Propositional Calculus

Transcription

1 Propositional Calculus Dr. Neil T. Dantam CSCI-498/598 RPM, Colorado School of Mines Spring 2018 Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

2 Calculus? Definition: Calculus A well defined method for mathematical reasoning employing axioms and rules of inference or transformation. A formal system or rewrite system. Examples: Differential calculus Lambda (λ) calculus Propositional calculus (Boolean/Propositional logic) Predicate calculus (First-Order logic) Etymology: from the Latin calx (limestone) + -ulus (dimutive). A pebble or stone used for counting. Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

3 Boolean Logic Outline Boolean Logic Forward and Backward Chaining Horn Clauses Forward Chaining Backward Chaining Satisfiability Conjunctive Normal Form Davis-Putnam-Logemann-Loveland Tools Logic Programming Constraint Solvers Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

4 Boolean Logic Boolean Variables (propositions) Values: B {0, 1} true: 1, T, false: 0, F, Variables: p B p 1,..., p n B n Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

5 Boolean Logic Boolean Operators Basic Unary: f : B B Binary: g : B B B Not 0 = 1 1 = 0 And 0 0 = = = = 1 Or 0 0 = = = = 1 Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

6 Boolean Logic Boolean Operators Extended Xor Implies Biconditional (iff) (a b) (a b) (a b) (a b) ( a b) 0 0 = = = = 0 (a = b) ( a b) (0 = 0) = 1 (0 = 1) = 1 (1 = 0) = 0 (1 = 1) = 1 (a b) (a = b) (b = a) (a b) (a b) ( a b) (0 0) = 1 (0 1) = 0 (1 0) = 0 (1 1) = 1 Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

7 Boolean Logic Example: Boolean Formulae as S-expressions a b a b (a b) ( a b) ( a) = (b c) (and a b) (and a (not b)) (and (or a b) (or (not a) b)) (implies (not a) (or b c)) = a b a b a b b a b c a Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

8 Boolean Logic Exercise: Boolean Formulae as S-expressions (a b) = c (a b) c a b c All these formulae are equivalent Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

9 Boolean Logic N-ary Boolean Operators Infix AND S-exp. α β = (AND α β) α β γ = (AND α β γ) α = (AND α)? = (AND) Infix OR S-exp. α β = (OR α β) α β γ = (OR α β γ) α = (OR α)? = (OR) Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

10 Boolean Logic Identity Element Arithmetic Generally: f (α, identity {}}{ χ ) = α Infix: Multiplication a χ = a χ = 1 S-exp.: ( a 0... a n 1) = ( a 0... a n ) Infix: Addition a + χ = a χ = 0 S-exp.: (+ a 0... a n 0) = (+ a 0... a n ) Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

11 Boolean Logic Identity Element Boolean Algebra Generally: f (α, identity {}}{ χ ) = α Infix: AND a χ = a χ = S-exp.: (AND a 0... a n ) = (AND a 0... a n ) Infix: OR a χ = a χ = S-exp.: (OR a 0... a n ) = (OR a 0... a n ) Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

12 Boolean Logic Identity Element Cancellation AND χ = OR χ = (AND α 0 α 1... α n 1 α n ) ( ) AND α 0 α 1... α n 1 α n (AND α 0 α 1... α n 1 ) ( AND α 0 α ) 1... α n01 (AND) (OR α 0 α 1... α n 1 α n ) ( ) OR α 0 α 1... α n 1 α n (OR α 0 α 1... α n 1 ) ( OR α 0 α ) 1... α n01 (OR) Remove canceled identity terms: base case is identity element Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

13 Boolean Logic Describing Expressions Literal: A single variable or its negation Conjunction: An AND ( ) expression Disjunction: An OR ( ) expression Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

14 Boolean Logic Definitions Literal Definition (Literal) A single variable or its negation. Positive Literal: p Negative Literal: p Examples Counter Examples positive: p i negative: p i p i p j p i p j (p i p j ) Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

15 Boolean Logic Definitions Conjunction Definition (Conjunction) An n-ary AND. True when ALL of its arguments are true. Examples p i = p i p i p j p i p j p k p i (p j p k ) Counter Examples p i p j p i (p j p k ) (p i p j ) p k Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

16 Boolean Logic Definitions Disjunction Definition (Disjunction) An n-ary OR. True when ANY of its arguments are true. Examples p i = p i p i p j p i p j p k p i (p j p k ) Counter Examples p i p j p i p j p k p i (p j p k ) Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

17 Forward and Backward Chaining Outline Boolean Logic Forward and Backward Chaining Horn Clauses Forward Chaining Backward Chaining Satisfiability Conjunctive Normal Form Davis-Putnam-Logemann-Loveland Tools Logic Programming Constraint Solvers Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

