This document discusses binary decision diagrams (BDDs) and their efficient implementation. It begins by contrasting canonical and non-canonical data structures for representing Boolean functions. It then introduces reduced ordered binary decision diagrams (ROBDDs) which represent functions as directed acyclic graphs. The document details how ROBDDs use unique and computed tables along with variable ordering to efficiently represent and operate on Boolean functions. It provides examples and algorithms for ROBDD operations like ITE, compose, and dynamic variable reordering.
The document discusses reduced ordered binary decision diagrams (ROBDDs), which are a compact data structure for representing Boolean functions. It explains that ROBDDs are derived from binary decision diagrams (BDDs) and Shannon's expansion. An ROBDD is constructed by first building an ordered binary decision tree (OBDT) and then applying reduction rules to remove redundant tests and merge isomorphic subgraphs, resulting in a reduced, acyclic graph. The document provides examples of constructing ROBDDs from truth tables and discusses properties like canonical representation and efficient manipulation.
The document discusses Binary Decision Diagrams (BDDs) and Ordered BDDs (OBDDs) which provide a more compact representation of Boolean functions compared to truth tables. It describes algorithms for reducing, applying logical operations, restricting variables, and checking satisfiability on BDDs/OBDDs. OBDDs ensure variables appear in the same order on all paths, allowing efficient equivalence checking. The document concludes with applications of OBDDs in symbolic model checking where sets of states are represented as OBDDs.
The document discusses reduced ordered binary decision diagrams (ROBDDs), which provide a canonical representation of Boolean functions. Some key points:
1) An ROBDD is an efficient way to represent Boolean functions and has a unique representation up to isomorphism for any given variable ordering.
2) Two Boolean functions are logically equivalent if their ROBDDs are isomorphic under the same variable ordering.
3) ROBDDs allow efficient implementation of Boolean operations like checking equivalence due to their canonical form.
The document discusses directed acyclic graphs (DAGs) and how they can be used to represent basic blocks of code. It describes how a DAG is constructed from three-address statements, with nodes labeled by variables, operators, or unique identifiers. Interior nodes represent computed values and leaves represent variables or constants. The DAG construction process creates nodes and links them based on the statements. DAGs are useful for detecting common subexpressions, determining which variables are used in a block, and which statements compute values used outside the block. Array accesses, pointers, and procedure calls require additional rules when constructing DAGs to properly capture dependencies.
The document summarizes a seminar presentation on using directed acyclic graphs (DAGs) to represent and optimize basic blocks in compiler design. DAGs can be constructed from three-address code to identify common subexpressions and eliminate redundant computations. Rules for DAG construction include creating a node only if it does not already exist, representing identifiers as leaf nodes and operators as interior nodes. DAGs allow optimizations like common subexpression elimination and dead code elimination to improve performance of local optimizations on basic blocks. Examples show how DAGs identify common subexpressions and avoid recomputing the same values.
The document discusses various optimization techniques that can be applied to basic blocks in code including:
- Common subexpression elimination and dead code elimination to remove redundant computations.
- Algebraic transformations like applying arithmetic identities to simplify expressions and reduce strength of operations.
- Representing the basic block as a directed acyclic graph (DAG) to help identify common subexpressions and apply transformations using properties like commutativity.
Implementation of Low-Complexity Redundant Multiplier Architecture for Finite...ijcisjournal
In the present work, a low-complexity Digit-Serial/parallel Multiplier over Finite Field is proposed. It is
employed in applications like cryptography for data encryption and decryptionto deal with discrete
mathematical andarithmetic structures. The proposedmultiplier utilizes a redundant representation because
of their free squaring and modular reduction. The proposed 10-bit multiplier is simulated and synthesized
using Xilinx VerilogHDL. It is evident from the simulation results that the multiplier has significantly low
area and power when compared to the previous structures using the same representation.
Boolean algebra deals with logical operations on binary variables that have two possible values, typically represented as 1 and 0. George Boole first introduced Boolean algebra in 1854. Boolean algebra uses logic gates like AND, OR, and NOT as basic building blocks. Positive logic represents 1 as high and 0 as low, while negative logic uses the opposite. Boolean algebra laws and Karnaugh maps are used to simplify logical expressions. Don't care conditions allow for groupings in K-maps that further reduce expressions.
The document discusses reduced ordered binary decision diagrams (ROBDDs), which are a compact data structure for representing Boolean functions. It explains that ROBDDs are derived from binary decision diagrams (BDDs) and Shannon's expansion. An ROBDD is constructed by first building an ordered binary decision tree (OBDT) and then applying reduction rules to remove redundant tests and merge isomorphic subgraphs, resulting in a reduced, acyclic graph. The document provides examples of constructing ROBDDs from truth tables and discusses properties like canonical representation and efficient manipulation.
The document discusses Binary Decision Diagrams (BDDs) and Ordered BDDs (OBDDs) which provide a more compact representation of Boolean functions compared to truth tables. It describes algorithms for reducing, applying logical operations, restricting variables, and checking satisfiability on BDDs/OBDDs. OBDDs ensure variables appear in the same order on all paths, allowing efficient equivalence checking. The document concludes with applications of OBDDs in symbolic model checking where sets of states are represented as OBDDs.
The document discusses reduced ordered binary decision diagrams (ROBDDs), which provide a canonical representation of Boolean functions. Some key points:
1) An ROBDD is an efficient way to represent Boolean functions and has a unique representation up to isomorphism for any given variable ordering.
2) Two Boolean functions are logically equivalent if their ROBDDs are isomorphic under the same variable ordering.
3) ROBDDs allow efficient implementation of Boolean operations like checking equivalence due to their canonical form.
The document discusses directed acyclic graphs (DAGs) and how they can be used to represent basic blocks of code. It describes how a DAG is constructed from three-address statements, with nodes labeled by variables, operators, or unique identifiers. Interior nodes represent computed values and leaves represent variables or constants. The DAG construction process creates nodes and links them based on the statements. DAGs are useful for detecting common subexpressions, determining which variables are used in a block, and which statements compute values used outside the block. Array accesses, pointers, and procedure calls require additional rules when constructing DAGs to properly capture dependencies.
