about the polynomial ideals, which can be exploited to overcome the complexity. Using our algorithm design and implementations, we demonstrate how to perform reachability analysis of finite-state machines purely at the word level. Using this concept, we perform scalable verification of sequential arithmetic circuits. As contemporary approaches make use of resolution proofs and unsatisfiable cores for state-space abstraction, we introduce the algebraic geometry analog of unsatisfiable cores, and present algorithms to extract and refine unsatisfiable cores of polynomial ideals. Experiments are performed to demonstrate the efficacy of our approaches

Formal verification of a fully IEEE compliant floating point unit

In this thesis we describe the formal verification of a fully IEEE compliant floating point unit (FPU). The hardware is verified on the gate-level against a formalization of the IEEE standard. The verification is performed using the theorem proving system PVS. The FPU supports both single and double precision floating point numbers, normal and denormal numbers, all four IEEE rounding modes, and exceptions as required by the standard. Beside the verification of the combinatorial correctness of the FPUs we pipeline the FPUs to allow the integration into an out-of-order processor. We formally define the correctness criterion the pipelines must obey in order to work properly within the processor. We then describe a new methodology based on combining model checking and theorem proving for the verification of the pipelines.Die vorliegende Arbeit behandelt die formale Verifikation einer vollständig IEEE konformen Floating Point Unit (FPU). Die Hardware wird auf Gatter-Ebene gegen eine Formalisierung des IEEE Standards verifiziert. Zur Verifikation wird das Beweis-System PVS benutzt. Die FPU unterstützt Fließkommazahlen mit einfacher und doppelter Genauigkeit, normale und denormale Zahlen, alle vier Rundungsmodi und alle Exception-Signale. Neben der Verifikation der kombinatorischen Schaltkreise werden die FPUs gepipelined, um sie in einen Out-of-order Prozessor zu integrieren. Die Korrektheits- Kriterien, die die gepipelineten FPUs befolgen müssen, um im Prozessor korrekt zu arbeiten, werden formal definiert. Es wird eine neue Methode zur Verifikation solcher Pipelines beschrieben. Die Methode beruht auf der Kombination von Model-Checking und Theorem-Proving

Doctor of Philosophy

dissertationWith the spread of internet and mobile devices, transferring information safely and securely has become more important than ever. Finite fields have widespread applications in such domains, such as in cryptography, error correction codes, among many others. In most finite field applications, the field size - and therefore the bit-width of the operands - can be very large. The high complexity of arithmetic operations over such large fields requires circuits to be (semi-) custom designed. This raises the potential for errors/bugs in the implementation, which can be maliciously exploited and can compromise the security of such systems. Formal verification of finite field arithmetic circuits has therefore become an imperative. This dissertation targets the problem of formal verification of hardware implementations of combinational arithmetic circuits over finite fields of the type F2k . Two specific problems are addressed: i) verifying the correctness of a custom-designed arithmetic circuit implementation against a given word-level polynomial specification over F2k ; and ii) gate-level equivalence checking of two different arithmetic circuit implementations. This dissertation proposes polynomial abstractions over finite fields to model and represent the circuit constraints. Subsequently, decision procedures based on modern computer algebra techniques - notably, Gr¨obner bases-related theory and technology - are engineered to solve the verification problem efficiently. The arithmetic circuit is modeled as a polynomial system in the ring F2k [x1, x2, · · · , xd], and computer algebrabased results (Hilbert's Nullstellensatz) over finite fields are exploited for verification. Using our approach, experiments are performed on a variety of custom-designed finite field arithmetic benchmark circuits. The results are also compared against contemporary methods, based on SAT and SMT solvers, BDDs, and AIG-based methods. Our tools can verify the correctness of, and detect bugs in, up to 163-bit circuits in F2163 , whereas contemporary approaches are infeasible beyond 48-bit circuits

Formal verification of a fully IEEE compliant floating point unit

In this thesis we describe the formal verification of a fully IEEE compliant floating point unit (FPU). The hardware is verified on the gate-level against a formalization of the IEEE standard. The verification is performed using the theorem proving system PVS. The FPU supports both single and double precision floating point numbers, normal and denormal numbers, all four IEEE rounding modes, and exceptions as required by the standard. Beside the verification of the combinatorial correctness of the FPUs we pipeline the FPUs to allow the integration into an out-of-order processor. We formally define the correctness criterion the pipelines must obey in order to work properly within the processor. We then describe a new methodology based on combining model checking and theorem proving for the verification of the pipelines.Die vorliegende Arbeit behandelt die formale Verifikation einer vollständig IEEE konformen Floating Point Unit (FPU). Die Hardware wird auf Gatter-Ebene gegen eine Formalisierung des IEEE Standards verifiziert. Zur Verifikation wird das Beweis-System PVS benutzt. Die FPU unterstützt Fließkommazahlen mit einfacher und doppelter Genauigkeit, normale und denormale Zahlen, alle vier Rundungsmodi und alle Exception-Signale. Neben der Verifikation der kombinatorischen Schaltkreise werden die FPUs gepipelined, um sie in einen Out-of-order Prozessor zu integrieren. Die Korrektheits- Kriterien, die die gepipelineten FPUs befolgen müssen, um im Prozessor korrekt zu arbeiten, werden formal definiert. Es wird eine neue Methode zur Verifikation solcher Pipelines beschrieben. Die Methode beruht auf der Kombination von Model-Checking und Theorem-Proving

Unbounded Scalable Hardware Verification.

