This document discusses non-deterministic finite automata (NFAs). It provides examples of NFA transition graphs and explains how NFAs can accept input strings in a non-deterministic manner, meaning there may be multiple possible computations for a given input. It also defines the extended transition function for NFAs and explains that a string is accepted by an NFA if at least one computation of the NFA leads to an accepting state while consuming all input symbols. The language accepted by an NFA is the set of all strings that have an accepting computation. Finally, it notes that NFAs and deterministic finite automata (DFAs) have equivalent computational power since any NFA can be converted to an equivalent DFA.
The document discusses non-deterministic finite automata (NFAs) and how they are equivalent to deterministic finite automata (DFAs). It shows examples of NFAs accepting various strings and languages. It then proves that NFAs and DFAs have equal computational power by showing that any language accepted by an NFA is also accepted by a DFA, and vice versa. This is done by describing a procedure to convert any NFA into an equivalent DFA, demonstrating that the languages they accept are the same. Therefore, NFAs and DFAs recognize the same class of formal languages called regular languages.
The document describes finite automata. A finite automaton consists of a finite number of internal states and a transition function that determines the next state based on the current state and input. It takes an input string over a given alphabet and produces an output of "accept" or "reject". A deterministic finite automaton (DFA) is formally defined as a 5-tuple consisting of a set of states, an input alphabet, a transition function, an initial state, and a set of final states. The transition function of a DFA is represented by a transition graph with states as vertices and transitions as edges between states.
Finite automata are computational models that can be used to recognize regular languages. A finite automaton consists of a finite set of states, an input alphabet, transition functions between states based on input symbols, a start state, and accept states. It accepts a string by starting in the start state and following the transitions based on symbols in the string until it reaches an accept state or no possible transition. The behavior of a finite automaton can be formally defined as a 5-tuple. Regular operations like union, concatenation, and star can be used to combine regular languages while preserving their regularity.
VHDL constructs include sequential and concurrent statements. Sequential statements include if-then-else, case-when, for-loops, and while-loops. Concurrent statements allow signal assignments and use logical operators. Signal assignments in processes update signals at the end of the process based on initial values, while concurrent statements update signals immediately. Processes are sensitive to signals in their sensitivity list and are evaluated in simulation cycles that advance based on scheduled signal updates.
The document discusses Deterministic Finite Automata (DFAs) and Nondeterministic Finite Automata (NFAs). It defines the key components of a DFA/NFA including states, alphabet, transition function, initial state, and accepting states. It provides examples of DFAs and NFAs and their transition diagrams. It also discusses how to determine if a string is accepted by a DFA/NFA and the language recognized by a DFA/NFA.
This document summarizes Chapter 5 of a textbook on nondeterministic finite automata (NFAs). It discusses how NFAs relax the requirement that deterministic finite automata (DFAs) have exactly one transition from every state on every input symbol. NFAs allow multiple transitions from a state on a single symbol, making them more flexible but also nondeterministic. The chapter defines key concepts like spontaneous transitions, nondeterminism, the 5-tuple representation of an NFA, and how the language an NFA accepts is defined in terms of all possible sequences of transitions. It provides examples to illustrate these concepts.
simple problem to convert NFA with epsilon to without epsilonkanikkk
This document discusses the steps to construct the transition table for an NFA with epsilon transitions. It begins by taking the epsilon closure of each state. It then determines the output states for each input symbol applied to each state/closure set by taking the epsilon closure. This information is used to construct the transition table and diagram. The transition table shows the output state(s) for each input applied to each state. The transition diagram visually depicts the transitions.
The document discusses non-deterministic finite automata (NFAs) and how they are equivalent to deterministic finite automata (DFAs). It shows examples of NFAs accepting various strings and languages. It then proves that NFAs and DFAs have equal computational power by showing that any language accepted by an NFA is also accepted by a DFA, and vice versa. This is done by describing a procedure to convert any NFA into an equivalent DFA, demonstrating that the languages they accept are the same. Therefore, NFAs and DFAs recognize the same class of formal languages called regular languages.
The document describes finite automata. A finite automaton consists of a finite number of internal states and a transition function that determines the next state based on the current state and input. It takes an input string over a given alphabet and produces an output of "accept" or "reject". A deterministic finite automaton (DFA) is formally defined as a 5-tuple consisting of a set of states, an input alphabet, a transition function, an initial state, and a set of final states. The transition function of a DFA is represented by a transition graph with states as vertices and transitions as edges between states.
Finite automata are computational models that can be used to recognize regular languages. A finite automaton consists of a finite set of states, an input alphabet, transition functions between states based on input symbols, a start state, and accept states. It accepts a string by starting in the start state and following the transitions based on symbols in the string until it reaches an accept state or no possible transition. The behavior of a finite automaton can be formally defined as a 5-tuple. Regular operations like union, concatenation, and star can be used to combine regular languages while preserving their regularity.
VHDL constructs include sequential and concurrent statements. Sequential statements include if-then-else, case-when, for-loops, and while-loops. Concurrent statements allow signal assignments and use logical operators. Signal assignments in processes update signals at the end of the process based on initial values, while concurrent statements update signals immediately. Processes are sensitive to signals in their sensitivity list and are evaluated in simulation cycles that advance based on scheduled signal updates.
The document discusses Deterministic Finite Automata (DFAs) and Nondeterministic Finite Automata (NFAs). It defines the key components of a DFA/NFA including states, alphabet, transition function, initial state, and accepting states. It provides examples of DFAs and NFAs and their transition diagrams. It also discusses how to determine if a string is accepted by a DFA/NFA and the language recognized by a DFA/NFA.
This document summarizes Chapter 5 of a textbook on nondeterministic finite automata (NFAs). It discusses how NFAs relax the requirement that deterministic finite automata (DFAs) have exactly one transition from every state on every input symbol. NFAs allow multiple transitions from a state on a single symbol, making them more flexible but also nondeterministic. The chapter defines key concepts like spontaneous transitions, nondeterminism, the 5-tuple representation of an NFA, and how the language an NFA accepts is defined in terms of all possible sequences of transitions. It provides examples to illustrate these concepts.
simple problem to convert NFA with epsilon to without epsilonkanikkk
This document discusses the steps to construct the transition table for an NFA with epsilon transitions. It begins by taking the epsilon closure of each state. It then determines the output states for each input symbol applied to each state/closure set by taking the epsilon closure. This information is used to construct the transition table and diagram. The transition table shows the output state(s) for each input applied to each state. The transition diagram visually depicts the transitions.
This document discusses reliable data transfer protocols including Go-Back-N (GBN). GBN allows a sender to transmit multiple packets without waiting for acknowledgements, up to a maximum window size of N. The sender bases retransmissions on a timeout for the oldest unacknowledged packet. The receiver discards out-of-order packets and sends cumulative acknowledgements for the highest in-order packet received. Pipelining helps increase utilization over stop-and-wait protocols but requires numbering packets, buffering, and handling retransmissions and duplicate packets.
