- The document discusses logic concepts including propositional calculus, propositional logic, and natural deduction systems.
- Propositional calculus uses logical operators like conjunction, disjunction, negation, implication, and biconditional to combine atomic propositions into compound propositions. Truth tables are used to determine the truth values of propositions.
- Propositional logic represents statements using propositional variables and logical connectives. It has limitations and cannot represent relations. Natural deduction systems and axiomatic systems provide formal rules for deducing conclusions.
This document provides an overview of first-order logic in artificial intelligence:
- First-order logic extends propositional logic by adding objects, relations, and functions to represent knowledge. Objects can include people and numbers, while relations include concepts like "brother of" and functions like "father of".
- A sentence in first-order logic contains a predicate and a subject, represented by a variable. For example, "tall(John)" asserts that John is tall. Quantifiers like "forall" and "exists" are used to structure sentences.
- First-order logic contains constants, variables, predicates, functions, connectives, equality, and quantifiers as its basic elements.
Knowledge representation In Artificial IntelligenceRamla Sheikh
facts, information, and skills acquired through experience or education; the theoretical or practical understanding of a subject.
Knowledge = information + rules
EXAMPLE
Doctors, managers.
Turing machines are abstract machines that can simulate any modern computer. They are very powerful and can solve problems by answering yes or no to any input. A Turing machine consists of a finite control, tape, and tape head. The tape is infinite and divided into cells containing symbols. The machine operates by reading/writing symbols on the tape and moving the tape head left or right according to a transition function based on its current state and tape symbol. Variations of Turing machines like those with storage, multiple tracks or tapes are equivalent to basic Turing machines. Turing machines can recognize formal languages and perform computations like arithmetic.
The document discusses sources and approaches to handling uncertainty in artificial intelligence. It provides examples of uncertain inputs, knowledge, and outputs in AI systems. Common methods for representing and reasoning with uncertain data include probability, Bayesian belief networks, hidden Markov models, and temporal models. Effectively handling uncertainty through probability and inference allows AI to make rational decisions with imperfect knowledge.
This document discusses propositional logic and knowledge representation. It introduces propositional logic as the simplest form of logic that uses symbols to represent facts that can then be joined by logical connectives like AND and OR. Truth tables are presented as a way to determine the truth value of propositions connected by these logical operators. The document also discusses concepts like models of formulas, satisfiable and valid formulas, and rules of inference like modus ponens and disjunctive syllogism that allow deducing new facts from initial propositions. Examples are provided to illustrate each concept.
The document discusses first-order logic (FOL) and its advantages over propositional logic for representing knowledge. It introduces the basic elements of FOL syntax, such as constants, predicates, functions, variables, and connectives. It provides examples of FOL expressions and discusses how objects and relations between objects can be represented. It also covers quantification in FOL using universal and existential quantifiers.
Object Automation Software Solutions Pvt Ltd in collaboration with SRM Ramapuram delivered Workshop for Skill Development on Artificial Intelligence.
Uncertain Knowledge and reasoning by Mr.Abhishek Sharma, Research Scholar from Object Automation.
This document provides an overview of first-order logic in artificial intelligence:
- First-order logic extends propositional logic by adding objects, relations, and functions to represent knowledge. Objects can include people and numbers, while relations include concepts like "brother of" and functions like "father of".
- A sentence in first-order logic contains a predicate and a subject, represented by a variable. For example, "tall(John)" asserts that John is tall. Quantifiers like "forall" and "exists" are used to structure sentences.
- First-order logic contains constants, variables, predicates, functions, connectives, equality, and quantifiers as its basic elements.
Knowledge representation In Artificial IntelligenceRamla Sheikh
facts, information, and skills acquired through experience or education; the theoretical or practical understanding of a subject.
Knowledge = information + rules
EXAMPLE
Doctors, managers.
Turing machines are abstract machines that can simulate any modern computer. They are very powerful and can solve problems by answering yes or no to any input. A Turing machine consists of a finite control, tape, and tape head. The tape is infinite and divided into cells containing symbols. The machine operates by reading/writing symbols on the tape and moving the tape head left or right according to a transition function based on its current state and tape symbol. Variations of Turing machines like those with storage, multiple tracks or tapes are equivalent to basic Turing machines. Turing machines can recognize formal languages and perform computations like arithmetic.
The document discusses sources and approaches to handling uncertainty in artificial intelligence. It provides examples of uncertain inputs, knowledge, and outputs in AI systems. Common methods for representing and reasoning with uncertain data include probability, Bayesian belief networks, hidden Markov models, and temporal models. Effectively handling uncertainty through probability and inference allows AI to make rational decisions with imperfect knowledge.
This document discusses propositional logic and knowledge representation. It introduces propositional logic as the simplest form of logic that uses symbols to represent facts that can then be joined by logical connectives like AND and OR. Truth tables are presented as a way to determine the truth value of propositions connected by these logical operators. The document also discusses concepts like models of formulas, satisfiable and valid formulas, and rules of inference like modus ponens and disjunctive syllogism that allow deducing new facts from initial propositions. Examples are provided to illustrate each concept.
The document discusses first-order logic (FOL) and its advantages over propositional logic for representing knowledge. It introduces the basic elements of FOL syntax, such as constants, predicates, functions, variables, and connectives. It provides examples of FOL expressions and discusses how objects and relations between objects can be represented. It also covers quantification in FOL using universal and existential quantifiers.
Object Automation Software Solutions Pvt Ltd in collaboration with SRM Ramapuram delivered Workshop for Skill Development on Artificial Intelligence.
Uncertain Knowledge and reasoning by Mr.Abhishek Sharma, Research Scholar from Object Automation.
The document discusses inference in first-order logic. It provides a brief history of reasoning and logic. It then discusses reducing first-order inference to propositional inference using techniques like universal instantiation and existential instantiation. It introduces the concepts of unification and generalized modus ponens to perform inference in first-order logic. Forward chaining and resolution are also discussed as algorithms for performing inference in first-order logic.
Polygon clipping involves taking a polygon and clipping it against another shape to produce one or more smaller polygons. The Sutherland-Hodgman algorithm handles polygon clipping by testing each edge of the clipping polygon against each edge of the clip shape. There are four cases for how an edge can be clipped - wholly inside, exit, wholly outside, enter - and the algorithm saves or discards vertices based on these cases. Repeatedly clipping against each edge of the clip shape handles all cases and produces the final clipped polygon(s).
This document summarizes a chapter on perception in artificial intelligence. It discusses how perception provides information about the world through sensors. There are two main approaches to perception - feature extraction and model-based. Feature extraction detects key features in sensory input while model-based reconstructs a model of the world from sensory stimuli. The chapter then covers topics like image formation, image processing techniques including filtering, edge detection and segmentation. It also discusses representation and description of images as well as object recognition methods.
This document provides an overview and introduction to the course "Knowledge Representation & Reasoning" taught by Ms. Jawairya Bukhari. It discusses the aims of developing skills in knowledge representation and reasoning using different representation methods. It outlines prerequisites like artificial intelligence, logic, and programming. Key topics covered include symbolic and non-symbolic knowledge representation methods, types of knowledge, languages for knowledge representation like propositional logic, and what knowledge representation encompasses.
The document discusses inference rules for quantifiers in first-order logic. It describes the rules of universal instantiation and existential instantiation. Universal instantiation allows inferring sentences by substituting terms for variables, while existential instantiation replaces a variable with a new constant symbol. The document also introduces unification, which finds substitutions to make logical expressions identical. Generalized modus ponens is presented as a rule that lifts modus ponens to first-order logic by using unification to substitute variables.