The document summarizes a seminar presentation on using directed acyclic graphs (DAGs) to represent and optimize basic blocks in compiler design. DAGs can be constructed from three-address code to identify common subexpressions and eliminate redundant computations. Rules for DAG construction include creating a node only if it does not already exist, representing identifiers as leaf nodes and operators as interior nodes. DAGs allow optimizations like common subexpression elimination and dead code elimination to improve performance of local optimizations on basic blocks. Examples show how DAGs identify common subexpressions and avoid recomputing the same values.
The document discusses various optimization techniques that can be applied to basic blocks in code including:
- Common subexpression elimination and dead code elimination to remove redundant computations.
- Algebraic transformations like applying arithmetic identities to simplify expressions and reduce strength of operations.
- Representing the basic block as a directed acyclic graph (DAG) to help identify common subexpressions and apply transformations using properties like commutativity.
Implementation of Low-Complexity Redundant Multiplier Architecture for Finite...ijcisjournal
In the present work, a low-complexity Digit-Serial/parallel Multiplier over Finite Field is proposed. It is
employed in applications like cryptography for data encryption and decryptionto deal with discrete
mathematical andarithmetic structures. The proposedmultiplier utilizes a redundant representation because
of their free squaring and modular reduction. The proposed 10-bit multiplier is simulated and synthesized
using Xilinx VerilogHDL. It is evident from the simulation results that the multiplier has significantly low
area and power when compared to the previous structures using the same representation.
Boolean algebra deals with logical operations on binary variables that have two possible values, typically represented as 1 and 0. George Boole first introduced Boolean algebra in 1854. Boolean algebra uses logic gates like AND, OR, and NOT as basic building blocks. Positive logic represents 1 as high and 0 as low, while negative logic uses the opposite. Boolean algebra laws and Karnaugh maps are used to simplify logical expressions. Don't care conditions allow for groupings in K-maps that further reduce expressions.
This document describes a K-Map software tool that simplifies Boolean equations. The tool reads in a Boolean expression with up to 4 variables in sum-of-products or product-of-sums form, generates a Karnaugh map, and uses it to minimize the expression. Algorithms are provided for solving 2, 3, and 4 variable maps. The tool could aid in designing sequential circuits and simplifying expressions frequently in other applications. Its use of different input forms and deductive reasoning achieves simplified output.
Lec6 Intro to Computer Engineering by Hsien-Hsin Sean Lee Georgia Tech -- Can...Hsien-Hsin Sean Lee, Ph.D.
This document discusses canonical (standard) forms for Boolean functions, including sum-of-products (SOP) and product-of-sums (POS) forms. It defines key concepts like Boolean variables, literals, product/sum terms, minterms/maxterms. It also provides examples of converting between a Boolean expression and its canonical SOP and POS forms, and how the canonical SOP and POS forms are complements of each other.
This document presents the design and implementation of an FPGA-based BCH decoder. It discusses BCH codes, which are binary error-correcting codes used in wireless communications. The implemented decoder is for a (15, 5, 3) BCH code, meaning it can correct up to 3 errors in a block of 15 bits. The decoder uses a serial input/output architecture and is implemented using VHDL on a FPGA device. It performs BCH decoding through syndrome calculation, running the Berlekamp-Massey algorithm to solve the key equation, and using Chien search to find error locations. The simulation result verifies correct decoding operation.
This document provides information about minimizing Boolean functions using Karnaugh maps. It discusses how Karnaugh maps can be used to simplify Boolean expressions into sums of products. Different examples are provided to demonstrate how to minimize functions with 2, 3, 4, and 5 variables using Karnaugh maps. Additional topics covered include don't care conditions, implementing logic with NAND and NOR gates, and exclusive OR functions.
Lec7 Intro to Computer Engineering by Hsien-Hsin Sean Lee Georgia Tech -- Kar...Hsien-Hsin Sean Lee, Ph.D.
This document provides an overview of Karnaugh maps and their use in simplifying Boolean functions. It defines key concepts like Hamming distance, unit-distance codes, Gray codes, implicants, prime implicants, essential and non-essential prime implicants. Examples are given to show how to identify implicants on a K-map and use them to find a minimum sum-of-products or product-of-sums expression. The use of don't care conditions to simplify expressions is also demonstrated. Finally, it discusses extending K-maps to multiple variables using stacked or composite maps.
FYBSC IT Digital Electronics Unit II Chapter II Minterm, Maxterm and Karnaugh...Arti Parab Academics
Minterm, Maxterm and Karnaugh Maps:
Introduction, minterms and sum of minterm form, maxterm and Product
of maxterm form, Reduction technique using Karnaugh maps –
2/3/4/5/6 variable K-maps, Grouping of variables in K-maps, K-maps
for product of sum form, minimize Boolean expression using K-map
and obtain K-map from Boolean expression, Quine Mc Cluskey
Method.
This document discusses logic simplification using Karnaugh maps. It begins with an overview of Boolean algebra simplification techniques. It then covers standard forms such as sum-of-products (SOP) and product-of-sums (POS), and how to convert between different forms. The document also discusses mapping logic expressions to Karnaugh maps and using K-map rules for simplification. Truth tables and determining logic expressions from truth tables are also covered.
This document provides an overview of digital electronics and Boolean algebra topics, including:
- Boolean algebra deals with binary variables and logical operations. It originated from George Boole's 1854 book.
- Logic gates are basic building blocks of digital systems. Common logic gates include AND, OR, NOT, NAND, NOR gates.
- Boolean laws like commutative, associative, distributive, De Morgan's theorems are used to simplify logic expressions.
- Karnaugh maps are used to minimize logic expressions into sum of products or product of sums form. Don't care conditions allow for further simplification.
- Universal gates like NAND and NOR can be used to construct all other logic gates
Ch4 Boolean Algebra And Logic Simplication1Qundeel
The document provides an overview of Boolean algebra and its application to logic circuits and digital design. It defines basic Boolean operations like AND, OR, NOT. It describes laws and identities of Boolean algebra including commutative, associative, distributive, Demorgan's theorems. It discusses ways to simplify Boolean expressions using these laws and identities. It also covers standard forms like Sum of Products and Product of Sums and how to convert between them. Truth tables are presented as a way to represent Boolean functions. Programmable logic devices like PALs and GALs are also briefly mentioned.
The document summarizes the instruction formats used by the 8086 microprocessor. It discusses the different parts of an instruction, including the opcode, addressing mode byte, registers, and displacement fields. It explains that instructions can be 1 to 7 bytes long depending on the addressing mode. Several examples of ADD instructions are provided to illustrate how the different fields specify the operands and destination.
