Theory of Computer Science
|
|
|
- Antony French
- 5 years ago
- Views:
Transcription
1 Theory of Computer Science B2. Propositional Logic II Gabriele Röger University of Basel March 7, 2018
2 Logic: Overview Logic Propositional Logic Predicate Logic
3 The Story So Far propositional logic based on atomic propositions syntax: which formulas are well-formed? semantics: when is a formula true? interpretations: important basis of semantics satisfiability and validity: important properties of formulas truth tables: systematically consider all interpretations equivalences: describe when formulas are semantically indistinguishable
4 Simplified Notation
5 Parentheses Associativity: ((ϕ ψ) χ) (ϕ (ψ χ)) ((ϕ ψ) χ) (ϕ (ψ χ)) Placement of parentheses for a conjunction of conjunctions does not influence whether an interpretation is a model. ditto for disjunctions of disjunctions can omit parentheses and treat this as if parentheses placed arbitrarily Example: (A 1 A 2 A 3 A 4 ) instead of ((A 1 (A 2 A 3 )) A 4 ) Example: ( A (B C) D) instead of (( A (B C)) D)
6 Parentheses Does this mean we can always omit all parentheses and assume an arbitrary placement? No!
7 Parentheses Does this mean we can always omit all parentheses and assume an arbitrary placement? No! ((ϕ ψ) χ) (ϕ (ψ χ))
8 Parentheses Does this mean we can always omit all parentheses and assume an arbitrary placement? No! ((ϕ ψ) χ) (ϕ (ψ χ)) What should ϕ ψ χ mean?
9 Placement of Parentheses by Convention Often parentheses can be dropped in specific cases and an implicit placement is assumed: binds more strongly than binds more strongly than binds more strongly than or cf. PEMDAS/ Punkt vor Strich
10 Placement of Parentheses by Convention Often parentheses can be dropped in specific cases and an implicit placement is assumed: binds more strongly than binds more strongly than binds more strongly than or cf. PEMDAS/ Punkt vor Strich Example A C B A D stands for A C B A D
11 Placement of Parentheses by Convention Often parentheses can be dropped in specific cases and an implicit placement is assumed: binds more strongly than binds more strongly than binds more strongly than or cf. PEMDAS/ Punkt vor Strich Example A C B A D stands for A ( C B) A D
12 Placement of Parentheses by Convention Often parentheses can be dropped in specific cases and an implicit placement is assumed: binds more strongly than binds more strongly than binds more strongly than or cf. PEMDAS/ Punkt vor Strich Example A C B A D stands for (A ( C B)) (A D)
13 Placement of Parentheses by Convention Often parentheses can be dropped in specific cases and an implicit placement is assumed: binds more strongly than binds more strongly than binds more strongly than or cf. PEMDAS/ Punkt vor Strich Example A C B A D stands for ((A ( C B)) (A D))
14 Placement of Parentheses by Convention Often parentheses can be dropped in specific cases and an implicit placement is assumed: binds more strongly than binds more strongly than binds more strongly than or cf. PEMDAS/ Punkt vor Strich Example A C B A D stands for ((A ( C B)) (A D)) often harder to read error-prone not used in this course
15 Short Notations for Conjunctions and Disjunctions Short notation for addition: n i=1 x i = x 1 + x x n
16 Short Notations for Conjunctions and Disjunctions Short notation for addition: n i=1 x i = x 1 + x x n Analogously: ( n ϕ ) i = (ϕ1 ϕ 2 ϕ n ) i=1 ( n ϕ ) i = (ϕ1 ϕ 2 ϕ n ) i=1
17 Short Notations for Conjunctions and Disjunctions Short notation for addition: n Analogously: i=1 x i = x 1 + x x n x {x 1,...,x n} x = x 1 + x x n ( n ϕ ) i = (ϕ1 ϕ 2 ϕ n ) i=1 ( n ϕ ) i = (ϕ1 ϕ 2 ϕ n ) i=1
18 Short Notations for Conjunctions and Disjunctions Short notation for addition: n i=1 x i = x 1 + x x n x {x 1,...,x n} x = x 1 + x x n Analogously (possible because of commutativity of and ): ( n ϕ ) i = (ϕ1 ϕ 2 ϕ n ) i=1 ( n ϕ ) i = (ϕ1 ϕ 2 ϕ n ) i=1 ( ϕ X ϕ) = (ϕ 1 ϕ 2 ϕ n ) ( ϕ X ϕ) = (ϕ 1 ϕ 2 ϕ n ) for X = {ϕ 1,..., ϕ n }
19 Short Notation: Corner Cases Is I = ψ true for ψ = ( ϕ) and ψ = ( ϕ X if X = or X = {χ}? ϕ X ϕ)
20 Short Notation: Corner Cases Is I = ψ true for ψ = ( if X = or X = {χ}? ϕ X ϕ) and ψ = ( convention: ( ϕ ϕ) is tautology. ( ϕ ϕ) is unsatisfiable. ( ϕ {χ} ϕ) = ( ϕ {χ} ϕ) = χ ϕ X ϕ)
21 Short Notation: Corner Cases Is I = ψ true for ψ = ( if X = or X = {χ}? ϕ X ϕ) and ψ = ( convention: ( ϕ ϕ) is tautology. ( ϕ ϕ) is unsatisfiable. ( ϕ {χ} ϕ) = ( ϕ {χ} ϕ) = χ Why? ϕ X ϕ)
22 Questions Questions?
23 Normal Forms
24 Logic: Overview Syntax Semantics Properties Logic Propositional Logic Predicate Logic Equivalences Normal Forms Logical Consequence Inference Resolution
25 Why Normal Forms? A normal form is a representation with certain syntactic restrictions. condition for reasonable normal form: every formula must have a logically equivalent formula in normal form advantages: can restrict proofs to formulas in normal form can define algorithms only for formulas in normal form German: Normalform
26 Literals, Clauses and Monomials A literal is an atomic proposition or the negation of an atomic proposition (e. g., A and A). A clause is a disjunction of literals (e. g., (Q P S R)). A monomial is a conjunction of literals (e. g., (Q P S R)).
27 Literals, Clauses and Monomials A literal is an atomic proposition or the negation of an atomic proposition (e. g., A and A). A clause is a disjunction of literals (e. g., (Q P S R)). A monomial is a conjunction of literals (e. g., (Q P S R)).
