Model Checking for Propositions CS477 Formal Software Dev Methods

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1 S477 Formal Software Dev Methods Elsa L Gunter 2112 S, UIU egunter@illinois.edu Slides based in part on previous lectures by Mahesh Vishwanathan, and by Gul gha January 31, 2018 Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17

2 Note: general algorithm to answer the last can be used to answer the second and vice versa. Note: general algorithm to answer the last can be used to answer the second and vice versa. Difficulty: nswering if P is satisfiable is NP-complete Note: general algorithm to answer the last can be used to answer the second and vice versa. Difficulty: nswering if P is satisfiable is NP-complete lgorithms exist with good performance in general practice Note: general algorithm to answer the last can be used to answer the second and vice versa. Difficulty: nswering if P is satisfiable is NP-complete lgorithms exist with good performance in general practice DDs are one such Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17

3 Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 Internal nodes have exactly two out edges Internal nodes have exactly two out edges Left edges labeled false and right edges labeled true. Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 Internal nodes have exactly two out edges Left edges labeled false and right edges labeled true. Think 0 and 1 Internal nodes have exactly two out edges Left edges labeled false and right edges labeled true. Think 0 and 1 For each path (branch) in the tree, each atomic proposition may label at most one vertex of that path. Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17

4 Is a model (true, the tree accepts the valuation) Is a model (true, the tree accepts the valuation) Or not a model (false, the tree rejects the valuation) Is a model (true, the tree accepts the valuation) Or not a model (false, the tree rejects the valuation) Each valuation matches exactly one branch

5 Is a model (true, the tree accepts the valuation) Or not a model (false, the tree rejects the valuation) Each valuation matches exactly one branch More than one valuation may (will) match a given branch false false true true false ( ) ( ) true true false true false true false true false true true Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 Different Variable Ordering - Different Tree ( ) ( ) Many Logically Equivalent Trees ( ) ( ) false false true true true true true true true false true false false false false true false true true false true true false true false false true Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 lternate Syntax for Propositional Logic Still have constants {T, F} Still have countable set P of propositional a.k.a. atomic propositions Only one ternary connective: the conditional if then else First argument only a variable Second and third arguments propositions Example if then if then if then T else F else F else T Represents the last tree above Semantics for onditional Propositional Logic Define when a valuation v satisfies a conditional proposition by v = T v = F v = if then P t else P f iff v() = true and v = P t or v() = false and v = P f let v = { true, true, true} v = if then if then if then T else F else F else T Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17

6 Semantics for onditional Propositional Logic Define when a valuation v satisfies a conditional proposition by v = T v = F v = if then P t else P f iff v() = true and v = P t or v() = false and v = P f Semantics for onditional Propositional Logic Define when a valuation v satisfies a conditional proposition by v = T v = F v = if then P t else P f iff v() = true and v = P t or v() = false and v = P f let v = { true, true, true} let v = { true, true, true} v = if then if then if then T else F else F else T v() = true and v = if then if then T else F else F v = if then if then if then T else F else F else T v() = true and v = if then if then T else F else F v() = true and v = if then T else F Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 Semantics for onditional Propositional Logic Translating Original Propositions into if then else Define when a valuation v satisfies a conditional proposition by v = T v = F v = if then P t else P f iff v() = true and v = P t or v() = false and v = P f let v = { true, true, true} v = if then if then if then T else F else F else T v() = true and v = if then if then T else F else F v() = true and v = if then T else F v() = true andv = T Start with proposition P 0 with v 1,... v n P[c/v] is the proposition resulting from replacing all occurrences of variable v with constant c Let P be the result of evaluating every subexpression of P containing no Let P 1 = if v 1 then P 0 [T/v 1 ] else P 0 [F/v 1 ] Let P i = if v i then P i 1 [T/v i ] else P i 1 [F/v i ] P n is logically equivalent to P, but only uses if then else. Valuation satisfies P if and only if it satisfies P n P n depends on the order of v 1,... v n P n directly corresponds to a binary decision tree Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 P = ( ) ( ), {,, }, < < P 0 = ( ) ( ) P = ( ) ( ), {,, }, < < P 0 = ( ) ( ) P 1 = if then (T ) ( ) else (F ) ( ) Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17

7 P = ( ) ( ), {,, }, < < P 0 = ( ) ( ) P 1 = if then (T ) ( ) else (F ) ( ) P 2 = if then (if then (T T) ( ) else (F T) ( )) else (if then (T F) ( ) else (F F) ( )) P = ( ) ( ), {,, }, < < P 0 = ( ) ( ) P 1 = if then (T ) ( ) else (F ) ( ) P 2 = if then (if then (T T) ( ) else (F T) ( )) else (if then (T F) ( ) else (F F) ( )) P 2 = if then (if then T ( ) else F ( )) else (if then F ( ) else F ( )) Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 P = ( ) ( ), {,, }, < < P 0 = ( ) ( ) P 1 = if then (T ) ( ) else (F ) ( ) P 2 = if then (if then (T T) ( ) else (F T) ( )) else (if then (T F) ( ) else (F F) ( )) P 2 = if then (if then T ( ) else F ( )) else (if then F ( ) else F ( )) P 3 = if then (if then (if then T ( T) else F ( T)) else (if then F ( T) else F ( T))) else (if then (if then T ( F) else F ( F)) else (if then F ( F) else F ( F))) P = ( ) ( ), {,, }, < < P 0 = ( ) ( ) P 1 = if then (T ) ( ) else (F ) ( ) P 2 = if then (if then (T T) ( ) else (F T) ( )) else (if then (T F) ( ) else (F F) ( )) P 2 = if then (if then T ( ) else F ( )) else (if then F ( ) else F ( )) P 3 = if then (if then (if then T ( T) else F ( T)) else (if then F ( T) else F ( T))) else (if then (if then T ( F) else F ( F)) else (if then F ( F) else F ( F))) P 3 = if then (if then (if then T else F) else (if then F else F)) else (if then (if then T else T) else (if then T else T)) Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 Example, cont. inary Decision Diagram P 3 = if then (if then (if then T else F) else (if then F else F)) else (if then (if then T else T) else (if then T else T)) P 3 corresponds to second binary decision tree given earlier ny proposition in strict if then else form corresponds directly to a binary decision tree that accepts exactly the valuations that satisfy (model) the proposition. inary decision trees may contain (much) redundancy inary Decision Diagram (DD): Replace trees by (rooted) directed acyclic graphs Require all other conditions still hold Generalization of binary decision trees llows for sharing of common subtrees. ccepts / rejects valuations as with binary decision trees. Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17

8 Example Reduced Ordered inary Decision Diagrams Problem: given proposition may correspond to many different DDs How to create a (compact) canonical DD for a proposition such that two different propositions are logically equivalent if and only if they have the same (isomorphic) canonical DD Start: order propositional v i < v j. ryant showed you can obtain such a canonical DD by requiring Variables should appear in order on each path for root to leaf No distinct duplicate (isomorphic) subtrees (including leaves) Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 chieving anonical Form Example Start with an Ordered DD (all edges in correct order) Repeat following until none apply Remove duplicate leaves: Eliminate all but one leaf with a given label and redirect all edges to the eliminated leaves to the remaining one Remove duplicate nonterminals: If node n and m have the same variable label, their left edges point to the same node and their right edges point to the same node, remove one and redirect edges that pointed to it to the other Remove redundant tests: If both out edges of node n point to node m, eliminate n and redirect all edges coming into n to m ryant gave procedure to do the above that terminates in linear time Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17 Elsa L Gunter S477 Formal Software Dev Methods January 31, / 17