18 Forward and Backward Chaining Horn Clauses Horn Clauses Definition: Horn Clause An implication whose premise (body) is a conjunction ( ) of positive literals and whose conclusion (head) is a single positive literals: body {}}{ (b 0 b 1... b n ) = head {}}{ h Equivalently, a disjunction with at most one positive literal: body {}}{ b 0 b 1... b n Horn clause reasoning is more efficient than general Boolean formulae. head {}}{ h Many real-world domains can be expressed with only Horn clauses. Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

19 Forward and Backward Chaining Horn Clauses Examples: Horn Clauses Examples Counterexamples a = = a a b = a = b a = b = a b = b = a a b = (a b) = a b = a = b a b c = a = (b c) Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

20 Forward and Backward Chaining Horn Clauses Example: Engine Troubleshooting Four-cycle Engine Operation Intake Compression Power Exhaust close in. valve spark open ex. valve close ex. valve / open in. valve ignition (fuel compression spark) Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

21 Forward and Backward Chaining Horn Clauses Example: Engine Troubleshooting Troubleshooting Knowledge Base ignition (fuel compression spark) fuel (full-tank clean-carbs) compression (clean-air-filter good-piston-rings) spark (battery-charged good-connection good-plugs) turns-over = battery-charged Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

22 Forward and Backward Chaining Horn Clauses Exercise: Course Prerequisites Calculus I MATH-111 Calculus II MATH-112 Prob. & Stats. MATH-201 Calculus III MATH-213 Discrete Math CSCI-358 Programming Concepts CSCI-261 Data Structures CSCI-262 Human Ctr. Robo. CSCI-473 AI CSCI-404 Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

23 Forward and Backward Chaining Horn Clauses Exercise: Course Prerequisites continued Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

24 Forward and Backward Chaining Forward Chaining Overview: Forward Chaining 1. Start with known propositions, = p 2. Derive new propositions and add to knowledgebase 2.1 For (p 0... p n ) = p h 2.2 When all p 0... p n are known true 2.3 p h must be true 3. Terminate when either we prove the query proposition true or we can derive no more Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

25 Forward and Backward Chaining Forward Chaining Example: Forward Chaining The engine turns over but won t start. I just cleaned the carbs and filled the gas tank. Should I check the spark plugs? ignition (fuel compression spark) fuel (full-tank clean-carbs) compression (clean-air-filter good-piston-rings) spark (battery-charged good-connection good-plugs) turns-over = battery-charged turns-over full-tank clean-carbs ignition ( (fuel compression spark) fuel full-tank ) clean-carbs compression (clean-air-filter good-piston-rings) spark (battery-charged good-connection good-plugs) turns-over = battery-charged turns-over full-tank clean-carbs Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

26 Forward and Backward Chaining Forward Chaining Example: Forward Chaining continued ignition (fuel compression spark) fuel ( full-tank clean-carbs ) compression (clean-air-filter good-piston-rings) spark (battery-charged good-connection good-plugs) turns-over = battery-charged ( ignition fuel ) compression spark fuel ( full-tank clean-carbs ) compression (clean-air-filter good-piston-rings) spark ( ) battery-charged good-connection good-plugs turns-over = battery-charged turns-over full-tank clean-carbs fuel battery-charged turns-over full-tank clean-carbs fuel battery-charged Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

27 Forward and Backward Chaining Forward Chaining Algorithm: Forward Chaining Procedure forward-chain(k,q) 1 W {c K c is a literal}; 2 V ; 3 while empty(w) do 4 let x = pop(w) in 5 if head(c) = q then return ; 6 else if x V then 7 V V {x}; 8 foreach c K where x body(c) do 9 Remove x from body (c); 10 if empty (body(c)) then 11 W push (head(c), W ) Efficiency Notes Index K by variables in its body Track remaining (unproven) terms in body with count Initialize count to length of body Decrement count each time we remove a term When 0, head is true 12 return ; Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

28 Forward and Backward Chaining Forward Chaining Exercise: Forward Chaining I want to take Human-Centered Robotics (CSCI-473). Do I need Calculus III (MATH-213)? Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

29 Forward and Backward Chaining Forward Chaining Exercise: Forward Chaining Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

30 Forward and Backward Chaining Forward Chaining Exercise: Forward Chaining Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

31 Forward and Backward Chaining Backward Chaining Overview: Backward Chaining 1. Start with query proposition q 2. Recursively follow clauses that imply q: 2.1 p 0... p n = q 2.2 l 0... l n = p etc. 3. Terminate recursion when: 3.1 we arrive at a know proposition ( ) 3.2 or there are no more clauses that imply the current proposition ( ) Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