28 Literals, Clauses and Monomials A literal is an atomic proposition or the negation of an atomic proposition (e. g., A and A). A clause is a disjunction of literals (e. g., (Q P S R)). A monomial is a conjunction of literals (e. g., (Q P S R)).
29 Literals, Clauses and Monomials A literal is an atomic proposition or the negation of an atomic proposition (e. g., A and A). A clause is a disjunction of literals (e. g., (Q P S R)). A monomial is a conjunction of literals (e. g., (Q P S R)). The terms clause and monomial are also used for the corner case with only one literal.
30 Literals, Clauses and Monomials A literal is an atomic proposition or the negation of an atomic proposition (e. g., A and A). A clause is a disjunction of literals (e. g., (Q P S R)). A monomial is a conjunction of literals (e. g., (Q P S R)). The terms clause and monomial are also used for the corner case with only one literal. German: Literal, Klausel, Monom
31 Terminology: Examples Examples ( Q R) (P Q) ((P Q) P) P (P Q) (P P) P
32 Terminology: Examples Examples ( Q R) is a monomial (P Q) ((P Q) P) P (P Q) (P P) P
33 Terminology: Examples Examples ( Q R) is a monomial (P Q) is a clause ((P Q) P) P (P Q) (P P) P
34 Terminology: Examples Examples ( Q R) is a monomial (P Q) is a clause ((P Q) P) is neither literal nor clause nor monomial P (P Q) (P P) P
35 Terminology: Examples Examples ( Q R) is a monomial (P Q) is a clause ((P Q) P) is neither literal nor clause nor monomial P is a literal, a clause and a monomial (P Q) (P P) P
36 Terminology: Examples Examples ( Q R) is a monomial (P Q) is a clause ((P Q) P) is neither literal nor clause nor monomial P is a literal, a clause and a monomial (P Q) is neither literal nor clause nor monomial (but ( P Q) is a clause!) (P P) P
37 Terminology: Examples Examples ( Q R) is a monomial (P Q) is a clause ((P Q) P) is neither literal nor clause nor monomial P is a literal, a clause and a monomial (P Q) is neither literal nor clause nor monomial (but ( P Q) is a clause!) (P P) is a clause, but not a literal or monomial P
38 Terminology: Examples Examples ( Q R) is a monomial (P Q) is a clause ((P Q) P) is neither literal nor clause nor monomial P is a literal, a clause and a monomial (P Q) is neither literal nor clause nor monomial (but ( P Q) is a clause!) (P P) is a clause, but not a literal or monomial P is neither literal nor clause nor monomial
39 Conjunctive Normal Form Definition (Conjunctive Normal Form) A formula is in conjunctive normal form (CNF) if it is a conjunction of clauses, i. e., if it has the form n m i i=1 j=1 with n, m i > 0 (for 1 i n), where the L ij are literals. German: konjunktive Normalform (KNF) Example (( P Q) R (P S)) is in CNF. L ij
40 Disjunctive Normal Form Definition (Disjunctive Normal Form) A formula is in disjunctive normal form (DNF) if it is a disjunction of monomials, i. e., if it has the form n m i i=1 j=1 with n, m i > 0 (for 1 i n), where the L ij are literals. German: disjunktive Normalform (DNF) Example (( P Q) R (P S)) is in DNF. L ij
41 CNF and DNF: Examples Examples ((P Q) P) ((R Q) P (R S)) (P ( Q R)) ((P Q) P) P
42 CNF and DNF: Examples Examples ((P Q) P) is in CNF ((R Q) P (R S)) (P ( Q R)) ((P Q) P) P
43 CNF and DNF: Examples Examples ((P Q) P) is in CNF ((R Q) P (R S)) is in CNF (P ( Q R)) ((P Q) P) P
44 CNF and DNF: Examples Examples ((P Q) P) is in CNF ((R Q) P (R S)) is in CNF (P ( Q R)) is in DNF ((P Q) P) P
45 CNF and DNF: Examples Examples ((P Q) P) is in CNF ((R Q) P (R S)) is in CNF (P ( Q R)) is in DNF ((P Q) P) is neither in CNF nor in DNF P
46 CNF and DNF: Examples Examples ((P Q) P) is in CNF ((R Q) P (R S)) is in CNF (P ( Q R)) is in DNF ((P Q) P) is neither in CNF nor in DNF P is in CNF and in DNF
47 Construction of CNF (and DNF) Algorithm to Construct CNF 1 Replace abbreviations and by their definitions (( )-elimination and ( )-elimination). formula structure: only,, 2 Move negations inside using De Morgan and double negation. formula structure: only,, literals 3 Distribute over with distributivity (strictly speaking also with commutativity). formula structure: CNF 4 optionally: Simplify the formula at the end or at intermediate steps (e. g., with idempotence). Note: For DNF, distribute over instead. Question: runtime complexity?