The document discusses deterministic finite automata (DFA) and regular languages. It defines a DFA as a 5-tuple (Q, Σ, δ, q0, F) where Q is a finite set of states, Σ is an input alphabet, δ is the transition function, q0 is the initial state, and F is a set of accepting states. A language is regular if there exists a DFA that accepts it. The document provides several examples of DFAs and the regular languages they accept.
A 4-bit Johnson counter uses 4 D flip-flops connected in a loop. On each clock pulse, the value shifts from one flip-flop to the next in a circular fashion, incrementing the counter. If an illegal state occurs, correction gates block the invalid input and force the next flip-flop to the correct state to maintain the proper counting sequence. The Johnson counter allows for all possible state combinations and self-corrects any illegal states through the use of correction gates.
The document describes the workings of an SR latch circuit. An SR latch consists of two cross-coupled NOR or NAND gates with inputs named S (Set) and R (Reset). The circuit can be in one of two states: the set state where output Q=1 and Q'=0, or the reset state where Q=0 and Q'=1. When S=1 and R=0, the circuit enters the set state by forcing Q to 1 and Q' to 0. When R=1 and S=0, the circuit enters the reset state with Q=0 and Q'=1. Once set or reset, the state will be maintained even if the input changing it toggles again.
The document discusses sequential circuits and latches. It explains that sequential circuits have memory so their outputs depend not only on current inputs but also on the stored state. Latches are described as basic memory units that can store a single bit. Different types of latches like SR, D, and JK latches are presented along with their truth tables and operation. State diagrams are introduced as a way to represent sequential circuits. The use of latches with an ALU to increment a stored value is given as an example, but the timing issue of when to disable the latches is noted as a potential problem.
Sequential circuits have memory and their output depends on both the current inputs and past outputs. They contain combinational circuits and feedback loops using latches and flip-flops. There are two main types of sequential circuits - asynchronous which can change state anytime the inputs change, and synchronous which only change on a clock signal.
Latches continuously track inputs and can change output anytime, while flip-flops only change output on a clock signal. Common flip-flop types include SR, D, T, and JK. Counters are sequential circuits that cycle through a sequence of states on each clock pulse and are used to count events.
This document provides an overview of continuous variable quantum cryptography (CVQKD). It discusses how CVQKD works at a medium range of ~25km with medium rates of a few kbit/s, offering less security than single-photon QKD but more potential for improvement. The document reviews the theoretical basis of CVQKD in terms of field quadratures, homodyne detection, information theory, and how the protocol encodes a secret key. It also summarizes the progress made in improving CVQKD protocols and increasing their security over the last 10 years based on theoretical work. Finally, it mentions the development of 1st generation CVQKD experimental demonstrators.
1. The document discusses sequential logic circuits and various types of flip-flops including SR, D, JK, and T flip-flops. It explains the operation of each flip-flop through truth tables and timing diagrams.
2. Master-slave JK flip-flops are described as using two SR flip-flops in a cascade configuration to avoid unwanted output changes from glitches in the clock signal.
3. Other topics covered include latches, triggering methods, and uses of different flip-flop types in applications such as registers and counters.
Good news, everybody! Guile 2.2 performance notes (FOSDEM 2016)Igalia
By Andy Wingo.
With the new compiler and virtual machine in Guile 2.2, Guile hackers need to update their mental performance models. This talk will give a bit of a state of the union of Guile performance, with an updated overview of the cost of various kinds of abstractions. Sometimes abstraction is free!
(c) 2016 FOSDEM VZW
CC BY 2.0 BE
https://archive.fosdem.org/2016/
The document discusses sequential circuits and their components. It begins with an overview of sequential circuits and finite state machines. It then covers different types of flip-flops like D flip-flops and their usage. Counters and sequencers are presented as examples of sequential circuits. Details about designing a 3-bit up counter like its state table and logic equations are provided. Finally, registers are discussed including an example of a 4-bit register with parallel load.
This document discusses different types of latches used in computing and data storage. It describes latches as basic bistable elements that use feedback to retain information. The document outlines several types of latches including SR latches, D latches, JK latches, and T latches. It explains their circuit designs and behaviors. Examples are provided of how latches are used to encode binary data and in synchronous and asynchronous systems. Advantages of latches like flexibility and power efficiency are contrasted with disadvantages like potential race conditions and metastability issues.
The document describes a method called the "Four Russians method" to speed up Bayesian Hidden Markov Model (HMM) classification by exploiting repetition in long observation sequences. The key ideas are to break the observation sequence into blocks of length k and compute the forward variables only at block boundaries, and to sample the hidden state sequence block-by-block from the backward-forward distribution rather than the full backward distribution. This reduces the computational complexity from O(TN^2) to O(TNk/k^2) = O(TN/k).
Turing Machines are a simple mathematical model of a general purpose computer invented by Alan Turing in 1936. A Turing Machine consists of an infinite tape divided into cells, a head that reads and writes symbols on the tape, a finite set of states, and transition rules determining the behavior of the machine. The machine operates by reading a symbol on the tape, updating the symbol according to its transition rules, moving the head left or right, and transitioning to a new state. Turing Machines can simulate any algorithm and are capable of performing any calculation that can be performed by any computing machine.
1) The document describes the procedure for deriving the small signal model and transfer functions of an ideal boost converter. Key steps include defining state variables, writing state equations, averaging, linearizing, and taking the Laplace transform.
2) Transfer functions are derived from the small signal model matrix for the ideal case, including control-to-output, control-to-inductor current, and inductor current-to-output.
3) The process is then repeated for a non-ideal boost converter that includes resistances in the inductor and capacitor. Additional state equations are written to account for the resistances.
Pushdown automata (PDA) are machines that process input strings and use a stack to determine transitions between states. A PDA consists of states, input symbols, stack symbols, transition functions, an initial state, initial stack symbol, and final states. The transition function specifies how the PDA moves between states based on the current state, input symbol, and top stack symbol, and may involve operations like replacing, pushing, or popping symbols from the stack. PDAs can recognize languages like {anbn: n ≥ 0} by pushing a symbol for each a and popping for each b.
I have been receiving multiple queries on what is clk-to-q delay, how's it different from library setup time and library hold time, etc. I mentioned in my discussions, that the videos on CMOS digital circuit will be uploaded soon, but looks like, it might take some time, and hence decided to uploaded few images from my CMOS course, to explain the difference between all of them.
This document discusses latches and their design process. It begins by defining a latch as a circuit that has two stable states and can store state information. It then describes the different types of latches including asynchronous and synchronous latches. The RS latch is examined in more detail with diagrams of its logic structure and a truth table. Key properties of the RS latch are that it uses two inputs called Set and Reset to store a 1 or 0 without a clock, and it can immediately change its output when the inputs change.