This document discusses uncertainty and probability theory. It begins by explaining sources of uncertainty for autonomous agents from limited sensors and an unknown future. It then covers representing uncertainty with probabilities and Bayes' rule for updating beliefs. Examples show inferring diagnoses from symptoms using conditional probabilities. Independence is described as reducing the information needed for joint distributions. The document emphasizes probability theory and Bayesian reasoning for handling uncertainty.
The document discusses knowledge representation using propositional logic and predicate logic. It begins by explaining the syntax and semantics of propositional logic for representing problems as logical theorems to prove. Predicate logic is then introduced as being more versatile than propositional logic for representing knowledge, as it allows quantifiers and relations between objects. Examples are provided to demonstrate how predicate logic can formally represent statements involving universal and existential quantification.
The document provides an overview of knowledge representation techniques. It discusses propositional logic, including syntax, semantics, and inference rules. Propositional logic uses atomic statements that can be true or false, connected with operators like AND and OR. Well-formed formulas and normal forms are explained. Forward and backward chaining for rule-based reasoning are summarized. Examples are provided to illustrate various concepts.
This document provides an introduction to First Order Predicate Logic (FOPL). It discusses the differences between propositional logic and FOPL, the parts and syntax of FOPL including terms, atomic sentences, quantifiers and rules of inference. The semantics of FOPL are also explained. Pros and cons are provided, such as FOPL's ability to represent individual entities and generalizations compared to propositional logic. Applications include using FOPL as a framework for formulating theories.
Problem Decomposition: Goal Trees, Rule Based Systems, Rule Based Expert Systems. Planning:
STRIPS, Forward and Backward State Space Planning, Goal Stack Planning, Plan Space Planning,
A Unified Framework For Planning. Constraint Satisfaction : N-Queens, Constraint Propagation,
Scene Labeling, Higher order and Directional Consistencies, Backtracking and Look ahead
Strategies.
The document discusses problem solving by searching. It describes problem solving agents and how they formulate goals and problems, search for solutions, and execute solutions. Tree search algorithms like breadth-first search, uniform-cost search, and depth-first search are described. Example problems discussed include the 8-puzzle, 8-queens, and route finding problems. The strategies of different uninformed search algorithms are explained.
This document discusses various applications of computer graphics including computer-aided design (CAD), visualization, animation, and computer games. It then describes the frame buffer, which stores pixel information for the screen in memory. Finally, it explains two basic line drawing algorithms - the digital differential analyzer (DDA) line drawing algorithm and Bresenham's line drawing algorithm. The DDA algorithm calculates pixel coordinates by incrementing x or y values based on the slope of the line, while Bresenham's algorithm optimizes for integer coordinates.
John likes all foods, apples and chicken are foods, anything that does not kill someone who eats it is a food, Bill eats peanuts and is still alive so peanuts are food, and Sue eats everything that Bill eats. This document translates statements about people and foods into logical forms using predicates and quantifiers, and then expresses them in conjunctive normal form.
This document discusses different types of intelligent agents. It describes four basic types of agent programs: simple reflex agents, model-based reflex agents, goal-based agents, and utility-based agents. Simple reflex agents select actions based only on the current percept, while model-based reflex agents maintain an internal model of the world. Goal-based agents use goals to determine desirable situations. Utility-based agents maximize an internal utility function that represents the performance measure. The document also discusses agent functions, percepts, environments, and the PEAS properties of task environments.
The document discusses different types of intelligent agents based on their architecture and programs. It describes five classes of agents: 1) Simple Reflex agents which react solely based on current percepts; 2) Model-based agents which use an internal model of the world; 3) Goal-based agents which take actions to reduce distance to a goal; 4) Utility-based agents which choose actions to maximize utility; and 5) Learning agents which are able to learn from experiences. The document also outlines different forms of learning like rote learning, learning from instruction, and reinforcement learning.
The document discusses the 2D viewing pipeline. It describes how a 3D world coordinate scene is constructed and then transformed through a series of steps to 2D device coordinates that can be displayed. These steps include converting to viewing coordinates using a window-to-viewport transformation, then mapping to normalized and finally device coordinates. It also covers techniques for clipping objects and lines that fall outside the viewing window including Cohen-Sutherland line clipping and Sutherland-Hodgeman polygon clipping.
Projection is the transformation of a 3D object into a 2D plane by mapping points from the 3D object to the projection plane. There are two main types of projection: perspective projection and parallel projection. Perspective projection uses lines that converge to a single point, while parallel projection uses parallel lines. Perspective projection includes one-point, two-point, and three-point perspectives. Parallel projection includes orthographic projection, which projects lines perpendicular to the plane, and oblique projection, where lines are parallel but not perpendicular to the plane.
This document discusses the 0/1 knapsack problem and how it can be solved using backtracking. It begins with an introduction to backtracking and the difference between backtracking and branch and bound. It then discusses the knapsack problem, giving the definitions of the profit vector, weight vector, and knapsack capacity. It explains how the problem is to find the combination of items that achieves the maximum total value without exceeding the knapsack capacity. The document constructs state space trees to demonstrate solving the knapsack problem using backtracking and fixed tuples. It concludes with examples problems and references.
Artificial Intelligence: Introduction, Typical Applications. State Space Search: Depth Bounded
DFS, Depth First Iterative Deepening. Heuristic Search: Heuristic Functions, Best First Search,
Hill Climbing, Variable Neighborhood Descent, Beam Search, Tabu Search. Optimal Search: A
*
algorithm, Iterative Deepening A*
, Recursive Best First Search, Pruning the CLOSED and OPEN
Lists
First-order logic allows for more expressive power than propositional logic by representing objects, relations, and functions in the world. It includes constants like names, predicates that relate objects, functions, variables, logical connectives, equality, and quantifiers. Relations can represent properties of single objects or facts about multiple objects. Models represent interpretations of first-order logic statements graphically. Terms refer to objects as constants or functions. Atomic sentences make statements about objects using predicates. Complex sentences combine atomic sentences with connectives. Universal quantification asserts something is true for all objects, while existential quantification asserts something is true for at least one object.
- Logic is the study of principles of reasoning and determining valid inferences. Propositional logic deals with propositions that can be either true or false.
- A propositional calculus defines rules for combining propositions using logical operators like conjunction, disjunction, negation, implication, and biconditional.
- Truth tables define the meanings of logical operators by listing their truth values under all combinations of true and false propositions.
- Natural deduction is a system to derive logical consequences through inference rules like introduction and elimination rules for logical operators. It mimics natural patterns of reasoning.
This document provides an overview of propositional logic concepts. It discusses:
- Propositional logic deals with validity and satisfiability of logical formulas using truth tables or deduction systems.
- Natural deduction is a deductive system that mimics natural reasoning using inference rules. Propositional formulas can be derived or proven in this system.
- Axiomatic systems use only axioms and rules of inference to prove formulas, requiring creative use of the minimal structure.
The document gives examples of proving formulas using these logical systems.
The document discusses inference in first-order logic. It provides a brief history of reasoning and logic. It then discusses reducing first-order inference to propositional inference using techniques like universal instantiation and existential instantiation. It introduces the concepts of unification and generalized modus ponens to perform inference in first-order logic. Forward chaining and resolution are also discussed as algorithms for performing inference in first-order logic.
