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 Push Down Automata (PDA). It defines PDA as a finite state machine with a stack. A PDA is formally defined as a 7-tuple that includes states, input symbols, stack symbols, transition function, initial state, initial stack symbol, and accepting states. The document provides examples of modeling PDA behavior using instantaneous descriptions that represent the state, remaining input, and stack contents. It also discusses different methods for determining if a PDA accepts an input string, such as checking if the string reduces the stack to empty or reaches an accepting state. Finally, it covers extensions of PDA like non-deterministic and multi-stack models, and how to convert context-free grammars to equivalent
The document describes pushdown automata (PDA). A PDA has a tape, stack, finite control, and transition function. It accepts or rejects strings by reading symbols on the tape, pushing/popping symbols on the stack, and changing state according to the transition function. The transition function defines the possible moves of the PDA based on the current state, tape symbol, and stack symbol. If the PDA halts in a final state with an empty stack, the string is accepted. PDAs can recognize any context-free language. Examples are given of PDAs for specific languages.
This document discusses binary search trees and efficient binary trees. It begins by explaining what a binary search tree is and how it allows for efficient searching in O(log n) time. It then discusses how to create, search, insert, delete, determine the height and number of nodes in a binary search tree. The document also covers mirrored binary trees, threaded binary trees, and AVL trees, which are self-balancing binary search trees that ensure searching remains O(log n) time.
Push Down Automata (PDA) | TOC (Theory of Computation) | NPDA | DPDAAshish Duggal
Push Down Automata (PDA) is part of TOC (Theory of Computation)
From this presentation you will get all the information related to PDA also it will help you to easily understand this topic. There is also one example.
This PPT is very helpful for Computer science and Computer Engineer
(B.C.A., M.C.A., B.TECH. , M.TECH.)
Here is the NFA in formal notation:
Q = {q0, q1, q2}
Σ = {a, b}
q0 = q1 (the initial state)
F = {q0} (the single accepting state)
q0
The transition function δ is:
a
q1 → {q0, q2}
q2 → {q0}
b
This NFA accepts ε, a, baba, baa, and aa since there is a path from the initial state q1 to the accepting state q0 under those inputs. It does not accept b, bb, or babba since there is no path from
John von Neumann invented the merge sort algorithm in 1945. Merge sort follows the divide and conquer paradigm by dividing the unsorted list into halves, recursively sorting each half through merging, and then merging the sorted halves back into a single sorted list. The time complexity of merge sort is O(n log n) in all cases (best, average, worst) due to its divide and conquer approach, while its space complexity is O(n) to store the temporary merged list.
The document discusses Breadth First Search (BFS) and Depth First Search (DFS) algorithms for graphs. It defines key graph terminology and describes how to represent graphs using adjacency lists and matrices. It then explains how BFS and DFS work by exploring the graph in different ways, keeping track of vertex colors and paths/trees produced. Both algorithms run in O(V+E) time and have different applications like routing, sorting, and finding connected components.
The document describes Push Down Automata (PDA). It defines PDA as a finite state machine with a stack. A PDA is formally defined as a 7-tuple that includes states, input symbols, stack symbols, transition function, initial state, initial stack symbol, and accepting states. The document provides examples of modeling PDA behavior using instantaneous descriptions that represent the state, remaining input, and stack contents. It also discusses different methods for determining if a PDA accepts an input string, such as checking if the string reduces the stack to empty or reaches an accepting state. Finally, it covers extensions of PDA like non-deterministic and multi-stack models, and how to convert context-free grammars to equivalent
The document describes pushdown automata (PDA). A PDA has a tape, stack, finite control, and transition function. It accepts or rejects strings by reading symbols on the tape, pushing/popping symbols on the stack, and changing state according to the transition function. The transition function defines the possible moves of the PDA based on the current state, tape symbol, and stack symbol. If the PDA halts in a final state with an empty stack, the string is accepted. PDAs can recognize any context-free language. Examples are given of PDAs for specific languages.
This document discusses binary search trees and efficient binary trees. It begins by explaining what a binary search tree is and how it allows for efficient searching in O(log n) time. It then discusses how to create, search, insert, delete, determine the height and number of nodes in a binary search tree. The document also covers mirrored binary trees, threaded binary trees, and AVL trees, which are self-balancing binary search trees that ensure searching remains O(log n) time.
Push Down Automata (PDA) | TOC (Theory of Computation) | NPDA | DPDAAshish Duggal
Push Down Automata (PDA) is part of TOC (Theory of Computation)
From this presentation you will get all the information related to PDA also it will help you to easily understand this topic. There is also one example.
This PPT is very helpful for Computer science and Computer Engineer
(B.C.A., M.C.A., B.TECH. , M.TECH.)
