A Sound and Complete Axiomatization of Majority- Logic

Amarù, Luca; Gaillardon, Pierre-Emmanuel; Chattopadhyay, Anupam; Micheli, Giovanni De

doi:10.1109/tc.2015.2506566

Cited by 33 publications

(15 citation statements)

References 31 publications

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“…The set of axioms that defines the majority algebra is represented by Ω and can be divided into axioms of Commutativity, Associativity, Distribution, Inverter Propagation and Majority [3]. The Commutativity axiom (Ω.C), represented in Equation 2, determines that the input order doesn't change the output value.…”

Section: Axiomatization Of Majority Functions (ω)mentioning

confidence: 99%

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Synthesis of Majority Expressions Through Primitive Function Manipulation

Ferraz

Muniz

Silva

et al. 2019

Advanced Boolean Techniques

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Due to technology advancements and circuits miniaturization, the study of logic systems that can be applied to nanotechnology has been progressing steadily. Among the creation of nanoeletronic circuits reversible and majority logic stand out. This paper proposes the M P C (Majority Primitives Combination) algorithm, used for majority logic synthesis. The algorithm receives a truth table as input and returns a majority function that covers the same set of minterms. The formulation of a valid output function is made with the combination of previously optimized functions. As cost criteria the algorithm searches for a function with the least number of levels, followed by the least number of gates, inverters, and gate inputs. In this paper it's also presented a comparison between the M P C and the exact mig, currently considered the best algorithm for majority synthesis. The exact mig encode the exact synthesis of majority functions using the number of levels and gates as cost criteria. The M P C considers two additional cost criteria, the number of inverters and the number of gate inputs, with the goal to further improve exact mig results. Tests have shown that both algorithms return optimal solutions for all functions with 3 input variables. For functions with 4 inputs, the M P C is able to further improve 42,987 (66%) functions and achieves equal results for 7,198 (11%). For functions with 5 input variables, out of a sample of 1,000 randomly generated functions, the M P C further improved 477 (48%) functions and achieved equal results for 112 (11%).

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Section: Axiomatization Of Majority Functions (ω)mentioning

confidence: 99%

“…The Associativity axiom (Ω.A) states that the exchange of variables between two functions is possible, as long as they are at subsequent levels and have one variable in common. An example of an Ω.A application is presented in Equation 3.…”

Section: Axiomatization Of Majority Functions (ω)mentioning

confidence: 99%

Synthesis of Majority Expressions Through Primitive Function Manipulation

Ferraz

Muniz

Silva

et al. 2019

Advanced Boolean Techniques

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“…For a general case of M n , where n > 3, a full axiomatic system is recently developed and proposed in [30]. Our goal is slightly different in the sense that we want to explore a mix of diverse majority nodes.…”

Section: Heterogeneous Majority Logic Networkmentioning

confidence: 99%

“…Recently, a logic optimization package based on the MIG datastructure, with 3-input majority node (M3) has been proposed [2], [30]. It is demonstrated to have efficient area-delay-power results compared to state-of-the-art logic optimization packages.…”

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confidence: 99%

Notes on Majority Boolean Algebra

Chattopadhyay

Amarù

Soeken

et al. 2016

2016 IEEE 46th International Symposium on Multiple-Valued Logic (ISMVL)

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Abstract-A Majority-Inverter Graph (MIG) is a homogeneous logic network, where each node represents the majority function. Recently, a logic optimization package based on the MIG datastructure, with 3-input majority node (M3) has been proposed [2], [30]. It is demonstrated to have efficient area-delay-power results compared to state-of-the-art logic optimization packages. In this paper, the Boolean algebraic transformations based on majority logic, i.e., majority Boolean algebra is studied. In the first part of this paper, we summarize a range of identities for majority Boolean algebra with their corresponding proofs. In the second part, we venture towards heterogeneous logic network and provide reversible logic mapping of majority nodes.

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“…In recent years, we have seen a shift from complex heterogeneous logic representations to simpler homogeneous networks such as And-Inverter Graphs (AIGs) [5], [6] and MajorityInverter Graphs (MIGs) [7]- [9]. These simpler representations enable more efficient logic representation and optimization, requiring less memory and allowing better run times [10].…”

Section: Introductionmentioning

confidence: 99%

A novel basis for logic rewriting

Haaswijk

Soeken

Amarù

et al. 2017

2017 22nd Asia and South Pacific Design Automation Conference (ASP-DAC)

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Abstract-Given a set of logic primitives and a Boolean function, exact synthesis finds the optimum representation (e.g., depth or size) of the function in terms of the primitives. Due to its high computational complexity, the use of exact synthesis is limited to small networks. Some logic rewriting algorithms use exact synthesis to replace small subnetworks by their optimum representations. However, conventional approaches have two major drawbacks. First, their scalability is limited, as Boolean functions are enumerated to precompute their optimum representations. Second, the strategies used to replace subnetworks are not satisfactory. We show how the use of exact synthesis for logic rewriting can be improved. To this end, we propose a novel method that includes various improvements over conventional approaches: (i) we improve the subnetwork selection strategy, (ii) we show how enumeration can be avoided, allowing our method to scale to larger subnetworks, and (iii) we introduce XOR Majority Graphs (XMGs) as compact logic representations that make exact synthesis more efficient. We show a 45.8% geometric mean reduction (taken over size, depth, and switching activity), a 6.5% size reduction, and depth · size reductions of 8.6%, compared to the academic state-of-the-art. Finally, we outperform 3 over 9 of the best known size results for the EPFL benchmark suite, reducing size by up to 11.5% and depth up to 46.7%.

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A Sound and Complete Axiomatization of Majority- Logic

Cited by 33 publications

References 31 publications

Synthesis of Majority Expressions Through Primitive Function Manipulation

Synthesis of Majority Expressions Through Primitive Function Manipulation

Notes on Majority Boolean Algebra

A novel basis for logic rewriting

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