48 Constructing CNF: Example Construction of Conjunctive Normal Form Given: ϕ = (((P Q) R) (P (S T)))
49 Constructing CNF: Example Construction of Conjunctive Normal Form Given: ϕ = (((P Q) R) (P (S T))) ϕ ( ((P Q) R) P (S T)) [Step 1]
50 Constructing CNF: Example Construction of Conjunctive Normal Form Given: ϕ = (((P Q) R) (P (S T))) ϕ ( ((P Q) R) P (S T)) [Step 1] (( (P Q) R) P (S T)) [Step 2]
51 Constructing CNF: Example Construction of Conjunctive Normal Form Given: ϕ = (((P Q) R) (P (S T))) ϕ ( ((P Q) R) P (S T)) [Step 1] (( (P Q) R) P (S T)) [Step 2] ((( P Q) R) P (S T)) [Step 2]
52 Constructing CNF: Example Construction of Conjunctive Normal Form Given: ϕ = (((P Q) R) (P (S T))) ϕ ( ((P Q) R) P (S T)) [Step 1] (( (P Q) R) P (S T)) [Step 2] ((( P Q) R) P (S T)) [Step 2] ((( P Q) R) P (S T)) [Step 2]
53 Constructing CNF: Example Construction of Conjunctive Normal Form Given: ϕ = (((P Q) R) (P (S T))) ϕ ( ((P Q) R) P (S T)) [Step 1] (( (P Q) R) P (S T)) [Step 2] ((( P Q) R) P (S T)) [Step 2] ((( P Q) R) P (S T)) [Step 2] ((( P Q) R) P ( S T)) [Step 2]
54 Constructing CNF: Example Construction of Conjunctive Normal Form Given: ϕ = (((P Q) R) (P (S T))) ϕ ( ((P Q) R) P (S T)) [Step 1] (( (P Q) R) P (S T)) [Step 2] ((( P Q) R) P (S T)) [Step 2] ((( P Q) R) P (S T)) [Step 2] ((( P Q) R) P ( S T)) [Step 2] (( P Q P ( S T)) ( R P ( S T))) [Step 3]
55 Constructing CNF: Example Construction of Conjunctive Normal Form Given: ϕ = (((P Q) R) (P (S T))) ϕ ( ((P Q) R) P (S T)) [Step 1] (( (P Q) R) P (S T)) [Step 2] ((( P Q) R) P (S T)) [Step 2] ((( P Q) R) P (S T)) [Step 2] ((( P Q) R) P ( S T)) [Step 2] (( P Q P ( S T)) ( R P ( S T))) [Step 3] ( R P ( S T)) [Step 4]
56 Constructing CNF: Example Construction of Conjunctive Normal Form Given: ϕ = (((P Q) R) (P (S T))) ϕ ( ((P Q) R) P (S T)) [Step 1] (( (P Q) R) P (S T)) [Step 2] ((( P Q) R) P (S T)) [Step 2] ((( P Q) R) P (S T)) [Step 2] ((( P Q) R) P ( S T)) [Step 2] (( P Q P ( S T)) ( R P ( S T))) [Step 3] ( R P ( S T)) [Step 4] (( R P S) ( R P T)) [Step 3]
57 Construct DNF: Example Construction of Disjunctive Normal Form Given: ϕ = (((P Q) R) (P (S T))) ϕ ( ((P Q) R) P (S T)) [Step 1] (( (P Q) R) P (S T)) [Step 2] ((( P Q) R) P (S T)) [Step 2] ((( P Q) R) P (S T)) [Step 2] ((( P Q) R) P ( S T)) [Step 2] (( P R) (Q R) P ( S T)) [Step 3]
58 Existence of an Equivalent Formula in Normal Form Theorem For every formula ϕ there is a logically equivalent formula in CNF and a logically equivalent formula in DNF. There is a always means there is at least one. Otherwise we would write there is exactly one. Intuition: algorithm to construct normal form works with any given formula and only uses equivalence rewriting. actual proof would use induction over structure of formula
59 Existence of an Equivalent Formula in Normal Form Theorem For every formula ϕ there is a logically equivalent formula in CNF and a logically equivalent formula in DNF. There is a always means there is at least one. Otherwise we would write there is exactly one. Intuition: algorithm to construct normal form works with any given formula and only uses equivalence rewriting. actual proof would use induction over structure of formula
60 Existence of an Equivalent Formula in Normal Form Theorem For every formula ϕ there is a logically equivalent formula in CNF and a logically equivalent formula in DNF. There is a always means there is at least one. Otherwise we would write there is exactly one. Intuition: algorithm to construct normal form works with any given formula and only uses equivalence rewriting. actual proof would use induction over structure of formula
61 Existence of an Equivalent Formula in Normal Form Theorem For every formula ϕ there is a logically equivalent formula in CNF and a logically equivalent formula in DNF. There is a always means there is at least one. Otherwise we would write there is exactly one. Intuition: algorithm to construct normal form works with any given formula and only uses equivalence rewriting. actual proof would use induction over structure of formula
62 More Theorems Theorem A formula in CNF is a tautology iff every clause is a tautology. Theorem A formula in DNF is satisfiable iff at least one of its monomials is satisfiable. both proved easily with semantics of propositional logic
63 Questions Questions?
64 Logical Consequences
65 Logic: Overview Syntax Semantics Properties Logic Propositional Logic Predicate Logic Equivalences Normal Forms Logical Consequence Inference Resolution
66 Knowledge Bases: Example If not DrinkBeer, then EatFish. If EatFish and DrinkBeer, then not EatIceCream. If EatIceCream or not DrinkBeer, then not EatFish. KB = {( DrinkBeer EatFish), ((EatFish DrinkBeer) EatIceCream), ((EatIceCream DrinkBeer) EatFish)} Exercise from U. Schöning: Logik für Informatiker Picture courtesy of graur razvan ionut / FreeDigitalPhotos.net
67 Models for Sets of Formulas Definition (Model for Knowledge Base) Let KB be a knowledge base over A, i. e., a set of propositional formulas over A. A truth assignment I for A is a model for KB (written: I = KB) if I is a model for every formula ϕ KB. German: Wissensbasis, Modell
68 Properties of Sets of Formulas A knowledge base KB is satisfiable if KB has at least one model unsatisfiable if KB is not satisfiable valid (or a tautology) if every interpretation is a model for KB falsifiable if KB is no tautology German: erfüllbar, unerfüllbar, gültig, gültig/eine Tautologie, falsifizierbar
69 Example I Which of the properties does KB = {(A B), (B A)} have? KB is unsatisfiable: For every model I with I = (A B) we have I(A) = 1. This means I = (B A) and thus I = (B A). This directly implies that KB is falsifiable, not satisfiable and no tautology.
70 Example I Which of the properties does KB = {(A B), (B A)} have? KB is unsatisfiable: For every model I with I = (A B) we have I(A) = 1. This means I = (B A) and thus I = (B A). This directly implies that KB is falsifiable, not satisfiable and no tautology.
71 Example I Which of the properties does KB = {(A B), (B A)} have? KB is unsatisfiable: For every model I with I = (A B) we have I(A) = 1. This means I = (B A) and thus I = (B A). This directly implies that KB is falsifiable, not satisfiable and no tautology.
