# Research

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"You see things; and you say, 'Why?' But I dream things that never were; and I say, 'Why not?'"–– George Bernard Shaw (1856 –1950)

Our research spans different disciplines ranging from digital circuit design, to algorithms, to mathematics, to synthetic biology. It tends to be inductive (as opposed to deductive) and conceptual (as opposed to applied). A recurring theme is building systems that compute in novel or unexpected ways with new and emerging technologies. Often, the task of analyzing the way things work in a new technology is straightforward; however the task of synthesizing new computational constructs is more challenging.

## Computing with Random Bit Streams

"To invent, all you need is a pile of junk and a good imagination." –– Thomas A. Edison (1847–1931)

Humans are accustomed to counting in a positional number system – decimal radix. Nearly all computer systems operate on another positional number system – binary radix. From the standpoint of representation, such positional systems are compact: given a radix b, one can represent bn distinct numbers with n digits. However, from the standpoint of computation, positional systems impose a burden: for each operation such as addition or multiplication, the signal must be "decoded", with each digit weighted according to its position. The result must be "encoded" back in positional form. Any student who has designed a binary multiplier in a course on logic design can appreciate all the complexity that goes into wiring up such an operation.

#### Logic that Operates on Probabilities

We advocate an alternative representation: random bit streams where the signal value is encoded by the probability of obtaining a one versus a zero. This representation is much less compact than binary radix. However, complex operations can be performed with very simple logic. For instance, multiplication can be performed with a single AND gate; scaled addition can be performed with a multiplexer (MUX).

 Multiplication with an AND gate. Here the variables represents the probabilities of obtaining a 1 versus a 0 in stochastic bit streams. The AND gate produces an output probability that is the product of the of the input probabilities and . Scaled addition with a multiplexer (MUX). Given input probabilities , and , the MUX produces an output probability .

We have developed a general method for synthesizing digital circuitry that computes on such stochastic bit streams. Our method can be used to synthesize arbitrary polynomial functions. Through polynomial approximations, it can also be used to synthesize non-polynomial functions. Because the representation is uniform, with all bits weighted equally, the resulting circuits are highly tolerant of soft errors (i.e., bit flips).

 title: Synthesizing Logical Computation on Stochastic Bit Streams authors: Weikang Qian and Marc Riedel appeared as: Techincal Report, UMN
 title: Logical Computation on Stochastic Bit Streams with Linear Finite State Machines authors: Peng Li, David Lilja, Weikang Qian,Kia Bazargan and Marc Riedel appeared in: IEEE Transactions on Computers, Vol. 63, No. 6., pp. 1474–1486, 2014 presented at: IEEE/ACM Asia and South Pacific Design Automation Conference,Sydney, Australia, 2012

#### Logic that Generates Probabilities

Schemes for probabilistic computation can exploit physical sources to generate random values in the form of bit streams. Generally, each source has a fixed bias and so provides bits that have a specific probability of being one versus zero. If many different probability values are required, it can be difficult or expensive to generate all of these directly from physical sources. In this work, we demonstrate novel techniques for synthesizing combinational logic that transforms a set of source probabilities into different target probabilities.

Given a set S of source probabilities {0.4, 0.5}, we can synthesize a combinational circuit to generate an arbitrary decimal output probability. The example shows how to generate 0.119. Each AND gate performs a multiplication and each inverter performs a "one-minus" operation.
 title: Transforming Probabilities with Combinational Logic authors: Weikang Qian, Marc Riedel, Hongchao Zhou, and Jehoshua Bruck will appear in: IEEE Trans. on Computer-Aided Design of Integrated Circuits and Systems, 2012. presented at: International Conference on Computer-Aided Design, San Jose, 2009(nominated for IEEE/ACM William J. McCalla ICCAD Best Paper Award).

#### Computing with Crappy Clocks

Clock distribution networks are a significant source of power consumption and a major design bottleneck for high-performance circuits. We have proposed a radically new approach: splitting clock domains at a very fine level, with domains consisting of only a handful of gates each. These domains are synchrnonized by "crappy clocks", generated locally with inverter rings. This is feasible if one adopts the paradigm of computing on randomized bit streams.

Stochastic multiplication using an AND gate with unsynchronized random bit streams. The stochastic paradigm can tolerate arbitrarly high clock skew. Accordingly, one can replace an expensive global clock distribution network with cheap local clocks, generated by inverter rings – "crappy clocks".
 title: Polysynchronous Stochastic Circuits authors: M. Hassan_Najafi, David Lilja, Marc Riedel, and Kia Bazargan to appear in: IEEE/ACM Asia and South Pacific Design Automation Conference, 2016

Please see our "Publications" page for more of our papers on these topics.