Here is the NFA in formal notation:
Q = {q0, q1, q2}
Σ = {a, b}
q0 = q1 (the initial state)
F = {q0} (the single accepting state)
q0
The transition function δ is:
a
q1 → {q0, q2}
q2 → {q0}
b
This NFA accepts ε, a, baba, baa, and aa since there is a path from the initial state q1 to the accepting state q0 under those inputs. It does not accept b, bb, or babba since there is no path from
John von Neumann invented the merge sort algorithm in 1945. Merge sort follows the divide and conquer paradigm by dividing the unsorted list into halves, recursively sorting each half through merging, and then merging the sorted halves back into a single sorted list. The time complexity of merge sort is O(n log n) in all cases (best, average, worst) due to its divide and conquer approach, while its space complexity is O(n) to store the temporary merged list.
The document discusses Breadth First Search (BFS) and Depth First Search (DFS) algorithms for graphs. It defines key graph terminology and describes how to represent graphs using adjacency lists and matrices. It then explains how BFS and DFS work by exploring the graph in different ways, keeping track of vertex colors and paths/trees produced. Both algorithms run in O(V+E) time and have different applications like routing, sorting, and finding connected components.
The document discusses stacks and converting expressions between infix, postfix, and prefix notation. It provides examples and algorithms for converting infix expressions to postfix. The algorithms involve using an operator stack to determine the order of operations based on precedence rules. Operands are added to the output string and operators are pushed to the stack or popped and added to the output string based on their precedence.
The document discusses techniques for converting non-deterministic finite automata (NFAs) to deterministic finite automata (DFAs) in three steps:
1) Using subset construction to determinize an NFA by considering sets of reachable states from each transition as single states in the DFA.
2) Minimizing the number of states in the resulting DFA using an algorithm that merges equivalent states that have identical transitions for all inputs.
3) Computing equivalent state sets using partition refinement, which iteratively partitions states based on their transitions for each input symbol.
This document discusses pushdown automata (PDA) and how they can be used to accept context-free languages. It defines a PDA as a 7-tuple that includes a finite set of states, input symbols, stack symbols, initial state, initial stack symbol, set of final states, and a transition function. A context-free grammar is also defined as a 4-tuple that includes variables, terminals, production rules, and a start symbol. The document then shows how to construct a PDA that is equivalent to a given context-free grammar by defining transition rules based on the grammar productions. An example of converting a grammar to a PDA using this construction method is also provided.
The document discusses depth-first search (DFS) and breadth-first search (BFS) graph traversal algorithms. It provides details on how DFS uses a stack to search deeper in the graph first by exploring edges from the most recently discovered vertex. BFS uses a queue to search outward from the starting node to neighboring nodes first before searching further. An example of DFS running on a graph is shown with each step of the algorithm.
This document discusses deterministic finite automata (DFA) minimization. It defines the components of a DFA and provides an example of a non-minimized DFA that accepts strings with 'a' or 'b'. The document then introduces an algorithm to minimize a DFA by identifying redundant states that are not necessary to recognize the language. The algorithm works by iteratively labeling states as distinct or equivalent based on their transitions and whether they are accepting states. This process combines equivalent states to produce a minimized DFA with the smallest number of states.
This document discusses various problems that can be solved using backtracking, including graph coloring, the Hamiltonian cycle problem, the subset sum problem, the n-queen problem, and map coloring. It provides examples of how backtracking works by constructing partial solutions and evaluating them to find valid solutions or determine dead ends. Key terms like state-space trees and promising vs non-promising states are introduced. Specific examples are given for problems like placing 4 queens on a chessboard and coloring a map of Australia.
The document discusses several shortest path algorithms for graphs, including Dijkstra's algorithm, Bellman-Ford algorithm, and Floyd-Warshall algorithm. Dijkstra's algorithm finds the shortest path from a single source node to all other nodes in a graph with non-negative edge weights. Bellman-Ford can handle graphs with negative edge weights but is slower. Floyd-Warshall can find shortest paths in a graph between all pairs of nodes.
This document discusses Hasse diagrams. It begins by stating the learning outcome is for students to illustrate Hasse diagrams. It then provides background on Hasse diagrams, noting they were originally devised to represent partially ordered sets and were created by Helmut Hasse. The document gives examples of drawing Hasse diagrams representing dividing relationships between numbers. It notes there can be multiple ways to draw a Hasse diagram for a given problem. In conclusion, it restates the topic covered was Hasse diagrams.
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.
Dijkstra's algorithm is used to find the shortest path between nodes in a weighted graph. It works by assigning initial distances to all nodes from the starting node and updating them as shorter paths are found, extracting the node with the lowest distance, and updating distances for neighboring unvisited nodes. The algorithm returns the shortest distance and path between the starting node and all other nodes in the graph.
Here are the steps to construct an NFA-ε equivalent to the given regular expressions:
1. ε + a b (a+b)*
- States: q0, q1, q2
- Transitions: q0 -> q1 on ε, q0 -> q2 on a, q1 -> q1 on a/b, q1 -> q2 on b
- Start state: q0
- Final states: q2
2. (0+1)* (00+11)
- States: q0, q1, q2, q3
- Transitions: q0 -> q0 on 0/1, q0 -> q1 on 0/1
1) The pumping lemma can be used to prove that a language is not regular or context-free.
2) For regular languages, the pumping lemma states that for any string x longer than n, x can be broken into uvw such that pumping v any number of times results in strings still in the language.