72 Example II Which of the properties does KB = {( DrinkBeer EatFish), ((EatFish DrinkBeer) EatIceCream), ((EatIceCream DrinkBeer) EatFish)} have? satisfiable, e. g. with I = {EatFish 1, DrinkBeer 1, EatIceCream 0} thus not unsatisfiable falsifiable, e. g. with I = {EatFish 0, DrinkBeer 0, EatIceCream 1} thus not valid
73 Example II Which of the properties does KB = {( DrinkBeer EatFish), ((EatFish DrinkBeer) EatIceCream), ((EatIceCream DrinkBeer) EatFish)} have? satisfiable, e. g. with I = {EatFish 1, DrinkBeer 1, EatIceCream 0} thus not unsatisfiable falsifiable, e. g. with I = {EatFish 0, DrinkBeer 0, EatIceCream 1} thus not valid
74 Logical Consequences: Motivation What s the secret of your long life? Claim: the woman drinks beer to every meal. How can we prove this? I am on a strict diet: If I don t drink beer to a meal, then I always eat fish. Whenever I have fish and beer with the same meal, I abstain from ice cream. When I eat ice cream or don t drink beer, then I never touch fish. Exercise from U. Schöning: Logik für Informatiker Picture courtesy of graur razvan ionut/freedigitalphotos.net
75 Logical Consequences Definition (Logical Consequence) Let KB be a set of formulas and ϕ a formula. We say that KB logically implies ϕ (written as KB = ϕ) if all models of KB are also models of ϕ. also: KB logically entails ϕ, ϕ logically follows from KB, ϕ is a logical consequence of KB German: KB impliziert ϕ logisch, ϕ folgt logisch aus KB, ϕ ist logische Konsequenz von KB Attention: the symbol = is overloaded : KB = ϕ vs. I = ϕ. What if KB is unsatisfiable or the empty set?
76 Logical Consequences Definition (Logical Consequence) Let KB be a set of formulas and ϕ a formula. We say that KB logically implies ϕ (written as KB = ϕ) if all models of KB are also models of ϕ. also: KB logically entails ϕ, ϕ logically follows from KB, ϕ is a logical consequence of KB German: KB impliziert ϕ logisch, ϕ folgt logisch aus KB, ϕ ist logische Konsequenz von KB Attention: the symbol = is overloaded : KB = ϕ vs. I = ϕ. What if KB is unsatisfiable or the empty set?
77 Logical Consequences Definition (Logical Consequence) Let KB be a set of formulas and ϕ a formula. We say that KB logically implies ϕ (written as KB = ϕ) if all models of KB are also models of ϕ. also: KB logically entails ϕ, ϕ logically follows from KB, ϕ is a logical consequence of KB German: KB impliziert ϕ logisch, ϕ folgt logisch aus KB, ϕ ist logische Konsequenz von KB Attention: the symbol = is overloaded : KB = ϕ vs. I = ϕ. What if KB is unsatisfiable or the empty set?
78 Logical Consequences: Example Let ϕ = DrinkBeer and KB = {( DrinkBeer EatFish), Show: KB = ϕ Proof sketch. ((EatFish DrinkBeer) EatIceCream), ((EatIceCream DrinkBeer) EatFish)}. Proof by contradiction: assume I = KB, but I = DrinkBeer. Then it follows that I = DrinkBeer. Because I is a model of KB, we also have I = ( DrinkBeer EatFish) and thus I = EatFish. (Why?) With an analogous argumentation starting from I = ((EatIceCream DrinkBeer) EatFish) we get I = EatFish and thus I = EatFish. Contradiction!
79 Logical Consequences: Example Let ϕ = DrinkBeer and KB = {( DrinkBeer EatFish), Show: KB = ϕ Proof sketch. ((EatFish DrinkBeer) EatIceCream), ((EatIceCream DrinkBeer) EatFish)}. Proof by contradiction: assume I = KB, but I = DrinkBeer. Then it follows that I = DrinkBeer. Because I is a model of KB, we also have I = ( DrinkBeer EatFish) and thus I = EatFish. (Why?) With an analogous argumentation starting from I = ((EatIceCream DrinkBeer) EatFish) we get I = EatFish and thus I = EatFish. Contradiction!
80 Important Theorems about Logical Consequences Theorem (Deduction Theorem) KB {ϕ} = ψ iff KB = (ϕ ψ) German: Deduktionssatz Theorem (Contraposition Theorem) KB {ϕ} = ψ iff KB {ψ} = ϕ German: Kontrapositionssatz Theorem (Contradiction Theorem) KB {ϕ} is unsatisfiable iff KB = ϕ German: Widerlegungssatz (without proof)
81 Questions Questions?
82 Summary
83 Summary CNF: formula is a conjunction of clauses DNF: formula is a disjunction of monomials every formula has equivalent formulas in DNF and in CNF knowledge base: set of formulas describing given information; satisfiable, valid etc. used like for individual formulas logical consequence KB = ϕ means that ϕ is true whenever (= in all models where) KB is true
Theory of Computer Science
Theory of Computer Science B1. Propositional Logic I Malte Helmert University of Basel February 29, 2016 Motivation Exercise from Last Lecture What s the secret of your long life? Simplify this advice!
B1.1 Motivation. Theory of Computer Science. Theory of Computer Science. B1.1 Motivation. B1.2 Syntax. B1.3 Semantics
Theory of Computer Science February 29, 2016 B1. Propositional Logic I Theory of Computer Science B1. Propositional Logic I Malte Helmert University of Basel February 29, 2016 B1.1 Motivation B1.2 Syntax
Description Logics. Foundations of Propositional Logic. franconi. Enrico Franconi
(1/27) Description Logics Foundations of Propositional Logic Enrico Franconi franconi@cs.man.ac.uk http://www.cs.man.ac.uk/ franconi Department of Computer Science, University of Manchester (2/27) Knowledge
Logic: Propositional Logic (Part I)
Logic: Propositional Logic (Part I) Alessandro Artale Free University of Bozen-Bolzano Faculty of Computer Science http://www.inf.unibz.it/ artale Descrete Mathematics and Logic BSc course Thanks to Prof.