Polygon clipping involves taking a polygon and clipping it against another shape to produce one or more smaller polygons. The Sutherland-Hodgman algorithm handles polygon clipping by testing each edge of the clipping polygon against each edge of the clip shape. There are four cases for how an edge can be clipped - wholly inside, exit, wholly outside, enter - and the algorithm saves or discards vertices based on these cases. Repeatedly clipping against each edge of the clip shape handles all cases and produces the final clipped polygon(s).
This document summarizes a chapter on perception in artificial intelligence. It discusses how perception provides information about the world through sensors. There are two main approaches to perception - feature extraction and model-based. Feature extraction detects key features in sensory input while model-based reconstructs a model of the world from sensory stimuli. The chapter then covers topics like image formation, image processing techniques including filtering, edge detection and segmentation. It also discusses representation and description of images as well as object recognition methods.
This document provides an overview and introduction to the course "Knowledge Representation & Reasoning" taught by Ms. Jawairya Bukhari. It discusses the aims of developing skills in knowledge representation and reasoning using different representation methods. It outlines prerequisites like artificial intelligence, logic, and programming. Key topics covered include symbolic and non-symbolic knowledge representation methods, types of knowledge, languages for knowledge representation like propositional logic, and what knowledge representation encompasses.
The document discusses inference rules for quantifiers in first-order logic. It describes the rules of universal instantiation and existential instantiation. Universal instantiation allows inferring sentences by substituting terms for variables, while existential instantiation replaces a variable with a new constant symbol. The document also introduces unification, which finds substitutions to make logical expressions identical. Generalized modus ponens is presented as a rule that lifts modus ponens to first-order logic by using unification to substitute variables.
This document discusses uncertainty and probability theory. It begins by explaining sources of uncertainty for autonomous agents from limited sensors and an unknown future. It then covers representing uncertainty with probabilities and Bayes' rule for updating beliefs. Examples show inferring diagnoses from symptoms using conditional probabilities. Independence is described as reducing the information needed for joint distributions. The document emphasizes probability theory and Bayesian reasoning for handling uncertainty.
The document discusses knowledge representation using propositional logic and predicate logic. It begins by explaining the syntax and semantics of propositional logic for representing problems as logical theorems to prove. Predicate logic is then introduced as being more versatile than propositional logic for representing knowledge, as it allows quantifiers and relations between objects. Examples are provided to demonstrate how predicate logic can formally represent statements involving universal and existential quantification.
The document provides an overview of knowledge representation techniques. It discusses propositional logic, including syntax, semantics, and inference rules. Propositional logic uses atomic statements that can be true or false, connected with operators like AND and OR. Well-formed formulas and normal forms are explained. Forward and backward chaining for rule-based reasoning are summarized. Examples are provided to illustrate various concepts.
This document provides an introduction to First Order Predicate Logic (FOPL). It discusses the differences between propositional logic and FOPL, the parts and syntax of FOPL including terms, atomic sentences, quantifiers and rules of inference. The semantics of FOPL are also explained. Pros and cons are provided, such as FOPL's ability to represent individual entities and generalizations compared to propositional logic. Applications include using FOPL as a framework for formulating theories.
Problem Decomposition: Goal Trees, Rule Based Systems, Rule Based Expert Systems. Planning:
STRIPS, Forward and Backward State Space Planning, Goal Stack Planning, Plan Space Planning,
A Unified Framework For Planning. Constraint Satisfaction : N-Queens, Constraint Propagation,
Scene Labeling, Higher order and Directional Consistencies, Backtracking and Look ahead
Strategies.
The document discusses problem solving by searching. It describes problem solving agents and how they formulate goals and problems, search for solutions, and execute solutions. Tree search algorithms like breadth-first search, uniform-cost search, and depth-first search are described. Example problems discussed include the 8-puzzle, 8-queens, and route finding problems. The strategies of different uninformed search algorithms are explained.
This document discusses various applications of computer graphics including computer-aided design (CAD), visualization, animation, and computer games. It then describes the frame buffer, which stores pixel information for the screen in memory. Finally, it explains two basic line drawing algorithms - the digital differential analyzer (DDA) line drawing algorithm and Bresenham's line drawing algorithm. The DDA algorithm calculates pixel coordinates by incrementing x or y values based on the slope of the line, while Bresenham's algorithm optimizes for integer coordinates.
John likes all foods, apples and chicken are foods, anything that does not kill someone who eats it is a food, Bill eats peanuts and is still alive so peanuts are food, and Sue eats everything that Bill eats. This document translates statements about people and foods into logical forms using predicates and quantifiers, and then expresses them in conjunctive normal form.
This document discusses different types of intelligent agents. It describes four basic types of agent programs: simple reflex agents, model-based reflex agents, goal-based agents, and utility-based agents. Simple reflex agents select actions based only on the current percept, while model-based reflex agents maintain an internal model of the world. Goal-based agents use goals to determine desirable situations. Utility-based agents maximize an internal utility function that represents the performance measure. The document also discusses agent functions, percepts, environments, and the PEAS properties of task environments.
The document discusses different types of intelligent agents based on their architecture and programs. It describes five classes of agents: 1) Simple Reflex agents which react solely based on current percepts; 2) Model-based agents which use an internal model of the world; 3) Goal-based agents which take actions to reduce distance to a goal; 4) Utility-based agents which choose actions to maximize utility; and 5) Learning agents which are able to learn from experiences. The document also outlines different forms of learning like rote learning, learning from instruction, and reinforcement learning.
The document discusses the 2D viewing pipeline. It describes how a 3D world coordinate scene is constructed and then transformed through a series of steps to 2D device coordinates that can be displayed. These steps include converting to viewing coordinates using a window-to-viewport transformation, then mapping to normalized and finally device coordinates. It also covers techniques for clipping objects and lines that fall outside the viewing window including Cohen-Sutherland line clipping and Sutherland-Hodgeman polygon clipping.
Projection is the transformation of a 3D object into a 2D plane by mapping points from the 3D object to the projection plane. There are two main types of projection: perspective projection and parallel projection. Perspective projection uses lines that converge to a single point, while parallel projection uses parallel lines. Perspective projection includes one-point, two-point, and three-point perspectives. Parallel projection includes orthographic projection, which projects lines perpendicular to the plane, and oblique projection, where lines are parallel but not perpendicular to the plane.
This document discusses the 0/1 knapsack problem and how it can be solved using backtracking. It begins with an introduction to backtracking and the difference between backtracking and branch and bound. It then discusses the knapsack problem, giving the definitions of the profit vector, weight vector, and knapsack capacity. It explains how the problem is to find the combination of items that achieves the maximum total value without exceeding the knapsack capacity. The document constructs state space trees to demonstrate solving the knapsack problem using backtracking and fixed tuples. It concludes with examples problems and references.
Artificial Intelligence: Introduction, Typical Applications. State Space Search: Depth Bounded
DFS, Depth First Iterative Deepening. Heuristic Search: Heuristic Functions, Best First Search,
Hill Climbing, Variable Neighborhood Descent, Beam Search, Tabu Search. Optimal Search: A
*
algorithm, Iterative Deepening A*
, Recursive Best First Search, Pruning the CLOSED and OPEN
Lists
First-order logic allows for more expressive power than propositional logic by representing objects, relations, and functions in the world. It includes constants like names, predicates that relate objects, functions, variables, logical connectives, equality, and quantifiers. Relations can represent properties of single objects or facts about multiple objects. Models represent interpretations of first-order logic statements graphically. Terms refer to objects as constants or functions. Atomic sentences make statements about objects using predicates. Complex sentences combine atomic sentences with connectives. Universal quantification asserts something is true for all objects, while existential quantification asserts something is true for at least one object.