32 Forward and Backward Chaining Backward Chaining Example: Backward Chaining The engine turns over but won t start. I just cleaned the carbs and filled the gas tank. Should I check the spark plugs? ignition (fuel compression spark) fuel (full-tank clean-carbs) compression (clean-air-filter good-piston-rings) good-plugs spark (battery-charged good-connection good-plugs) turns-over = battery-charged spark = good-plugs nil battery-charged good-connection good-plugs = spark nil turns-over = battery-charged = turns-over spark = good-connection cycle Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

33 Forward and Backward Chaining Backward Chaining Algorithm: Backward Chaining Simple Procedure backward-chain(k,q) 1 function visit(v, x) is 2 if x V then return ; // circular definition 3 else // c K, ((head(c) = x) (body(c))) 4 foreach c K where x = head(c) do 5 foreach b body(c) do // b body(c), b 6 if visit (V {x}, b) then return ; 7 return ; 8 return ; 9 return visit (, q); Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

34 Forward and Backward Chaining Backward Chaining Algorithm: Backward Chaining Memoizing Procedure backward-chain(k,q) 1 M ; // Cached set of true propositions 2 function visit(v, x) is 3 if x M then return ; // cached result 4 else if x V then return ; // circular definition 5 else // c K, ((head(c) = x) (body(c))) 6 foreach c K where x = head(c) do 7 foreach b body(c) do // b body(c), b 8 if visit (V {x}, b) then return ; 9 M M {x}; 10 return ; 11 return ; 12 return visit (, q); Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

35 Forward and Backward Chaining Backward Chaining Exercise: Backward Chaining I want to take Human-Centered Robotics (CSCI-473). Do I need Calculus III (MATH-213)? Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

36 Forward and Backward Chaining Backward Chaining Knowledge Base Indexing Procedure forward-chain(k,q) 1 W {c K c is a literal}; 2 V ; 3 while empty(w) do 4 let x = pop(w) in 5 if head(c) = q then return ; 6 else if x V then 7 V V {x}; /* Naive: O(n) to select matching clauses */ 8 foreach c K where x body(c) do 9 Remove x from body (c); 10 if empty (body(c)) then 11 W push (head(c), W ) 12 return ; Procedure backward-chain(k,q) 1 M ; 2 function visit(v, x) is 3 if x M then return ; 4 else if x V then return ; 5 else /* Naive: O(n) to select matching clauses */ 6 foreach c K where x = head(c) do 7 foreach b body(c) do 8 if visit (V {x}, b) then return ; 9 M M {x}; 10 return ; 11 return ; 12 return visit (, q); Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

37 Forward and Backward Chaining Backward Chaining Exercise: Knowledge base Indexing Forward Chaining foreach c K where x body(c) Procedure forward-chain-index(k) Backward Chaining foreach c K where x = head(c) Procedure backward-chain-index(k) Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

38 Satisfiability Outline Boolean Logic Forward and Backward Chaining Horn Clauses Forward Chaining Backward Chaining Satisfiability Conjunctive Normal Form Davis-Putnam-Logemann-Loveland Tools Logic Programming Constraint Solvers Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

39 Satisfiability SAT Problem Given: A Boolean formula: Variables P = p 1... p n Formula φ : B n B Find: Is φ(p) satisfiable? P, (φ(p) = 1) What is P? Solution: Davis-Putnam-Logeman-Loveland (DPLL) Backtracking Search General and efficient reasoning over propositional logic. Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

40 Satisfiability So what? Software verification AI / Robot Planning Combinatorial Design Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

41 Satisfiability Conjunctive Normal Form Conjunctive Normal Form S-Expression Definition (Conjunctive Normal Form) A conjunction of disjunction of literals: (AND (OR l 0,0 l 0,1... l 0,n ) (OR l 1,0 l 1,1... l 1,n )... (OR l n,0 l n,1... l n,n )) where each l i,j is a literal, that is one of p, (NOT p). Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

42 Satisfiability Conjunctive Normal Form Conjunctive Normal Form infix Definition (Conjunctive Normal Form) A conjunction of disjunction of literals Examples p i (AND (OR p i )) p i p j (AND (OR (NOT p i ) p j )) p i (p j p k ) (p i p j ) ( p i p k ) Counter Examples p i (p j p k ) (p i p j ) p i = p j Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

43 Satisfiability Conjunctive Normal Form Conversion to CNF 1. Eliminate : α β (α = β) (β = α) 2. Eliminate = : α = β α β 3. Eliminate : a b (a b) (a b) 4. Move in : ( α) α (α β) ( α β) (α β) ( α β) 5. Distribute over : α (β γ) (α β) (α γ) Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

44 Satisfiability Conjunctive Normal Form Example: CNF Conversions a b Elim. (a = b) (b = a) Elim. = ( a b) ( b a) a = (b c) Elim. = a (b c) move a ( b c) dist. ( a b) ( a c) Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