Part 1: Propositional Logic
Part 1: Propositional Logic Literature (also for first-order logic) Schöning: Logik für Informatiker, Spektrum Fitting: First-Order Logic and Automated Theorem Proving, Springer 1 Last time 1.1 Syntax
Language of Propositional Logic
Logic A logic has: 1. An alphabet that contains all the symbols of the language of the logic. 2. A syntax giving the rules that define the well formed expressions of the language of the logic (often called
Theory of Computer Science
Theory of Computer Science A3. Proof Techniques Malte Helmert University of Basel February 22, 2017 Introduction Mathematical Statements Mathematical Statement A mathematical statement consists of a set
Foundations of Artificial Intelligence
Foundations of Artificial Intelligence 7. Propositional Logic Rational Thinking, Logic, Resolution Wolfram Burgard, Maren Bennewitz, and Marco Ragni Albert-Ludwigs-Universität Freiburg Contents 1 Agents
Foundations of Artificial Intelligence
Foundations of Artificial Intelligence 7. Propositional Logic Rational Thinking, Logic, Resolution Joschka Boedecker and Wolfram Burgard and Bernhard Nebel Albert-Ludwigs-Universität Freiburg May 17, 2016
CS206 Lecture 03. Propositional Logic Proofs. Plan for Lecture 03. Axioms. Normal Forms
CS206 Lecture 03 Propositional Logic Proofs G. Sivakumar Computer Science Department IIT Bombay siva@iitb.ac.in http://www.cse.iitb.ac.in/ siva Page 1 of 12 Fri, Jan 03, 2003 Plan for Lecture 03 Axioms
Inference in Propositional Logic
Inference in Propositional Logic Deepak Kumar November 2017 Propositional Logic A language for symbolic reasoning Proposition a statement that is either True or False. E.g. Bryn Mawr College is located
Propositional and Predicate Logic - II
Propositional and Predicate Logic - II Petr Gregor KTIML MFF UK WS 2016/2017 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - II WS 2016/2017 1 / 16 Basic syntax Language Propositional logic
Propositional Logic Language
Propositional Logic Language A logic consists of: an alphabet A, a language L, i.e., a set of formulas, and a binary relation = between a set of formulas and a formula. An alphabet A consists of a finite
7. Propositional Logic. Wolfram Burgard and Bernhard Nebel
Foundations of AI 7. Propositional Logic Rational Thinking, Logic, Resolution Wolfram Burgard and Bernhard Nebel Contents Agents that think rationally The wumpus world Propositional logic: syntax and semantics
Foundations of Artificial Intelligence
Foundations of Artificial Intelligence 7. Propositional Logic Rational Thinking, Logic, Resolution Joschka Boedecker and Wolfram Burgard and Frank Hutter and Bernhard Nebel Albert-Ludwigs-Universität Freiburg
The Calculus of Computation: Decision Procedures with Applications to Verification. Part I: FOUNDATIONS. by Aaron Bradley Zohar Manna
The Calculus of Computation: Decision Procedures with Applications to Verification Part I: FOUNDATIONS by Aaron Bradley Zohar Manna 1. Propositional Logic(PL) Springer 2007 1-1 1-2 Propositional Logic(PL)
Tautologies, Contradictions, and Contingencies
Section 1.3 Tautologies, Contradictions, and Contingencies A tautology is a proposition which is always true. Example: p p A contradiction is a proposition which is always false. Example: p p A contingency
CS156: The Calculus of Computation
CS156: The Calculus of Computation Zohar Manna Winter 2010 It is reasonable to hope that the relationship between computation and mathematical logic will be as fruitful in the next century as that between
Advanced Topics in LP and FP
Lecture 1: Prolog and Summary of this lecture 1 Introduction to Prolog 2 3 Truth value evaluation 4 Prolog Logic programming language Introduction to Prolog Introduced in the 1970s Program = collection
Part 1: Propositional Logic
Part 1: Propositional Logic Literature (also for first-order logic) Schöning: Logik für Informatiker, Spektrum Fitting: First-Order Logic and Automated Theorem Proving, Springer 1 Last time 1.1 Syntax
Propositional logic (revision) & semantic entailment. p. 1/34
Propositional logic (revision) & semantic entailment p. 1/34 Reading The background reading for propositional logic is Chapter 1 of Huth/Ryan. (This will cover approximately the first three lectures.)
Supplementary exercises in propositional logic
Supplementary exercises in propositional logic The purpose of these exercises is to train your ability to manipulate and analyze logical formulas. Familiarize yourself with chapter 7.3-7.5 in the course
CS 512, Spring 2017, Handout 10 Propositional Logic: Conjunctive Normal Forms, Disjunctive Normal Forms, Horn Formulas, and other special forms
CS 512, Spring 2017, Handout 10 Propositional Logic: Conjunctive Normal Forms, Disjunctive Normal Forms, Horn Formulas, and other special forms Assaf Kfoury 5 February 2017 Assaf Kfoury, CS 512, Spring
Introduction to Sets and Logic (MATH 1190)
Introduction to Sets Logic () Instructor: Email: shenlili@yorku.ca Department of Mathematics Statistics York University Sept 18, 2014 Outline 1 2 Tautologies Definition A tautology is a compound proposition
Title: Logical Agents AIMA: Chapter 7 (Sections 7.4 and 7.5)
B.Y. Choueiry 1 Instructor s notes #12 Title: Logical Agents AIMA: Chapter 7 (Sections 7.4 and 7.5) Introduction to Artificial Intelligence CSCE 476-876, Fall 2018 URL: www.cse.unl.edu/ choueiry/f18-476-876
COMP9414: Artificial Intelligence Propositional Logic: Automated Reasoning
COMP9414, Monday 26 March, 2012 Propositional Logic 2 COMP9414: Artificial Intelligence Propositional Logic: Automated Reasoning Overview Proof systems (including soundness and completeness) Normal Forms
Propositional Resolution
Artificial Intelligence Propositional Resolution Marco Piastra Propositional Resolution 1] Deductive systems and automation Is problem decidible? A deductive system a la Hilbert (i.e. derivation using
Propositional Logic: Models and Proofs
Propositional Logic: Models and Proofs C. R. Ramakrishnan CSE 505 1 Syntax 2 Model Theory 3 Proof Theory and Resolution Compiled at 11:51 on 2016/11/02 Computing with Logic Propositional Logic CSE 505
Propositional Calculus: Formula Simplification, Essential Laws, Normal Forms
P Formula Simplification, Essential Laws, Normal Forms Lila Kari University of Waterloo P Formula Simplification, Essential Laws, Normal CS245, Forms Logic and Computation 1 / 26 Propositional calculus
Knowledge base (KB) = set of sentences in a formal language Declarative approach to building an agent (or other system):
Logic Knowledge-based agents Inference engine Knowledge base Domain-independent algorithms Domain-specific content Knowledge base (KB) = set of sentences in a formal language Declarative approach to building
Outline. Formale Methoden der Informatik First-Order Logic for Forgetters. Why PL1? Why PL1? Cont d. Motivation
Outline Formale Methoden der Informatik First-Order Logic for Forgetters Uwe Egly Vienna University of Technology Institute of Information Systems Knowledge-Based Systems Group Motivation Syntax of PL1
Logic: First Order Logic
Logic: First Order Logic Raffaella Bernardi bernardi@inf.unibz.it P.zza Domenicani 3, Room 2.28 Faculty of Computer Science, Free University of Bolzano-Bozen http://www.inf.unibz.it/~bernardi/courses/logic06
Section 1.2 Propositional Equivalences. A tautology is a proposition which is always true. A contradiction is a proposition which is always false.