- Logic is the study of principles of reasoning and determining valid inferences. Propositional logic deals with propositions that can be either true or false.
- A propositional calculus defines rules for combining propositions using logical operators like conjunction, disjunction, negation, implication, and biconditional.
- Truth tables define the meanings of logical operators by listing their truth values under all combinations of true and false propositions.
- Natural deduction is a system to derive logical consequences through inference rules like introduction and elimination rules for logical operators. It mimics natural patterns of reasoning.
This document provides an overview of propositional logic concepts. It discusses:
- Propositional logic deals with validity and satisfiability of logical formulas using truth tables or deduction systems.
- Natural deduction is a deductive system that mimics natural reasoning using inference rules. Propositional formulas can be derived or proven in this system.
- Axiomatic systems use only axioms and rules of inference to prove formulas, requiring creative use of the minimal structure.
The document gives examples of proving formulas using these logical systems.
The document discusses propositional logic including:
- Propositional logic uses propositions that can be either true or false and logical connectives to connect propositions.
- It introduces syntax of propositional logic including atomic and compound propositions.
- Logical connectives like negation, conjunction, disjunction, implication, and biconditional are explained along with their truth tables and significance.
- Other concepts discussed include precedence of connectives, logical equivalence, properties of operators, and limitations of propositional logic.
- Examples are provided to illustrate propositional logic concepts like truth tables, logical equivalence, and translating English statements to symbolic form.
The document provides an introduction to discrete structures and mathematical reasoning. It discusses key concepts like propositions, logical operators, quantification, and proof techniques. Propositions can be combined using logical operators like negation, conjunction, disjunction, etc. Quantifiers like universal and existential are used to represent statements about all or some elements. Mathematical reasoning involves using axioms, rules of inference, and deductive proofs to establish theorems from given conditions.
The document discusses formal systems and logic. It begins by introducing formal systems, which consist of an axiomatic theory, symbols for constructing formulas, grammar rules, axioms, inference rules, and theorems deduced from axioms and rules. Propositional and predicate logic are then covered, including their languages, semantics using interpretation functions, axiomatic theories, and properties like soundness and completeness. Resolution principles like normal forms and the resolution rule are also summarized. The document provides examples and explanations throughout.
The document discusses models and formalisms in logic. It introduces formal systems as consisting of an axiomatic theory, symbols for constructing formulas, grammar rules, axioms, inference rules, and properties like consistency. Propositional and predicate logic are examined, including their model theories, axiomatic theories, properties like completeness and soundness, and resolution principles. Normal forms and the resolution rule are defined as ways to deduce theorems in a formal system.
This document provides an overview of propositional logic concepts including:
- Logic is used to distinguish correct from incorrect reasoning and explicate laws of thought.
- A proposition is a declarative statement that is either true or false. Propositional logic uses logical operators to combine propositions into compound statements.
- Truth tables are used to determine the truth values of propositional statements under different variable assignments. Various rules of inference like modus ponens are discussed.
- Propositional logic deals with validity, satisfiability, and logical consequence through truth table analysis and the application of equivalence laws. Examples are provided to illustrate logical reasoning techniques.
Artificial Intelligence (AI) | Prepositional logic (PL)and first order predic...Ashish Duggal
The following are the topics in this presentation Prepositional Logic (PL) and First-order Predicate Logic (FOPL) is used for knowledge representation in artificial intelligence (AI).
There are also sub-topics in this presentation like logical connective, atomic sentence, complex sentence, and quantifiers.
This PPT is very helpful for Computer science and Computer Engineer
(B.C.A., M.C.A., B.TECH. , M.TECH.)
The document is a thesis on fuzzy implications submitted for a Master of Science in Applied Mathematics. It discusses fuzzy sets and fuzzy logic as extensions of classical set theory and logic to account for imprecise and uncertain information. It defines fuzzy implications and various basic fuzzy implications functions. It also discusses properties of fuzzy implication lattices, including meet, join and other properties like commutativity, associativity and absorption. An example is provided to demonstrate these properties for specific fuzzy implication functions.
The document discusses propositional logic as a knowledge representation language. It defines key concepts in propositional logic including: syntax, semantics, validity, satisfiability, interpretation, models, and entailment. It explains that propositional logic uses symbols to represent facts about the world and connectives to combine symbols into sentences. Sentences can then be evaluated based on the truth values assigned to symbols to determine if the overall sentence is true or false. Propositional logic allows new sentences to be deduced from existing sentences through inference rules while maintaining logical validity.
The document provides an introduction to formal logic. It discusses how to formulate valid arguments through propositional logic and syllogistic logic. Propositional logic uses truth tables to evaluate combinations of propositions and operators like negation and conjunction. Syllogistic logic examines implications of general statements using domains and categories. The key rules of inference for valid arguments are hypothetical syllogism, modus ponens, and modus tollens.
This document discusses properties that a good domain description for reasoning about actions should have beyond mere consistency. It introduces the concept of modularity for action theories, where the different types of laws (static, effect, executability, inexecutability) are arranged in separate components with limited interaction. Violations of the proposed postulates about modularity can lead to unexpected conclusions from logically consistent theories. The document outlines algorithms to check whether an action theory satisfies the postulates of modularity.
The document discusses logic agents and logical reasoning. It provides background on logic, including syntax, semantics, models, and inference rules. It then discusses how logic can be used to represent knowledge in knowledge-based agents and systems. The agents use a knowledge base and inference engine, where the inference engine derives new knowledge by applying inference rules to the knowledge base.
This document discusses different techniques for knowledge representation in artificial intelligence, including logical representation, semantic networks, frames, and production rules. It focuses on logical representation using propositional logic and first-order logic. Propositional logic uses atomic and compound propositions connected by logical operators like negation, conjunction, disjunction, implication, and biconditional. First-order logic extends propositional logic by adding quantifiers and predicates to represent objects and relations. Knowledge representation enables AI systems to understand and utilize knowledge to solve complex problems.
Propositional logic is a good vehicle to introduce basic properties of logicpendragon6626
Propositional logic uses symbols and logical connectives to evaluate the validity of compound statements based on the validity of atomic statements. Natural deduction and resolution are deductive systems that use inference rules to prove statements. Natural deduction is sound and complete, while resolution is also complete. Propositional resolution can check validity by constructing a refutation tree, and linear resolution with Horn clauses is efficient for this task like the logic programming language Prolog.
1) Logic and inferences are important aspects of artificial intelligence as they allow systems to think and act rationally by making decisions based on available information and drawing conclusions.
2) Inference is the process of generating conclusions from facts and evidence. Formal logic represents knowledge through logical sentences using propositional or first-order logic.
3) Propositional logic uses symbolic variables to represent propositions that can be either true or false. Compound propositions combine simpler propositions using logical connectives like "and" and "or". Truth tables define the values of logical connectives.
The document discusses machine learning algorithms for learning first-order logic rules from examples. It introduces the FOIL algorithm, which extends propositional rule learning algorithms like sequential covering to learn more general first-order Horn clauses. FOIL uses an iterative specialization approach, starting with a general rule and greedily adding literals to better fit examples while avoiding negative examples. The document also discusses how combining inductive learning with analytical domain knowledge can improve learning, such as in the KBANN approach where a neural network is initialized based on the domain theory before training on examples.
