AUTOMATED REASONING. Agostino Dovier. Udine, November Università di Udine CLPLAB
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1 AUTOMATED REASONING Agostino Dovier Università di Udine CLPLAB Udine, Novemer 2017 AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
2 Semntics of Logic Progrms AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
3 UNIVERSES AND INTERPRETATIONS p(). p(). A progrm (or in generl, first-order theory) P is uilt from list of symols. In this cse constnts nd, nd one unry predicte symol p We would like to ssign mening (interprettion) to these symols on universe of ojects. AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
4 UNIVERSES AND INTERPRETATIONS p(). p(). A progrm (or in generl, first-order theory) P is uilt from list of symols. In this cse constnts nd, nd one unry predicte symol p We would like to ssign mening (interprettion) to these symols on universe of ojects. AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
5 UNIVERSES AND INTERPRETATIONS p(). p(). A progrm (or in generl, first-order theory) P is uilt from list of symols. In this cse constnts nd, nd one unry predicte symol p We would like to ssign mening (interprettion) to these symols on universe of ojects. AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
6 UNIVERSES AND INTERPRETATIONS p(). p(). A progrm (or in generl, first-order theory) P is uilt from list of symols. In this cse constnts nd, nd one unry predicte symol p We would like to ssign mening (interprettion) to these symols on universe of ojects. AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
7 UNIVERSES AND INTERPRETATIONS Different interprettions for constnt symols induce different interprettions for first-order formuls (with equlity) X (X X ) = AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
8 UNIVERSES AND INTERPRETATIONS p(). p(s(x)) p(x). q(g(x,y)) p(x),p(y). Constnt nd function symols s/1 nd g/2. Function symols must e interpreted s functions (g(x, y) = z mens (x, y, z) G, where G is the interprettion viewed s set of triples) AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
9 UNIVERSES AND INTERPRETATIONS p(). p(s(x)) p(x). q(g(x,y)) p(x),p(y). Constnt nd function symols s/1 nd g/2. Function symols must e interpreted s functions (g(x, y) = z mens (x, y, z) G, where G is the interprettion viewed s set of triples) AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
10 UNIVERSES AND INTERPRETATIONS p(). p(s(x)) p(x). q(g(x,y)) p(x),p(y). Constnt nd function symols s/1 nd g/2. Function symols must e interpreted s functions (g(x, y) = z mens (x, y, z) G, where G is the interprettion viewed s set of triples) s AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
11 UNIVERSES AND INTERPRETATIONS p(). p(s(x)) p(x). q(g(x,y)) p(x),p(y). Constnt nd function symols s/1 nd g/2. Function symols must e interpreted s functions (g(x, y) = z mens (x, y, z) G, where G is the interprettion viewed s set of triples) s AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
12 UNIVERSES AND INTERPRETATIONS p(). p(s(x)) p(x). q(g(x,y)) p(x),p(y). Constnt nd function symols s/1 nd g/2. Function symols must e interpreted s functions (g(x, y) = z mens (x, y, z) G, where G is the interprettion viewed s set of triples) s AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
13 UNIVERSES AND INTERPRETATIONS p(). p(s(x)) p(x). q(g(x,y)) p(x),p(y). Constnt nd function symols s/1 nd g/2. Function symols must e interpreted s functions (g(x, y) = z mens (x, y, z) G, where G is the interprettion viewed s set of triples) s AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
14 UNIVERSES AND INTERPRETATIONS p(). p(s(x)) p(x). q(g(x,y)) p(x),p(y). Constnt nd function symols s/1 nd g/2. Function symols must e interpreted s functions (g(x, y) = z mens (x, y, z) G, where G is the interprettion viewed s set of triples) g AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
15 UNIVERSES AND INTERPRETATIONS Predicte symols (e.g. p/1, q/n) should e interpreted (s 1-ry, n-ry reltions). AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