3) For context-free languages, the pumping lemma similarly states that any string x longer than n can be broken into five parts such that pumping the second and fourth parts any number of times results in strings still in the language.
BFS is the most commonly used approach. BFS is a traversing algorithm where you should start traversing from a selected node (source or starting node) and traverse the graph layerwise thus exploring the neighbor nodes (nodes which are directly connected to the source node.
1. The document discusses the equivalence between different models of regular languages including finite state automata (FSA), regular grammars, and regular expressions.
2. It explains how to convert between an FSA and regular grammar by establishing a one-to-one correspondence between the states of the FSA and symbols in the grammar.
3. Regular languages are closed under operations like concatenation - the concatenation of two regular languages is also regular.
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
The document discusses transaction concepts in database systems. It defines a transaction as a series of actions performed by a single user or application to access and modify database contents. Transactions must follow the ACID properties - atomicity, consistency, isolation, and durability. Concurrency control techniques like locking are used to ensure transactions execute correctly in a concurrent environment without interfering with each other. Serializability is discussed as a way to identify which non-serial schedules maintain consistency. Conflict serializability and precedence graphs are described as a technique to test schedules for serializability.
A lattice is a partially ordered set where every pair of elements has a supremum (least upper bound) and infimum (greatest lower bound). A lattice must satisfy the properties that (1) any two elements have a supremum and infimum and (2) the supremum of two elements is their join and the infimum is their meet. Common examples of lattices include the natural numbers under the divisibility relation and sets under the subset relation. Lattice theory has applications in many areas of computer science and engineering such as distributed computing, concurrency theory, and programming language semantics.
Let's analyze the remainder term R6 using the geometry series method:
|tj+1| = (j+1)π-2 ≤ π-2 = k|tj| for all j ≥ 6 (where 0 < k = π-2 < 1)
Then, |R6| ≤ t7(1 + k + k2 + k3 + ...)
= t7/(1-k)
= 7π-2/(1-π-2)
So the estimated upper bound of the truncation error |R6| is 7π-2/(1-π-2)
The Laplace transform is an integral transform that converts a function of time into a function of complex frequency. It is defined as the integral of the function multiplied by e-st from 0 to infinity. The Laplace transform is used to solve differential equations by converting them to algebraic equations. Some key properties of the Laplace transform include linearity, shifting theorems, differentiation and integration formulas, and methods for periodic and anti-periodic functions.
The document provides an overview of topics related to the Laplace transform and its applications. It defines the Laplace transform, discusses properties like linearity and examples of transforms of elementary functions. It also covers the inverse Laplace transform, differentiation and integration of transforms, evaluation of integrals using transforms, and applications to differential equations.
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 provides an overview of pushdown automata (PDA). It defines a PDA as a finite automaton with an additional memory stack. This stack allows two operations - push, which adds a new symbol to the top of the stack, and pop, which removes and reads the top symbol. The document then discusses the formal definition of a PDA as a septuple and provides an example of a PDA that accepts the language of strings with an equal number of 0s and 1s. It concludes with an explanation of the state operations of replace, push, pop and no change and conditions for PDA acceptance and rejection.
The document discusses stacks and converting expressions between infix, postfix, and prefix notation. It provides examples and algorithms for converting infix expressions to postfix. The algorithms involve using an operator stack to determine the order of operations based on precedence rules. Operands are added to the output string and operators are pushed to the stack or popped and added to the output string based on their precedence.
The document discusses techniques for converting non-deterministic finite automata (NFAs) to deterministic finite automata (DFAs) in three steps:
1) Using subset construction to determinize an NFA by considering sets of reachable states from each transition as single states in the DFA.
2) Minimizing the number of states in the resulting DFA using an algorithm that merges equivalent states that have identical transitions for all inputs.
3) Computing equivalent state sets using partition refinement, which iteratively partitions states based on their transitions for each input symbol.
This document discusses pushdown automata (PDA) and how they can be used to accept context-free languages. It defines a PDA as a 7-tuple that includes a finite set of states, input symbols, stack symbols, initial state, initial stack symbol, set of final states, and a transition function. A context-free grammar is also defined as a 4-tuple that includes variables, terminals, production rules, and a start symbol. The document then shows how to construct a PDA that is equivalent to a given context-free grammar by defining transition rules based on the grammar productions. An example of converting a grammar to a PDA using this construction method is also provided.
The document discusses depth-first search (DFS) and breadth-first search (BFS) graph traversal algorithms. It provides details on how DFS uses a stack to search deeper in the graph first by exploring edges from the most recently discovered vertex. BFS uses a queue to search outward from the starting node to neighboring nodes first before searching further. An example of DFS running on a graph is shown with each step of the algorithm.
This document discusses deterministic finite automata (DFA) minimization. It defines the components of a DFA and provides an example of a non-minimized DFA that accepts strings with 'a' or 'b'. The document then introduces an algorithm to minimize a DFA by identifying redundant states that are not necessary to recognize the language. The algorithm works by iteratively labeling states as distinct or equivalent based on their transitions and whether they are accepting states. This process combines equivalent states to produce a minimized DFA with the smallest number of states.