Shift registers allow data to be transported serially by cascading flip flops. They can transfer data either serially or in parallel and are used to implement serial communication and arithmetic operations. A universal shift register uses a multiplexer structure to provide a programmable register that can perform different operations like shifting, loading, or clearing through control signals.
Sequential circuits consist of combinational logic and memory elements like latches and flip-flops. There are different types of latches and flip-flops that differ in their trigger mechanisms and outputs, including SR latches, D latches, and edge-triggered flip-flops like SR, D, and JK flip-flops. Asynchronous inputs can directly set or reset flip-flop outputs independent of the clock signal.
This document discusses sequential logic circuits and memory elements such as latches and flip-flops. It describes different types of latches including the S-R latch, gated S-R latch, and gated D latch. It also covers various types of flip-flops including the S-R, D, J-K, and T flip-flops. It explains the differences between latches and flip-flops and their applications in synchronous and asynchronous logic circuits.
Ethernet has taken the lead as the preferred platform for connecting to the cloud, but not all Ethernet cloud connections (CloudE) are created equal. Providers and users alike are calling for the industry to establish standards of excellence, and the Cloud Ethernet Forum is responding. Read about CloudE 1.0 and what its standards of excellence could mean for CloudE.
This document discusses reliable data transfer protocols including Go-Back-N (GBN). GBN allows a sender to transmit multiple packets without waiting for acknowledgements, up to a maximum window size of N. The sender bases retransmissions on a timeout for the oldest unacknowledged packet. The receiver discards out-of-order packets and sends cumulative acknowledgements for the highest in-order packet received. Pipelining helps increase utilization over stop-and-wait protocols but requires numbering packets, buffering, and handling retransmissions and duplicate packets.
The document discusses deterministic finite automata (DFA) and regular languages. It defines a DFA as a 5-tuple (Q, Σ, δ, q0, F) where Q is a finite set of states, Σ is an input alphabet, δ is the transition function, q0 is the initial state, and F is a set of accepting states. A language is regular if there exists a DFA that accepts it. The document provides several examples of DFAs and the regular languages they accept.
A 4-bit Johnson counter uses 4 D flip-flops connected in a loop. On each clock pulse, the value shifts from one flip-flop to the next in a circular fashion, incrementing the counter. If an illegal state occurs, correction gates block the invalid input and force the next flip-flop to the correct state to maintain the proper counting sequence. The Johnson counter allows for all possible state combinations and self-corrects any illegal states through the use of correction gates.
The document describes the workings of an SR latch circuit. An SR latch consists of two cross-coupled NOR or NAND gates with inputs named S (Set) and R (Reset). The circuit can be in one of two states: the set state where output Q=1 and Q'=0, or the reset state where Q=0 and Q'=1. When S=1 and R=0, the circuit enters the set state by forcing Q to 1 and Q' to 0. When R=1 and S=0, the circuit enters the reset state with Q=0 and Q'=1. Once set or reset, the state will be maintained even if the input changing it toggles again.
The document discusses sequential circuits and latches. It explains that sequential circuits have memory so their outputs depend not only on current inputs but also on the stored state. Latches are described as basic memory units that can store a single bit. Different types of latches like SR, D, and JK latches are presented along with their truth tables and operation. State diagrams are introduced as a way to represent sequential circuits. The use of latches with an ALU to increment a stored value is given as an example, but the timing issue of when to disable the latches is noted as a potential problem.
Sequential circuits have memory and their output depends on both the current inputs and past outputs. They contain combinational circuits and feedback loops using latches and flip-flops. There are two main types of sequential circuits - asynchronous which can change state anytime the inputs change, and synchronous which only change on a clock signal.
Latches continuously track inputs and can change output anytime, while flip-flops only change output on a clock signal. Common flip-flop types include SR, D, T, and JK. Counters are sequential circuits that cycle through a sequence of states on each clock pulse and are used to count events.
This document provides an overview of continuous variable quantum cryptography (CVQKD). It discusses how CVQKD works at a medium range of ~25km with medium rates of a few kbit/s, offering less security than single-photon QKD but more potential for improvement. The document reviews the theoretical basis of CVQKD in terms of field quadratures, homodyne detection, information theory, and how the protocol encodes a secret key. It also summarizes the progress made in improving CVQKD protocols and increasing their security over the last 10 years based on theoretical work. Finally, it mentions the development of 1st generation CVQKD experimental demonstrators.
1. The document discusses sequential logic circuits and various types of flip-flops including SR, D, JK, and T flip-flops. It explains the operation of each flip-flop through truth tables and timing diagrams.
2. Master-slave JK flip-flops are described as using two SR flip-flops in a cascade configuration to avoid unwanted output changes from glitches in the clock signal.
3. Other topics covered include latches, triggering methods, and uses of different flip-flop types in applications such as registers and counters.
Good news, everybody! Guile 2.2 performance notes (FOSDEM 2016)Igalia
By Andy Wingo.
With the new compiler and virtual machine in Guile 2.2, Guile hackers need to update their mental performance models. This talk will give a bit of a state of the union of Guile performance, with an updated overview of the cost of various kinds of abstractions. Sometimes abstraction is free!
(c) 2016 FOSDEM VZW
CC BY 2.0 BE
https://archive.fosdem.org/2016/
The document discusses sequential circuits and their components. It begins with an overview of sequential circuits and finite state machines. It then covers different types of flip-flops like D flip-flops and their usage. Counters and sequencers are presented as examples of sequential circuits. Details about designing a 3-bit up counter like its state table and logic equations are provided. Finally, registers are discussed including an example of a 4-bit register with parallel load.
This document discusses different types of latches used in computing and data storage. It describes latches as basic bistable elements that use feedback to retain information. The document outlines several types of latches including SR latches, D latches, JK latches, and T latches. It explains their circuit designs and behaviors. Examples are provided of how latches are used to encode binary data and in synchronous and asynchronous systems. Advantages of latches like flexibility and power efficiency are contrasted with disadvantages like potential race conditions and metastability issues.
The document describes a method called the "Four Russians method" to speed up Bayesian Hidden Markov Model (HMM) classification by exploiting repetition in long observation sequences. The key ideas are to break the observation sequence into blocks of length k and compute the forward variables only at block boundaries, and to sample the hidden state sequence block-by-block from the backward-forward distribution rather than the full backward distribution. This reduces the computational complexity from O(TN^2) to O(TNk/k^2) = O(TN/k).
Turing Machines are a simple mathematical model of a general purpose computer invented by Alan Turing in 1936. A Turing Machine consists of an infinite tape divided into cells, a head that reads and writes symbols on the tape, a finite set of states, and transition rules determining the behavior of the machine. The machine operates by reading a symbol on the tape, updating the symbol according to its transition rules, moving the head left or right, and transitioning to a new state. Turing Machines can simulate any algorithm and are capable of performing any calculation that can be performed by any computing machine.