Section 1.2 Propositional Equivalences A tautology is a proposition which is always true. Classic Example: P P A contradiction is a proposition which is always false. Classic Example: P P A contingency
Truth-Functional Logic
Truth-Functional Logic Syntax Every atomic sentence (A, B, C, ) is a sentence and are sentences With ϕ a sentence, the negation ϕ is a sentence With ϕ and ψ sentences, the conjunction ϕ ψ is a sentence
COMP219: Artificial Intelligence. Lecture 20: Propositional Reasoning
COMP219: Artificial Intelligence Lecture 20: Propositional Reasoning 1 Overview Last time Logic for KR in general; Propositional Logic; Natural Deduction Today Entailment, satisfiability and validity Normal
UNIT-I: Propositional Logic
1. Introduction to Logic: UNIT-I: Propositional Logic Logic: logic comprises a (formal) language for making statements about objects and reasoning about properties of these objects. Statements in a logical
Theory of Computer Science
Theory of Computer Science B4. Predicate Logic II Malte Helmert University of Basel March 9, 2016 Free and Bound Variables Free and Bound Variables: Motivation Question: Consider a signature with variable
2. The Logic of Compound Statements Summary. Aaron Tan August 2017
2. The Logic of Compound Statements Summary Aaron Tan 21 25 August 2017 1 2. The Logic of Compound Statements 2.1 Logical Form and Logical Equivalence Statements; Compound Statements; Statement Form (Propositional
A brief introduction to Logic. (slides from
A brief introduction to Logic (slides from http://www.decision-procedures.org/) 1 A Brief Introduction to Logic - Outline Propositional Logic :Syntax Propositional Logic :Semantics Satisfiability and validity
Automated Program Verification and Testing 15414/15614 Fall 2016 Lecture 2: Propositional Logic
Automated Program Verification and Testing 15414/15614 Fall 2016 Lecture 2: Propositional Logic Matt Fredrikson mfredrik@cs.cmu.edu October 17, 2016 Matt Fredrikson Propositional Logic 1 / 33 Propositional
PROPOSITIONAL LOGIC. VL Logik: WS 2018/19
PROPOSITIONAL LOGIC VL Logik: WS 2018/19 (Version 2018.2) Martina Seidl (martina.seidl@jku.at), Armin Biere (biere@jku.at) Institut für Formale Modelle und Verifikation BOX Game: Rules 1. The game board
02 Propositional Logic
SE 2F03 Fall 2005 02 Propositional Logic Instructor: W. M. Farmer Revised: 25 September 2005 1 What is Propositional Logic? Propositional logic is the study of the truth or falsehood of propositions or
https://vu5.sfc.keio.ac.jp/slide/
1 FUNDAMENTALS OF LOGIC NO.3 NORMAL FORMS Tatsuya Hagino hagino@sfc.keio.ac.jp lecture URL https://vu5.sfc.keio.ac.jp/slide/ 2 So Far What is Logic? mathematical logic symbolic logic Proposition A statement
Propositional Logic. Methods & Tools for Software Engineering (MTSE) Fall Prof. Arie Gurfinkel
Propositional Logic Methods & Tools for Software Engineering (MTSE) Fall 2017 Prof. Arie Gurfinkel References Chpater 1 of Logic for Computer Scientists http://www.springerlink.com/content/978-0-8176-4762-9/
Propositional Logic. Testing, Quality Assurance, and Maintenance Winter Prof. Arie Gurfinkel
Propositional Logic Testing, Quality Assurance, and Maintenance Winter 2018 Prof. Arie Gurfinkel References Chpater 1 of Logic for Computer Scientists http://www.springerlink.com/content/978-0-8176-4762-9/
Propositional Logic: Review
Propositional Logic: Review Propositional logic Logical constants: true, false Propositional symbols: P, Q, S,... (atomic sentences) Wrapping parentheses: ( ) Sentences are combined by connectives:...and...or
Tecniche di Verifica. Introduction to Propositional Logic
Tecniche di Verifica Introduction to Propositional Logic 1 Logic A formal logic is defined by its syntax and semantics. Syntax An alphabet is a set of symbols. A finite sequence of these symbols is called
Normal Forms of Propositional Logic
Normal Forms of Propositional Logic Bow-Yaw Wang Institute of Information Science Academia Sinica, Taiwan September 12, 2017 Bow-Yaw Wang (Academia Sinica) Normal Forms of Propositional Logic September
2.5.2 Basic CNF/DNF Transformation
2.5. NORMAL FORMS 39 On the other hand, checking the unsatisfiability of CNF formulas or the validity of DNF formulas is conp-complete. For any propositional formula φ there is an equivalent formula in
CSC Discrete Math I, Spring Propositional Logic
CSC 125 - Discrete Math I, Spring 2017 Propositional Logic Propositions A proposition is a declarative sentence that is either true or false Propositional Variables A propositional variable (p, q, r, s,...)