16 UNIVERSES AND INTERPRETATIONS Predicte symols (e.g. p/1, q/n) should e interpreted (s 1-ry, n-ry reltions). AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
17 UNIVERSES AND INTERPRETATIONS Predicte symols (e.g. p/1, q/n) should e interpreted (s 1-ry, n-ry reltions). AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
18 UNIVERSES AND INTERPRETATIONS Predicte symols (e.g. p/1, q/n) should e interpreted (s 1-ry, n-ry reltions). AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
19 UNIVERSES AND INTERPRETATIONS Different interprettions for predicte symols induce different interprettions for first-order formuls (with equlity) X p(x) X p(x) X ( p(x)) X p(x) X Y (X Y p(x) p(y )) X Y (X Y p(x) p(y )) AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
20 MODELS Some of the vrious interprettions of constnt, function, nd predicte symols cn e models of the progrm (or of the theory) P. p(). q(). r(x) p(x). Let us denote =tringle nd =squre. Let use denote with P, Q, R the interprettions of the predicte symols p, q, r. An interprettion tht stisfies the logicl mening of ll the formuls of P is model. P = {}, Q = {}, R = {, } is model. P = {, }, Q = {}, R = {} is NOT model. An tom q(t 1,..., t n ) is logicl consequence of progrm/theory P if (t 1,..., t n ) Q in ll (interprettions tht re) models of P. We sy tht P = q(t 1,..., t n ). AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
21 MODELS Some of the vrious interprettions of constnt, function, nd predicte symols cn e models of the progrm (or of the theory) P. p(). q(). r(x) p(x). Let us denote =tringle nd =squre. Let use denote with P, Q, R the interprettions of the predicte symols p, q, r. An interprettion tht stisfies the logicl mening of ll the formuls of P is model. P = {}, Q = {}, R = {, } is model. P = {, }, Q = {}, R = {} is NOT model. An tom q(t 1,..., t n ) is logicl consequence of progrm/theory P if (t 1,..., t n ) Q in ll (interprettions tht re) models of P. We sy tht P = q(t 1,..., t n ). AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
22 MODELS Some of the vrious interprettions of constnt, function, nd predicte symols cn e models of the progrm (or of the theory) P. p(). q(). r(x) p(x). Let us denote =tringle nd =squre. Let use denote with P, Q, R the interprettions of the predicte symols p, q, r. An interprettion tht stisfies the logicl mening of ll the formuls of P is model. P = {}, Q = {}, R = {, } is model. P = {, }, Q = {}, R = {} is NOT model. An tom q(t 1,..., t n ) is logicl consequence of progrm/theory P if (t 1,..., t n ) Q in ll (interprettions tht re) models of P. We sy tht P = q(t 1,..., t n ). AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
23 MODELS The set of logicl consequences seems to e wht we expect from the progrm. P = q(t 1,..., t n ) An tom q(t 1,..., t n ) is logicl consequence of progrm/theory P if (t 1,..., t n ) Q in ll (interprettions tht re) models of P. The questions re: Does it exist lwys? If yes, how to compute it? AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
24 HERBRAND INTERPRETATIONS Let us consider the set of ll ground terms tht cn e uilt with constnt nd function symols in progrm P. This set cn e used s Universe for interprettions (the Herrnd Universe or H P ). Ground terms re interpreted s themselves 0 s(0) s(s(0)) s(s(s(0))) 0 s(0) s(s(0)) s(s(s(0))) AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
25 HERBRAND MODELS Interprettions on the Herrnd Universe cn e (or not) models (Herrnd models) p(). q(). r(x) p(x). Now, = nd =. Let us denote with P, Q, R the interprettions of the predicte symols p, q, r. 1 P = {}, Q = {}, R = {, } is model. 2 P = {, }, Q = {}, R = {} is NOT model. Herrnd interprettions nd models cn e represented uniquely y set of toms: 1 {p(), q(), r(), r()} (model) 2 {p(), p(), q(), r()} (not model) AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
26 HERBRAND MODELS Interprettions on the Herrnd Universe cn e (or not) models (Herrnd models) p(). q(). r(x) p(x). Now, = nd =. Let us denote with P, Q, R the interprettions of the predicte symols p, q, r. 1 P = {}, Q = {}, R = {, } is model. 2 P = {, }, Q = {}, R = {} is NOT model. Herrnd interprettions nd models cn e represented uniquely y set of toms: 1 {p(), q(), r(), r()} (model) 2 {p(), p(), q(), r()} (not model) AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