This document discusses simplifying Boolean expressions using Boolean algebra. It explains how to simplify expressions by applying rules like distribution, idempotency, etc. It also covers converting expressions to standard forms, including sum-of-products (SOP) and product-of-sums (POS). Standard forms make expressions easier to evaluate, simplify and implement. The document provides examples of simplifying expressions and converting between SOP and POS form.
Ec2203 digital electronics questions anna university by www.annaunivedu.organnaunivedu
EC2203 Digital Electronics Anna University Important Questions for 3rd Semester ECE , EC2203 Digital Electronics Important Questions, 3rd Sem Question papers,
http://www.annaunivedu.org/digital-electronics-ec-2203-previous-year-question-paper-for-3rd-sem-ece-anna-univ-question/
The document contains 37 multiple choice questions related to computer science topics such as Boolean logic, data structures, algorithms, computer architecture, operating systems, and computer networks. The questions cover a wide range of fundamental concepts and require applying core CS principles to analyze problems and choose the best answer.
Principles of Combinational Logic: Definition of combinational logic, canonical forms, Generation of switching equations from truth tables, Karnaugh maps-3,4,5 variables, Incompletely specified functions (Don‘t care terms) Simplifying Max term equations
This document provides an overview of Boolean algebra and logic gates. It begins with an introduction to Boolean algebra, which deals with binary logic and is used in designing computer circuits. George Boole is identified as the founder of Boolean algebra. The document then covers logical operators like AND, OR, and NOT; Boolean functions; truth tables; canonical forms using minterms and maxterms; and how Boolean functions can be implemented using logic gates. Key concepts are illustrated with examples throughout.
Boolean algebra simplification and combination circuitsJaipal Dhobale
This document discusses Boolean algebra simplification and combinational circuits. It covers objectives like simplifying Boolean functions using K-maps and Quine-McCluskey method. It also discusses adders, subtractors, multipliers, dividers, ALUs, encoders, decoders, comparators, multiplexers and demultiplexers. Finally, it covers standard forms of Boolean expressions like Sum of Products and Product of Sums forms and how to convert between them.
The document discusses Boolean algebra and logic gates. It defines logic gates, explains their operations, and provides their logic symbols and truth tables. The types of logic gates covered are AND, OR, NOT, NOR, NAND, XOR, and XNOR. It also discusses sequential logic circuits like flip-flops, providing details on SR, JK, T, and D flip-flops including how to build them using logic gates. Additional topics covered include the difference between combinational and sequential logic circuits, Boolean theorems, sum-of-products and product-of-sums expressions, and the Karnaugh map method for simplifying logic expressions.
FYBSC IT Digital Electronics Unit II Chapter I Boolean Algebra and Logic GatesArti Parab Academics
Boolean Algebra and Logic Gates:
Introduction, Logic (AND OR NOT), Boolean theorems, Boolean
Laws, De Morgan’s Theorem, Perfect Induction, Reduction of Logic
expression using Boolean Algebra, Deriving Boolean expression from
given circuit, exclusive OR and Exclusive NOR gates, Universal Logic
gates, Implementation of other gates using universal gates, Input
bubbled logic, Assertion level.
This document provides information about getting fully solved assignments. It instructs students to send their semester and specialization name to the email address "help.mbaassignments@gmail.com" or call the phone number 08263069601. Mailing is preferred but calling can be done in an emergency. It then provides an example of an assignment for the subject "Logic Design" including questions about converting hexadecimal to other number systems, constructing logic gates from NAND gates, expanding Boolean functions, simplifying a Boolean function using a K-map, explaining differences between sequential and combinational circuits, and describing a serial in serial out shift register.
This document contains a 20 question multiple choice exam on topics in computer science such as algorithms, data structures, automata theory, and programming. Some example questions are about the number of states in a deterministic finite automaton for a specific language, properties of regular languages, time complexity of sorting algorithms, and topological ordering of directed acyclic graphs. The exam also contains a section matching scheduling algorithms to applications and classifying statements about threads as true or false.
This document discusses Boolean functions and logic circuits. It begins by defining Boolean functions as mappings from a Boolean space B^n to Boolean values {0,1}. It then covers various representations of Boolean functions including truth tables, sum of products, Karnaugh maps, and Boolean formulas. The document also discusses Boolean operations like AND, OR, and complement. It defines Boolean circuits and covers topics like gates, fanin, fanout, and cyclic circuits. Finally, it discusses different representations used for Boolean reasoning like binary decision diagrams and how Boolean reasoning engines are implemented and interfaced.
The document discusses query optimization in databases. It explains that the goal of query optimization is to determine the most efficient execution plan for a query to minimize the time needed. It outlines the typical steps in query optimization, including parsing/translation, applying relational algebra, and optimizing the query plan. It also discusses techniques like generating alternative execution plans using equivalence rules, estimating plan costs based on statistical data, and using heuristics or dynamic programming to choose the optimal plan.
This document describes a K-Map software tool that simplifies Boolean equations. The tool reads in a Boolean expression with up to 4 variables in sum-of-products or product-of-sums form, generates a Karnaugh map, and uses it to minimize the expression. Algorithms are provided for solving 2, 3, and 4 variable maps. The tool could aid in designing sequential circuits and simplifying expressions frequently in other applications. Its use of different input forms and deductive reasoning achieves simplified output.
Lec6 Intro to Computer Engineering by Hsien-Hsin Sean Lee Georgia Tech -- Can...Hsien-Hsin Sean Lee, Ph.D.
This document discusses canonical (standard) forms for Boolean functions, including sum-of-products (SOP) and product-of-sums (POS) forms. It defines key concepts like Boolean variables, literals, product/sum terms, minterms/maxterms. It also provides examples of converting between a Boolean expression and its canonical SOP and POS forms, and how the canonical SOP and POS forms are complements of each other.
This document presents the design and implementation of an FPGA-based BCH decoder. It discusses BCH codes, which are binary error-correcting codes used in wireless communications. The implemented decoder is for a (15, 5, 3) BCH code, meaning it can correct up to 3 errors in a block of 15 bits. The decoder uses a serial input/output architecture and is implemented using VHDL on a FPGA device. It performs BCH decoding through syndrome calculation, running the Berlekamp-Massey algorithm to solve the key equation, and using Chien search to find error locations. The simulation result verifies correct decoding operation.