This document is the preface to a textbook on discrete mathematics. It introduces the topics that will be covered in the book, including mathematical logic, proofs, set theory, relations, functions, algorithms analysis, counting, probability, and graph theory. It encourages students to work through all the exercises to fully understand the material. It also thanks a professor who provided feedback that improved the book.
This document provides an overview of probabilistic reasoning and uncertainty in knowledge representation. It discusses:
1) Using probability theory to represent uncertainty quantitatively rather than logical rules with certainty factors.
2) Key concepts in probability theory including random variables, probability distributions, joint probabilities, marginal probabilities, and conditional probabilities.
3) Representing a problem domain as a probabilistic model with a sample space of possible variable states.
4) Independence of variables allowing simpler computation of probabilities.
5) The document is an introduction to probabilistic reasoning concepts to be covered in more detail later, including Bayesian networks.
Design and optimization of ion propulsion dronebjmsejournal
Electric propulsion technology is widely used in many kinds of vehicles in recent years, and aircrafts are no exception. Technically, UAVs are electrically propelled but tend to produce a significant amount of noise and vibrations. Ion propulsion technology for drones is a potential solution to this problem. Ion propulsion technology is proven to be feasible in the earth’s atmosphere. The study presented in this article shows the design of EHD thrusters and power supply for ion propulsion drones along with performance optimization of high-voltage power supply for endurance in earth’s atmosphere.
Digital Twins Computer Networking Paper Presentation.pptxaryanpankaj78
A Digital Twin in computer networking is a virtual representation of a physical network, used to simulate, analyze, and optimize network performance and reliability. It leverages real-time data to enhance network management, predict issues, and improve decision-making processes.
Electric vehicle and photovoltaic advanced roles in enhancing the financial p...IJECEIAES
Climate change's impact on the planet forced the United Nations and governments to promote green energies and electric transportation. The deployments of photovoltaic (PV) and electric vehicle (EV) systems gained stronger momentum due to their numerous advantages over fossil fuel types. The advantages go beyond sustainability to reach financial support and stability. The work in this paper introduces the hybrid system between PV and EV to support industrial and commercial plants. This paper covers the theoretical framework of the proposed hybrid system including the required equation to complete the cost analysis when PV and EV are present. In addition, the proposed design diagram which sets the priorities and requirements of the system is presented. The proposed approach allows setup to advance their power stability, especially during power outages. The presented information supports researchers and plant owners to complete the necessary analysis while promoting the deployment of clean energy. The result of a case study that represents a dairy milk farmer supports the theoretical works and highlights its advanced benefits to existing plants. The short return on investment of the proposed approach supports the paper's novelty approach for the sustainable electrical system. In addition, the proposed system allows for an isolated power setup without the need for a transmission line which enhances the safety of the electrical network
Rainfall intensity duration frequency curve statistical analysis and modeling...bijceesjournal
Using data from 41 years in Patna’ India’ the study’s goal is to analyze the trends of how often it rains on a weekly, seasonal, and annual basis (1981−2020). First, utilizing the intensity-duration-frequency (IDF) curve and the relationship by statistically analyzing rainfall’ the historical rainfall data set for Patna’ India’ during a 41 year period (1981−2020), was evaluated for its quality. Changes in the hydrologic cycle as a result of increased greenhouse gas emissions are expected to induce variations in the intensity, length, and frequency of precipitation events. One strategy to lessen vulnerability is to quantify probable changes and adapt to them. Techniques such as log-normal, normal, and Gumbel are used (EV-I). Distributions were created with durations of 1, 2, 3, 6, and 24 h and return times of 2, 5, 10, 25, and 100 years. There were also mathematical correlations discovered between rainfall and recurrence interval.
Findings: Based on findings, the Gumbel approach produced the highest intensity values, whereas the other approaches produced values that were close to each other. The data indicates that 461.9 mm of rain fell during the monsoon season’s 301st week. However, it was found that the 29th week had the greatest average rainfall, 92.6 mm. With 952.6 mm on average, the monsoon season saw the highest rainfall. Calculations revealed that the yearly rainfall averaged 1171.1 mm. Using Weibull’s method, the study was subsequently expanded to examine rainfall distribution at different recurrence intervals of 2, 5, 10, and 25 years. Rainfall and recurrence interval mathematical correlations were also developed. Further regression analysis revealed that short wave irrigation, wind direction, wind speed, pressure, relative humidity, and temperature all had a substantial influence on rainfall.
Originality and value: The results of the rainfall IDF curves can provide useful information to policymakers in making appropriate decisions in managing and minimizing floods in the study area.
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELijaia
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
AI for Legal Research with applications, toolsmahaffeycheryld
AI applications in legal research include rapid document analysis, case law review, and statute interpretation. AI-powered tools can sift through vast legal databases to find relevant precedents and citations, enhancing research accuracy and speed. They assist in legal writing by drafting and proofreading documents. Predictive analytics help foresee case outcomes based on historical data, aiding in strategic decision-making. AI also automates routine tasks like contract review and due diligence, freeing up lawyers to focus on complex legal issues. These applications make legal research more efficient, cost-effective, and accessible.
2. Artificial Intelligence
Unit – III
Logic Concepts: Introduction, propositional calculus, propositional logic, natural
deduction system, axiomatic system, semantic tableau system in propositional logic,
resolution refutation in propositional logic, predicate logic.
3. Aditya Engineering College (A)
LOGIC CONCEPTS
Logic helps in investigating and classifying the structure of statements and
arguments through the system of formal study of inference.
Logic is a study of principles used to
– distinguish correct from incorrect reasoning.
– Logical system should possesses properties such as consistency, soundness, and
completeness.
– Consistency implies that none of the theorems of the system should contradict each other.
– Soundness means that the inference rules shall never allow a false inference from true
premises.
Formally it deals with
– the notion of truth in an abstract sense and is concerned with the principles of valid
inferencing.
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Logic…
A proposition in logic is a declarative statements which are either true or
false (but not both) in a given context. For example,
– “Jack is a male”,
– "Jack loves Mary" etc.
Givensome propositions to be true ina given context,
– logic helps in inferencing new proposition, which is also true in the
same context.
Suppose we are given a set of propositions as
– “It is hot today" and
– “If it is hot it will rain", then
– we can infer that
“It will rain today".
5. Propositional Calculus
Aditya Engineering College (A)
Propositional Calculus is a language of propositions basically
refers
– to set of rules used to combine the propositions to form compound
propositions using logical operators often called connectives such as
or dot, V, ~, or ( ),
1.Well-Formed Formula
Well-formed formula is defined as:
– An atom is a well-formed formula.
– If is a well-formed formula, then ~ is a well-formed
formula.
– If and are well formed formulae, then ( ), ( V ), ( ),
( ) are also well-formed formulae.
– A propositional expression is a well-formed formula
if and only if it can be obtained by using above conditions.
6. Propositional Calculus
Aditya Engineering College (A)
Well-Formed Formula Examples
R1-An atom is a well-formed formula.
R2-If is a WFF, then ~ is a
well-formed formula.
R3-If and are well formed formulae, then ( ),
( V ), ( ),( ) are also well-formed
formulae.