This document discusses various problems that can be solved using backtracking, including graph coloring, the Hamiltonian cycle problem, the subset sum problem, the n-queen problem, and map coloring. It provides examples of how backtracking works by constructing partial solutions and evaluating them to find valid solutions or determine dead ends. Key terms like state-space trees and promising vs non-promising states are introduced. Specific examples are given for problems like placing 4 queens on a chessboard and coloring a map of Australia.
The document discusses several shortest path algorithms for graphs, including Dijkstra's algorithm, Bellman-Ford algorithm, and Floyd-Warshall algorithm. Dijkstra's algorithm finds the shortest path from a single source node to all other nodes in a graph with non-negative edge weights. Bellman-Ford can handle graphs with negative edge weights but is slower. Floyd-Warshall can find shortest paths in a graph between all pairs of nodes.
This document discusses Hasse diagrams. It begins by stating the learning outcome is for students to illustrate Hasse diagrams. It then provides background on Hasse diagrams, noting they were originally devised to represent partially ordered sets and were created by Helmut Hasse. The document gives examples of drawing Hasse diagrams representing dividing relationships between numbers. It notes there can be multiple ways to draw a Hasse diagram for a given problem. In conclusion, it restates the topic covered was Hasse diagrams.
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.
Dijkstra's algorithm is used to find the shortest path between nodes in a weighted graph. It works by assigning initial distances to all nodes from the starting node and updating them as shorter paths are found, extracting the node with the lowest distance, and updating distances for neighboring unvisited nodes. The algorithm returns the shortest distance and path between the starting node and all other nodes in the graph.
Here are the steps to construct an NFA-ε equivalent to the given regular expressions:
1. ε + a b (a+b)*
- States: q0, q1, q2
- Transitions: q0 -> q1 on ε, q0 -> q2 on a, q1 -> q1 on a/b, q1 -> q2 on b
- Start state: q0
- Final states: q2
2. (0+1)* (00+11)
- States: q0, q1, q2, q3
- Transitions: q0 -> q0 on 0/1, q0 -> q1 on 0/1
1) The pumping lemma can be used to prove that a language is not regular or context-free.
2) For regular languages, the pumping lemma states that for any string x longer than n, x can be broken into uvw such that pumping v any number of times results in strings still in the language.
3) For context-free languages, the pumping lemma similarly states that any string x longer than n can be broken into five parts such that pumping the second and fourth parts any number of times results in strings still in the language.
BFS is the most commonly used approach. BFS is a traversing algorithm where you should start traversing from a selected node (source or starting node) and traverse the graph layerwise thus exploring the neighbor nodes (nodes which are directly connected to the source node.
1. The document discusses the equivalence between different models of regular languages including finite state automata (FSA), regular grammars, and regular expressions.
2. It explains how to convert between an FSA and regular grammar by establishing a one-to-one correspondence between the states of the FSA and symbols in the grammar.
3. Regular languages are closed under operations like concatenation - the concatenation of two regular languages is also regular.
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
The document discusses transaction concepts in database systems. It defines a transaction as a series of actions performed by a single user or application to access and modify database contents. Transactions must follow the ACID properties - atomicity, consistency, isolation, and durability. Concurrency control techniques like locking are used to ensure transactions execute correctly in a concurrent environment without interfering with each other. Serializability is discussed as a way to identify which non-serial schedules maintain consistency. Conflict serializability and precedence graphs are described as a technique to test schedules for serializability.
A lattice is a partially ordered set where every pair of elements has a supremum (least upper bound) and infimum (greatest lower bound). A lattice must satisfy the properties that (1) any two elements have a supremum and infimum and (2) the supremum of two elements is their join and the infimum is their meet. Common examples of lattices include the natural numbers under the divisibility relation and sets under the subset relation. Lattice theory has applications in many areas of computer science and engineering such as distributed computing, concurrency theory, and programming language semantics.
Let's analyze the remainder term R6 using the geometry series method:
|tj+1| = (j+1)π-2 ≤ π-2 = k|tj| for all j ≥ 6 (where 0 < k = π-2 < 1)
Then, |R6| ≤ t7(1 + k + k2 + k3 + ...)
= t7/(1-k)
= 7π-2/(1-π-2)
So the estimated upper bound of the truncation error |R6| is 7π-2/(1-π-2)
The Laplace transform is an integral transform that converts a function of time into a function of complex frequency. It is defined as the integral of the function multiplied by e-st from 0 to infinity. The Laplace transform is used to solve differential equations by converting them to algebraic equations. Some key properties of the Laplace transform include linearity, shifting theorems, differentiation and integration formulas, and methods for periodic and anti-periodic functions.
The document provides an overview of topics related to the Laplace transform and its applications. It defines the Laplace transform, discusses properties like linearity and examples of transforms of elementary functions. It also covers the inverse Laplace transform, differentiation and integration of transforms, evaluation of integrals using transforms, and applications to differential equations.
