Tableaux.dfw: Semantic Tableaux for Propositional Classical Logic with Derive November, 2009

Transcription

1 Tableaux.dfw: Semantic Tableaux fr Prpsitinal Classical Lgic with Derive Nvember, 2009 Gabriel Aguilera Venegas Jsé Luis Galán García María Ángeles Galán García Antni Gálvez Galian Ylanda Padilla Dmínguez Pedr Rdríguez Ciels Department f Applied Mathematic University f Malaga (Spain) The fllwing functins have been develped in this utility/dem dfw file t use Derive as a Autmated Therem Prver fr Prpsitinal Classical Lgic by means f Semantic Tableaux algrithm. SemanticTableaux SemanticTableaux(f) t check the satsfiability f the frmula f using Semantic. If f is unsatisfiable then [] is return. If f is satisfiable then a list f literal crrespnding t a mdel f A is returned. InferenceSemanticTableaux(h,f) t check if the frmula f (cnclusin) can be deduced frm the list f frmulae h (hypthesis) using Semantic. In case that the inference were nn-valid a cuntermdel is returned. TAUTSemanticTableaux(f) t check if the frmula f is a valid frmula using Semantic. In case that the frmula were nn-valid a cuntermdel is returned. SATSemanticTableaux(f) t check if the frmula f is a satisfiable frmula using Semantic.

2 Semantic Tableaux Using SemanticTableaux Syntax: Example: SemanticTableaux(f) SemanticTableaux(" (p p)") return a mdel fr the frmula (p p) using Semantic, empty list mdel fr unsatisfiable frmulas. #1: SemanticTableaux( (p p)) #2: [] Example: SemanticTableaux("p q") return a mdel fr the frmula p q using Semantic, empty list mdel fr unsatisfiable frmulas. #3: SemanticTableaux(p q) #4: [1] Checking the validity f a inference using Semantic Tableaux Syntax: InferenceSemanticTableaux(h,f) Example: InferenceSemanticTableaux(["p q", "p r", " r"]," q") t check the validity f the inference {p q, p r, r} = q using Semantic

3 #5: InferenceSemanticTableaux([p q, p r, r], q) #6: VALID INFERENCE Example: InferenceSemanticTableaux(["p q", "p r", " r"],"q") t check the validity f the inference {p q, p r, r} = q using Semantic #7: InferenceSemanticTableaux([p q, p r, r], q) #8: NON VALID INFERENCE. COUNTERMODEL: I(p) = 0, I(q) = 0, I(r) = 1 Checking the validity f a frmula using Semantic Tableaux Syntax: TAUTSemanticTableaux(h,f) Example: TAUTSemanticTableaux("(p IMP q) r (q imp p)") t check the validity f the frmula (p q) (q p) using Semantic #9: TAUTSemanticTableaux((p IMP q) r (q imp p)) #10: VALID FORMULA Example: SATSemanticTableaux("(p IMP q) AND (q imp p)") t check the validity f the frmula (p q) (q p) using Semantic #11: TAUTSemanticTableaux((p IMP q) AND (q imp p)) #12: NON VALID FORMULA. COUNTERMODEL: I(q) = 0, I(p) = 1

4 Checking the satisfiability f a frmula using Semantic Tableaux Syntax: Example: SATSemanticTableaux(f) SATSemanticTableaux("(p imp q) and (q imp p)") t check the satisfiability f the frmula (p q) (q p) using SemanticTableux methd #13: SATSemanticTableaux((p imp q) and (q imp p)) #14: SATISFIABLE FRORMULA. MODEL: I(q) = 0, I(p) = 0 Example: SATSemanticTableaux("(p r q) IFF (nt p and NOT q)") t check the satisfiability f the frmula (p q) ( p q) using SemanticTableux methd #15: SATSemanticTableaux((p r q) IFF (nt p and NOT q)) #16: UNSATISFIABLE Examples with Graphical Apprach #17: SemanticTableaux( (p p), true) ( (p p)) [] px[p] #18: []

5 #19: SemanticTableaux(p q, true) (p q) [] q [q] #20: [1] #21: InferenceSemanticTableaux([p q, p r, r], q, true) (p q) [] (p r) [] r [ r] q [q, r] (p q) [q, r] (q p) [q, r] p [ p,q, r] rx[q, r] p [ p,q, r] q [ p,q, r] qx[ p,q, r] px[ p,q, r] qx[ p,q, r] px[ p,q, r] #22: VALID INFERENCE #23: InferenceSemanticTableaux([p r, p], r, true) (p r) [] p [ p] r [r, p] p [r, p] r [r, p] #24: NON VALID INFERENCE. COUNTERMODEL: I(r) = 1, I(p) = 0 #25: TAUTSemanticTableaux((p IMP q) r (q imp p), true) ( ((p q) (q p))) [] ( (p q)) [] ( (q p)) [] q [ q,p] qx[ q,p] #26: VALID FORMULA

6 #27: TAUTSemanticTableaux((p IMP q) AND (q imp p), true) ( ((p q) (q p))) [] ( (p q)) [] ( (q p)) [] q [ q,p] q [q] p [ p,q] #28: NON VALID FORMULA. COUNTERMODEL: I(q) = 0, I(p) = 1 #29: SATSemanticTableaux((p imp q) and (q imp p), true) ((p q) (q p)) [] (p q) [] (q p) [] p [ p] q [q] q [ q, p] px[ p] qx[q] p [p,q] #30: SATISFIABLE FORMULA. MODEL: I(q) = 0, I(p) = 0