This document provides information about minimizing Boolean functions using Karnaugh maps. It discusses how Karnaugh maps can be used to simplify Boolean expressions into sums of products. Different examples are provided to demonstrate how to minimize functions with 2, 3, 4, and 5 variables using Karnaugh maps. Additional topics covered include don't care conditions, implementing logic with NAND and NOR gates, and exclusive OR functions.
Lec7 Intro to Computer Engineering by Hsien-Hsin Sean Lee Georgia Tech -- Kar...Hsien-Hsin Sean Lee, Ph.D.
This document provides an overview of Karnaugh maps and their use in simplifying Boolean functions. It defines key concepts like Hamming distance, unit-distance codes, Gray codes, implicants, prime implicants, essential and non-essential prime implicants. Examples are given to show how to identify implicants on a K-map and use them to find a minimum sum-of-products or product-of-sums expression. The use of don't care conditions to simplify expressions is also demonstrated. Finally, it discusses extending K-maps to multiple variables using stacked or composite maps.
FYBSC IT Digital Electronics Unit II Chapter II Minterm, Maxterm and Karnaugh...Arti Parab Academics
Minterm, Maxterm and Karnaugh Maps:
Introduction, minterms and sum of minterm form, maxterm and Product
of maxterm form, Reduction technique using Karnaugh maps –
2/3/4/5/6 variable K-maps, Grouping of variables in K-maps, K-maps
for product of sum form, minimize Boolean expression using K-map
and obtain K-map from Boolean expression, Quine Mc Cluskey
Method.
This document discusses logic simplification using Karnaugh maps. It begins with an overview of Boolean algebra simplification techniques. It then covers standard forms such as sum-of-products (SOP) and product-of-sums (POS), and how to convert between different forms. The document also discusses mapping logic expressions to Karnaugh maps and using K-map rules for simplification. Truth tables and determining logic expressions from truth tables are also covered.
This document provides an overview of digital electronics and Boolean algebra topics, including:
- Boolean algebra deals with binary variables and logical operations. It originated from George Boole's 1854 book.
- Logic gates are basic building blocks of digital systems. Common logic gates include AND, OR, NOT, NAND, NOR gates.
- Boolean laws like commutative, associative, distributive, De Morgan's theorems are used to simplify logic expressions.
- Karnaugh maps are used to minimize logic expressions into sum of products or product of sums form. Don't care conditions allow for further simplification.
- Universal gates like NAND and NOR can be used to construct all other logic gates
Ch4 Boolean Algebra And Logic Simplication1Qundeel
The document provides an overview of Boolean algebra and its application to logic circuits and digital design. It defines basic Boolean operations like AND, OR, NOT. It describes laws and identities of Boolean algebra including commutative, associative, distributive, Demorgan's theorems. It discusses ways to simplify Boolean expressions using these laws and identities. It also covers standard forms like Sum of Products and Product of Sums and how to convert between them. Truth tables are presented as a way to represent Boolean functions. Programmable logic devices like PALs and GALs are also briefly mentioned.
The document summarizes the instruction formats used by the 8086 microprocessor. It discusses the different parts of an instruction, including the opcode, addressing mode byte, registers, and displacement fields. It explains that instructions can be 1 to 7 bytes long depending on the addressing mode. Several examples of ADD instructions are provided to illustrate how the different fields specify the operands and destination.
This document discusses simplifying Boolean expressions using Boolean algebra. It explains how to simplify expressions by applying rules like distribution, idempotency, etc. It also covers converting expressions to standard forms, including sum-of-products (SOP) and product-of-sums (POS). Standard forms make expressions easier to evaluate, simplify and implement. The document provides examples of simplifying expressions and converting between SOP and POS form.
Ec2203 digital electronics questions anna university by www.annaunivedu.organnaunivedu
EC2203 Digital Electronics Anna University Important Questions for 3rd Semester ECE , EC2203 Digital Electronics Important Questions, 3rd Sem Question papers,
http://www.annaunivedu.org/digital-electronics-ec-2203-previous-year-question-paper-for-3rd-sem-ece-anna-univ-question/
The document contains 37 multiple choice questions related to computer science topics such as Boolean logic, data structures, algorithms, computer architecture, operating systems, and computer networks. The questions cover a wide range of fundamental concepts and require applying core CS principles to analyze problems and choose the best answer.
Principles of Combinational Logic: Definition of combinational logic, canonical forms, Generation of switching equations from truth tables, Karnaugh maps-3,4,5 variables, Incompletely specified functions (Don‘t care terms) Simplifying Max term equations
This document provides an overview of Boolean algebra and logic gates. It begins with an introduction to Boolean algebra, which deals with binary logic and is used in designing computer circuits. George Boole is identified as the founder of Boolean algebra. The document then covers logical operators like AND, OR, and NOT; Boolean functions; truth tables; canonical forms using minterms and maxterms; and how Boolean functions can be implemented using logic gates. Key concepts are illustrated with examples throughout.
Boolean algebra simplification and combination circuitsJaipal Dhobale
This document discusses Boolean algebra simplification and combinational circuits. It covers objectives like simplifying Boolean functions using K-maps and Quine-McCluskey method. It also discusses adders, subtractors, multipliers, dividers, ALUs, encoders, decoders, comparators, multiplexers and demultiplexers. Finally, it covers standard forms of Boolean expressions like Sum of Products and Product of Sums forms and how to convert between them.
The document discusses Boolean algebra and logic gates. It defines logic gates, explains their operations, and provides their logic symbols and truth tables. The types of logic gates covered are AND, OR, NOT, NOR, NAND, XOR, and XNOR. It also discusses sequential logic circuits like flip-flops, providing details on SR, JK, T, and D flip-flops including how to build them using logic gates. Additional topics covered include the difference between combinational and sequential logic circuits, Boolean theorems, sum-of-products and product-of-sums expressions, and the Karnaugh map method for simplifying logic expressions.
FYBSC IT Digital Electronics Unit II Chapter I Boolean Algebra and Logic GatesArti Parab Academics
Boolean Algebra and Logic Gates:
Introduction, Logic (AND OR NOT), Boolean theorems, Boolean
Laws, De Morgan’s Theorem, Perfect Induction, Reduction of Logic
expression using Boolean Algebra, Deriving Boolean expression from
given circuit, exclusive OR and Exclusive NOR gates, Universal Logic
gates, Implementation of other gates using universal gates, Input
bubbled logic, Assertion level.