1) The document describes the procedure for deriving the small signal model and transfer functions of an ideal boost converter. Key steps include defining state variables, writing state equations, averaging, linearizing, and taking the Laplace transform.
2) Transfer functions are derived from the small signal model matrix for the ideal case, including control-to-output, control-to-inductor current, and inductor current-to-output.
3) The process is then repeated for a non-ideal boost converter that includes resistances in the inductor and capacitor. Additional state equations are written to account for the resistances.
Pushdown automata (PDA) are machines that process input strings and use a stack to determine transitions between states. A PDA consists of states, input symbols, stack symbols, transition functions, an initial state, initial stack symbol, and final states. The transition function specifies how the PDA moves between states based on the current state, input symbol, and top stack symbol, and may involve operations like replacing, pushing, or popping symbols from the stack. PDAs can recognize languages like {anbn: n ≥ 0} by pushing a symbol for each a and popping for each b.
I have been receiving multiple queries on what is clk-to-q delay, how's it different from library setup time and library hold time, etc. I mentioned in my discussions, that the videos on CMOS digital circuit will be uploaded soon, but looks like, it might take some time, and hence decided to uploaded few images from my CMOS course, to explain the difference between all of them.
This document discusses latches and their design process. It begins by defining a latch as a circuit that has two stable states and can store state information. It then describes the different types of latches including asynchronous and synchronous latches. The RS latch is examined in more detail with diagrams of its logic structure and a truth table. Key properties of the RS latch are that it uses two inputs called Set and Reset to store a 1 or 0 without a clock, and it can immediately change its output when the inputs change.
Shift registers allow data to be transported serially by cascading flip flops. They can transfer data either serially or in parallel and are used to implement serial communication and arithmetic operations. A universal shift register uses a multiplexer structure to provide a programmable register that can perform different operations like shifting, loading, or clearing through control signals.
Sequential circuits consist of combinational logic and memory elements like latches and flip-flops. There are different types of latches and flip-flops that differ in their trigger mechanisms and outputs, including SR latches, D latches, and edge-triggered flip-flops like SR, D, and JK flip-flops. Asynchronous inputs can directly set or reset flip-flop outputs independent of the clock signal.
This document discusses sequential logic circuits and memory elements such as latches and flip-flops. It describes different types of latches including the S-R latch, gated S-R latch, and gated D latch. It also covers various types of flip-flops including the S-R, D, J-K, and T flip-flops. It explains the differences between latches and flip-flops and their applications in synchronous and asynchronous logic circuits.
Ethernet has taken the lead as the preferred platform for connecting to the cloud, but not all Ethernet cloud connections (CloudE) are created equal. Providers and users alike are calling for the industry to establish standards of excellence, and the Cloud Ethernet Forum is responding. Read about CloudE 1.0 and what its standards of excellence could mean for CloudE.
The document discusses different scheduling algorithms used in operating systems. It describes embedded and autonomous schedulers, priority scheduling, and common scheduling algorithms like first-in-first-out (FIFO), shortest job first (SJF), round-robin (RR), and earliest deadline first (EDF). It also compares different scheduling methods and discusses how scheduling policies are determined by factors like priority functions and decision modes.
This document provides an overview of centralized (client-server) and decentralized (peer-to-peer) network architectures. It begins by classifying computer systems and network paradigms. It then defines and describes key aspects of the peer-to-peer and client-server architectures, including advantages and disadvantages of each. The document considers attempting to replace all client-server systems with peer-to-peer systems and identifies limits and challenges with such an approach. It focuses on comparing representative file sharing applications of each paradigm from an economic perspective.
Building for success and failure with DisqusJonathon Hill
The document discusses using caching strategies to improve performance when accessing dynamic data from the Disqus API. It recommends using a "burst cache" like Memcached for fast access and a "failover cache" like MongoDB to handle cache misses or failures. A health check is also suggested to monitor the API and gradually increase the failover cache expiration times if issues arise. Code examples and links are provided for implementing these caching techniques using Guzzle, Memcached, MongoDB and other tools.
The system unit contains the main electronic components of the computer. It includes a processor, memory, adapter cards, drive bays, and a power supply. The processor interprets and carries out instructions, and includes components like the control unit and arithmetic logic unit. Memory temporarily stores instructions, data, and results, and can be volatile RAM or nonvolatile ROM/flash memory. Additional components allow for expansion and connection of external devices like ports, buses, and bays. Regular cleaning of the system unit is important for preventing overheating and corrosion.
Midiendo la calidad de código en WTF/Min (Revisado EUI Abril 2014)David Gómez García
The document discusses various examples of poor code quality, such as unnecessary comments, overly complex code, poor naming conventions, and unnecessary code. It provides examples of real code snippets that demonstrate these issues. It also discusses principles of good code quality like keeping code simple, avoiding duplication, and separation of concerns. Finally, it discusses tools and techniques for measuring and ensuring code quality like unit testing, code reviews, quality metrics, and issue tracking dashboards.
The document discusses input/output (I/O) systems and device management. It describes a hierarchical model for I/O with an abstract interface and device-dependent drivers. It also covers various I/O techniques like programmed I/O with polling and interrupts, direct memory access, buffering, error handling, disk scheduling, and device sharing.
32 Ways a Digital Marketing Consultant Can Help Grow Your BusinessBarry Feldman
How can a digital marketing consultant help your business? In this resource we'll count the ways. 24 additional marketing resources are bundled for free.
This document describes non-deterministic finite automata (NFAs). It begins by presenting examples of NFAs with different transition diagrams and explanations. It then provides a formal definition of NFAs, including their components like states, alphabet, transition function, initial state, and final states. The document explains how the transition function works and how to determine the language accepted by an NFA using the extended transition function. It proves that NFAs and deterministic finite automata (DFAs) are equivalent by showing that the languages accepted by each are equal subsets of each other. Finally, it demonstrates how to convert an NFA to an equivalent DFA by constructing the DFA states as power sets of the NFA states and defining
1. The document discusses deterministic finite automata (DFAs) and nondeterministic finite automata (NFAs). DFAs have a single transition function, while NFAs have a transition function that can map a state-symbol pair to multiple possible next states.
2. Examples are given of DFAs and NFAs that accept certain languages over various alphabets. The DFA examples use transition diagrams to represent the transition functions, while the NFA examples explicitly define the transition functions.
3. Key properties of DFAs and NFAs are summarized, including their definitions as 5-tuples and how languages are accepted by looking for paths from the starting to a final state.
This document discusses deterministic finite automata (DFAs) and nondeterministic finite automata (NFAs). It provides examples of DFAs that accept certain languages over various alphabets. It also defines NFAs formally and provides examples of NFAs that accept languages. The key differences between DFAs and NFAs are that NFA transition functions can map a state-symbol pair to multiple possible next states, whereas DFA transition functions map to exactly one next state.
