One-way permutations, computational asymmetry and distortion

Computational asymmetry, i.e., the discrepancy between the complexity of transformations and the complexity of their inverses, is at the core of one-way transformations. We introduce a computational asymmetry function that measures the amount of one-wayness of permutations. We also introduce the word-length asymmetry function for groups, which is an algebraic analogue of computational asymmetry. We relate boolean circuits to words in a Thompson monoid, over a fixed generating set, in such a way that circuit size is equal to word-length. Moreover, boolean circuits have a representation in terms of elements of a Thompson group, in such a way that circuit size is polynomially equivalent to word-length. We show that circuits built with gates that are not constrained to have fixed-length inputs and outputs, are at most quadratically more compact than circuits built from traditional gates (with fixed-length inputs and outputs). Finally, we show that the computational asymmetry function is closely related to certain distortion functions: The computational asymmetry function is polynomially equivalent to the distortion of the path length in Schreier graphs of certain Thompson groups, compared to the path length in Cayley graphs of certain Thompson monoids. We also show that the results of Razborov and others on monotone circuit complexity lead to exponential lower bounds on certain distortions.Comment: 33 page

Behavioral synthesis from VHDL using structured modeling

This dissertation describes work in behavioral synthesis involving the development of a VHDL Synthesis System VSS which accepts a VHDL behavioral input specification and performs technology independent synthesis to generate a circuit netlist of generic components. The VHDL language is used for input and output descriptions. An intermediate representation which incorporates signal typing and component attributes simplifies compilation and facilitates design optimization.A Structured Modeling methodology has been developed to suggest standard VHDL modeling practices for synthesis. Structured modeling provides recommendations for the use of available VHDL description styles so that optimal designs will be synthesized.A design composed of generic components is synthesized from the input description through a process of Graph Compilation, Graph Criticism, and Design Compilation. Experiments were performed to demonstrate the effects of different modeling styles on the quality of the design produced by VSS. Several alternative VHDL models were examined for each benchmark, illustrating the improvements in design quality achieved when Structured Modeling guidelines were followed

Circuit Size Lower Bounds and #SAT Upper Bounds Through a General Framework

Most of the known lower bounds for binary Boolean circuits with unrestricted depth are proved by the gate elimination method. The most efficient known algorithms for the #SAT problem on binary Boolean circuits use similar case analyses to the ones in gate elimination. Chen and Kabanets recently showed that the known case analyses can also be used to prove average case circuit lower bounds, that is, lower bounds on the size of approximations of an explicit function. In this paper, we provide a general framework for proving worst/average case lower bounds for circuits and upper bounds for #SAT that is built on ideas of Chen and Kabanets. A proof in such a framework goes as follows. One starts by fixing three parameters: a class of circuits, a circuit complexity measure, and a set of allowed substitutions. The main ingredient of a proof goes as follows: by going through a number of cases, one shows that for any circuit from the given class, one can find an allowed substitution such that the given measure of the circuit reduces by a sufficient amount. This case analysis immediately implies an upper bound for #SAT. To~obtain worst/average case circuit complexity lower bounds one needs to present an explicit construction of a function that is a disperser/extractor for the class of sources defined by the set of substitutions under consideration. We show that many known proofs (of circuit size lower bounds and upper bounds for #SAT) fall into this framework. Using this framework, we prove the following new bounds: average case lower bounds of 3.24n and 2.59n for circuits over U_2 and B_2, respectively (though the lower bound for the basis B_2 is given for a quadratic disperser whose explicit construction is not currently known), and faster than 2^n #SAT-algorithms for circuits over U_2 and B_2 of size at most 3.24n and 2.99n, respectively. Here by B_2 we mean the set of all bivariate Boolean functions, and by U_2 the set of all bivariate Boolean functions except for parity and its complement

Constraint solving over multi-valued logics - application to digital circuits

Due to usage conditions, hazardous environments or intentional causes, physical and virtual systems are subject to faults in their components, which may affect their overall behaviour. In a ‘black-box’ agent modelled by a set of propositional logic rules, in which just a subset of components is externally visible, such faults may only be recognised by examining some output function of the agent. A (fault-free) model of the agent’s system provides the expected output given some input. If the real output differs from that predicted output, then the system is faulty. However, some faults may only become apparent in the system output when appropriate inputs are given. A number of problems regarding both testing and diagnosis thus arise, such as testing a fault, testing the whole system, finding possible faults and differentiating them to locate the correct one. The corresponding optimisation problems of finding solutions that require minimum resources are also very relevant in industry, as is minimal diagnosis. In this dissertation we use a well established set of benchmark circuits to address such diagnostic related problems and propose and develop models with different logics that we formalise and generalise as much as possible. We also prove that all techniques generalise to agents and to multiple faults. The developed multi-valued logics extend the usual Boolean logic (suitable for faultfree models) by encoding values with some dependency (usually on faults). Such logics thus allow modelling an arbitrary number of diagnostic theories. Each problem is subsequently solved with CLP solvers that we implement and discuss, together with a new efficient search technique that we present. We compare our results with other approaches such as SAT (that require substantial duplication of circuits), showing the effectiveness of constraints over multi-valued logics, and also the adequacy of a general set constraint solver (with special inferences over set functions such as cardinality) on other problems. In addition, for an optimisation problem, we integrate local search with a constructive approach (branch-and-bound) using a variety of logics to improve an existing efficient tool based on SAT and ILP