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Propositional Calculus…
2.Truth Table
Truth table gives us operational definitions of
important logical operators.
– By using truth table, the truth values of well-formed
formulae are calculated.
● Truth table elaborates all possible truth values of a
formula.
● The meanings of the logical operators are given by
the following truth table.
P Q ~P P Q P V Q P Q P Q
T T F T T T T
T F F F T F F
F T T F T T F
F F T F F T T
Q) Compute the TT: (AVB) ∧ (~B->A)
Negation ~
Conjunction ^ or dot
Disjunction V
If-then(Implication) ->
iff(Biconditional) <->
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3.Equivalence Laws
1. P Q Q P
2. P V Q Q V P
Commutation
Association
1. P (Q R) (P Q) R
2. P V (Q V R) (P V Q) V R
Double Negation
~ (~ P)
Distributive Laws
P
1. P ( Q V R) (P Q) V (P R)
2. P V ( Q R) (P V Q) (P V R)
Morgan’s Laws
1. ~ (P Q) ~ P V ~ Q
2. ~ (P V Q) ~ P ~ Q
De
Law of Excluded Middle
P V ~ P
Law of Contradiction
P ~ P
T (true)
F (false)
Propositional Calculus…
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Propositional Logic
• Knowledge Representation can be done with
1)Propositional Logic 2)First Order Logic
• Propositional logic (PL) is the simplest form of logic where all the
statements are made by propositions.
• A proposition is a declarative statement which is either true or false.
It is a technique of knowledge representation in logical and
mathematical form.
• Propositional logic is also called Boolean logic as it works on 0 and
• Machine In AI can be given input in terms of knowledge but machine cannot understand this
knowledge which is framed using English sentences.
• So this knowledge must be represented ina way understandable to machine , this is called
knowledge representation.
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Propositional Logic
• In propositional logic, we use symbolic variables to represent the logic, and we can
use any symbol for a representing a proposition, such A, B, C, P, Q, R, etc.
• Propositions can be either true or false, but it cannot be both.
• Propositional logic consists of an object, relations or function, and logical
connectives.
• These connectives are also called logical operators.
• The propositions and connectives are the basic elements of the propositional logic.
• Connectives can be said as a logical operator which connects two sentences.
• A proposition formula which is always true is called tautology, and it is also called a
valid sentence.
•A proposition formula which is always false is called Contradiction.
•A proposition formula which has both true and false values is called.
•Statements which are questions, commands, or opinions are not propositions
such as "Where is ", "How are you", "What is your name", are not propositions.
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Propositional Logic
Syntax of propositional logic:
• The syntax of propositional logic defines the allowable sentences for the knowledge
representation. There are two types of Propositions:
Atomic Propositions:
Atomic Proposition: Atomic propositions are the simple propositions. It consists of a
single proposition symbol. These are the sentences which must be either true or false.
a) 2+2 is 4, it is an atomic proposition as it is a true fact.
b) "The Sun is cold" is also a proposition as it is a false fact.
Compound propositions
•Compound propositions are constructed by combining simpler or atomic
propositions, using parenthesis and logical connectives.
Example:
a) "It is raining today, and street is wet."
b) “Raj is a doctor, and his clinic is in Mumbai."
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Propositional Logic
Logical Connectives
Negation ~
Conjunction ^
Disjunction V
If-then(Implication) ->
iff(Biconditional) <->
Logical connectives are used to connect two simpler propositions or representing a sentence
logically. We can create compound propositions with the help of logical connectives.
Example:
X: It is cold
Y: It is sunny
Z:It is breezy
1. It is not cold ~X
2. It is cold and it is breezy X^Z
3. It is cold or it is breezy XvZ
4. If it is breezy then it is cold Z->X
5. If it is breezy and cold then it is not sunny Z^X -> ~Y
6. It will be cold iff it is breezy X <-> Z
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Propositional Logic
Examples:
Raj did not read book
~ read(Raj,book)
Raj watches Amazon or Netflix
Watches(Raj,Amazon) v Watches(Raj,Netflix)
If Raj buys mobile then color is black
Buys(Raj,Mobile) -> Color( Mobile,black)
Raj becomes happy if and only if Raj eats dairy milk.
becomes(Raj,happy) <-> eats( Raj,Dairy milk)
14. Limitations of Propositional logic:
•We cannot represent relations like ALL, some, or none with propositional
logic. Example:
• All the boys are intelligent.
• Some apples are sweet.
•Propositional logic has limited expressive power.
•In propositional logic, we cannot describe statements in terms of their
properties or logical relationships.
Propositional Logic
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Natural Deduction System
● ND is based on the set of few deductive inference rules.
● The name natural deductive system is given because it mimics
the pattern of natural reasoning.
● It has about 10 deductive inference rules.
Conventions:
– E for Elimination, I for Introducing.
– P, Pk , (1 k n) are atoms.
– k, (1 k n) and are formulae.
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Natural Deduction System
ND RULES:
Rule 1: I- (Introducing )
I- : If P1, P2, …, Pn then P1 P2 … Pn
Interpretation: If we have hypothesized or proved P1, P2, … and Pn , then their conjunction
P1 P2 … Pn is also proved or derived.
Rule 2: E- ( Eliminating )
E- : If P1 P2 … Pn then Pi ( 1 i n)
Interpretation: If we have proved P1 P2 … Pn , then any Pi is also proved or derived.
This rule shows that can be eliminated to yield one of its conjuncts.
Rule 3: I-V (Introducing V)
I-V : If Pi ( 1 i n) then P1V P2 V …V Pn
Interpretation: If any Pi (1 i n) is proved, then P1V … V Pn is also proved.
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Natural Deduction System
ND RULES…
Rule 4: E-V ( Eliminating V)
E-V : If P1 V … V Pn, P1 P, … , Pn P then P
Interpretation: If P1 V … V Pn, P1 P, … , and Pn P are proved, then P is proved.
Rule 5: I- (Introducing )
I- : If from 1, …, n infer is proved then
1 … n is proved
Interpretation: If given 1, 2, …and n to be proved and from these we deduce then 1 2 … n
is also proved.
Rule 6: E- (Eliminating ) - Modus Ponen
E- : If P1 P, P1 then P
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Natural Deduction System
ND RULES…
Rule 7: I- (Introducing )
I- : If P1 P2, P2 P1 then P1 P2
Rule 8: E- (Elimination )
E- : If P1 P2 then P1 P2 , P2 P1
Rule 9:I- ~ (Introducing ~)
I- ~ : If from P infer P1 ~ P1 is proved then
~P is proved
Rule 10: E- ~ (Eliminating ~)
E- ~ : If from ~ P infer P1 ~ P1 is proved
then P is proved
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Natural Deduction System
Cont…
● If a formula is derived / proved from a set of
premises / hypotheses { 1,…, n },
– then one can write it as from 1, …, n
● In natural deductive system,
infer .
– a theorem to be proved should have
from 1, …, n infer .
a form
● Theorem infer means that
– there are no premises and is true under all
interpretations i.e., is a tautology or valid.
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Natural Deduction System
● If we assume that is a premise, then we
conclude that is proved if is given i.e.,
– if ‘from infer ’ is a theorem then is concluded.
– The converse of this is also true.
Deduction Theorem: Infer (1 2 … n )
is a theorem of natural deductive system if and
only if
from 1, 2,… ,n infer is a theorem.
Useful tips: To prove a formula 1 2 … n
, it is sufficient to prove a theorem
from 1, 2, …, n infer .
