History of Logic Synthesis
The roots of logic synthesis can be traced to the treatment of logic by George Boole (1815 to 1864), in what is now termed Boolean algebra. In 1938, Claude Shannon showed that the two-valued Boolean algebra can describe the operation of switching circuits. In the early days, logic design involved manipulating the truth table representations as Karnaugh maps. The Karnaugh map-based minimization of logic is guided by a set of rules on how entries in the maps can be combined. A human designer can only work with Karnaugh maps containing four to six variables.
The first step toward automation of logic minimization was the introduction of the Quine-McCluskey algorithm that could be implemented on a computer. This exact minimization technique presented the notion of prime implicants and minimum cost covers that would become the cornerstone of two-level minimization. Another area of early research was in state minimization and encoding of finite state machines (FSMs), a task that was the bane of designers. The applications for logic synthesis lay primarily in digital computer design. Hence, IBM and Bell Labs played a pivotal role in the early automation of logic synthesis. The evolution from discrete logic components to programmable logic arrays (PLAs) hastened the need for efficient two-level minimization, since minimizing terms in a two-level representation reduces the area in a PLA.
However, two-level logic circuits are of limited importance in a very-large-scale integration (VLSI) design; most designs use multiple levels of logic. An early system that was used to design multilevel circuits was LSS from IBM. It used local transformations to simplify logic. Work on LSS and the Yorktown Silicon Compiler spurred rapid research progress in logic synthesis in the 1980s. Several universities contributed by making their research available to the public; most notably, MIS  from University of California, Berkeley and BOLD from University of Colorado, Boulder. Within a decade, the technology migrated to commercial logic synthesis products offered by electronic design automation companies.
With the goal of increasing designer productivity, there has been a significant amount of research on synthesis of circuits specified at the behavioral level using a hardware description language (HDL). The goal of behavioral synthesis is to transform a behavioral HDL specification into a register transfer level (RTL) specification, which can be used as input to a gate-level logic synthesis flow. Behavioral optimization decisions are guided by cost functions that are based on the number of hardware resources and states required. These cost functions provide a coarse estimate of the combinational and sequential circuitry required to implement the design.
The tasks of scheduling, resource allocation, and sharing generate the FSM and the datapath of the RTL description of the design. Scheduling assigns operations to points in time, while allocation assigns each operation or variable to a hardware resource. Given a schedule, the allocation operation optimizes the amount of hardware required to implement the design.
Multi Level Logic Minimization
Typical practical implementations of a logic function utilize a multilevel network of logic elements. Starting from an RTL description of a design, the synthesis tool constructs a corresponding multilevel Boolean network.
Next, this network is optimized using several technology-independent techniques before technology-dependent optimizations are performed. The typical cost function during technology-independent optimizations is total literal count of the factored representation of the logic function (which correlates quite well with circuit area).
Finally, technology-dependent optimization transforms the technology-independent circuit into a network of gates in a given technology. The simple cost estimates are replaced by more concrete, implementation-driven estimates during and after technology mapping. Mapping is constrained by factors such as the available gates (logic functions) in the technology library, the drive sizes for each gate, and the delay, power, and area characteristics of each gate.
Commercial logic synthesis
Examples of software tools for logic synthesis are Design Compiler from Synopsys and the humorously named BuildGates, from Cadence Design Systems. Both of these target ASICs. Example of FPGA synthesis tools include Synplify from Synplicity, Leonardo and Precision from Mentor Graphics and BlastFPGA from Magma Design Automation.