This document provides information about getting fully solved assignments. It instructs students to send their semester and specialization name to the email address "help.mbaassignments@gmail.com" or call the phone number 08263069601. Mailing is preferred but calling can be done in an emergency. It then provides an example of an assignment for the subject "Logic Design" including questions about converting hexadecimal to other number systems, constructing logic gates from NAND gates, expanding Boolean functions, simplifying a Boolean function using a K-map, explaining differences between sequential and combinational circuits, and describing a serial in serial out shift register.
This document contains a 20 question multiple choice exam on topics in computer science such as algorithms, data structures, automata theory, and programming. Some example questions are about the number of states in a deterministic finite automaton for a specific language, properties of regular languages, time complexity of sorting algorithms, and topological ordering of directed acyclic graphs. The exam also contains a section matching scheduling algorithms to applications and classifying statements about threads as true or false.
This document discusses Boolean functions and logic circuits. It begins by defining Boolean functions as mappings from a Boolean space B^n to Boolean values {0,1}. It then covers various representations of Boolean functions including truth tables, sum of products, Karnaugh maps, and Boolean formulas. The document also discusses Boolean operations like AND, OR, and complement. It defines Boolean circuits and covers topics like gates, fanin, fanout, and cyclic circuits. Finally, it discusses different representations used for Boolean reasoning like binary decision diagrams and how Boolean reasoning engines are implemented and interfaced.
The document discusses query optimization in databases. It explains that the goal of query optimization is to determine the most efficient execution plan for a query to minimize the time needed. It outlines the typical steps in query optimization, including parsing/translation, applying relational algebra, and optimizing the query plan. It also discusses techniques like generating alternative execution plans using equivalence rules, estimating plan costs based on statistical data, and using heuristics or dynamic programming to choose the optimal plan.
This document discusses various techniques for simplifying Boolean functions including K-maps, don't care conditions, and implementing Boolean functions as logic circuits. It covers:
1) Using K-maps to solve 3-variable functions and simplify sums.
2) How don't care conditions can be represented on K-maps to simplify structures.
3) Converting Boolean functions to logic diagrams using only NAND or NOR gates as universal gates. Steps are provided to convert functions to NAND and NOR gate implementations.
4) Equivalents for NOT, AND and OR gates using only NAND or NOR gates.
The document discusses numerical methods for finding roots of equations and integrating functions. It covers root-finding algorithms like the bisection method, Regula Falsi method, modified Regula Falsi, and secant method. These algorithms iteratively find roots by narrowing the interval that contains the root. The document also discusses numerical integration techniques like the trapezoidal rule to approximate the area under a curve without having a closed-form solution. It notes the tradeoffs between different root-finding algorithms in terms of speed, accuracy, and ability to guarantee convergence.
Spark 4th Meetup Londond - Building a Product with Sparksamthemonad
This document discusses common technical problems encountered when building products with Spark and provides solutions. It covers Spark exceptions like out of memory errors and shuffle file problems. It recommends increasing partitions and memory configurations. The document also discusses optimizing Spark code using functional programming principles like strong and weak pipelining, and leveraging monoid structures to reduce shuffling. Overall it provides tips to debug issues, optimize performance, and productize Spark applications.
I am Anne L. I am an Algorithms Design Homework Expert at programminghomeworkhelp.com. I hold a Ph.D. in Programming, Auburn University, USA. I have been helping students with their homework for the past 8 years. I solve homework related to Algorithms Design.
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Lecture 2: Data-Intensive Computing for Text Analysis (Fall 2011)Matthew Lease
Data-Intensive Computing for Text Analysis CS395T / INF385T / LIN386M
University of Texas at Austin, Fall 2011
Lecture 2 September 1, 2011
Jason Baldridge and Matt Lease
https://sites.google.com/a/utcompling.com/dicta-f11/
Big Data Day LA 2016/ Hadoop/ Spark/ Kafka track - Iterative Spark Developmen...Data Con LA
This presentation will explore how Bloomberg uses Spark, with its formidable computational model for distributed, high-performance analytics, to take this process to the next level, and look into one of the innovative practices the team is currently developing to increase efficiency: the introduction of a logical signature for datasets.
The document discusses data structures and algorithms. It defines key concepts like primitive data types, data structures, static vs dynamic structures, abstract data types, algorithm design, analysis of time and space complexity, recursion, stacks and common stack operations like push and pop. Examples are provided to illustrate factorial calculation using recursion and implementation of a stack.
The document discusses data structures and algorithms. It defines key concepts like primitive data types, data structures, static vs dynamic structures, abstract data types, algorithm design, analysis of time and space complexity, and recursion. It provides examples of algorithms and data structures like stacks and using recursion to calculate factorials. The document covers fundamental topics in data structures and algorithms.
The document discusses data structures and algorithms. It defines key concepts like primitive data types, data structures, static vs dynamic structures, abstract data types, algorithm design, analysis of time and space complexity, and recursion. It provides examples of algorithms and data structures like arrays, stacks and the factorial function to illustrate recursive and iterative implementations. Problem solving techniques like defining the problem, designing algorithms, analyzing and testing solutions are also covered.
The document discusses data structures and algorithms. It defines key concepts like primitive data types, data structures, static vs dynamic structures, abstract data types, algorithm analysis including time and space complexity, and common algorithm design techniques like recursion. It provides examples of algorithms and data structures like stacks and using recursion to calculate factorials. The document covers fundamental topics in data structures and algorithms.
This document discusses data structures and algorithms. It begins by defining data structures as the logical organization of data and primitive data types like integers that hold single pieces of data. It then discusses static versus dynamic data structures and abstract data types. The document outlines the main steps in problem solving as defining the problem, designing algorithms, analyzing algorithms, implementing, testing, and maintaining solutions. It provides examples of space and time complexity analysis and discusses analyzing recursive algorithms through repeated substitution and telescoping methods.
The document discusses data structures and algorithms. It defines key concepts like primitive data types, data structures, static vs dynamic structures, abstract data types, algorithm design, analysis of time and space complexity, and recursion. It provides examples of algorithms and data structures like stacks and using recursion to calculate factorials. The document covers fundamental topics in data structures and algorithms.