## Computing with Molecules

If I can’t create it, I don’t understand it.” –– Richard Feynman (1918–1988)

The theory of mass-action kinetics underpins our understanding of biological and chemical systems. It is a simple and elegant formalism: molecular reactions define rules according to which reactants form products; each rule fires at a rate that is proportional to the quantities of the corresponding reactants that are present. Just as electronic systems implement computation in terms of voltage (energy per unit charge), we can conceive of molecular systems that compute in terms of chemical concentrations (molecules per unit volume). We are studying techniques for implementing a variety of computational constructs with molecular reactions such as logic, memory, arithmetic, and signal processing. Although conceptual, we target DNA Strand Displacement as our experimental chassis.

 Molecular reactions define rules according to which reactants form products. Here molecules of type A combine with molecules of type B to form molecules of type C, at a rate proportional to the molecular concentrations of A and B as well as a rate constant k. We map abstract molecular reactions to DNA reactions. Through a process called DNA strand displacement, single strands of DNA displace parts of double strands, releasing other single strands.

Computational Constructs

We have developed a strategy for implementing digital logic with molecular reactions. Based on a bistable mechanism for representing bits, we implement a constituent set of logical components, including combinational components such as AND, OR, and XOR gates, as well as sequential components such as D latches and D flip-flops. Using these components, we build full-fledged digital circuits such as a binary counters and linear feedback shift registers.

 title: Digital Logic with Molecular Reactions authors: Hua Jiang, Marc Riedel, Keshab Parhi presented at: The International Conference on Computer-Aided Design, San Jose, CA, 2013.

We have developed a strategy for implementing arithmetic with molecular reactions – operations such as increments & decrements, multiplication, logarithms, and exponentiation. Try out our compiler: it translates arbitrary constructs from a C-like language into a robust implementation with molecular reactions.

 title: Rate-Independent Constructs for Chemical Computation authors: Phil Senum and Marc Riedel appeared in: PLoS ONE, Vol. 6, No. 6, 2011. Supplementary Information

We have developed a strategy for implementing signal processing with molecular reactions including operations such as filtering. We have demonstrated robust designs for Finite-Impulse Response (FIR), Infinite-Impulse Response (IIR) filters, and Fast Fourier Transforms (FFTs).

 title: Discrete-Time Signal Processing with DNA authors: Hua Jiang, Ahmed Salehi, Marc Riedel and Keshab Parhi appeared in: ACS Synthetic Biology, Vol. 2 no. 5, pp. 245–254, 2013. Supplementary Information: List of Reactions appeared in: IEEE Design & Test of Computers, Vol. 29, No. 3, pp. 21–31, 2012. presented at: IEEE/ACM International Conference on Computer-Aided Design, San Jose, CA, 2010. presented at: IEEE Workshop on Signal Processing Systems, San Francisco, 2010

 Simulations of DNA implementation of logic gates. The input signals are molecular concentrations X and Y; the output signal is a molecular concentration Z. (A) AND gate. (B) OR gate. (C) NOR gate. (D) XOR gate. Simulations of DNA implementation of a moving-average FIR filter. This filter removes the high-frequency component from an input signal, producing an output signal consisting of only the low-frequency component. Here the "signals" are molecular concentrations.

The impetus for this research is not computation per se. Molecular computation will never compete with conventional computers made of silicon integrated circuits for tasks such as number crunching. Chemical systems are inherently slow and messy, taking minutes or even hours to finish, and producing fragmented results. Rather, the goal is to create “embedded controllers” – viruses and bacteria that are engineered to perform useful molecular computation in situ where it is needed, for instance for drug delivery and biochemical sensing.

Molecular computation is applicable to the design of embedded controllers: engineered bacteria and viruses for tasks such as drug delivery.

Please see our "Publications" page for more of our papers on these topics.

## Computing with Nanoscale Lattices

"Listen to the technology; find out what it’s telling you.” –– Carver Mead (1934–  )

In his seminal Master's Thesis, Claude Shannon made the connection between Boolean algebra and switching circuits. He considered two-terminal switches corresponding to electromagnetic relays. A Boolean function can be implemented in terms of connectivity across a network of switches, often arranged in a series/parallel configuration. We have developed a method for synthesizing Boolean functions with networks of four-terminal switches. Our model is applicable for variety of nanoscale technologies, such as nanowire crossbar arrays, as molecular switch-based structures.

 Shannon's model: two-terminal switches. Each switch is either ON (closed) or OFF (open). A Boolean function is implemented in terms of connectivity across a network of switches, between the source S and the drain D. Our model: four-terminal switches. Each switch is either mutually connected to its neighbors (ON) or disconnected (OFF). A Boolean function is implemented in terms of connectivity between the top and bottom plates. This network implements the same function as the two-terminal network on the left.
 title: Logic Synthesis for Switching Lattices authors: Mustafa Altun and Marc Riedel will appear in: IEEE Transactions on Computers, 2011. presented at: Design Automation Conference, Anaheim, CA, 2010.