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 provides an overview of pushdown automata (PDA). It defines a PDA as a finite automaton with an additional memory stack. This stack allows two operations - push, which adds a new symbol to the top of the stack, and pop, which removes and reads the top symbol. The document then discusses the formal definition of a PDA as a septuple and provides an example of a PDA that accepts the language of strings with an equal number of 0s and 1s. It concludes with an explanation of the state operations of replace, push, pop and no change and conditions for PDA acceptance and rejection.
1. The document discusses various concepts in automata theory including finite automata, deterministic finite automata (DFA), pushdown automata (PDA), Mealy machines, and Moore machines.
2. It provides examples of alphabets, strings, and languages in automata. It also explains the key components of finite automata, DFAs, Mealy machines, Moore machines, and PDAs.
3. The document solves examples of DFAs that do or do not accept certain strings and compares the differences between Mealy machines and Moore machines.
This document defines deterministic finite automata (DFAs) and provides examples of how they work. It begins by defining the key components of a DFA: states, symbols/alphabet, transition function, start state, and accepting states. It then shows how a DFA is mathematically represented as a 5-tuple and how the transition function maps state-symbol pairs to states. The document provides an example DFA and explains how to represent it with a transition table. It then gives more examples of building DFAs to recognize specific languages and explains how the transition function can be extended to strings. The document also discusses minimizing DFAs, functional representations, products of automata, and complement/intersection operations.
Deterministic finite automata (DFAs) are mathematical models of computation that can be used to represent regular languages. A DFA consists of: (1) a finite set of states, (2) a finite input alphabet, (3) a transition function mapping a state and input to another state, (4) a start state, and (5) a set of accepting states. DFAs can be represented visually using state transition diagrams or mathematically as a 5-tuple. Operations like union, intersection, and complement of languages can be modeled using operations like union, product, and complement of DFAs.
This document defines deterministic finite automata (DFAs) and discusses their components, representations, operations, and properties. A DFA consists of a finite set of states, a finite alphabet, a transition function, a start state, and a set of accepting/final states. DFAs can be represented by transition tables or diagrams. The product and complement operations on DFAs are defined, and it is shown that these preserve regularity of languages. The accessible and minimal parts of a DFA are also discussed.
The document defines deterministic finite automata (DFAs) and describes their key components: states, symbols, transition function, start state, and accepting states. It provides examples of how to represent DFAs using transition tables and diagrams. The document also discusses extending the transition function to strings, minimizing DFAs, representing DFAs functionally, automatic theorem proving using DFAs, and the product and complement operations on DFAs.
This document introduces finite automata and regular languages. It discusses that regular languages can be described by finite automata, regular expressions, and regular grammars. It then defines deterministic finite automata (DFAs) and nondeterministic finite automata (NFAs) formally, provides examples of each, and shows their equivalence in terms of the languages they accept. It also discusses extending the transition function to strings, regular expressions, converting between DFAs/NFAs and regular expressions, and properties of regular expressions.
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.
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.
This document provides an introduction to automata theory and finite automata. It defines an automaton as an abstract computing device that follows a predetermined sequence of operations automatically. A finite automaton has a finite number of states and can be deterministic or non-deterministic. The document outlines the formal definitions and representations of finite automata. It also discusses related concepts like alphabets, strings, languages, and the conversions between non-deterministic and deterministic finite automata. Methods for minimizing deterministic finite automata using Myhill-Nerode theorem and equivalence theorem are also introduced.
Formal Languages and Automata Theory unit 3Srimatre K
This document provides an overview of context-free grammars and pushdown automata. It begins with definitions and examples of context-free grammars, including their components, derivations, parse trees, ambiguity, and applications. It then covers pushdown automata, defining them as finite state machines with a stack. It discusses the two types of acceptability in PDAs - final state and empty stack acceptability. Examples of languages recognized by PDAs are provided. The document concludes with how to convert between context-free grammars and pushdown automata, including examples of converting a CFG to a PDA and vice versa. It also briefly discusses deterministic pushdown automata.
This document provides an overview of automata and formal languages. It defines key terms like automata, finite automata, deterministic finite automaton (DFA), non-deterministic finite automaton (NDFA), and grammars. It also explains the differences between DFAs and NDFAs, and between Mealy and Moore machines. Formal definitions of these concepts are given using common notation like tuples.
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.
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.
A pushdown automaton (PDA) is a nondeterministic finite automaton that uses a stack to process input symbols. It has a finite set of states, input symbols, stack symbols, and transition rules that can push symbols onto or pop symbols off the stack. The transition function defines the state and stack changes for each input symbol. A PDA accepts a language if any path through its transitions leads to an accepting state.
This document discusses finite automata and regular languages. It defines finite automata using a 5-tuple notation and provides examples of automata and their accepted languages. It also covers nondeterministic finite automata and shows regular languages are closed under operations like union, concatenation and star.
The document contains details of 7 experiments on designing programs for finite state machines:
1. A program is designed to accept strings ending with 101.
2. A program accepts strings with three consecutive 1s.
3. A program for a machine that accepts strings with a remainder of 0 when divided by 3.
4. A program accepts decimal numbers divisible by 2.
5. A program accepts strings with equal numbers of 0s and 1s.
6. A program counts the numbers of 0s and 1s in a given string.
7. A program finds the 2's complement of a given binary number.
Thinking of getting a dog? Be aware that breeds like Pit Bulls, Rottweilers, and German Shepherds can be loyal and dangerous. Proper training and socialization are crucial to preventing aggressive behaviors. Ensure safety by understanding their needs and always supervising interactions. Stay safe, and enjoy your furry friends!