27 HERBRAND MODELS Interprettions on the Herrnd Universe cn e (or not) models (Herrnd models) p(). q(). r(x) p(x). Now, = nd =. Let us denote with P, Q, R the interprettions of the predicte symols p, q, r. 1 P = {}, Q = {}, R = {, } is model. 2 P = {, }, Q = {}, R = {} is NOT model. Herrnd interprettions nd models cn e represented uniquely y set of toms: 1 {p(), q(), r(), r()} (model) 2 {p(), p(), q(), r()} (not model) AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
28 A LATTICE OF INTERPRETATIONS Given progrm P, the corresponding Herrnd Universe H P is determined uniquely p(). q(). r(x) p(x). nt(0). nt(s(x)) nt(x). 0 0 s(0) s(s(0)) s(s(s(0))) s(0) s(s(0)) s(s(s(0))) Let B P = {p(t 1,..., t n ) : p is predicte symol in P, t i s re ground terms mde with constnt nd function symols in P } B P is clled the Herrnd se. Any suset of B P uniquely determines n Herrnd Interprettion (some of them cn e models) ( (B P ), ) forms complete lttice AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
29 THE FUNDAMENTAL THEOREM A cluse is formul of the form X(A 0 A n ) where A i s re positive or negtive literls uilt on the vriles X. Oserve tht A 0 A 1 A n is A 0 A 1 A n. The notions given for progrms in the previous slides pply to conjunction of cluses s well. If T is conjunction of cluses: H T denotes the Herrnd Universe nd B T the Herrnd Bse. THEOREM Let T e conjunction of cluses. Then T hs model if nd only if T hs n Herrnd model. THEOREM Let T e conjunction of cluses nd A B T e ground tom. Then T = A if nd only if A is true in ll Herrnd models of A. AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
30 THE FUNDAMENTAL THEOREM A cluse is formul of the form X(A 0 A n ) where A i s re positive or negtive literls uilt on the vriles X. Oserve tht A 0 A 1 A n is A 0 A 1 A n. The notions given for progrms in the previous slides pply to conjunction of cluses s well. If T is conjunction of cluses: H T denotes the Herrnd Universe nd B T the Herrnd Bse. THEOREM Let T e conjunction of cluses. Then T hs model if nd only if T hs n Herrnd model. THEOREM Let T e conjunction of cluses nd A B T e ground tom. Then T = A if nd only if A is true in ll Herrnd models of A. AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
31 NON HORN CLAUSES EXAMPLE Let T = p() p(). There re 4 Herrnd interprettions: {p()} {p(), p()} {p()} 3 of them re models. There is no A such tht T = A. EXAMPLE Let T e (p() p()) ( p() p()) (p() p()) ( p() p()). Sme 4 interprettions s ove. No one of them is model. AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
32 NON HORN CLAUSES EXAMPLE Let T = p() p(). There re 4 Herrnd interprettions: {p()} {p(), p()} {p()} 3 of them re models. There is no A such tht T = A. EXAMPLE Let T e (p() p()) ( p() p()) (p() p()) ( p() p()). Sme 4 interprettions s ove. No one of them is model. AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
33 THE FUNDAMENTAL THEOREM (2) Definite cluses hve exctly one positive literls. The rule: p(a) q(a, B), r(b). is the cluse A B (p(a) q(a, B) r(b)) Progrms re conjunctions of definite cluses. THEOREM Let P e (definite cluse) progrm. Then P dmits (unique) minimum Herrnd model M P (M P is the semntics of P). (i.e., if I is Herrnd model of P, then M P I). COROLLARY Let P e (definite cluse) progrm nd A B P e ground tom. Then P = A if nd only if A M P. AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
34 THE FUNDAMENTAL THEOREM (2) Definite cluses hve exctly one positive literls. The rule: p(a) q(a, B), r(b). is the cluse A B (p(a) q(a, B) r(b)) Progrms re conjunctions of definite cluses. THEOREM Let P e (definite cluse) progrm. Then P dmits (unique) minimum Herrnd model M P (M P is the semntics of P). (i.e., if I is Herrnd model of P, then M P I). COROLLARY Let P e (definite cluse) progrm nd A B P e ground tom. Then P = A if nd only if A M P. AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, NOVEMBER / 15
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