Deductive Systems. Lecture - 3
Deductive Systems Lecture - 3 Axiomatic System Axiomatic System (AS) for PL 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
Logic for Computer Scientists
Logic for Computer Scientists Pascal Hitzler http://www.pascal-hitzler.de CS 499/699 Lecture, Winter Quarter 2011 Wright State University, Dayton, OH, U.S.A. [final version: 03/10/2011] Contents 1 Propositional
7 LOGICAL AGENTS. OHJ-2556 Artificial Intelligence, Spring OHJ-2556 Artificial Intelligence, Spring
109 7 LOGICAL AGENS We now turn to knowledge-based agents that have a knowledge base KB at their disposal With the help of the KB the agent aims at maintaining knowledge of its partially-observable environment
The Importance of Being Formal. Martin Henz. February 5, Propositional Logic
The Importance of Being Formal Martin Henz February 5, 2014 Propositional Logic 1 Motivation In traditional logic, terms represent sets, and therefore, propositions are limited to stating facts on sets
Introduction to Artificial Intelligence. Logical Agents
Introduction to Artificial Intelligence Logical Agents (Logic, Deduction, Knowledge Representation) Bernhard Beckert UNIVERSITÄT KOBLENZ-LANDAU Winter Term 2004/2005 B. Beckert: KI für IM p.1 Outline Knowledge-based
Logic for Computer Scientists
Logic for Computer Scientists Pascal Hitzler http://www.pascal-hitzler.de CS 499/699 Lecture, Spring Quarter 2010 Wright State University, Dayton, OH, U.S.A. Final version. Contents 1 Propositional Logic
MAI0203 Lecture 7: Inference and Predicate Calculus
MAI0203 Lecture 7: Inference and Predicate Calculus Methods of Artificial Intelligence WS 2002/2003 Part II: Inference and Knowledge Representation II.7 Inference and Predicate Calculus MAI0203 Lecture
Warm-Up Problem. Is the following true or false? 1/35
Warm-Up Problem Is the following true or false? 1/35 Propositional Logic: Resolution Carmen Bruni Lecture 6 Based on work by J Buss, A Gao, L Kari, A Lubiw, B Bonakdarpour, D Maftuleac, C Roberts, R Trefler,
Propositional Resolution Introduction
Propositional Resolution Introduction (Nilsson Book Handout) Professor Anita Wasilewska CSE 352 Artificial Intelligence Propositional Resolution Part 1 SYNTAX dictionary Literal any propositional VARIABLE
Propositional Logic Basics Propositional Equivalences Normal forms Boolean functions and digital circuits. Propositional Logic.
Propositional Logic Winter 2012 Propositional Logic: Section 1.1 Proposition A proposition is a declarative sentence that is either true or false. Which ones of the following sentences are propositions?
3 Propositional Logic
3 Propositional Logic 3.1 Syntax 3.2 Semantics 3.3 Equivalence and Normal Forms 3.4 Proof Procedures 3.5 Properties Propositional Logic (25th October 2007) 1 3.1 Syntax Definition 3.0 An alphabet Σ consists
Equivalences. Proposition 2.8: The following equivalences are valid for all formulas φ, ψ, χ: (φ φ) φ. Idempotency Idempotency Commutativity
Substitution Theorem Proposition 2.7: Let φ 1 and φ 2 be equivalent formulas, and ψ[φ 1 ] p be a formula in which φ 1 occurs as a subformula at position p. Then ψ[φ 1 ] p is equivalent to ψ[φ 2 ] p. 84
Sec$on Summary. Tautologies, Contradictions, and Contingencies. Logical Equivalence. Normal Forms (optional, covered in exercises in text)
Section 1.3 1 Sec$on Summary Tautologies, Contradictions, and Contingencies. Logical Equivalence Important Logical Equivalences Showing Logical Equivalence Normal Forms (optional, covered in exercises
Comp487/587 - Boolean Formulas
Comp487/587 - Boolean Formulas 1 Logic and SAT 1.1 What is a Boolean Formula Logic is a way through which we can analyze and reason about simple or complicated events. In particular, we are interested
Logic: Propositional Logic Truth Tables
Logic: Propositional Logic Truth Tables Raffaella Bernardi bernardi@inf.unibz.it P.zza Domenicani 3, Room 2.28 Faculty of Computer Science, Free University of Bolzano-Bozen http://www.inf.unibz.it/~bernardi/courses/logic06
CS156: The Calculus of Computation Zohar Manna Autumn 2008
Page 3 of 52 Page 4 of 52 CS156: The Calculus of Computation Zohar Manna Autumn 2008 Lecturer: Zohar Manna (manna@cs.stanford.edu) Office Hours: MW 12:30-1:00 at Gates 481 TAs: Boyu Wang (wangboyu@stanford.edu)
Review of Predicate Logic
Review of Predicate Logic Martin Held FB Computerwissenschaften Universität Salzburg A-5020 Salzburg, Austria held@cosy.sbg.ac.at 19. Jänner 2016 COMPUTERWISSENSCHAFTEN Legal Fine Print and Disclaimer
Intelligent Agents. First Order Logic. Ute Schmid. Cognitive Systems, Applied Computer Science, Bamberg University. last change: 19.
Intelligent Agents First Order Logic Ute Schmid Cognitive Systems, Applied Computer Science, Bamberg University last change: 19. Mai 2015 U. Schmid (CogSys) Intelligent Agents last change: 19. Mai 2015
Propositional logic. Programming and Modal Logic
Propositional logic Programming and Modal Logic 2006-2007 4 Contents Syntax of propositional logic Semantics of propositional logic Semantic entailment Natural deduction proof system Soundness and completeness
CS 380: ARTIFICIAL INTELLIGENCE
CS 380: RTIFICIL INTELLIGENCE PREDICTE LOGICS 11/8/2013 Santiago Ontañón santi@cs.drexel.edu https://www.cs.drexel.edu/~santi/teaching/2013/cs380/intro.html Summary of last day: Logical gents: The can
Announcements. CS311H: Discrete Mathematics. Propositional Logic II. Inverse of an Implication. Converse of a Implication
Announcements CS311H: Discrete Mathematics Propositional Logic II Instructor: Işıl Dillig First homework assignment out today! Due in one week, i.e., before lecture next Wed 09/13 Remember: Due before
Propositional Logic and Semantics
Propositional Logic and Semantics English is naturally ambiguous. For example, consider the following employee (non)recommendations and their ambiguity in the English language: I can assure you that no
Intelligent Systems. Propositional Logic. Dieter Fensel and Dumitru Roman. Copyright 2008 STI INNSBRUCK
Intelligent Systems Propositional Logic Dieter Fensel and Dumitru Roman www.sti-innsbruck.at Copyright 2008 STI INNSBRUCK www.sti-innsbruck.at Where are we? # Title 1 Introduction 2 Propositional Logic
Logical Agents (I) Instructor: Tsung-Che Chiang
Logical Agents (I) Instructor: Tsung-Che Chiang tcchiang@ieee.org Department of Computer Science and Information Engineering National Taiwan Normal University Artificial Intelligence, Spring, 2010 編譯有誤
Logic Overview, I. and T T T T F F F T F F F F
Logic Overview, I DEFINITIONS A statement (proposition) is a declarative sentence that can be assigned a truth value T or F, but not both. Statements are denoted by letters p, q, r, s,... The 5 basic logical
Kecerdasan Buatan M. Ali Fauzi
Kecerdasan Buatan M. Ali Fauzi Artificial Intelligence M. Ali Fauzi Logical Agents M. Ali Fauzi In which we design agents that can form representations of the would, use a process of inference to derive
Propositional Logic Part 1
Propositional Logic Part 1 Yingyu Liang yliang@cs.wisc.edu Computer Sciences Department University of Wisconsin, Madison [Based on slides from Louis Oliphant, Andrew Moore, Jerry Zhu] slide 1 5 is even
CS1021. Why logic? Logic about inference or argument. Start from assumptions or axioms. Make deductions according to rules of reasoning.