NFA or Non deterministic finite automatadeepinderbedi
An NFA (non-deterministic finite automaton) can have multiple transitions from a single state on a given input symbol, whereas a DFA (deterministic finite automaton) has exactly one transition from each state on each symbol. The document discusses NFAs and how they differ from DFAs, provides examples of NFA diagrams, and describes how to convert an NFA to an equivalent DFA.
The document describes non-deterministic finite automata (NFAs). It provides examples of NFAs accepting or rejecting various strings. An NFA may have multiple possible state transitions for a given input symbol, allowing different computations. A string is accepted if any computation reaches a final state with the entire input consumed. A string is rejected if no computation satisfies these conditions. The language accepted by an NFA consists of all strings for which there exists an accepting computation.
The document discusses finite state automata (FSA). It defines FSA as an abstract mathematical model with discrete inputs and outputs that can recognize the simplest languages (regular languages). It distinguishes between deterministic finite state automata (DFSA) and nondeterministic finite state automata (NFSA). For DFSA, there is a single target state for each current state and input, while NFSA can have multiple target states. The document provides examples of DFSA and NFSA and discusses their formal definitions, transition functions, extended transition functions, accepted languages, and equivalence between DFSA and NFSA models.
The document discusses automata and formal languages. It begins by defining an automaton as a theoretical self-propelled computing device that follows a predetermined sequence of operations automatically. It then defines a language as a set of strings chosen from an alphabet. There are two types of finite automata: deterministic finite automata (DFA) and nondeterministic finite automata (NDFA). A DFA is defined by a 5-tuple including states, transitions, start and accepting states. The document provides examples of DFAs and how strings are processed. It also discusses epsilon-NFAs and provides steps to convert an NFA to a DFA.
This document provides an overview of finite automata, including deterministic finite automata (DFAs) and non-deterministic finite automata (NFAs). It defines what a finite automaton is, describes the components of a DFA and NFA, and how they process input strings. It also discusses the relationship between DFAs and NFAs, showing that any language recognizable by an NFA is also recognizable by a DFA through subset construction. Examples are provided to illustrate DFA and NFA design.
This document provides an overview of finite automata and regular languages. It defines deterministic finite automata (DFAs) and non-deterministic finite automata (NFAs) as state machines that can recognize regular languages. A DFA has a single active state at any time and deterministic transitions, while an NFA can have multiple active states and non-deterministic transitions. The document shows that NFAs and DFAs are equivalent in their expressive power by describing how to convert any NFA to an equivalent DFA using subset construction.
Deterministic Finite State Automata (DFAs) are machines that read input strings and determine whether to accept or reject them based on their state transitions. A DFA is defined as a 5-tuple (Q, Σ, δ, q0, F) where Q is a finite set of states, Σ is a finite input alphabet, q0 is the starting state, F is the set of accepting states, and δ is the transition function that maps a state and input symbol to the next state. The language accepted by a DFA is the set of strings that cause the DFA to enter an accepting state. Nondeterministic Finite State Automata (NFAs) are similar but δ maps to sets of states rather
Deterministic Finite State Automata (DFAs) are machines that read input strings and determine whether to accept or reject them based on their state transitions. A DFA is defined as a 5-tuple (Q, Σ, δ, q0, F) where Q is a finite set of states, Σ is a finite input alphabet, q0 is the starting state, F is the set of accepting states, and δ is the transition function that maps a state and input symbol to the next state. The language accepted by a DFA is the set of strings that cause the DFA to enter an accepting state. Nondeterministic Finite State Automata (NFAs) are similar but δ maps to sets of states rather
The document discusses the construction of a deterministic finite automaton (DFA) that is equivalent to a given nondeterministic finite automaton (NFA). It presents a step-by-step process for transforming an NFA into an equivalent DFA by removing non-determinism. The key steps are: 1) assigning arcs from NFA transitions to DFA states, 2) eliminating epsilon transitions, 3) handling undefined transitions, and 4) determining accepting states. The construction ensures that for every string accepted by the DFA, there is an accepting path in the NFA, and vice versa.
The document describes pushdown automata (PDA) which are analogous to context-free languages in the same way that finite automata are analogous to regular languages. A PDA has states, input symbols, stack symbols, transition functions, an initial state, initial stack symbol, and accepting states. The transition function specifies state transitions based on the current state, input symbol, and top of stack symbol and can modify the stack. The document provides examples of PDAs for languages of the form wwr and balanced parentheses and discusses how PDAs work by changing their instantaneous descriptions as the input is processed and stack is modified.
This document discusses finite automata, including:
- Finite automata are machines that recognize patterns in input sequences.
- There are two main types: deterministic finite automata (DFAs) and non-deterministic finite automata (NFAs).
- DFAs have a single transition between states for each input, while NFAs may have multiple possible next states for each input.
This document describes pushdown automata (PDA) and how they are used to recognize context-free languages. It provides definitions of PDA, including their components and transition function. An example PDA is given for the language of balanced parentheses. The document also discusses how PDA can accept by final state or empty stack, and how PDA are equivalent to context-free grammars. It describes how to convert between PDA that accept by final state vs empty stack, and how to construct a PDA from a given context-free grammar.
The document describes how to convert a given NFA-ε into an equivalent DFA. It finds the ε-closure of each state in the NFA to create the states of the DFA. It then determines the transitions between these DFA states on each input symbol by taking the ε-closure of the NFA state transitions. This results in a DFA transition table and diagram that is equivalent to the original NFA.
This document introduces flip-flops, an important building block for sequential circuits. It defines the basic SR latch and investigates its properties. It then introduces clocks and shows how they can synchronize latches to create gated latches. Finally, it develops a more stable clocking technique called dynamic clocks to create flip-flops. The document discusses the unstable behavior of SR latches if inputs change simultaneously and how gated latches and flip-flops avoid this issue. It provides examples of gated SR latches, gated D latches, and positive edge-triggered D flip-flops.
This document discusses Turing machines and the hierarchy of formal languages. It provides examples of deterministic Turing machines that recognize specific formal languages. The key points are:
1) Turing machines can model any computable function and generalize classes of formal languages from regular to recursively enumerable.
2) A deterministic Turing machine is formally defined as a 7-tuple with states, alphabets, transitions, and acceptance conditions.
3) Examples show deterministic Turing machines that recognize the languages of strings ending in 0 and strings with equal numbers of 0s and 1s.
This document summarizes key aspects of file systems and distributed file systems. It discusses hierarchical file system models, file directories and operations on directories. It also covers physical file organization methods like contiguous, linked and indexed organization. For distributed file systems, it discusses directory structures, location transparency and implementing a global directory across multiple servers.
The document discusses file systems and their implementation. It covers hierarchical directory structures, operations on files and directories, physical storage methods like contiguous and linked allocation, and distributed file system structures with global or local naming and caching techniques. Distributed file systems must address consistency across replicated file copies.