45 Satisfiability Conjunctive Normal Form Exercise: CNF Conversions 0 (a b) = c (a b) (a c) Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

46 Satisfiability Conjunctive Normal Form Exercise: CNF Conversions 1 (a b) (a c) Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

47 Satisfiability Davis-Putnam-Logemann-Loveland DPLL Outline For φ in conjunctive normal form: Base Case: Recursive Case: 1. If φ has all true clauses ( ), return true. 2. If φ has any false clauses ( ), return false. 1. Propagate values from unit (single-variable) clauses. 2. Choose a branching variable v. 3. Branch (recurse) for v = 1 or v = 0. Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

48 Satisfiability Davis-Putnam-Logemann-Loveland Unit Propagation Procedure unit-propagate(φ) 1 if φ has some unit clause with variable v then 2 φ replace v in φ with value to make unit clause true; 3 return unit-propagate(φ); 4 else 5 return φ; Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

49 Satisfiability Davis-Putnam-Logemann-Loveland Example: Unit Propagation unit {}}{ a (a b) ( b c) a=1 1 1 (1 b) ( b c) simplify ( b c) Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

50 Satisfiability Davis-Putnam-Logemann-Loveland Exercise: Unit Propagation unit {}}{ a (a b c) (a b) b (b c) (a b c) (b c) c a ( a b) ( a b) Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

51 Satisfiability Davis-Putnam-Logemann-Loveland DPLL Algorithm Procedure DPLL(φ) 1 φ unit-propagate(φ); 2 if φ = true then 3 return true; 4 else if φ = false then 5 return false; 6 else // Recursive case 7 v choose-variable(φ ); 8 if DPLL(φ v) then 9 return true; 10 else 11 return DPLL(φ v); Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

52 Satisfiability Davis-Putnam-Logemann-Loveland Example 0: DPLL (a b) ( a b) choose a = 1 1 (1 b) ( 1 0 b) propagate b = 1 1 SAT Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

53 Satisfiability Davis-Putnam-Logemann-Loveland Example 1: DPLL (a b) ( a b) ( a b) 1 (1 b) ( 1 0 b) ( 1 0 b) propagate b = 1 choose a = choose a = 0 (0 b) 1 ( 0 b) 1 ( 0 b) propagate b = SAT Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

54 Satisfiability Davis-Putnam-Logemann-Loveland Exercise 0: DPLL Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

55 Satisfiability Davis-Putnam-Logemann-Loveland Exercise 1: DPLL Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

56 Satisfiability Davis-Putnam-Logemann-Loveland Exercise 2: DPLL Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

57 Satisfiability Davis-Putnam-Logemann-Loveland Exercise 3: DPLL Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

58 Satisfiability Davis-Putnam-Logemann-Loveland Exercise 4: DPLL Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

59 Tools Outline Boolean Logic Forward and Backward Chaining Horn Clauses Forward Chaining Backward Chaining Satisfiability Conjunctive Normal Form Davis-Putnam-Logemann-Loveland Tools Logic Programming Constraint Solvers Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

60 Tools Logic Programming and Constraint Solvers Statements: Logic Programming Logical Expressions Query Execution: Logical Inference Output: Query true/false Constraint Solvers Algorithm: DPLL + heuristics Input: Set of variables Constraint equations (assertions) Output: Satisfying variable assignment Different view, similar operation and capabilities Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

61 Tools Logic Programming Prolog Statements are Horn clauses Horn: body = head Prolog: head :- body Example: Horn: (a b) = c Prolog: c :- a, b. Evaluation: Query:?x Backward Chaining Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

62 Tools Logic Programming Answer-Set Programming Horn Clauses: Horn: (a b) = c ASP: c :- a, b. Choice Rule: Math: p = (s t) ASP: {s,t} :- p. Constraint: Math: (s t) = = (s t) = s t ASP: :- s, not t. Evaluation: DPLL Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

63 Tools Constraint Solvers Satisfiability Solvers Math DIMACS Output (a b) (a c) c Problem D e f i n i t i o n ( p ) : c problem type ( c n f ) c v a r i a b l e count ( 3 ) c c l a u s e count ( 2 ) p cn f s SATISFIABLE v Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64

64 Tools Constraint Solvers Satisfiability Modulo Theories Math (a b) (a c) (SMT also handles non-boolean types.) SMTLib ( d e c l a r e fun a ( ) Bool ) ( d e c l a r e fun b ( ) Bool ) ( d e c l a r e fun c ( ) Bool ) ( a s s e r t ( or a b ) ) ( a s s e r t ( or a ( not c ) ) ) ( check sat ) ( get v a l u e ( a b c ) ) Output sat ( ( a f a l s e ) ( b true ) ( c f a l s e ) ) Dantam (Mines CSCI, RPM) Propositional Calculus Spring / 64