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
-------------------------------------------------------------------------------
Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
-------------------------------------------------------------------------------
For more information about PECB:
Website: https://pecb.com/
LinkedIn: https://www.linkedin.com/company/pecb/
Facebook: https://www.facebook.com/PECBInternational/
Slideshare: http://www.slideshare.net/PECBCERTIFICATION
2. 2
PDA - the automata for CFLs
What is?
FA to Reg Lang, PDA is to CFL
PDA == [ -NFA + “a stack” ]
Why a stack?
-NFA
A stack filled with “stack symbols”
Input
string
Accept/reject
3. 3
Pushdown Automata -
Definition
A PDA P := ( Q,∑,, δ,q0,Z0,F ):
Q: states of the -NFA
∑: input alphabet
: stack symbols
δ: transition function
q0: start state
Z0: Initial stack top symbol
F: Final/accepting states
4. δ : The Transition Function
δ(q,a,X) = {(p,Y), …}
1. state transition from q to p
2. a is the next input symbol
3. X is the current stack top symbol
4. Y is the replacement for X;
it is in * (a string of stack
symbols)
i. Set Y = for: Pop(X)
ii. If Y=X: stack top is
unchanged
iii. If Y=Z1Z2…Zk: X is popped
and is replaced by Y in
reverse order (i.e., Z1 will be
the new stack top)
4
old state Stack top
input symb. new state(s)new Stack top(s)
δ : Q x ∑ x => Q x
q
a X
p
Y
Y = ? Action
i) Y= Pop(X)
ii) Y=X Pop(X)
Push(X)
iii) Y=Z1Z2..Zk Pop(X)
Push(Zk)
Push(Zk-1)
…
Push(Z2)
Push(Z1)
5. 5
Example
Let Lwwr = {wwR | w is in (0+1)*}
CFG for Lwwr : S==> 0S0 | 1S1 |
PDA for Lwwr :
P := ( Q,∑, , δ,q0,Z0,F )
= ( {q0, q1, q2},{0,1},{0,1,Z0},δ,q0,Z0,{q2})
6. 6
PDA for Lwwr
1. δ(q0,0, Z0)={(q0,0Z0)}
2. δ(q0,1, Z0)={(q0,1Z0)}
3. δ(q0,0, 0)={(q0,00)}
4. δ(q0,0, 1)={(q0,01)}
5. δ(q0,1, 0)={(q0,10)}
6. δ(q0,1, 1)={(q0,11)}
7. δ(q0, , 0)={(q1, 0)}
8. δ(q0, , 1)={(q1, 1)}
9. δ(q0, , Z0)={(q1, Z0)}
10. δ(q1,0, 0)={(q1, )}
11. δ(q1,1, 1)={(q1, )}
12. δ(q1, , Z0)={(q2, Z0)}
First symbol push on stack
Grow the stack by pushing
new symbols on top of old
(w-part)
Switch to popping mode, nondeterministically
(boundary between w and wR)
Shrink the stack by popping matching
symbols (wR-part)
Enter acceptance state
Z0
Initial state of the PDA:
q0
Stack
top
7. 7
PDA as a state diagram
qi qj
a, X / Y
Next
input
symbol
Current
state
Current
stack
top
Stack
Top
Replacement
(w/ string Y)
Next
state
δ(qi,a, X)={(qj,Y)}
8. 8
PDA for Lwwr: Transition Diagram
q0 q1 q2
0, Z0/0Z0
1, Z0/1Z0
0, 0/00
0, 1/01
1, 0/10
1, 1/11
0, 0/
1, 1/
, Z0/Z0
, 0/0
, 1/1
, Z0/Z0
Grow stack
Switch to
popping mode
Pop stack for
matching symbols
Go to acceptance
∑ = {0, 1}
= {Z0, 0, 1}
Q = {q0,q1,q2}
, Z0/Z0
This would be a non-deterministic PDA
9. 9
Example 2: language of
balanced paranthesis
q0 q1 q2
(, Z0 / ( Z0
, Z0 / Z0
, Z0 / Z0
Grow stack
Switch to
popping mode
Pop stack for
matching symbols
Go to acceptance (by final state)
when you see the stack bottom symbo
∑ = { (, ) }
= {Z0, ( }
Q = {q0,q1,q2}
(, ( / ( (
), ( /
), ( /
To allow adjacent
blocks of nested paranthesis
(, ( / ( (
(, Z0 / ( Z0
, Z0 / Z0
11. 11
PDA’s Instantaneous
Description (ID)
A PDA has a configuration at any given instance:
(q,w,y)
q - current state
w - remainder of the input (i.e., unconsumed part)
y - current stack contents as a string from top to bottom
of stack
If δ(q,a, X)={(p, A)} is a transition, then the following are also true:
(q, a, X ) |--- (p,,A)
(q, aw, XB ) |--- (p,w,AB)
|--- sign is called a “turnstile notation” and represents
one move
|---* sign represents a sequence of moves
12. 12
How does the PDA for Lwwr
work on input “1111”?