3: Logic Why logic? Logic about inference or argument Start from assumptions or axioms Make deductions according to rules of reasoning Logic 3-1 Why logic? (continued) If I don t buy a lottery ticket on
CS 380: ARTIFICIAL INTELLIGENCE PREDICATE LOGICS. Santiago Ontañón
CS 380: RTIFICIL INTELLIGENCE PREDICTE LOGICS Santiago Ontañón so367@drexeledu Summary of last day: Logical gents: The can reason from the knowledge they have They can make deductions from their perceptions,
Equivalents of Mingle and Positive Paradox
Eric Schechter Equivalents of Mingle and Positive Paradox Abstract. Relevant logic is a proper subset of classical logic. It does not include among itstheoremsanyof positive paradox A (B A) mingle A (A
Propositional and Predicate Logic - V
Propositional and Predicate Logic - V Petr Gregor KTIML MFF UK WS 2016/2017 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - V WS 2016/2017 1 / 21 Formal proof systems Hilbert s calculus
Propositional Reasoning
Propositional Reasoning CS 440 / ECE 448 Introduction to Artificial Intelligence Instructor: Eyal Amir Grad TAs: Wen Pu, Yonatan Bisk Undergrad TAs: Sam Johnson, Nikhil Johri Spring 2010 Intro to AI (CS
Propositional Logic: Part II - Syntax & Proofs 0-0
Propositional Logic: Part II - Syntax & Proofs 0-0 Outline Syntax of Propositional Formulas Motivating Proofs Syntactic Entailment and Proofs Proof Rules for Natural Deduction Axioms, theories and theorems
Agenda. Artificial Intelligence. Reasoning in the Wumpus World. The Wumpus World
Agenda Artificial Intelligence 10. Propositional Reasoning, Part I: Principles How to Think About What is True or False 1 Introduction Álvaro Torralba Wolfgang Wahlster 2 Propositional Logic 3 Resolution
Learning Goals of CS245 Logic and Computation
Learning Goals of CS245 Logic and Computation Alice Gao April 27, 2018 Contents 1 Propositional Logic 2 2 Predicate Logic 4 3 Program Verification 6 4 Undecidability 7 1 1 Propositional Logic Introduction
Propositional Language - Semantics
Propositional Language - Semantics Lila Kari University of Waterloo Propositional Language - Semantics CS245, Logic and Computation 1 / 41 Syntax and semantics Syntax Semantics analyzes Form analyzes Meaning
PHIL12A Section answers, 16 February 2011
PHIL12A Section answers, 16 February 2011 Julian Jonker 1 How much do you know? 1. Show that the following sentences are equivalent. (a) (Ex 4.16) A B A and A B A B (A B) A A B T T T T T T T T T T T F
ECE473 Lecture 15: Propositional Logic
ECE473 Lecture 15: Propositional Logic Jeffrey Mark Siskind School of Electrical and Computer Engineering Spring 2018 Siskind (Purdue ECE) ECE473 Lecture 15: Propositional Logic Spring 2018 1 / 23 What
Class Assignment Strategies
Class Assignment Strategies ì Team- A'ack: Team a'ack ì Individualis2c: Search for possible ì Poli2cal: look at others and make decision based on who is winning, who is loosing, and conversa;on ì Emo2on
Logic and Discrete Mathematics. Section 3.5 Propositional logical equivalence Negation of propositional formulae
Logic and Discrete Mathematics Section 3.5 Propositional logical equivalence Negation of propositional formulae Slides version: January 2015 Logical equivalence of propositional formulae Propositional
Artificial Intelligence. Propositional Logic. Copyright 2011 Dieter Fensel and Florian Fischer
Artificial Intelligence Propositional Logic Copyright 2011 Dieter Fensel and Florian Fischer 1 Where are we? # Title 1 Introduction 2 Propositional Logic 3 Predicate Logic 4 Reasoning 5 Search Methods
Foundations of Artificial Intelligence
Foundations of Artificial Intelligence 31. Propositional Logic: DPLL Algorithm Malte Helmert and Gabriele Röger University of Basel April 24, 2017 Propositional Logic: Overview Chapter overview: propositional
CS:4420 Artificial Intelligence
CS:4420 Artificial Intelligence Spring 2018 Propositional Logic Cesare Tinelli The University of Iowa Copyright 2004 18, Cesare Tinelli and Stuart Russell a a These notes were originally developed by Stuart
2.2: Logical Equivalence: The Laws of Logic
Example (2.7) For primitive statement p and q, construct a truth table for each of the following compound statements. a) p q b) p q Here we see that the corresponding truth tables for two statement p q
Propositional logic II.
Lecture 5 Propositional logic II. Milos Hauskrecht milos@cs.pitt.edu 5329 ennott quare Propositional logic. yntax yntax: ymbols (alphabet) in P: Constants: True, False Propositional symbols Examples: P
Propositional Equivalence
Propositional Equivalence Tautologies and contradictions A compound proposition that is always true, regardless of the truth values of the individual propositions involved, is called a tautology. Example:
Logic Part I: Classical Logic and Its Semantics
Logic Part I: Classical Logic and Its Semantics Max Schäfer Formosan Summer School on Logic, Language, and Computation 2007 July 2, 2007 1 / 51 Principles of Classical Logic classical logic seeks to model
Artificial Intelligence Knowledge Representation I
Artificial Intelligence Knowledge Representation I Agents that reason logically knowledge-based approach implement agents that know about their world and reason about possible courses of action needs to
Artificial Intelligence. Propositional logic
Artificial Intelligence Propositional logic Propositional Logic: Syntax Syntax of propositional logic defines allowable sentences Atomic sentences consists of a single proposition symbol Each symbol stands
Propositional Resolution Part 1. Short Review Professor Anita Wasilewska CSE 352 Artificial Intelligence
Propositional Resolution Part 1 Short Review Professor Anita Wasilewska CSE 352 Artificial Intelligence SYNTAX dictionary Literal any propositional VARIABLE a or negation of a variable a, a VAR, Example