The document discusses code and data sharing in operating systems. It explains that sharing allows multiple processes to access a single copy of code or data in memory, reducing memory usage. This is implemented through techniques like static and dynamic linking, where processes can call and use shared code and data segments. The document provides an example of simple code sharing between two processes that call and access the same AddMul2 function.
The document discusses different approaches to linking and sharing memory between processes, including static linking, dynamic linking, and distributed shared memory. It covers:
- Static linking resolves references before execution, while dynamic linking does so during execution.
- Sharing in systems without virtual memory uses base registers to point to shared code and private stacks/data.
- Paging and segmentation allow processes to share code and data pages/segments by mapping to the same physical memory frames.
- Dynamic linking uses transfer vectors to lazily resolve external references.
- Distributed shared memory aims to provide a single shared memory illusion across distributed physical memories.
The document summarizes key concepts about virtual memory, including:
1) Virtual memory allows processes to execute even if not entirely in physical memory by automatically allocating storage upon request, creating the illusion of large contiguous memory spaces.
2) Common virtual memory implementations include paging, segmentation, and paging with segmentation. Paging divides memory into fixed-size pages while segmentation uses multiple logical segments.
3) Issues in virtual memory design include address mapping, placement, replacement, load control, and sharing. Translation lookaside buffers help speed up address translation.
This document discusses principles and implementations of virtual memory. It describes how virtual memory is implemented using paging, with each process' memory divided into fixed-size pages. It covers address translation from virtual to physical addresses using a page table. Demand paging is described, where pages are loaded on demand when accessed. Global and local page replacement algorithms are summarized, including LRU, FIFO and working set models. Implementation techniques like segmentation, multi-level page tables, and translation lookaside buffers are also briefly outlined.
The document discusses different memory partitioning schemes used in operating systems. It describes fixed partitions where memory is divided into predetermined sized partitions at initialization time. It also describes variable partitions where memory is not pre-partitioned and is allocated on demand, which can cause external fragmentation. Dynamic binding is discussed where the logical to physical address mapping occurs at execution time with hardware support.
The document discusses several topics related to physical memory management:
1) Program transformations like compilation, linking, and loading prepare a program for execution by assigning physical addresses through static or dynamic binding.
2) Memory can be partitioned into fixed or variable partitions, with tradeoffs between the two approaches. Variable partitions use allocation strategies like first-fit to allocate memory.
3) Insufficient memory can be addressed through memory compaction, swapping, or overlays which load only needed program sections into physical memory.
The document discusses deadlocks in operating systems, focusing on deadlocks that can occur with reusable resources. It defines deadlocks and distinguishes between reusable and consumable resources. It then covers various approaches to handling deadlocks, including detection methods like resource graphs and reduction, and prevention techniques like deadlock avoidance and prevention.
The document discusses different approaches to handling deadlocks in systems, including detection and recovery from deadlocks. It describes deadlock detection methods like reducing resource graphs and detecting knots. Dynamic deadlock avoidance techniques like claim graphs and the Banker's algorithm are presented, tentatively granting requests to check for safe states. Deadlock prevention restricts resource acquisition ordering to eliminate circular waits.
The document discusses various process and thread scheduling methods. It covers the organization of schedulers, common scheduling algorithms like priority scheduling, and comparisons of scheduling methods. It also describes the priority inversion problem and solution of dynamic priority inheritance to prevent lower priority processes from indefinitely blocking higher priority processes.
The document discusses the implementation of processes and threads in an operating system kernel. It covers topics like process control blocks (PCBs) that store process state and metadata, data structures like queues that the kernel uses to manage processes and threads, and how the kernel implements key operations on processes and threads like creation, suspension, activation, and destruction through manipulating the PCBs. It also discusses implementing synchronization mechanisms like semaphores using primitives like test-and-set instructions.
The document summarizes key concepts related to operating system kernels including process and thread management, synchronization mechanisms like semaphores and monitors, interrupt handling, and communication primitives. It describes data structures like process control blocks and priority queues used to implement these concepts. It also explains algorithms for operations on processes, synchronization, clock management, and interrupt servicing.
The document discusses higher-level synchronization and communication techniques in operating systems, including monitors, condition variables, and message-based communication. Monitors provide mutual exclusion and condition variables that allow processes to wait for and signal events. They present a higher-level alternative to low-level semaphores. An example monitor uses condition variables to synchronize access to a bounded buffer queue.
The document discusses various techniques for synchronization in shared memory and distributed systems, including monitors, message passing, remote procedure calls (RPC), and logical clocks. It describes monitors as a synchronization primitive for shared memory that uses condition variables and mutual exclusion. For distributed synchronization it discusses message passing with channels/ports/mailboxes and RPC/rendezvous. Classical synchronization problems like the readers-writers problem and dining philosophers problem are presented along with their solutions. Logical clocks are introduced as a way to order events in a distributed system when physical clocks may be skewed.
This document discusses processes and their interaction in operating systems. It defines a process as a program in execution that includes code, data, threads of execution, and resources. It describes how operating systems virtualize CPUs and memory to allow multiple processes to run concurrently. It discusses different types of process interactions including competition over shared resources, which can cause race conditions, and cooperation through synchronization and message passing. It presents several algorithms to solve the critical section problem of mutual exclusion when processes access shared resources.
The document discusses processes and process interactions. It defines processes as activities executing on a CPU that conceptually run concurrently. Processes can compete for resources or cooperate through synchronization. Basic synchronization techniques discussed include semaphores, which are integers that processes operate on to synchronize access to shared resources through wait and signal operations. Producer-consumer problems are provided as examples where semaphores can be used for processes to cooperate. Event synchronization constructs like wait and post are also summarized for synchronizing on asynchronous events.
The document discusses properties of context-free languages. It states that context-free languages are closed under operations like union, concatenation, and reversal. However, they are not closed under intersection and complement, as these operations can result in non-context-free languages. The document also notes that the intersection of a context-free language and a regular language is always context-free.
This document discusses context-free grammars and ambiguity. It provides examples of context-free grammars and their derivations using derivation trees. A context-free grammar is ambiguous if some string in the language has two or more derivation trees. Ambiguity is undesirable for programming languages as it leads to parsing ambiguities. To remove ambiguity, an ambiguous grammar can be modified to produce a non-ambiguous grammar.
Let L = {anbn : n ≥ 0}
Assume L is regular. By the pumping lemma, there exists m ≥ 1 such that any string w ∈ L with |w| ≥ m can be written as w = xyz such that:
1. |xy| ≤ m
2. |y| ≥ 1
3. ∀i ≥ 0, xyiz ∈ L
Consider the string s = ambm where m is the pumping lemma constant. We can write s = xyz satisfying the properties above.
Since y must contain at least one symbol, pumping y up i times for i > 0 would yield a string not in L, since the number of a's and b's
2. Nondeterministic FA
As we know, any w *, then *(q0, w)
corresponds to a unique walk on the
transition graph of a DFA M.