MADlib Architecture and Functional Demo on How to Use MADlib/PivotalRPivotalOpenSourceHub
This document discusses the MADlib architecture for performing scalable machine learning and analytics on large datasets using massively parallel processing. It describes how MADlib implements algorithms like linear regression across distributed database segments to solve challenges like multiplying data across nodes. It also discusses how MADlib uses a convex optimization framework to iteratively solve machine learning problems and the use of streaming algorithms to compute analytics in a single data scan. Finally, it outlines how the MADlib architecture provides scalable machine learning capabilities to data scientists through interfaces like PivotalR.
This document provides a 3-sentence summary of the given document:
The document is a tutorial introduction to high-performance Haskell that covers topics like lazy evaluation, reasoning about space usage, benchmarking, profiling, and making Haskell code run faster. It explains concepts like laziness, thunks, and strictness and shows how to define tail-recursive functions, use foldl' for a strict left fold, and force evaluation of data constructor arguments to avoid space leaks. The goal is to help programmers optimize Haskell code and make efficient use of multiple processor cores.
The document provides details about the architecture of SIC and SIC/XE machines. It describes the memory, registers, data formats, instruction formats, addressing modes, instruction set, and input/output for both machines. It also provides example programs to illustrate different instructions and addressing modes. Additionally, it explains CISC machines in general and provides details about the VAX and Pentium Pro architectures as examples of CISC instruction set architectures.
Electronic Codebook Book (ECB) encrypts each message block independently without chaining blocks together. This can reveal patterns in the ciphertext if the plaintext has repetitive blocks. Cipher Block Chaining (CBC) chains blocks together by XORing the previous ciphertext block with the current plaintext block before encryption. Counter (CTR) mode encrypts a counter value rather than feedback from previous blocks, allowing parallel encryption. These modes of operation can be used to encrypt data blocks securely depending on needs such as bulk encryption or streaming data.
This document provides an overview of basic data structures concepts. It discusses bits and data types, different numeric representations like binary and decimal. It introduces common data types used in programming like integers and floats. It also covers abstract data types and provides examples like stacks, queues and lists. The document describes iterative and recursive algorithms for problems like factorial calculation, binary search and the Towers of Hanoi. It analyzes the time complexity of algorithms like selection sort.
1. 1
Courtesy RK Brayton (UCB)
and A Kuehlmann (Cadence)
Logic SynthesisLogic Synthesis
Binary Decision Diagrams
2. 2
Representing Boolean functionsRepresenting Boolean functions
• Fundamental trade-off
– canonical data structures
• data structure uniquely represents function
• Tautology decision procedure is trivial (e.g., just pointer
comparison)
• example: truth tables, Binary Decision Diagrams
• size of data structure is in general exponential
– noncanonical data structures
• covers, POS, formulas, logic circuits
• systematic search for satisfying assignment
• size of data structure is often small
• Tautology checking computationally expensive
3. 3
ROBDDsROBDDs
• General idea: Representation of a logic function as graph (DAG)
– use Shannon decomposition to build a decision tree representation
• Similar to what we saw in 2-level minimization
– difference: instead of exploring sub-cases by enumerating them in time
store sub-cases in memory
– Key to making efficient : two hashing mechanisms:
– unique table: find identical sub-cases and avoid replication
– computed table: reduce redundant computation of sub-cases
• Represent of a logic function as graph
– many logic functions can be represented compactly - usually better than
SOPs
• Many logic operations can be performed efficiently on BDDs
– usually linear in size of result - tautology and complement are constant time
• Size of BDD critically dependent on variable ordering
4. 4
ROBDDsROBDDs
• Directed acyclic graph (DAG)
• one root node, two terminals 0, 1
• each node, two children, and a variable
• Shannon co-factoring tree, except reduced and ordered (ROBDD)
– Reduced:
• any node with two identical children is removed
• two nodes with isomorphic BDD’s are merged
– Ordered:
• Co-factoring variables (splitting variables) always follow the
same order along all paths
xi1
< xi2
< xi3
< … < xin
6. 6
ROBDDROBDD
Ordered BDD (OBDD) Input variables are ordered - each path from root to
sink visits nodes with labels (variables) in ascending order.
a
c c
b
0 1
ordered
order = a,c,b
Reduced Ordered BDD (ROBDD) - reduction rules:
1. if the two children of a node are the same, the node is eliminated: f
= vf + vf
2. two nodes have isomorphic graphs => replace by one of them
These two rules make it so that each node represents a distinct logic
function.
a
b c
c
0 1
not
ordered
b
7. 7
Efficient Implementation of BDD’sEfficient Implementation of BDD’s
Unique Table:
• avoids duplication of existing nodes
– Hash-Table: hash-function(key) = value
– identical to the use of a hash-table in AND/INVERTER circuits
Computed Table:
• avoids re-computation of existing results
hash value
of key
collision
chain
hash value
of key
No collision chain
8. 8
Efficient Implementation of BDD’sEfficient Implementation of BDD’s
• BDDs is a compressed Shannon co-factoring tree:
• f = v fv + v fv
• leafs are constants “0” and “1”
• Three components make ROBDDs canonical (Proof: Bryant 1986):
– unique nodes for constant “0” and “1”
– identical order of case splitting variables along each paths
– hash table that ensures:
• (node(fv) = node(gv)) ∧ (node(fv) = node(gv)) ⇒ node(f) = node(g)
– provides recursive argument that node(f) is unique when using the
unique hash-table
v
0 1
f
fv fv
9. 9
Onset is Given by all Paths to “1”Onset is Given by all Paths to “1”
Notes:
• By tracing paths to the 1 node, we get a cover of pair wise disjoint cubes.
• The power of the BDD representation is that it does not explicitly
enumerate all paths; rather it represents paths by a graph whose size is
measures by its nodes and not paths.
• A DAG can represent an exponential number of paths with a linear
number of nodes.
• BDDs can be used to efficiently represent sets
– interpret elements of the onset as elements of the set
– f is called the characteristic function of that set
F = b’+a’c’ = ab’+a’cb’+a’c’ all paths to the 1 node
a
c
b
0 1
1
0
1
1
0
0
f
fa= b’
fa = cb’+c’
10. 10
ImplementationImplementation
Variables are totally ordered: If v < w then v occurs “higher” up in the ROBDD
Top variable of a function f is a variable associated with its root node.