Model checking is a formal verification method that has been successfully applied to real-world hardware and software designs. Model checking tools, however, encounter the so-called state-explosion problem, since the size of the state spaces of such designs is exponential in the number of their state elements. In this thesis, we address this problem by exploiting the power of two complementary approaches: (a) counterexample-guided abstraction and refinement (CEGAR) of the design's datapath; and (b) the recently-introduced incremental induction algorithms for approximate reachability. These approaches are well-suited for the verification of control-centric properties in hardware designs consisting of wide datapaths and complex control logic. They also handle most complex design errors in typical hardware designs. Datapath abstraction prunes irrelevant bit-level details of datapath elements, thus greatly reducing the size of the state space that must be analyzed and allowing the verification to be focused on the control logic, where most errors originate. The induction-based approximate reachability algorithms offer the potential of significantly reducing the number of iterations needed to prove/disprove given properties by avoiding the implicit or explicit enumeration of reachable states. Our implementation of this verification framework, which we call the Averroes system, extends the approximate reachability algorithms at the bit level to first-order logic with equality and uninterpreted functions. To facilitate this extension, we formally define the solution space and state space of the abstract transition system produced by datapath abstraction. In addition, we develop an efficient way to represent sets of abstract solutions involving present- and next-states and a systematic way to project such solutions onto the space of just the present-state variables. To further increase the scalability of the Averroes verification system, we introduce the notion of structural abstraction, which extends datapath abstraction with two optimizations for better classification of state variables as either datapath or control, and with efficient memory abstraction techniques. We demonstrate the scalability of this approach by showing that Averroes significantly outperforms bit-level verification on a number of industrial benchmarks.PhDComputer Science and EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133375/1/suholee_1.pd

Lossy Polynomial Datapath Synthesis

The design of the compute elements of hardware, its datapath, plays a crucial role in determining the speed, area and power consumption of a device. The building blocks of datapath are polynomial in nature. Research into the implementation of adders and multipliers has a long history and developments in this area will continue. Despite such efficient building block implementations, correctly determining the necessary precision of each building block within a design is a challenge. It is typical that standard or uniform precisions are chosen, such as the IEEE floating point precisions. The hardware quality of the datapath is inextricably linked to the precisions of which it is composed. There is, however, another essential element that determines hardware quality, namely that of the accuracy of the components. If one were to implement each of the official IEEE rounding modes, significant differences in hardware quality would be found. But in the same fashion that standard precisions may be unnecessarily chosen, it is typical that components may be constructed to return one of these correctly rounded results, where in fact such accuracy is far from necessary. Unfortunately if a lesser accuracy is permissible then the techniques that exist to reduce hardware implementation cost by exploiting such freedom invariably produce an error with extremely difficult to determine properties. This thesis addresses the problem of how to construct hardware to efficiently implement fixed and floating-point polynomials while exploiting a global error freedom. This is a form of lossy synthesis. The fixed-point contributions include resource minimisation when implementing mutually exclusive polynomials, the construction of minimal lossy components with guaranteed worst case error and a technique for efficient composition of such components. Contributions are also made to how a floating-point polynomial can be implemented with guaranteed relative error.Open Acces

Hardware Acceleration Using Functional Languages

Cílem této práce je prozkoumat možnosti využití funkcionálního paradigmatu pro hardwarovou akceleraci, konkrétně pro datově paralelní úlohy. Úroveň abstrakce tradičních jazyků pro popis hardwaru, jako VHDL a Verilog, přestáví stačit. Pro popis na algoritmické či behaviorální úrovni se rozmáhají jazyky původně navržené pro vývoj softwaru a modelování, jako C/C++, SystemC nebo MATLAB. Funkcionální jazyky se s těmi imperativními nemůžou měřit v rozšířenosti a oblíbenosti mezi programátory, přesto je předčí v mnoha vlastnostech, např. ve verifikovatelnosti, schopnosti zachytit inherentní paralelismus a v kompaktnosti kódu. Pro akceleraci datově paralelních výpočtů se často používají jednotky FPGA, grafické karty (GPU) a vícejádrové procesory. Praktická část této práce rozšiřuje existující knihovnu Accelerate pro počítání na grafických kartách o výstup do VHDL. Accelerate je možno chápat jako doménově specifický jazyk vestavěný do Haskellu s backendem pro prostředí NVIDIA CUDA. Rozšíření pro vysokoúrovňovou syntézu obvodů ve VHDL představené v této práci používá stejný jazyk a frontend.The aim of this thesis is to research how the functional paradigm can be used for hardware acceleration with an emphasis on data-parallel tasks. The level of abstraction of the traditional hardware description languages, such as VHDL or Verilog, is becoming to low. High-level languages from the domains of software development and modeling, such as C/C++, SystemC or MATLAB, are experiencing a boom for hardware description on the algorithmic or behavioral level. Functional Languages are not so commonly used, but they outperform imperative languages in verification, the ability to capture inherent paralellism and the compactness of code. Data-parallel task are often accelerated on FPGAs, GPUs and multicore processors. In this thesis, we use a library for general-purpose GPU programs called Accelerate and extend it to produce VHDL. Accelerate is a domain-specific language embedded into Haskell with a backend for the NVIDIA CUDA platform. We use the language and its frontend, and create a new backend for high-level synthesis of circuits in VHDL.

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