When we allow FA to act non-
deterministically, it implies *(q0, w) may
not correspond to a unique walk on the
transition graph of such FA anymore.
2
7. Example
q1 a q2 q1 a q2
a a
q0 q0
a a
q3 q3
All input is consumed & No transition for (q3, a),
stops at a final state. the automaton hangs, a
dead configuration
aa is “accepted” aa is “rejected”
7
8. An NFA accepts a string:
when there is one computation of the NFA
that accepts the string
such that
all the input is consumed AND
the automaton is in a final state
Thus aa is “accepted” by the NFA.
8
9. Therefore, an NFA rejects a string:
when there is NO computation of the NFA
that accepts the string:
• All the input is consumed and the
automaton is in a non final state
OR
• The input cannot be consumed
(dead configuration)
9
10. Example
a is rejected by the NFA:
“reject”
q1 a q2 q1 a q2
a a
q0 q0
a a
q3 “reject” q3
All possible computations lead to rejection
10
11. Another Example
aaa is rejected by the NFA:
No transition:
“reject” the automaton hangs
q1 a q2 q1 a q2
a a
q0 q0
a a
“reject”
q3 q3
All possible computations lead to rejection
11
14. transition is allowed for NFA
It implies that the NFA can move from
state qi to state qj without moving the
read head, i.e. without consuming any
input symbol.
14
15. Another NFA Example
Consider the input string ab:
q0 a q1 b q2 ab is “accepted”
q0 a q1 b q2 q3 “rejected”
q0 a q1 b q2 q3 q0 “rejected”
q0 a q1 b q2 q3
15
21. NFAs are interesting because we can
express languages easier than DFAs
Example: design a FA for {a} with ={a}.
Example: design a FA for
{awa: w {a, b}*} with ={a, b}.
21
22. Formal Definition of NFAs (p.49)
M Q, , , q0 , F
Q : Set of states, Remember:
The is not a symbol,
: Input alphabet, it never appears on the
: Transition function
q0 : Initial state
F : Final states
22
23. NFA’s Transition Function
:Qx( { }) 2Q
Note that Every DFA is an NFA.
When input is , the original state is always a
member of its transition output, i.e. q (q, )
23
27. :Qx( { }) 2Q
(q2 ,1) . So as (q0, 0) & (q2, 0)
0
q0 q1 0, 1 q2
1
Remark: Every DFA is an NFA.
27
28. Extended Transition
Function *
Similar as DFA, extended transition function is
defined to work with inputs to be strings.
*
:Qx( { } )* 2Q
*
or simply :Qx * 2Q
When input string is empty ( ), the original state
is always a member of its extended transition, i.e.
q * q,
28
35. Formally, for an NFA
There is a walk from qi to q j with label
w 1 2 ... k
qi w qj
if and only if q j * qi , w w 1 2 ... k
qi 1 2 k
qj
Def. 2.5 p.51
For an nfa, the extended transition function is
defined so that qj *( qi , w) iff there is a walk
in the transition graph from qi to qj labeled w.
35
36. Hw # 20 p.56
Show that for any nfa
*
(q, wv)
*
*
( p, v).
p ( q , w)
for all q Q, and all w, v *
36
37. The Language of an NFA M
aa
F q0 ,q5 q4 q5
a a
q0 a q1 b q2 q3
Consider the aa, or the walk from q0 with label
aa:
* q0 , aa q4 , q5 aa L(M )
F 37
38. ab
F q0 ,q5 q4 q5
a a
q0 a q1 b q2 q3
* q0 , ab q2 , q3 , q0 ab L M
F 38
39. abaa
F q0 ,q5 q4 q5
a a
q0 a q1 b q2 q3
* q0 , abaa q4 , q5 abaa L(M )
F 39
40. F q0 ,q5 q4 q5
a a
q0 a q1 b q2 q3
* q0 , aba q1 aba L M
F 40
41. q4 q5
a a
q0 a q1 b q2 q3
LM (ab) n aa : n 0
41
42. Formally
The language accepted by NFA M is:
LM w1, w2 , w3 ,...
where * (q0 , wm ) {qi , q j ,..., qk ,...}
and there is some qk F (final state)
42
43. w LM * (q0 , w)
qi
w
qk qk F
q0 w
w qj
L( M ) {w *: * ( q0, w) F }
43
44. An NFA Example
Consider the input string ab:
q0 a q1 b q2 * (q0 , ab) {q0 , q2 , q3}
q0 a q1 b q2 q3
q0 a q1 b q2 q3 q0
q0 a q1 b q2 q3
44
45. Consider the input string ab:
* (q0 , ab) {q0 , q2 , q3}
q0 a q1 b q2 ab is “accepted”
q0 a q1 b q2 q3
q0 a q1 b q2 q3 q0
q0 a q1 b q2 q3
45
47. q0 a q1 b q2 q3 q0 a q1 b q2
As long as there is one computation that
consumes all input symbols and stops at a final
state, aabb is accepted.
“accept”
q0 a q1 b q2 q3
47
49. NFAs are interesting because we can
express languages easier than DFAs
Example: design an FA for {a} with ={a, b}.
Example: design an FA for
{awa: w {a, b}*} with ={a, b}.
49
51. Hw # 20 p.56
Show that for any nfa
*
(q, wv) U
*
*
( p, v).
p ( q , w)
for all q Q, and all w, v *
51
52. Remarks for FA M = {Q, , , q0, F }:
•The symbol never appears on the
input alphabet
•Simple NFA:
M1 M2
q0 q0
L(M1 ) = {} L(M 2 ) = {λ}
52
53. The language accepted by NFA M is:
L( M ) {w *: * ( q0, w) F }
Is it true that for any NFA M ,
L(M ) {w *: * ( q0, w) F }
Is it true that for any NFA M,
L(M ) {w *: * (q0, w) (Q F ) }
Referring to Hw 2.3 # 5, 6. P.62 53
54. NFAs accept the
Regular Languages
Hw of 2.2(p.54) 1 – 19 not hand in
NFA 54
56. Example of equivalent
machines
0
L M1 {10} *
q0 q1
NFA M1 1
0,1
0
L M2 {10}* q1 1 q2
q0
1
DFA M2 0
56
57. Which is more powerful?
Every dfa is an nfa.
Thus, every language accepted by
a dfa is also accepted by an nfa.
Is nfa more powerful than dfa?