Example: f = ab + a’bc + a’bc’. Order is (a < b < c).
fa = b, fa = b
a
b
0 1
f b is top variable of f
b
0 1
f
reducedf does not depend on a,
since fa = fa .
Each node is written as a triple: f = (v,g,h) where g = fv and h = fv .
We read this triple as:
f = if v then g else h = ite (v,g,h) = vg+v ’ h
v
f
0 1
h g
1 0
f
v
g
h
mux
v is top variable of f
11. 11
ITE OperatorITE Operator
ITE operator can implement any two variable logic function. There are 16 such
functions corresponding to all subsets of vertices of B 2
:
Table Subset Expression Equivalent Form
0000 0 0 0
0001 AND(f, g) fg ite(f, g, 0)
0010 f > g fg ite(f,g, 0)
0011 f f f
0100 f < g fg ite(f, 0, g)
0101 g g g
0110 XOR(f, g) f ⊕ g ite(f,g, g)
0111 OR(f, g) f + g ite(f, 1, g)
1000 NOR(f, g) f + g ite(f, 0,g)
1001 XNOR(f, g) f ⊕ g ite(f, g,g)
1010 NOT(g) g ite(g, 0, 1)
1011 f ≥ g f + g ite(f, 1, g)
1100 NOT(f) f ite(f, 0, 1)
1101 f ≤ g f + g ite(f, g, 1)
1110 NAND(f, g) fg ite(f, g, 1)
1111 1 1 1
ite( , , )f g h fg f h= +
12. 12
Unique Table - Hash TableUnique Table - Hash Table
• Before a node (v, g, h ) is added to BDD data base, it is looked up in
the “unique-table”. If it is there, then existing pointer to node is used to
represent the logic function. Otherwise, a new node is added to the
unique-table and the new pointer returned.
• Thus a strong canonical form is maintained. The node for f = (v, g, h )
exists iff(v, g, h ) is in the unique-table. There is only one pointer for (v,
g, h ) and that is the address to the unique-table entry.
• Unique-table allows single multi-rooted DAG to represent all users’
functions:
hash index
of key
collision
chain
13. 13
Recursive Formulation of ITERecursive Formulation of ITE
v = top-most variable among the three BDDs f, g, h
Where A, B are pointers to results of ite(fv,gv,hv) and ite(fv’,gv’,hv’})
- merged if equal
14. 14
Recursive Formulation of ITERecursive Formulation of ITE
Algorithm ITE(f, g, h)
if(f == 1) return g
if(f == 0) return h
if(g == h) return g
if((p = HASH_LOOKUP_COMPUTED_TABLE(f,g,h)) return p
v = TOP_VARIABLE(f, g, h ) // top variable from f,g,h
fn = ITE(fv,gv,hv) // recursive calls
gn = ITE(fv,gv,hv)
if(fn == gn) return gn // reduction
if(!(p = HASH_LOOKUP_UNIQUE_TABLE(v,fn,gn)) {
p = CREATE_NODE(v,fn,gn) // and insert into UNIQUE_TABLE
}
INSERT_COMPUTED_TABLE(p,HASH_KEY{f,g,h})
return p
}
15. 15
ExampleExample
I = ite (F, G, H)
= (a, ite (Fa , Ga , Ha ), ite (Fa , Ga , Ha ))
= (a, ite (1, C, H), ite(B, 0, H ))
= (a, C, (b, ite (Bb , 0b , Hb ), ite (Bb , 0b , Hb ))
= (a, C, (b, ite (1, 0, 1), ite (0, 0, D)))
= (a, C, (b, 0, D))
=(a, C, J)
Check: F = a + b, G = ac, H = b + d
ite(F, G, H) = (a + b)(ac) + ab(b + d) = ac + abd
F,G,H,I,J,B,C,D
are pointers
b1
1
a
0
1 0
1 0
F
B
1
1
a
0
1 0
0
G
c 0C
1
b
0
1 0
0
H
d
D
1
1
0
a
1 0
0
I
b
J
1
C
D
16. 16
Computed TableComputed Table
Keep a record of (F, G, H ) triplets already computed by the ITE operator
– software cache ( “cache” table)
– simply hash-table without collision chain (lossy cache)
17. 17
Extension - Complement EdgesExtension - Complement Edges
Combine inverted functions by using complemented edge
– similar to circuit case
– reduces memory requirements
– BUT MORE IMPORTANT:
• makes some operations more efficient (NOT, ITE)
0 1
G
0 1
G
two different
DAGs
0 1
G G
only one DAG
using complement
pointer
18. 18
Extension - Complement EdgesExtension - Complement Edges
To maintain strong canonical form, need to resolve 4 equivalences:
VV VV VV VV
VV VV VV VV
Solution: Always choose one on left, i.e. the “then” leg must have no
complement edge.
19. 19
Ambiguities in Computed TableAmbiguities in Computed Table
Standard Triples: ite(F, F, G ) ⇒ ite(F, 1, G )
ite(F, G, F ) ⇒ ite(F, G, 0 )
ite(F, G,F ) ⇒ ite(F, G, 1 )
ite(F,F, G ) ⇒ ite(F, 0, G )
To resolve equivalences: ite(F, 1, G ) ≡ ite(G, 1, F )
ite(F, 0, G ) ≡ ite(G, 1,F )
ite(F, G, 0 ) ≡ ite(G, F, 0 )
ite(F, G, 1 ) ≡ ite(G,F, 1 )
ite(F, G,G ) ≡ ite(G, F,F )
To maximize matches on computed table:
1. First argument is chosen with smallest top variable.
2. Break ties with smallest address pointer. (breaks PORTABILITY!!!!!!!!!!)
Triples:
ite(F, G, H ) ≡ ite (F, H, G ) ≡ ite (F, G,H) ≡ ite (F, H, G)
Choose the one such that the first and second argument of ite should not be
complement edges(i.e. the first one above).