Cont..
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Natural Deduction System
Example
Example1: Prove that P(QVR) follows from PQ
Solution: This problem is restated in natural deductive system as "from P
Q infer P (Q V R)". The formal proof is given as follows:
{Theorem}
{ premise}
from P Q infer P (Q V R)
P Q (1)
{ E- , (1)} P (2)
{ E- , (1)} Q (3)
{ I-V , (3) } Q V R (4)
{ I-, ( 2, 4)} P (Q V R) Conclusion
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Axiomatic System (AS)
● AS is based on the set of only three axioms and one rule of
deduction.
– It is minimal in structure but as powerful as the truth table and natural
deduction approaches.
– The proofs of the theorems are often difficult and require a guess in
selection of appropriate axiom(s) and rules.
– These methods basically require forward chaining strategy where we start
with the given hypotheses and prove the goal.
– Only two logical operators not(~) and implies (->) are allowed.
(V , , <-> can be converted into the above operators).
Example:
A B = ~(A->~B)
A V B = ~A->B
A<-> B = (A->B) (B->A) = ~[((A->B) -> ~((B->A) ]
23. Aditya Engineering College (A)
Axiomatic System (AS)
● Three axioms and one rule of deduction.
Axiom1 (A1): ( )
Axiom2 (A2): ( ()) (( ) ( ))
Axiom3 (A3): (~ ~ ) ( )
Modus Ponen (MP) defined as follows:
Hypotheses: and Consequent:
24. Aditya Engineering College (A)
Axiomatic System (AS)
Definition: A deduction of a formula in Axiomatic
System for Propositional Logic is a sequence of well-
formed formulae 1, 2, ..., n such that for each i,
(1 i n), either
– Either i is an axiom or i is a hypothesis (given to be true)
– Or i is derived from j and k where j, k < i using modus
ponen inference rule.
We call i to be a deductive consequence of {1,
...,i-1 }.
It is denoted by {1, .. , i-1 } |- i. More formally,
deductive consequence is defined on next slide.
Cont…
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Examples
Establish the following:
Ex1:
{Q} |- (PQ) i.e.,PQ is a deductive consequence of {Q}.
{Hypothesis} Q (1)
{Axiom A1}
{MP, (1,2)}
Q
P
(P
Q
Q) (2)
proved
Axiomatic System (AS)
26. Aditya Engineering College (A)
Examples – Cont…
Ex2:
{ P Q, Q R } |- ( P R ) i.e., P R is a
deductive consequence of { P Q, Q R }.
P Q (1)
Q R
{Hypothesis}
{Hypothesis}
{Axiom A1}
{MP, (2, 3)}
{Axiom A2}
(2)
(Q R) (P (Q R)) (3)
P (Q R)
(P (Q R))
((P Q) (P R))
(4)
(5)
{MP , (4, 5)}
{MP, (1, 6)}
(P Q) (P R)
P R
(6)
proved
Axiomatic System (AS)
27. Aditya Engineering College (A)
Semantic Tableaux System in PL
● Earlier approaches require
– construction of proof of a formula from given set of formulae and
are called direct methods.
● In semantic tableaux,
– the set of rules are applied systematically on a formula or set of formulae
to establish its consistency or inconsistency.
● Semantic tableau
– binary tree constructed by using semantic rules formula as a root
● Assume and be any two formulae.
28. Aditya Engineering College (A)
Semantic Tableaux System…
● Semantic tableau
– binary tree constructed by using semantic rules formula as a root
● Assume and be any two formulae.
RULES
Let and be any two formulae.
Rule 1: A tableau for a formula ( ) is constructed by adding both and to
the same path (branch). This can be represented as follows:
Interpretation: is true if both and are true
29. Aditya Engineering College (A)
Semantic Tableaux System…
Rules - Cont…
Rule 2: A tableau for a formula ~ ( ) is
constructed by adding two alternative paths one
containing ~ and other containing ~
~ ( )
~ ~
Interpretation:
or ~ is true
~ ( ) is true if either ~
30. Aditya Engineering College (A)
Semantic Tableaux System…
Cont…
Rule 3: A tableau for a formula ( V ) is
constructed by adding two new paths one
containing and other containing .
V
Interpretation: V is true if either
or is true
31. Aditya Engineering College (A)
Semantic Tableaux System…
Rule 4: A tableau for a formula ~ ( V ) is
constructed by adding both ~ and ~ to the
same path. This can be expressed as follows:
~ ( V )
~
~
Rule 5: Semantic tableau for ~~
~~
34. Aditya Engineering College (A)
Artificial Intelligence Dr P Udayakumar
Semantic Tableaux System…
Consistency and Inconsistency: Satisfiability and Unsatisfiability
If an atom P and ~ P appear on a same path of a
semantic tableau,
– then inconsistency is indicated and such path is said to
be contradictory or closed (finished) path.
– Even if one path remains non contradictory or
unclosed (open), then the formula at the root of a
tableau is consistent.
35. Aditya Engineering College (A)
Artificial Intelligence Dr P Udayakumar
Semantic Tableaux System…
A valuation is said to be a model of (or
satisfies ) iff () = T.
In tableaux approach, model for a consistent formula is
constructed as follows:
– On an open path, assign truth values to atoms (positive or
negative) of which is at the root of a tableau such that is made
to be true.
– It is achieved by assigning truth value T to each atomic formula
(positive or negative) on that path.
Valuation
36. Aditya Engineering College (A)
Artificial Intelligence Dr P Udayakumar
Semantic Tableaux System…
Contradictory tableau (or finished tableau) is defined to
be a tableau in which all the paths are contradictory or
closed (finished).
Consistent and Inconsistent
If a tableau for a formula at the root is a contradictory
tableau, then a formula is said to be inconsistent.
A formula is consistent if there is at least on open path
in a tableau with root
Contradictory Tableau
37. Aditya Engineering College (A)
Artificial Intelligence Dr P Udayakumar
Semantic Tableaux System…
Example: Show that
: (P Q R) (~P S) Q ~ R ~ S is
inconsistent(Unsatisfiable) using tableaux method.
R) ( ~P S) Q ~ R ~ S
{T-root} (P Q
{Apply rule 1 to 1}
(1)
(2)
(3)
P Q R
~P S
Q
~ R
~ S
S
{Apply rule 6 to 3} P
{Apply rule 6 to 2)} ~ (P Q)
Closed: {S, ~ S} on the path
R
Closed { R, ~ R}
~P ~ Q
{P, ~ P}
Closed Closed{~ Q, Q}
38. Aditya Engineering College (A)
Artificial Intelligence Dr P Udayakumar
Semantic Tableaux System…
Problem: Show that : ( Q ~R) (R P) is
consistent( satisfiable) and find its model.
Solution:
{T-root} ( Q ~ R) ( R P)
{Apply rule 1 to 1}
(1)
(2)
(3)
(Q ~ R)
( R P)
{Apply rule 1 to 2} Q
{Apply rule 6 to 3} ~R
~ R P
open open
39. Aditya Engineering College (A)
Artificial Intelligence Dr P Udayakumar
Semantic Tableaux System…
Since tableau for has open paths, we conclude that is
consistent.
The models are constructed by assigning T to all atomic formulae
appearing on open paths.
– Assign Q = T and ~ R = T i.e., R = F.
• So { Q = T, R = F } is a model of .
– Assign Q = T and ~ R = T and P = T.