Quality and Quantity in Robustness-Checking Using Formal Techniques

Fault tolerance is one of the main challenges for future technology scaling to tolerate transient faults. Various techniques at design level are available to catch and handle transient faults, e.g., Triple Modular Redundancy. An important but missing step is to verify the implementation of those techniques since the implementation might be buggy itself. The thesis is focusing on formally verifying digital circuits with respect to fault-tolerant aspects. It considers transient faults and basically checks whether these faults can influence the output behavior of sequential circuits for any kind of scenarios. As a result the designer is pin-pointed directly to critical parts of the design and gets a prove about the absence of faulty behavior for non-critical parts. The focus of the verification is completeness with respect to the analysis. Three issues need to be adequately addressed: 1) cover all input stimuli, 2) all possible transient faults, and, 3) all possibly exponential long (wrt. to number of state bits) propagation paths. All three issues are addressed in different engines. A tool called RobuCheck has been implemented and evaluated on different academic benchmarks from ITC'99 and industrial benchmarks from IBM

On Co-Optimization Of Constrained Satisfiability Problems For Hardware Software Applications

Manufacturing technology has permitted an exponential growth in transistor count and density. However, making efficient use of the available transistors in the design has become exceedingly difficult. Standard design flow involves synthesis, verification, placement and routing followed by final tape out of the design. Due to the presence of various undesirable effects like capacitive crosstalk, supply noise, high temperatures, etc., verification/validation of the design has become a challenging problem. Therefore, having a good design convergence may not be possible within the target time, due to a need for a large number of design iterations. Capacitive crosstalk is one of the major causes of design convergence problems in deep sub-micron era. With scaling, the number of crosstalk violations has been increasing because of reduced inter-wire distances. Consequently only the most severe crosstalk faults are fixed pre-silicon while the rest are tested post-silicon. Testing for capacitive crosstalk involves generation of input patterns which can be applied post-silicon to the integrated circuit and comparison of the output response. These patterns are generated at the gate/ Register Transfer Level (RTL) of abstraction using Automatic Test Pattern Generation (ATPG) tools. In this dissertation, anInteger Linear Programming (ILP) based ATPG technique for maximizing crosstalk induced delay increase at the victim net, for multiple aggressor crosstalk faults, is presented. Moreover, various solutions for pattern generation considering both zero as well as unit delay models is also proposed. With voltage scaling, power supply switching noise has become one of the leading causes of signal integrity related failures in deep sub-micron designs. Hence, during power supply network design and analysis of power supply switching noise, computation of peak supply current is an essential step. Traditional peak current estimation approaches involve addition of peak current associated with all the CMOS gates which are switching in a combinational circuit. Consequently, this approach does not take the Boolean and temporal relationships of the circuit into account. This work presents an ILP based technique for generation of an input pattern pair which maximizes switching supply currents for a combinational circuit in the presence of integer gate delays. The input pattern pair generated using the above approach can be applied post-silicon for power droop testing. With high level of integration, Multi-Processor Systems on Chip (MPSoC) feature multiple processor cores and accelerators on the same die, so as to exploit the instruction level parallelism in the application. For hardware-software co-design, application programming model is based on a Task Graph, which represents task dependencies and execution/transfer times for various threads and processes within an application. Mapping an application to an MPSoC traditionally involves representing it in the form of a task graph and employing static scheduling in order to minimize the schedule length. Dynamic system behavior is not taken into consideration during static scheduling, while dynamic scheduling requires the knowledge of task graph at runtime. A run-time task graph extraction heuristic to facilitate dynamic scheduling is also presented here. A novel game theory based approach uses this extracted task graph to perform run-time scheduling in order to minimize total schedule length. With increase in transistor density, power density has gone up substantially. This has lead to generation of regions with very high temperature called Hotspots. Hotspots lead to reliability and performance issues and affect design convergence. In current generation Integrated Circuits (ICs) temperature is controlled by reducing power dissipation using Dynamic Thermal Management (DTM) techniques like frequency and/or voltage scaling. These techniques are reactive in nature and have detrimental effects on performance. Here, a look-ahead based task migration technique is proposed, in order to utilize the multitude of cores available in an MPSoC to eliminate thermal emergencies. Our technique is based on temperature prediction, leveraging upon a novel wavelet based thermal modeling approach. Hence, this work addresses several optimization problems that can be reduced to constrained max-satisfiability, involving integer as well as Boolean constraints in hardware and software domains. Moreover, it provides domain specific heuristic solutions for each of them