(q0,1111,Z0)
(q0,111,1Z0)
(q0,11,11Z0)
(q0,1,111Z0)
(q0,,1111Z0)
(q1, ,1111Z0) (q1, ,11Z0)
(q1,1,111Z0)
(q1,11,11Z0)
(q1,111,1Z0)
(q1,1111,Z0) Path dies…
Path dies…
(q1,1,1Z0)
(q1, ,Z0)
(q2, ,Z0)
Acceptance by
final state:
= empty input
AND
final state
All moves made by the non-deterministic PDA
Path dies…
Path dies…
13. 14
Acceptance by…
PDAs that accept by final state:
For a PDA P, the language accepted by P,
denoted by L(P) by final state, is:
{w | (q0,w,Z0) |---* (q,, A) }, s.t., q F
PDAs that accept by empty stack:
For a PDA P, the language accepted by P,
denoted by N(P) by empty stack, is:
{w | (q0,w,Z0) |---* (q, , ) }, for any q Q.
Checklist:
- input exhausted?
- in a final state?
Checklist:
- input exhausted?
- is the stack empty?
There are two types of PDAs that one can design:
those that accept by final state or by empty stack
Q) Does a PDA that accepts by empty stack
need any final state specified in the design?
14. Example: L of balanced
parenthesis
15
q0
(,Z0 / ( Z0
(,( / ( (
), ( /
start
q1
,Z0/ Z0
,Z0/ Z0
PDA that accepts by final state
q0
start
(,Z0 / ( Z0
(, ( / ( (
), ( /
,Z0 /
An equivalent PDA that
accepts by empty stack
,Z0/ Z0
PF: PN:
How will these two PDAs work on the input: ( ( ( ) ) ( ) ) ( )
15. 17
PDAs accepting by final state and empty
stack are equivalent
PF <= PDA accepting by final state
PF = (QF,∑, , δF,q0,Z0,F)
PN <= PDA accepting by empty stack
PN = (QN,∑, , δN,q0,Z0)
Theorem:
(PN==> PF) For every PN, there exists a PF s.t. L(PF)=L(PN)
(PF==> PN) For every PF, there exists a PN s.t. L(PF)=L(PN)
16. 18
PN==> PF construction
Whenever PN’s stack becomes empty, make PF go to
a final state without consuming any addition symbol
To detect empty stack in PN: PF pushes a new stack
symbol X0 (not in of PN) initially before simultating
PN
q0
… pf
p0
, X0/Z0X0
New
start
, X0/ X0
, X0/ X0
, X0/ X0
, X0/ X0
PN
PF:
PF = (QN U {p0,pf}, ∑, U {X0}, δF, p0, X0, {pf})
PN:
, X0 / X0
How to convert an empty stack PDA into a final state PDA?
18. 20
PF==> PN construction
Main idea:
Whenever PF reaches a final state, just make an -transition into a
new end state, clear out the stack and accept
Danger: What if PF design is such that it clears the stack midway
without entering a final state?
to address this, add a new start symbol X0 (not in of PF)
PN = (Q U {p0,pe}, ∑, U {X0}, δN, p0, X0)
p0
, X0/Z0X0
New
start
, any/
, any/
, any/
q0
… pe
, any/
PF
PN:
How to convert an final state PDA into an empty stack PDA?
20. 22
CFGs == PDAs ==> CFLs
CFG
PDA by
final state
PDA by
empty stack
?
≡
21. 23
Converting CFG to PDA
Main idea: The PDA simulates the leftmost derivation on a given
w, and upon consuming it fully it either arrives at acceptance (by
empty stack) or non-acceptance.
This is same as: “implementing a CFG using a PDA”
PDA
(acceptance
by empty
stack)
CFG
w
accept
reject
implements
INPUT
OUTPUT
22. 24
Converting a CFG into a PDA
Main idea: The PDA simulates the leftmost derivation on a given w,
and upon consuming it fully it either arrives at acceptance (by
empty stack) or non-acceptance.
Steps:
1. Push the right hand side of the production onto the stack,
with leftmost symbol at the stack top
2. If stack top is the leftmost variable, then replace it by all its
productions (each possible substitution will represent a
distinct path taken by the non-deterministic PDA)
3. If stack top has a terminal symbol, and if it matches with the
next symbol in the input string, then pop it
State is inconsequential (only one state is needed)
This is same as: “implementing a CFG using a PDA”
23. 25
Formal construction of PDA
from CFG
Given: G= (V,T,P,S)
Output: PN = ({q}, T, V U T, δ, q, S)
δ:
For all A V , add the following
transition(s) in the PDA:
δ(q, ,A) = { (q, ) | “A ==>” P}
For all a T, add the following
transition(s) in the PDA:
δ(q,a,a)= { (q, ) }
A
Before:
…
a
Before:
…
After:
…
a
After:
…
Note: Initial stack symbol (S)
same as the start variable
in the grammar
pop
a…
24. 26
Example: CFG to PDA
G = ( {S,A}, {0,1}, P, S)
P:
S ==> AS |
A ==> 0A1 | A1 | 01
PDA = ({q}, {0,1}, {0,1,A,S}, δ, q, S)
δ:
δ(q, , S) = { (q, AS), (q, )}
δ(q, , A) = { (q,0A1), (q,A1), (q,01) }
δ(q, 0, 0) = { (q, ) }
δ(q, 1, 1) = { (q, ) } How will this new PDA work?