20. 20
Use of Computed TableUse of Computed Table
• Often BDD packaged use optimized implementations for special
operations
– e.g. ITE_Constant (check whether the result would be a constant)
– AND_Exist (AND operation with existential quantification)
• All operations need a cache for decent performance
– local cache
• for one operation only - cache will be thrown away after
operation is finished (e.g. AND_Exist
• keep inter-operational (ITE, …)
– special cache for each operation
• does not need to store operation type
– shared cache for all operations
• better memory handling
• needs to store operation type
21. 21
Example: Tautology CheckingExample: Tautology Checking
Algorithm ITE_CONSTANT(f,g,h) { // returns 0,1, or NC
if(TRIVIAL_CASE(f,g,h) return result (0,1, or NC)
if((res = HASH_LOOKUP_COMPUTED_TABLE(f,g,h))) return res
v = TOP_VARIABLE(f,g,h)
i = ITE_CONSTANT(fv,gv,hv)
if(i == NC) {
INSERT_COMPUTED_TABLE(NC, HASH_KEY{f,g,h}) // special table!!
return NC
}
e = ITE_CONSTANT(fv,gv,hv)
if(e == NC) {
INSERT_COMPUTED_TABLE(NC, HASH_KEY{f,g,h})
return NC
}
if(e != i) {
INSERT_COMPUTED_TABLE(NC, HASH_KEY{f,g,h})
return NC
}
INSERT_COMPUTED_TABLE(e, HASH_KEY{f,g,h})
return i;
}
22. 22
ComposeCompose
Compose(F, v, G ) : F(v, x) → F( G(x), x), means substitute v by G(x)
Notes:
1. F1 is the 1-child of F, F0 the 0-child.
2. G , i, e are not functions of v
3. If TOP_VARIABLE of F is v, then ite (G , i, e ) does replacement of
v by G.
Algorithm COMPOSE(F,v,G) {
if(TOP_VARIABLE(F) > v) return F // F does not depend
on v
if(TOP_VARIABLE(F) == v) return ITE(G,F1,F0)
i = COMPOSE(F1,v,G)
e = COMPOSE(F0,v,G)
return ITE(TOP_VARIABLE(F),i,e) // Why not
CREATE_NODE...
}
23. 23
Variable OrderingVariable Ordering
• Static variable ordering
– variable ordering is computed up-front based on the problem
structure
– works very well for many combinational functions that come from
circuits we actually build
• general scheme: control variables first
• DFS order is pretty good for most cases
– work bad for unstructured problems
• e.g., using BDDs to represent arbitrary sets
– lots of research in ordering algorithms
• simulated annealing, genetic algorithms
• give better results but extremely costly
24. 24
Dynamic Variable OrderingDynamic Variable Ordering
• Changes the order in the middle of BDD applications
– must keep same global order
• Problem: External pointers reference internal nodes!!!
BDD Implementation:
...
...
...
...
External reference pointers attached
to application data structures
25. 25
Dynamic Variable OrderingDynamic Variable Ordering
Theorem (Friedman):
Permuting any top part of the variable order has no effect on the nodes
labeled by variables in the bottom part.
Permuting any bottom part of the variable order has no effect on the
nodes labeled by variables in the top part.
• Trick: Two adjacent variable layers can be exchanged by keeping the
original memory locations for the nodes
a
b b
c c c c
ff0
f1
f00 f01 f10
f11
bb b
c c c c
ff0
f1
f00 f01 f10
f11
aa
mem1
mem2
mem3
mem1
mem2
mem3
26. 26
Dynamic Variable OrderingDynamic Variable Ordering
• BDD sifting:
– shift each BDD variable to the top and then to the bottom and see
which position had minimal number of BDD nodes
– efficient if separate hash-table for each variable
– can stop if lower bound on size is worse then the best found so far
– shortcut:
• two layers can be swapped very cheaply if there is no
interaction between them
– expensive operation, sophisticated trigger condition to invoke it
• grouping of BDD variables:
– for many applications, pairing or grouping variables gives better
ordering
• e.g. current state and next state variables in state traversal
– grouping them for sifting explores ordering that are otherwise
skipped
27. 27
Garbage CollectionGarbage Collection
• Very important to free and ruse memory of unused BDD nodes
– explicitly freed by an external bdd_free operation
– BDD nodes that were temporary created during BDD operations
• Two mechanism to check whether a BDD is not referenced:
– Reference counter at each node
• increment whenever node gets one more reference (incl. External)
• decrement when node gets de-references (bdd_free from external, de-
reference from internal)
• counter-overflow -> freeze node
– Mark and Sweep algorithm
• does not need counter
• first pass, mark all BDDs that are referenced
• second pass, free the BDDs that are not marked
• need additional handle layer for external references
28. 28
Garbage CollectionGarbage Collection
• Timing is very crucial because garbage collection is expensive
– immediately when node gets freed
• bad because dead nodes get often reincarnated in next
operation
– regular garbage collections based on statistics collected during
BDD operations
– “death row” for nodes to keep them around for a bit longer
• Computed table must be cleared since not used in reference
mechanism
• Improving memory locality and therefore cache behavior:
– sort freed BDD nodes
29. 29
BDD DerivativesBDD Derivatives
• MDD: Multi-valued BDDs
– natural extension, have more then two branches
– can be implemented using a regular BDD package with binary encoding
• advantage that binary BDD variables for one MV variable do not have
to stay together -> potentially better ordering
• ADDs: (Analog BDDs) MTBDDs
– multi-terminal BDDs
– decision tree is binary
– multiple leafs, including real numbers, sets or arbitrary objects
– efficient for matrix computations and other non-integer applications
• FDDs: Free BDDs
– variable ordering differs
– not canonical anymore
• and many more …..
30. 30
Zero Suppressed BDDs - ZBDDsZero Suppressed BDDs - ZBDDs
ZBDD’s were invented by Minato to efficiently represent sparse sets. They
have turned out to be useful in implicit methods for representing primes
(which usually are a sparse subset of all cubes).
Different reduction rules:
• BDD: eliminate all nodes where then edge and else edge point to the
same node.
• ZBDD: eliminate all nodes where the then node points to 0. Connect
incoming edges to else node.
• For both: share equivalent nodes.
0 1
0 1 0 1
0 1
0
1
0 1
0
BDD:
ZBDD:
31. 31
CanonicityCanonicity
Theorem: (Minato) ZBDD’s are canonical given a variable ordering and the
support set.
x1
x2
01
BDD
x3
1
ZBDD if
support is
x1, x2, x3
1
ZBDD if
support is
x1, x2
Example:
x1
x2
01
BDD
x3
1
ZBDD if
support is
x1, x2 , x3
x1
x2
01
x3