• So { P = T , Q = T, R = F } is another model.
Useful Tip:
– Thumb rule for constructing a tableau is to apply non
branching rules before the branching rules in any order
Example -Cont…
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Artificial Intelligence Dr P Udayakumar
Semantic Tableaux System…
Theorem: (Soundness)
If is tableau provable ( |- ) , then is
valid ( |= ) i.e., |- |= .
Theorem: (Completeness)
If is valid, then is tableau provable i.e.,
|= |- .
Soundness and Completeness
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Artificial Intelligence Dr P Udayakumar
Semantic Tableaux System…
Example - Validity
Example: Show that : P ( Q P) is valid
Solution: In order to show that is a valid, we will try
to show that is tableau provable i.e., ~ is
inconsistent.
{T-root} ~ (P ( Q P)) (1)
P) (2)
{Apply rule 7 to 2}
{Apply rule7 to 1} P
~ ( Q
Q
~P
Closed {P, ~ P}
Hence P ( Q P) is valid.
42. Aditya Engineering College (A)
Artificial Intelligence Dr P Udayakumar
Resolution Refutation in PL
Resolution refutation is another simple method to prove a formula by
contradiction.
– Here negation of goal to be proved is added to given set of clauses.
– It is shown then that there is a refutation in new set using resolution principle.
Resolution: During this process we need to identify
– two clauses, one with positive atom (P) and other with negative atom (~P) for the
application of resolution rule.
Resolution is based on modus ponen inference rule.
– This method is most favoured for developing computer based theorem
provers.
– Automatic theorem provers using resolution are simple and efficient systems .
Resolution is performed on special types of formulae called clauses.
– Clause is propositional formula expressed using
• {V, ~ } operators.
43. Aditya Engineering College (A)
Artificial Intelligence Dr P Udayakumar
Resolution Refutation in PL
Conjunctive and Disjunctive Normal Forms
In Disjunctive Normal Form (DNF),
– a formula is represented in the form
– (L11 ….. L1n ) V ..… V (Lm1 ….. Lmk ), where all Lijare
literals. It is a disjunction of conjunction.
In Conjunctive Normal Form (CNF),
– a formula is represented in the form
– (L11 V ….. V L1n ) … … (Lp1 V ….. V Lpm ) ,
Lij
where all are literals. It is a
conjunction of disjunction.
A clause is a special formula expressed as disjunction of literals. If
a clause contains only one literal, then it is called unit clause.
44. Aditya Engineering College (A)
Artificial Intelligence Dr P Udayakumar
Resolution Refutation in PL
Conversion of a Formula to its CNF
Each formula in Propositional Logic can be easily
transformed into its equivalent DNF or CNF representation
using equivalence laws .
– Eliminate and by using the following
equivalence laws.
P Q
P Q
~ P V Q
( P Q) ( Q P)
– Eliminate double negation signs by using
~ ~ P P
45. Aditya Engineering College (A)
Artificial Intelligence Dr P Udayakumar
Resolution Refutation in PL
Cont…
Use De Morgan’s laws to push ~ (negation)
immediately before atomic formula.
~ ( P Q)
~ ( P V Q)
~ P V ~ Q
~ P ~ Q
Use distributive law to get CNF.
P V (Q R) (P V Q) (P V R)
We notice that CNF representation of a formula is
of the form
– (C1 ….. Cn ) , where each Ck , (1 k n ) is
a clause that is disjunction of literals.
46. Aditya Engineering College (A)
Artificial Intelligence Dr P Udayakumar
Resolution Refutation in PL
Resolution of Clauses
If two clauses C1 and C2 contain a complementary pair
of literals {L, ~L}, then
– these clauses can be resolved together by deleting L from C1
and ~ L from C2 and constructing a new clause by the
disjunction of the remaining literals in C1 and C2.
The new clause thus generated is called
resolvent of C1 and C2 .
– Here C1 and C2 are called parents of resolved clause.
– If the resolvent contains one or more set of
is
complementary pair of literals, then resolvent
always true.
47. Aditya Engineering College (A)
Artificial Intelligence Dr P Udayakumar
Resolution Refutation in PL
Resolution Tree
Inverted binary tree is generated with the last node of
the binary tree to be a resolvent.
This also called resolution tree.
Example: Find resolvent of:
C1 = P V Q V R
C2 = ~ Q V ~ W
C3 = ~ P V ~ W
48. Aditya Engineering College (A)
Artificial Intelligence Dr P Udayakumar
Resolution Refutation in PL
Example- Resolution Tree
P V Q V R ~ Q V ~W
{Q, ~ Q}
P V R V ~W
{P, ~P}
~ P V ~ W
R V ~W
Thus Resolvent(C1,C2, C3) = R V ~W
49. Aditya Engineering College (A)
Artificial Intelligence Dr P Udayakumar
Resolution Refutation in PL
Example: “Mary will get her degree if she registers as a
student and pass her exam. She has registered herself as a
student. She has passed her exam”. Show that she will get a
degree.
Solution: Symbolize above statements as follows: R: Mary
is a registered student
P: Mary has passed her exam D: Mary
gets her degree
The formulae corresponding to above listed sentences are as
follows:
50. Aditya Engineering College (A)
Artificial Intelligence Dr P Udayakumar
Resolution Refutation in PL
Mary will get her degree if she registers as a
student and pass her exam.
R P D (~ R V ~ P V D)
She has registered herself as a student.
R
She has passed her exam.
P
Conclude “Mary will get a degree”.
D
Cont…
51. Aditya Engineering College (A)
Artificial Intelligence Dr P Udayakumar
Resolution Refutation in PL
Set of clauses are:
– S = {~ R V ~ P V D , R, P }
Add negation of "Mary gets her degree (= D)" to S.
New set S' is:
– S' = {~ R V ~ P V D , R, P, ~ D}
We can easily see that empty clause is deduced
from above set.
Hence we can conclude that “Mary gets her degree”
Example – Cont…
52. Aditya Engineering College (A)
Artificial Intelligence Dr P Udayakumar
Resolution Refutation in PL
Deriving Contradiction
~ R V ~ P V D R
~ P V D P
D ~ D
53. Aditya Engineering College (A)
Artificial Intelligence Dr P Udayakumar
Resolution Refutation in PL
Exercises
I. Establish the following:
1. { P Q, Q R } |- ( P R )
2. { P Q} |- (R P) (R Q)
3. { P } |- (~ P Q)
4. { ~Q, P ( ~Q R) } |- P R
5. {P Q , ~ Q } |- ~ P. This is called Modus Tollen rule.
II. Prove the following theorems
1. |- (P P)
2. |- (~ P P) P
3. |- (P Q) (~ Q ~ P)
4. |- (P ~ Q) ( Q ~ P)
III. Give tableau proof of each of the following formulae and show that formulae are valid.
1. P ( Q P)
2. (P (Q V R) (( P Q) V ( P R))
3. ~ (P V Q) (~ P ~ Q)
IV. Are the following arguments valid?
1. If John lives in England then he lives in UK. John lives in England. Therefore, John lives in UK.
2. If John lives in England then he lives in UK. John lives in UK. Therefore, John lives in England.
3. If John lives in England then he lives in UK. John does not live in UK. Therefore, John does not live in England.
V. Prove by resolution refutation
1. {P Q , ~ P V R} |= Q V R
2. { P , Q R , P R} |= P R
3. {P Q R , P} |= R