Lets simulate string 0011
q
,S / S
1,1 /
0,0 /
,A/ 01
,A/ A1
,A/ 0A1
,S /
,S / AS
25. Simulating string 0011 on the
new PDA …
27
PDA (δ):
δ(q, , S) = { (q, AS), (q, )}
δ(q, , A) = { (q,0A1), (q,A1), (q,01) }
δ(q, 0, 0) = { (q, ) }
δ(q, 1, 1) = { (q, ) }
S
Stack moves (shows only the successful path):
S
A
S
1
A
0
S
1
A
0
S
1
1
0
S
1
1
0
S
1
1
S
1
Accept by
empty stack
q
,S / S
1,1 /
0,0 /
,A/ 01
,A/ A1
,A/ 0A1
,S /
,S / AS
S => AS
=> 0A1S
=> 0011S
=> 0011
Leftmost deriv.:
S =>AS =>0A1S =>0011S => 0011
26. 29
Converting a PDA into a CFG
Main idea: Reverse engineer the
productions from transitions
If δ(q,a,Z) => (p, Y1Y2Y3…Yk):
1. State is changed from q to p;
2. Terminal a is consumed;
3. Stack top symbol Z is popped and replaced with a
sequence of k variables.
Action: Create a grammar variable called
“[qZp]” which includes the following
production:
[qZp] => a[pY1q1] [q1Y2q2] [q2Y3q3]… [qk-1Ykqk]
Proof discussion (in the book)
27. 30
Example: Bracket matching
To avoid confusion, we will use b=“(“ and e=“)”
PN: ( {q0}, {b,e}, {Z0,Z1}, δ, q0, Z0 )
1. δ(q0,b,Z0) = { (q0,Z1Z0) }
2. δ(q0,b,Z1) = { (q0,Z1Z1) }
3. δ(q0,e,Z1) = { (q0, ) }
4. δ(q0, ,Z0) = { (q0, ) }
0. S => [q0Z0q0]
1. [q0Z0q0] => b [q0Z1q0] [q0Z0q0]
2. [q0Z1q0] => b [q0Z1q0] [q0Z1q0]
3. [q0Z1q0] => e
4. [q0Z0q0] =>
Let A=[q0Z0q0]
Let B=[q0Z1q0]
0. S => A
1. A => b B A
2. B => b B B
3. B => e
4. A =>
Simplifying,
0. S => b B S |
1. B => b B B | e
If you were to directly write a CFG:
S => b S e S |
28. 31
Two ways to build a CFG
Build a PDA Construct
CFG from PDA
Derive CFG directly
Derive a CFG Construct
PDA from CFG
Design a PDA directly
Similarly…
(indirect)
(direct)
(indirect)
(direct)
Two ways to build a PDA
30. 33
This PDA for Lwwr is non-deterministic
q0 q1 q2
0, Z0/0Z0
1, Z0/1Z0
0, 0/00
0, 1/01
1, 0/10
1, 1/11
0, 0/
1, 1/
, Z0/Z0
, 0/0
, 1/1
, Z0/Z0
Grow stack
Switch to
popping mode
Pop stack for
matching symbols
Accepts by final state
Why does it have
to be non-
deterministic?
To remove
guessing,
impose the user
to insert c in the
middle
31. 34
D-PDA for Lwcwr = {wcwR | c is some
special symbol not in w}
q0 q1 q2
0, Z0/0Z0
1, Z0/1Z0
0, 0/00
0, 1/01
1, 0/10
1, 1/11
0, 0/
1, 1/
c, Z0/Z0
c, 0/0
c, 1/1
, Z0/Z0
Grow stack
Switch to
popping mode
Pop stack for
matching symbols
Accepts by
final state
Note:
• all transitions have
become deterministic
Example shows that: Nondeterministic PDAs ≠ D-PDAs
32. 35
Deterministic PDA: Definition
A PDA is deterministic if and only if:
1. δ(q,a,X) has at most one member for any
a ∑ U {}
If δ(q,a,X) is non-empty for some a∑,
then δ(q, ,X) must be empty.
33. 36
PDA vs DPDA vs Regular
languages
Regular languages D-PDA
non-deterministic PDA
Lwwr
Lwcwr
34. 37
Summary
PDAs for CFLs and CFGs
Non-deterministic
Deterministic
PDA acceptance types
1. By final state
2. By empty stack
PDA
IDs, Transition diagram
Equivalence of CFG and PDA
CFG => PDA construction
PDA => CFG construction