Davide Anguita

Boolean Algebra and Combinational Logic

This chapter introduces to the idea of digitally representing analog quantities and goes step by step through the main concepts of the Boolean algebra: variables, functions, truth tables, operations, and properties. The chapter is quite detailed and accompanied by many examples and exercises in order to provide a precise framework of the fundamentals of digital design. It includes the theorems which constitute the foundation for the application of the Boolean algebra to logic networks, with a precise focus on their application for combinational network design.

Combinational Network Design

This chapter deals with the transition from Boolean algebra to the implementation of combinational networks. Karnaugh maps provide a simple and intuitive method to represent and minimize functions with a few variables. The Variable-Entered Maps extend their usefulness and overcome some of their limitation. Standard networks such as decoders, multiplexers, and demultiplexers provide a wider view of combinational circuits, where the random approach of classical synthesis is enriched with an architectural one that introduces the concepts of programmable logic. Finally, this chapter deals with time behavior of non-ideal components and its implications on the synthesis.

Introduction to Sequential Networks

The transition from combinational to sequential networks is explained step by step, starting from a simple gate with feedback and arriving to the structure and behavior of the principal types of flip-flops. They are classified according to their temporal response (direct command, level enabled, master–slave, and edge triggered) and the logical operation (SR, D, JK). The timing parameters of physically implemented devices are considered. The chapter introduces the concept and techniques for synchronization that will be further examined in the following ones.

Sequential Networks as Finite State Machines

The term Finite State Machine indicates a regular and standard structure that is able to describe and synthesize sequential networks in a general fashion. Algorithmic State Machine (ASM) is the tool adopted in the book and developed through the chapter. The attention is focused on synchronous stand-alone machines, their properties, and timing behavior.

Numeral Systems and Binary Arithmetic

The representation of numbers is essential for the digital logic design. In this chapter, positional number systems (decimal, binary, octal, hexadecimal), BCD and Gray codes are presented together with the rules for the conversion between numbers encoded in different bases and the representations of negative numbers. Then, the rules for the arithmetic operations and the circuits that execute them are presented. The addition of binary number is examined with particular attention, since it is the operation at the basis of all computational circuits. Alphanumeric codes and the concept of parity for error detection complete the chapter.

Introduction to FPGA and HDL Design

The last chapter deals with the practical implementation in hardware of systems similar to the ones presented in previous chapters and tested by simulation only. The devices that host the projects are Field-Programmable Gate Arrays (FPGAs), inserted on commercially available boards and managed by Deeds and proprietary tools. A short description of the devices and the associated tools is presented. An original, hands-on introduction of the VHDL hardware description language is included. A few exercises of digital system design and prototyping complete the chapter.

Flip-Flop-Based Synchronous Networks

The flip-flops are the building blocks of all sequential networks. A regular structure made of flip-flop and combinational networks can implement any sequential circuit. In this chapter, the structures are not designed but either assembled in an intuitive fashion or taken from standard building blocks. The presentation of counters and registers introduces progressively the real full-featured components that are available for design. Sequential network analysis in the time domain is the important skill that is developed at the end of the chapter.

The Finite State Machine as System Controller

The Finite State Machine can implement any algorithm, but it becomes over complicated when dealing with data. It is therefore convenient to include in digital systems other components that are more efficient to process and memorize data under the machines control. Such systems are called controller and datapath and described in this chapter. They optimize the sharing of duties between Finite State Machine and an external architecture. They are the first choice for the design of a wide variety of systems.

Complements in Combinational Network Design

In this chapters, we overcome the limitations of the Karnaugh maps, whose application becomes impractical when applied to expressions with more than four/five variables. We present the Quine–McCluskey method, the first algorithms for minimizing Boolean expressions developed by Willard V. Quine and improved by Edward J. McCluskey. We present both the methods for synthesize single and multiple functions at the same time.

Randomized learning: Generalization performance of old and new theoretically grounded algorithms

Abstract In the context of assessing the generalization abilities of a randomized model or learning algorithm, PAC-Bayes and Differential Privacy (DP) theories are the state-of-the-art tools. For this reason, in this paper, we will develop tight DP-based generalization bounds, which improve over the current state-of-the-art ones both in terms of constants and rate of convergence. Moreover, we will also prove that some old and new randomized algorithm, show better generalization performances with respect to their non private counterpart, if the DP is exploited for assessing their generalization ability. Results on a series of algorithms and real world problems show the practical validity of the achieved theoretical results.