Propositional logic & inference
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Propositional logic deals with propositions as units and the connectives that relate them. It has a syntax that defines allowable sentences using proposition symbols and logical connectives like conjunction, disjunction, implication and equivalence. Sentences are formed using Backus Naur Form grammar. Semantics specify how to compute the truth of sentences using truth tables and models. Knowledge bases can be represented as a set of sentences and inference is used to decide if conclusions are true in all models where the KB is true, such as using a truth table algorithm.
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This document summarizes key concepts in propositional logic. It discusses propositional logic, valid arguments, propositions, compound statements formed using logical connectives like conjunction and disjunction, truth tables, formal propositions using negation, conjunction, disjunction, examples of determining if statements are true or false, and conditional statements using implication. It provides examples and truth tables to illustrate these logical concepts and how propositional logic can be used to distinguish valid from invalid arguments.
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The document discusses inductive definitions and how they are used to define the syntax, semantics, and other properties of programming languages. Inductive definitions specify aspects of a language like its concrete syntax using context-free grammars (CFGs), abstract syntax using CFGs and ML data types, static semantics using typing rules, and dynamic semantics using evaluation rules. Evaluation rules are defined using judgments like e → e' to represent single-step evaluation, where an expression e evaluates to e' in one step. Properties proven about the language are done by induction over the structure of the inductive definitions.
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This document provides an overview of propositional logic and first-order logic. It begins with a review of propositional logic, including logical connectives, truth tables, and inference rules. It then discusses some limitations of propositional logic, such as its inability to represent properties, relations or generalizations. The document introduces first-order logic as a more expressive language that adds variables, quantifiers, relations and functions. It defines terms, atomic sentences, and well-formed formulas. Key concepts in first-order logic like universal and existential quantification are explained along with inference rules like universal instantiation.
Jarrar.lecture notes.aai.2011s.ch7.p logic
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.
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Logical agents use logical reasoning over a knowledge base to derive new information and make decisions. The document outlines key concepts in logic like syntax, semantics, entailment, and inference. It discusses representing the Wumpus world in propositional logic and using inference methods like forward and backward chaining to explore the world. Reasoning in the Wumpus world requires representing partial information and using propositional logic has limitations for complex, dynamic domains.
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This document provides an overview of knowledge representation in artificial intelligence. It discusses how knowledge representation is used to formalize problems so they can be solved through search techniques. Good knowledge representations reduce problems to search problems, while bad representations prevent problem solving. Logical expressions are presented as one way to represent states, actions, and allow an agent to deduce hidden properties and appropriate actions. The document then discusses the Wumpus World environment as an example problem domain. It describes the percepts, properties, and actions in the Wumpus World. Finally, it discusses using first-order logic to represent facts and inferences about the Wumpus World.
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Propositional logic & inference
- 1. Propositional Logic V.Saranya AP/CSE Sri Vidya College of Engineering and Technology, Virudhunagar
- 2. Definition • a branch of symbolic logic dealing with propositions (proposal, scheme, plan) as units and with their combinations and the connectives that relate them.
- 3. Syntax • Defines the allowable sentences. • Atomic Sentence: – Consist of single proposition symbol. – Either TRUE or FALSE • Rules: – Uppercase names used for symbols P,O,R – Names are arbitrary (uninformed or random) »Example: »W[1,3] Wumpus in [1,3]
- 4. Complex sentences • Constructed from simple sentences. • Using logical connectives. ...and [conjunction] ...or [disjunction] ...implies [implication / conditional] ..(if & only if)is equivalent [bi-conditional] ...not [negation]
- 5. BNF (Backus Naur Form) • Grammar of sentences in propositional logic Sentence Atomic Sentence | complex sentence Atomic sentence True|False|Symbol Symbol P, Q,R Complex Sentence ¬ sentence |Sentence ˄ Sentence |Sentence ˄ Sentence |Sentence Sentence |Sentence Sentence
- 6. • Every sentence constructed with binary connectives must be enclosed in parenthesis ((A ˄B) C) right form A ˄B C wrong one Multiplication has higher precedence than addition Order of precedence is , ˄,V, and (i) A ˄ B ˄ Cread as (A ˄B) ˄ C (or) A ˄(B ˄ C) (ii) ¬ P ˄Q˄ RS ((¬ P) ˄(Q˄ R))S
- 7. Semantics • Defines the rules. • Model fixes truth vales true or false for every propositional symbol. • Semantics specify how to compute the truth of sentences formed with each of 5 connectives. • Ex; (Wumpus World) M1= { P1,2 = False, P2,2 = False, P3,1= True}
- 8. • Atomic sentences are easy – True is true in every model – False is false in every model. • Complex Sentence – Using “ Truth Table”
- 9. Example 1: • Evaluate the sentence ¬ P1,2 ˄(P2,2 ˄ P3,1) (True ˄ (False ˄ True) Result= True Example 2: 5 is even implies sam is smart This sentence will be true if sam is smart P => Q is only FALSE when the Premise(p) is TRUE AND Consequence(Q) is FALSE. P => Q is always TRUE when the Premise(P) is FALSE OR the Consequence(Q) is TRUE.
- 10. Example 3: • B1,1 (P1,2 ˄P2,1) – B1,1 means breeze in [1,1] – P1,2 means pit in [1,2] – P2,1 means pit in [2,1] – So False False Now Result : True Example 3: • B1,1 (P1,2 ˄P2,1) • The result is true • But incomplete (violate the rules of wumpus world)
- 11. A Simple Knowledge Base • Take Pits alone • i,j values • Let Pi,j be true if there is a pit in [i,j] • Let Bi,j be true if there is a breeze in [i,j]
- 12. KB 1. There is no pit in [1,1] R1 : ¬P1,1 2. A square is breeze if and only if there is a pit in a neighboring square. R2 : B1,1 (P1,2 ˄P2,1) R3 : B2,1 (P1,1 ˄P1,2 ˄P3,1) 3. The above 2 sentences are true in all wumpus world. Now after visiting 2 squares R4 : ¬B1,1 R5 : B2,1
- 13. • KB consists of R1 to R5 Consider the all above in 5 single sentences R1 ˄ R2 ˄ R3 ˄ R4 ˄ R5 Concluded that all 5 sentences are True
- 14. Inference(conclusion, assumption..) • Used to decide whether α is true in every model in which KB is true. Example: Wumpus World B1,2 , B2,1 , P1,1 , P2,2 , P3,1, P1,2 , P2,1 So totally 27=128 models are possible
- 15. Truth table for the given KB
- 16. From the table KB is true if R1 through R5 is true in all 3 rows P1,2 is false so there is no pit in [1,2]. There may be or may not be pit in [2,2]
- 17. Truth Table Enumeration Algorithm
- 18. • Here TT truth table • This enumeration algorithm is sound and complete because it works for any KB and alpha and always terminates. • Complexity: – Time complexity O(2 power n) – Space complexity O(n) n symbols
- 19. Equivalence • 2 sentences are logically true in the same set of models then P Q. • Also P ˄Q and Q ˄ P are logically equivalence
- 20. Validity • A sentence is valid if it is true in all the models Example: • P ˄ ¬P is valid. • Valid is also know as tautologies.
- 21. Satisfiability • A sentence is true if it is true in some model. A sentence is satisfiable if it is true in some model e.g., A B, C A sentence is unsatisfiable if it is true in no models e.g., A A
- 22. • Validity and satisfiability are connected. • α is valid if α is satisfiable. • α is valid if ¬α is unsatisfiable. • ¬α is satisfiable if ¬α is not valid