The impetus for nanowire-based technology is the potential density, scalability and manufacturability. Many other novel and emerging technologies fit the general model of four-terminal switches. For instance, researchers are investigating spin waves. A common feature of many emerging technologies for switching networks is that they exhibit high defect rates.

 A nanowire crossbar switch. The connections between horizontal and vertical wires are FET-like junctions. When high or low voltages are applied to input nanowires, the FET-like junctions that cross these develop a high or low impedance, respectively. In a switching network with defects, percolation can be exploited to produce robust Boolean functionality. Unless the defect rate exceeds an error margin, with high probability no connection forms between the top and bottom plates for logical zero ("OFF"); with high probability, a connection forms for logical one ("ON").

We have devised a novel framework for digital computation with lattices of nanoscale switches with high defect rates, based on the mathematical phenomenon of percolation. With random connectivity, percolation gives rise to a sharp non-linearity in the probability of global connectivity as a function of the probability of local connectivity. We exploit this phenomenon to compute Boolean functions robustly in the presence of defects.

 title: Synthesizing Logic with Percolation in Nanoscale Lattices authors: Mustafa Altun and Marc Riedel appeared in: International Journal of Nanotechnology and Molecular Computation, Vol. 3, No. 2, pp. 12–30, 2011. presented at: Design Automation Conference, San Francisco, CA, 2009.

Please see our "Publications" page for more of our papers on these topics.

## Computing with Feedback

"A person with a new idea is a crank until the idea succeeds." –– Mark Twain (1835–1910)

The accepted wisdom is that combinational circuits (i.e., memoryless circuits) must have acyclic (i.e., loop-free or feed-forward) topologies. And yet simple examples suggest that this need not be so. We advocate the design of cyclic combinational circuits (i.e., circuits with loops or feedback paths). We have proposed a methodology for synthesizing such circuits and demonstrated that it produces significant improvements in area and in delay.

A circuit that has feedback and yet is combinational.
 title: Cyclic Boolean Circuits authors: Marc Riedel and Shuki Bruck appeared  in: Discrete Applied Mathematics, Vol. 160, No. 13–14, pp. 1877–1900, 2011. dissertation: Ph.D., Electrical Engineering, Caltech, 2004(winner of Charles H. Wilts Prize for the Best Ph.D. Dissertation in EE at Caltech). presented at: Design Automation Conference, Anahiem, CA, 2003(winner of DAC Best Paper Award).

Please see our Publications page for more of our papers on this topic.

## Algorithms and Data Structures

$\displaystyle \frac{1}{2} "''There are two kinds of people in the world: those who divide the world into two kinds of people, and those who don't.''" –– '''Robert Charles Benchley''' (1889–1945) Consider the task of designing a digital circuit with 256 inputs. From a mathematical standpoint, such a circuit performs mappings from a space of [itex]2^{256}$ Boolean input values to Boolean output values. (The number of rows in a truth table for such a function is approximately equal to the number of atoms in the universe rows versus atoms!) Verifying such a function, let alone designing the corresponding circuit, would seem to be an intractable problem. Circuit designers have succeeded in their endeavor largely as a result of innovations in the data structures and algorithms used to represent and manipulate Boolean functions. We have developed novel, efficient techniques for synthesizing functional dependencies based on so-called SAT-solving algorithms. We use Craig Interpolation to generate circuits from the corresponding Boolean functions.

 A circuit construct for SAT-based verification. A squid.
 title: Reduction of Interpolants For Logic Synthesis authors: John Backes and Marc Riedel presented at: The International Conference on Computer-Aided Design, San Jose, CA, 2010.

Please see our "Publications" page for more of our papers on this topic. (Papers on SAT-based circuit verification, that is, not on squids.)

## Mathematics

"Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." –– Bertrand Russell (1872–1970)

The great mathematician John von Neumann articulated the view that research should never meander too far down theoretical paths; it should always be guided by potential applications. This view was not based on concerns about the relevance of his profession; rather, in his judgment, real-world applications give rise to the most interesting problems for mathematicians to tackle. At their core, most of our research contributions are mathematical contributions. The tools of our trade are discrete math, including combinatorics and probability theory.

Mathematics, before the era of LaTeX.
 title: Uniform Approximation and Bernstein Polynomials withCoefficients in the Unit Interval authors: Weikang Qian, Marc Riedel, and Ivo Rosenberg appeared in: European Journal of Combinatorics, Vol. 32, No. 3, pp. 448–463, 2011.

Please see our "Publications" page for more of our papers on this topic.