57
58. We will prove:
Languages
Regular
accepted
Languages
by NFAs
Languages
accepted
by DFAs
NFAs and DFAs have the
same computation power
58
59. Step 1
Languages
Regular
accepted
Languages
by NFAs
Proof: Every DFA is trivially an NFA
Any language L accepted by a DFA
is also accepted by an NFA
59
60. Step 2
Languages
Regular
accepted
Languages
by NFAs
Proof: Any NFA can be converted to an
equivalent DFA
Any language L accepted by an NFA
is also accepted by a DFA 60
61. Convert NFA to DFA
a
NFA M N q0 a q1 q2
b
* a b
q0
q1
q2
An extended transition
function is very helpful
61
62. Convert NFA to DFA
NFA M N a
q0 a q1 q2
b
Let p0={q0} be the initial state for MD
DFA MD
q0
Back to Procedure NFA to DFA
62
63. Convert NFA to DFA
NFA M N a
q0 a q1 q2
b
Now, define D(p0,a) for a
DFA MD D(p0,a) = N*({q0},a) ={q1,q2} p1
q0 a
q1, q2
63
64. Convert NFA to DFA
NFA M N a
q0 a q1 q2
b
D(p0,b) = N*({q0},b) = p2
DFA MD
q0 a
q1, q2
b
Back to Procedure NFA to DFA
64
65. Convert NFA to DFA
NFA M N a
q0 a q1 q2
b
D(p1,a) = N*({q1,q2},a)= N*({q1},a) N*({q2},a)
= {q1,q2} = {q1,q2} = p1
DFA MD a
q0 a
q1, q2
b
65
66. Convert NFA to DFA
NFA M N a
q0 a q1 q2
b
D(p1,b)
= N*({q1,q2},b) = N*({q1},b) N*({q2},b)
= {q0} {q0} = {q0} = p0
DFA MD b a
q0 a
q1, q2
b
66
67. Convert NFA to DFA
NFA M N a
q0 a q1 q2
b
D(p2,a) = N*( ,a) = =p2= D(p2,b)
DFA MD b a
q0 a
q1, q2
b
Back to Procedure NFA to DFA
a, b
67
68. Convert NFA to DFA
NFA M N a
q0 a q1 q2
b
How about final state? Lastly, check if L(M)
b a
DFA MD
q0 a
q1, q2
b
a, b back
68
69. NFA to DFA: Remarks
We are given an NFA MN
We want to convert it
to an equivalent DFA MD
With L M N L( M D )
69
70. If the NFA has states
q0 , q1, q2 ,...
the DFA has states in the powerset
, q0 , q1 , q1, q2 , q3 , q4 , q7 ,....
70
71. Procedure NFA to DFA p.59
1. Initial state of NFA: q0
Initial state of DFA: p0 q0
To Conversion Example
71
72. Procedure NFA to DFA
2. For every DFA’s state pk {qi , q j ,..., qm }
Compute in the NFA
* qi , a ,
* q j,a , union {qi , q j ,..., qm}
...
Add transition to DFA To Convertion Example
D pk , a D {qi , q j ,..., qm }, a {qi , q j ,..., qm }
72
73. Procedure NFA to DFA
Repeat Step 2 for all letters in alphabet,
until
no more transitions can be added.
Remark:
In DFA, every vertex must have exactly | | outgoing
edges, each labeled with a different element of .
To Convertion Example
73
74. Procedure NFA to DFA
3. For any DFA state pk {qi , q j ,..., qm }
If some q j is a final state in the NFA
Then, make p k a final state in the DFA
Remark: If is accepted in the NFA then
p0 q0 should be a final state.
To Convertion Example
74
75. Theorem
Take NFA MN
Apply procedure to obtain DFA MD
Then MN and MD are equivalent :
L(MN) = L(MD)
75
77. First we show: L(MN) L(MD
)
Take arbitrary: w L(MN)
We will prove: w L(MD)
77
78. w L(MN)
MN q0 w qf
w 1 2 ... k
MN q0 1 2 k
qf
78
79. We will show that if w L(MN)
w 1 2 ... k
MN q0 1 2 k
qf
1 2 k
MD
{q0} {q f ,...}
w L(MD)
79
80. More generally, we will show that if in MN :
(arbitrary walk) v a1a2 ... an
a1 a2 an
MN q0 qi qj ql qm
a1 a2 an
MD
{q0} {qi ,...} {q j ,...} {ql ,...} {qm ,...}
80
81. Proof by induction on |v|
Induction Basis: v a1
a1
MN q0 qi
a1
MD
{q0} {qi ,...}
81
82. Induction hypothesis: 1 |v| k
v a1a2 ... ak
a1 a2 ak
MN q0 qi qj qc qd
a1 a2 ak
MD
{q0} {qi ,...} {q j ,...} {qc ,...} {qd ,...}
82
83. Induction Step: |v| k 1
v a1a2 ... ak ak 1 v ak 1
v
a1 a2 ak
MN q0 qi qj qc qd
v
a1 a2 ak
MD
{q0} {qi ,....} {q j ,...} {qc ,...} {qd ,...}
v 83
84. Induction Step: |v| k 1
v a1a2 ... ak ak 1 v ak 1
v
a1 a2 ak ak 1
MN q0 qi qj qc qd qe
v
a1 a2 ak ak 1
MD
{q0} {qi ,...} {q j ,...} {qc ,...} {qd ,...} {qe ,...}
v 84
85. Therefore if w L(MN)
w 1 2 ... k
MN q0 1 2 k
qf
1 2 k
MD
{q0} {q f ,...}
w L(MD)
85
86. We have shown: L(MN) L(MD
)
We also need to show: L(MN) L(MD
)
(proof is similar)
86
87. Example
a b
NFA
q0 , a, b q1 a, b q2
Convert NFA to DFA Simplified version of DFA
a a
a p1 p1
b a, b a, b
p0 b b
b p2 a, b p3 a p4 p2 a, b p3
Ex. 2.13 Fig. 2.14 (p.59)– in class (you
can eliminate the redundant state first) 87
88. Minimal DFAs
A dfa M’ is minimal if M is also a dfa such
that L(M’)=L(M) then |QM | |QM’|
88
89. Minimal DFAs
States p and q are called indistinguishable if
* ( p, w) F implies * (q, w) F
and
* ( p, w) F implies * (q, w) F
for all w *
States p and q are called distinguishable if
w * such that * ( p, w) F and * (q, w) F
89
90. Minimal DFAs
To obtain a minimal DFA M’ from a given DFA M:
Remove all inaccessible states, i.e. not reachable from the
initial state.
Reduce states until it has no more indistinguishable states.
Read procedures: mark (p.64) & reduce (p.66), and
theorem 2.3 (p.65) & 2.4 (p.67) for details.
Remove all inaccessible states can be accomplished
by enumerating all simple paths of the graph of the
dfa starting at the initial state, any state not part of
some path is inaccessible. 90
92. Minimal DFAs
Prove or Disprove:
If M = (Q, , , q0, F) is a minimal DFA for a
regular language L, then M’ = (Q, , , q0, Q-F)
is a minimal DFA for the complement of L.
92
93. Minimal DFAs
Prove or Disprove:
If M = (Q, , , q0, F) is a minimal DFA for a
regular language L, then M’ = (Q, , , q0,
Q-F) is a minimal DFA for the complement of
L.
93