designer can often repair design errors without changing the arrangement of wires. Therefore, in small volume products, programmable logic devices are often the preferred solution. They are usually designed by engineers using electronic design automation software. Integrated circuits consist of multiple transistors on one silicon chip, and are the least expensive way to make large number of interconnected logic gates. Integrated circuits are usually interconnected on a printed circuit board which is a board which holds electrical components, and connects them together with copper traces. Design Engineers use many methods to minimize logic redundancy in order to reduce the circuit complexity. Reduced complexity reduces component count and potential errors and therefore typically reduces cost. Logic redundancy can be removed by several well-known techniques, such as binary decision diagrams, Boolean algebra, Karnaugh maps, the Quine–McCluskey algorithm, and the heuristic computer method. These operations are typically performed within a computer-aided design system. Embedded systems with microcontrollers and programmable logic controllers are often used to implement digital logic for complex systems that don't require optimal performance. These systems are usually programmed by software engineers or by electricians, using ladder logic. Representation A digital circuit's input-output relationship can be represented as a truth table. An equivalent high-level circuit uses logic gates, each represented by a different shape (standardized by IEEE/ANSI 91-1984). A low-level representation uses an equivalent circuit of electronic switches (usually transistors). Most digital systems divide into combinational and sequential systems. The output of a combinational system depends only on the present inputs. However, a sequential system has some of its outputs fed back as inputs, so its output may depend on past inputs in addition to present inputs, to produce a sequence of operations. Simplified representations of their behavior called state machines facilitate design and test. Sequential systems divide into two further subcategories. "Synchronous" sequential systems change state all at once when a clock signal changes state. "Asynchronous" sequential systems propagate changes whenever inputs change. Synchronous sequential systems are made using flip flops that store inputted voltages as a bit only when the clock changes. Synchronous systems The usual way to implement a synchronous sequential state machine is to divide it into a piece of combinational logic and a set of flip flops called a state register. The state register represents the state as a binary number. The combinational logic produces the binary representation for the next state. On each clock cycle,