"On Herbrand's Theorem for Hybrid Logic"

Transcript

1. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic “On Herbrand’s Theorem for Hybrid Logic” Raquel Oliveira Federal University of Rio Grande do Norte Department of Informatics and Applied Mathematics DIM0809 – Logic 28 June 1/37
2. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic The Paper Diana Costa and Manuel A. Martins CIDMA - Center for R&D in Mathematics and Applications Dep. of Mathematics, University of Aveiro fdianafcosta,martinsg@ua.pt Jo˜ ao Marcos Department of Informatics and Applied Mathematics Federal University of Rio Grande do Norte jmarcos@dimap.ufrn.br 2/37
3. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Summary 1 Herbrand Theorem for first order logic 2 Modal Logic 3 Hybrid Logic 4 Algebraic Strong Priorean Logic 5 Conclusion 3/37
4. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Herbrand for first order logic Reduction of (sentences of) first-order logic to (set of sentences in) propositional logic. Original version X has a first-order proof if and only if some Xi is a tautology, where Xi comes from a sequence of quantifier-free formulas X1, X2, X3, ... associated with X. 4/37
5. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Herbrand for first order logic Reduction of (sentences of) first-order logic to (set of sentences in) propositional logic. Original version X has a first-order proof if and only if some Xi is a tautology, where Xi comes from a sequence of quantifier-free formulas X1, X2, X3, ... associated with X. Many versions Shawn Hedman Given p in SNF we find a (possibly infinite) set E(p) of sentences of propositional logic such that p is satisfiable if and only if E(p) is satisfiable. 4/37
6. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Modal Logic Expansion of propositional logic with the unary operators: ♦ and . The two operators are inter-definable by ϕ ::= ¬♦¬ϕ Useful for reasoning about relational structures. Example: Trees Knowledge/belief; obligation; possibility/necessity; time Default interpretation: Diamond: it is possible/I belive. Box: It is necessary/I know 5/37
7. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Hybrid logic Extension of propositional modal logic Rich underlying syntax new class of atomic formulas called nominals using a new operator, @, called hybrid component 6/37
8. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Hybrid Signature A hybrid similary type L is a tuple Prop, Nom where: Prop : is the denumerable set of propositional symbols symbol used to represent an element in Prop: p Nom : is the denumerable set of nominal symbols symbol used to represent an element in Nom: i ! The sets Prop, Nom are disjoint. Grammar of Form@ (L) - Hybrid formulas over L ϕ ::= i|p|¬ϕ|ϕ1 ∧ ϕ2|♦ϕ|@i ϕ The connectives ∨, →, and are defined as usual 7/37
9. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Hybrid Structure A hybrid structure M over L is a tuple W , R, N, V where: W : non-empty set called domain R : accessibility relation where R ⊆ W × W N : hybrid nomination where N : Nom → W V : hybrid valuation where V : Prop → Pow(W ) M is said to be a structure based on a frame F where F is defined by: F ::= W , R F is called a frame underlying M. 8/37
10. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Hybrid Satisfaction 1 M, w i iff w = N(i) 2 M, w p iff w ∈ V (p); 3 M, w ¬ϕ iff not M, w ϕ 4 M, w ϕ1 ∧ ϕ2 iff M, w ϕ1 and M, w ϕ2 5 M, w ♦ϕ iff ∃w ∈ W (wRw and M, w ϕ) 6 M, w @i ϕ iff M, w ϕ, where w = N(i) 9/37
11. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Hybrid Satisfaction 1 M, w i iff w = N(i) 2 M, w p iff w ∈ V (p); 3 M, w ¬ϕ iff not M, w ϕ 4 M, w ϕ1 ∧ ϕ2 iff M, w ϕ1 and M, w ϕ2 5 M, w ♦ϕ iff ∃w ∈ W (wRw and M, w ϕ) 6 M, w @i ϕ iff M, w ϕ, where w = N(i) M, w ϕ ϕ is satisfied in M at w M ϕ ϕ is satisfied in all states in a structure M F ϕ ϕ is valid on F ϕ ϕ is valid 9/37
12. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic More definitions At(L) : is the set of atomic satisfaction statements over L At(L) = {@i p, @i0 ♦i1} BCAt(L) : is the set of all (finite) boolean combinations of At(L) v : is the L-truth assignment given by the mapping At(L) → {T, F}. v : is the extension of an L-truth assignment v given by the mapping v : BCAt(L) → {T, F} thought the truth-functional interpretation of the propositional connectives. 10/37
13. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic More definitions At(L) : is the set of atomic satisfaction statements over L At(L) = {@i p, @i0 ♦i1} BCAt(L) : is the set of all (finite) boolean combinations of At(L) v : is the L-truth assignment given by the mapping At(L) → {T, F}. v : is the extension of an L-truth assignment v given by the mapping v : BCAt(L) → {T, F} thought the truth-functional interpretation of the propositional connectives. As the extension of v, v, is unique, we are going to use the same symbol: v. 10/37
14. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Equality axioms over nominals The set of all following formulas is denoted by Eq(L): Reflexivity : @i i, for i ∈ Nom Symmetry : @i0 i1 → @i1 i0, for some i0, i1 ∈ Nom Nom : (@i0 ϕ ∧ @i0 i1) → @i1 ϕ, for some i0, i1 ∈ Nom Bridge : (@i0 ♦i1 ∧ @i1 i2) → @i0 ♦i2, for some i0, i1, i2 ∈ Nom 11/37
15. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Equality axioms over nominals The set of all following formulas is denoted by Eq(L): Reflexivity : @i i, for i ∈ Nom Symmetry : @i0 i1 → @i1 i0, for some i0, i1 ∈ Nom Nom : (@i0 ϕ ∧ @i0 i1) → @i1 ϕ, for some i0, i1 ∈ Nom Bridge : (@i0 ♦i1 ∧ @i1 i2) → @i0 ♦i2, for some i0, i1, i2 ∈ Nom Following from Nom and Symmetry: Transitivity : @i0 i1 → (@i1 i2 → @i0 i2)), for some i0, i1, i2 ∈ Nom 11/37
16. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic More definitions Satisfiable A set of Φ of hybrid formulas is satisfiable iff there is a model M and a world w ∈ W such that M, w ϕ for all ϕ ∈ Φ. Propositionnaly satisfiable Let Φ ⊆ BCAt(L). Φ is propositionally satisfiable if there is an L-truth assignment that simultaneously satisfies every member of Φ. Φ is propositionally unsatisfiable if there is no such L-truth assignment. 12/37
17. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic (First) Herbrand-like Herbrand-like Let Φ be a set of satisfaction statements such that Eq(L) ⊆ Φ. Then Φ is propositionally unsatisfiable iff Φ is unsatisfiable. Strategy of the proof: ⇒ If Φ is propositionally unsatisfiable then Φ is unsatisfiable. ⇐ If Φ is unsatisfiable then Φ is propositionally unsatisfiable 13/37
18. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Investigating first Herbrand-like theorem ⇒ Theorem 1 Let Φ ⊆ BCAt(L). If Φ is propositionally unsatisfiable then Φ is unsatisfiable. Proof (contrapositive). Suppose Φ is satisfiable Hence, Φ is propositionally satisfiable. 14/37
19. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Investigating first Herbrand-like theorem ⇒ Theorem 1 Let Φ ⊆ BCAt(L). If Φ is propositionally unsatisfiable then Φ is unsatisfiable. Proof (contrapositive). Suppose Φ is satisfiable then there is a model M and a world w ∈ W , such that M, w ϕ for all ϕ ∈ Φ. Define a L-truth assignment over M: vM : At(L) → {T, F} where: vM(ψ) = T iff M, w ψ and vM(ψ) = F iff M, w ψ vM(ϕ) = T iff M, w ϕ follows immediatly by induction over the complexity of ϕ ∈ BCAt(M). ... We have that vM(ϕ) = T for any ϕ ∈ Φ. Hence, Φ is propositionally satisfiable. 14/37
20. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 1 Let L = (p, q), (i, j) and Φ = {@i p ∨ @i q,@j ¬q,@i ♦j, @i j} N(i) N(j)
21. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 1 Let L = (p, q), (i, j) and Φ = {@i p ∨ @i q,@j ¬q,@i ♦j, @i j} N(i) N(j) p ∨ q
22. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 1 Let L = (p, q), (i, j) and Φ = {@i p ∨ @i q, @j ¬q,@i ♦j, @i j} N(i) N(j) p ∨ q ¬q
23. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 1 Let L = (p, q), (i, j) and Φ = {@i p ∨ @i q, @j ¬q, @i ♦j, @i j} N(i) N(j) p ∨ q ¬q 15/37
24. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 1 Let L = (p, q), (i, j) and Φ = { @i p ∨ @i j,@j ¬q,@i ♦j, @i j} N(i) (p ∨ q), ¬q 16/37
25. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 1 Let L = (p, q), (i, j) and Φ = {@i p ∨ @i j,@j ¬q,@i ♦j, @i j} The set Φ is satisfiable, as there is a model M = (W , R, N, V ), such that W = {w}, R = {(i, i)}, N(i) = N(j) = w, V (p) = {w} and V (q) = ∅, where M, w Φ. Define vM : AtL → {T, F} by setting vM(ϕ) = T iff M, w ϕ. This implies that: vM(@i p) = T , vM(@i ♦j) = T, vM(@i j) = T, and for all the other atoms in L, vM assigns F. The extension of vM to vM is straightforward: M, w @i p ∨ @i q iff M, w @i p or M, w @i q iff vM(@i p) = T or vM(@i q) = T 17/37
26. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 1 Let L = (p, q), (i, j) and Φ = {@i p ∨ @i j,@j ¬q,@i ♦j, @i j} The set Φ is satisfiable, as there is a model M = (W , R, N, V ), such that W = {w}, R = {(i, i)}, N(i) = N(j) = w, V (p) = {w} and V (q) = ∅, where M, w Φ. Define vM : AtL → {T, F} by setting vM(ϕ) = T iff M, w ϕ. This implies that: vM(@i p) = T , vM(@i ♦j) = T, vM(@i j) = T, and for all the other atoms in L, vM assigns F. The extension of vM to vM is straightforward: M, w @i p ∨ @i q iff M, w @i p or M, w @i q iff vM(@i p) = T or vM(@i q) = T iff vM(@i p ∨ @i q) = T Thus Φ is propositionally satisfiable. w p 17/37
27. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Investigating first Herbrand-like theorem ⇐ Theorem 2 Let Φ ⊆ BCAt(L) be such that Eq(L) ⊆ Φ. If Φ is unsatisfiable then Φ is propositionally unsatisfiable. 18/37
28. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Generalization for satisfaction statement How to obtain a semantically equivalent formula ϕ◦ ∈ BCAt(L∗) Let ϕ be any satisfaction statement. Rewrite Rules 1 @i0 @i1 ϕ @i1 ϕ 2 @i ¬ϕ ¬@i ϕ 3 @i ϕ0 ∧ ϕ1 @i ϕ0 ∧ @i ϕ1 4 @i ♦ϕ @i ♦j ∧ @j ϕ, for j a fresh nominal (k ∈ Nom) 19/37
29. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Generalization for satisfaction statement Example Rewrite Rules 1 @i0 @i1 ϕ @i1 ϕ 2 @i ¬ϕ ¬@i ϕ 3 @i ϕ0 ∧ ϕ1 @i ϕ0 ∧ @i ϕ1 4 @i ♦ϕ @i ♦j ∧ @j ϕ, for j a fresh nominal (k ∈ Nom) Example: ϕ = @i @j ♦(p ∧ ¬q) in L where L = (p, q), (i, j) @i @j ♦(p ∧ ¬q) @j ♦(p ∧ ¬q) (applying 1) @j ♦k ∧ @k(p ∧ ¬q) (applying 4) @j ♦k ∧ (@kp ∧ @k¬q) (applying 3) @j ♦k ∧ (@kp ∧ ¬@kq) (applying 2) ϕ◦ = @j ♦k ∧ (@kp ∧ ¬@kq) in L∗ where L∗ = (p, q), (i, j, k) 19/37
30. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Quantified Hybrid logic Algebraic Strong Priorean Logic WVar is a denumerable set of world variabels Symbol used to represent the an element in WVar: s Algebraic Strong Priorean Logic is a hybrid logic enriched with operators over world variables 20/37
31. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Algebraic Signature A Algebraic similary type is a tuple Σ = F, σ , where: F : is a non-empty set of function symbols symbol used to represent an element in F : f σ : assigns to each function symbol its arity Grammar of Term@ (Σ, WVar) - Algebraic terms t ::= i|s|f (t0, . . . , tσ(f ) ) 21/37
32. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Hybrid Signature A hybrid similaty type is a tuple L = Prop, Nom, WVar ! The sets Prop, Nom, WVar are pairwise disjoint. Grammar of the hybrid language H(Σ, @, ∀) ϕ ::= p|t|¬ϕ|ϕ0 ∧ ϕ1|♦ϕ|@tϕ|∀sϕ|∃sϕ The connectives ∨, →, and are defined as usual Literals in H(Σ, @, ∀) A literal in H(Σ, @, ∀) is a formula of the form: @tp, @t¬p, @t♦t, @t¬♦t 22/37
33. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Hybrid Structure A hybrid structure H over L, Σ is a tuple W , R, f W )f ∈F , N, V , where: W : is a non-empty set called domain R : is accessibility relation where R ⊆ W × W N : is the hybrid nomination where N : Nom → W V : is the hybrid valuation where V : Prop → Pow(W ) (f W )f ∈F : set containing for each f ∈ F an interpretation f W : W σ(f ) → W 23/37
34. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic World Assignment World assignment g : WVar → W Two world assignments g and g are called s-variant iff g(u) = g (u), for all u ∈ WVar such u = s, in such case we write g s ∼ g . Extension of g into Term(Σ, WVar) is defined by: g(t) =      g(t), if t ∈ WVar N(t), if t ∈ Nom f W (g(t0), ..., g(tσ(f ) )), if t = f (t0, ..., tσ(f ) ) for some f ∈ F 24/37
35. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Hybrid Satisfaction Hybrid satisfaction between a hybrid structure, a world assignment, a state and a hybrid formula is recursively defined by: 1 H, g, w t iff w = g(t), for t ∈ Term(Σ, WVar) 2 H, g, w p iff w ∈ V (p), for p ∈ Prop 3 H, g, w ¬ϕ iff not H, g, w ϕ 4 H, g, w ϕ1 ∧ ϕ2 iff H, g, w ϕ1 and H, g, w ϕ2 5 H, g, w ♦ϕ iff ∃w ∈ W (wRw ) and H, g, w ϕ 6 H, g, w @i ϕ iff H, g, w ϕ, where w = g(t), for t ∈ Term(Σ, WVar) 7 H, g, w ∀sϕ iff H, g , w ϕ for all g such that g s ∼ g 8 H, g, w ∃sϕ iff H, g , w ϕ for some g such that g s ∼ g 25/37
36. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Hybrid Satisfaction Hybrid satisfaction between a hybrid structure, a world assignment, a state and a hybrid formula is recursively defined by: 1 H, g, w t iff w = g(t), for t ∈ Term(Σ, WVar) 2 H, g, w p iff w ∈ V (p), for p ∈ Prop 3 H, g, w ¬ϕ iff not H, g, w ϕ 4 H, g, w ϕ1 ∧ ϕ2 iff H, g, w ϕ1 and H, g, w ϕ2 5 H, g, w ♦ϕ iff ∃w ∈ W (wRw ) and H, g, w ϕ 6 H, g, w @i ϕ iff H, g, w ϕ, where w = g(t), for t ∈ Term(Σ, WVar) 7 H, g, w ∀sϕ iff H, g , w ϕ for all g such that g s ∼ g 8 H, g, w ∃sϕ iff H, g , w ϕ for some g such that g s ∼ g H, g, w ϕ means that ϕ is satisfied in M under the world assignment g at the state w. 25/37
37. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Satisfiability Problem Satisfiability A set Φ of hybrid formula in Form@,∀(L, Term(Σ, WVar) is said to be satisfiable if there exists a hybrid structure H = W , R, f W )f ∈F , N, V over L, Σ , a w ∈ W and a world assignment g : WVar → W such that: H, g, w ϕ, for all w ∈ Φ. Lemma Let ϕ be a hybrid formula in Form@,∀(H, Term(Σ, WVar)). Then ϕ is satisfiable iff @i ϕ is satisfiable where i is a fresh nominal. 26/37
38. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic (Second) Herbrand-like Herbrand-like Let L and Σ be, respectively, a hybrid and an algebraic similary type, and let Φ ⊆ Form@,∀(L, Term(Σ, WVar)). Then Φ is unsatisfiable iff some finite set Φ∗ of ground instances of Skolem forms of Φ ∪ Eq(L) is propositionally unsatisfiable. 27/37
39. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic (Second) Herbrand-like Herbrand-like Let L and Σ be, respectively, a hybrid and an algebraic similary type, and let Φ ⊆ Form@,∀(L, Term(Σ, WVar)). Then Φ is unsatisfiable iff some finite set Φ∗ of ground instances of Skolem forms of Φ ∪ Eq(L) is propositionally unsatisfiable. Strategy of the proof: A set of formulas Φ is unsatisfiable iff the set Ψ of its Skolem forms its unsatisfiable. ⇒ If Φ is unsatisfiable then some finite set Φ∗ of ground instances of Skolem forms of Φ ∪ Eq(L) is propositionally unsatisfiable. ⇐ If some finite set Φ∗ of ground instances of Skolem forms of Φ ∪ Eq(L) is propositionally unsatisfiable then Φ is unsatisfiable 27/37
40. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Skolem Form A formula of H(Σ, @, ∀) is in Skolem form if it is prenex normal form and its prefix contains only universal quantifiers Prenex Conjuntive Normal Form (PNF) A formula is in PNF when its quantifiers are flushed left, prefixing a quantifier-free part that is a conjunction of disjunctions of literals. 28/37
41. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Rectfied formula A formula is said to be rectified if no world variable occurs both bound and free and if all quantifiers in the formula refer to different world variables. Rectfied Existencial Closure (REC) Let s0, ..., sn be the world variables occurring free in ϕ. The REC of ϕ is the formula which results from rectifying ϕ and then existentially bounding its free variables, i.e., it is the formula ∃s0, ..., ∃snϕ 29/37
42. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Skolem Form Given a formula ϕ, how to obtain its Skolem form: 1 Labelling the formula 2 Retification and existencial closure of the new formula 3 Putting it in Prenex Conjutive Form 4 Perform Skolemization 30/37
43. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 1 Let ϕ1 = @i (♦p ∧ ¬@sp) ϕ1 = @i (♦p ∧ ¬@sp) 31/37
44. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 1 (Step 2) Retification Let ϕ1 = @i (♦p ∧ ¬@sp) ϕ1 = @i (♦p ∧ ¬@sp) ϕ1 = @i (♦p ∧ ¬@sp) 31/37
45. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 1 (Step 2) Existencial Closure Let ϕ1 = @i (♦p ∧ ¬@sp) ϕ1 = @i (♦p ∧ ¬@sp) ϕ1 = @i (♦p ∧ ¬@sp) ψ1 = ∃s@i (♦p ∧ ¬@sp) 31/37
46. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 1 (Step 3) Prenex Conjutive Form Let ϕ1 = @i (♦p ∧ ¬@sp) ϕ1 = @i (♦p ∧ ¬@sp) ϕ1 = @i (♦p ∧ ¬@sp) ψ1 = ∃s@i (♦p ∧ ¬@sp) ∃s(@i ♦p ∧ @i ¬@sp) ∃s(∃s1(@i ♦s1 ∧ @s1 p) ∧ @i @s¬p) ∃s(∃s1(@i ♦s1 ∧ @s1 p) ∧ @s¬p) 31/37
47. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 1 (Step 3) Prenex Conjutive Form Let ϕ1 = @i (♦p ∧ ¬@sp) ϕ1 = @i (♦p ∧ ¬@sp) ϕ1 = @i (♦p ∧ ¬@sp) ψ1 = ∃s@i (♦p ∧ ¬@sp) ∃s(@i ♦p ∧ @i ¬@sp) ∃s(∃s1(@i ♦s1 ∧ @s1 p) ∧ @i @s¬p) ∃s(∃s1(@i ♦s1 ∧ @s1 p) ∧ @s¬p) = ∃s∃s1(@i ♦s1 ∧ @s1 p ∧ @s¬p) 31/37
48. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 1 (Step 4) Skolemization Let ϕ1 = @i (♦p ∧ ¬@sp) ϕ1 = @i (♦p ∧ ¬@sp) ϕ1 = @i (♦p ∧ ¬@sp) ψ1 = ∃s@i (♦p ∧ ¬@sp) ∃s(@i ♦p ∧ @i ¬@sp) ∃s(∃s1(@i ♦s1 ∧ @s1 p) ∧ @i @s¬p) ∃s(∃s1(@i ♦s1 ∧ @s1 p) ∧ @s¬p) = ∃s∃s1(@i ♦s1 ∧ @s1 p ∧ @s¬p) = ∃s(@i ♦c1 ∧ @c1 p ∧ @s¬p) ψ1 = @i ♦c1 ∧ @c1 p ∧ @c2 ¬p 31/37
49. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 2 Let ϕ2 = @i (¬(∀s@s¬p ∧ ∃s@sp) ∧ @s¬p) ϕ = @i (¬(∀s@s¬p ∧ ∃s@sp) ∧ @s¬p) 32/37
50. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 2 (Step 2) Retification Let ϕ2 = @i (¬(∀s@s¬p ∧ ∃s@sp) ∧ @s¬p) ϕ = @i (¬(∀s@s¬p ∧ ∃s@sp) ∧ @s¬p) = @i (¬(∀s1@s1 ¬p ∧ ∃s@sp) ∧ @s¬p) ϕ2 = @i (¬(∀s1@s1 ¬p ∧ ∃s2@s2 p) ∧ @s¬p) 32/37
51. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 2 (Step 2) Existencial Closure Let ϕ2 = @i (¬(∀s@s¬p ∧ ∃s@sp) ∧ @s¬p) ϕ = @i (¬(∀s@s¬p ∧ ∃s@sp) ∧ @s¬p) = @i (¬(∀s1@s1 ¬p ∧ ∃s@sp) ∧ @s¬p) ϕ2 = @i (¬(∀s1@s1 ¬p ∧ ∃s2@s2 p) ∧ @s¬p) ψ2 = ∃s@i (¬(∀s1@s1 ¬p ∧ ∃s2@s2 p) ∧ @s¬p) 32/37
52. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 2 (Step 3) Prenex Conjutive Form Let ϕ2 = @i (¬(∀s@s¬p ∧ ∃s@sp) ∧ @s¬p) ϕ = @i (¬(∀s@s¬p ∧ ∃s@sp) ∧ @s¬p) = @i (¬(∀s1@s1 ¬p ∧ ∃s@sp) ∧ @s¬p) ϕ2 = @i (¬(∀s1@s1 ¬p ∧ ∃s2@s2 p) ∧ @s¬p) ψ2 = ∃s@i (¬(∀s1@s1 ¬p ∧ ∃s2@s2 p) ∧ @s¬p) ∃s@i ((¬∀s1@s1 ¬p ∨ ¬∃s2@s2 p) ∧ @s¬p) ∃s@i ((∃s1¬@s1 ¬p ∨ ∀s2¬@s2 p) ∧ @s¬p) ∃s(@i (∃s1@s1 ¬¬p ∨ ∀s2@s2 ¬p) ∧ @s¬p) ∃s((@i ∃s1@s1 p ∨ @i ∀s2@s2 ¬p) ∧ @s¬p) ∃s((∃s1@i @s1 p ∨ ∀s2@i @s2 ¬p) ∧ @s¬p) ∃s((∃s1@s1 p ∨ ∀s2@s2 ¬p) ∧ @s¬p) 32/37
53. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 2 (Step 3) Prenex Conjutive Form Let ϕ2 = @i (¬(∀s@s¬p ∧ ∃s@sp) ∧ @s¬p) ϕ = @i (¬(∀s@s¬p ∧ ∃s@sp) ∧ @s¬p) = @i (¬(∀s1@s1 ¬p ∧ ∃s@sp) ∧ @s¬p) ϕ2 = @i (¬(∀s1@s1 ¬p ∧ ∃s2@s2 p) ∧ @s¬p) ψ2 = ∃s@i (¬(∀s1@s1 ¬p ∧ ∃s2@s2 p) ∧ @s¬p) ∃s@i ((¬∀s1@s1 ¬p ∨ ¬∃s2@s2 p) ∧ @s¬p) ∃s@i ((∃s1¬@s1 ¬p ∨ ∀s2¬@s2 p) ∧ @s¬p) ∃s(@i (∃s1@s1 ¬¬p ∨ ∀s2@s2 ¬p) ∧ @s¬p) ∃s((@i ∃s1@s1 p ∨ @i ∀s2@s2 ¬p) ∧ @s¬p) ∃s((∃s1@i @s1 p ∨ ∀s2@i @s2 ¬p) ∧ @s¬p) ∃s((∃s1@s1 p ∨ ∀s2@s2 ¬p) ∧ @s¬p) ∃s∃s1∀s2((@s1 p ∨ @s2 ¬p) ∧ @s¬p) 32/37
54. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 2 (Step 4) Skolemization Let ϕ2 = @i (¬(∀s@s¬p ∧ ∃s@sp) ∧ @s¬p) ϕ = @i (¬(∀s@s¬p ∧ ∃s@sp) ∧ @s¬p) = @i (¬(∀s1@s1 ¬p ∧ ∃s@sp) ∧ @s¬p) ϕ2 = @i (¬(∀s1@s1 ¬p ∧ ∃s2@s2 p) ∧ @s¬p) ψ2 = ∃s@i (¬(∀s1@s1 ¬p ∧ ∃s2@s2 p) ∧ @s¬p) ∃s@i ((¬∀s1@s1 ¬p ∨ ¬∃s2@s2 p) ∧ @s¬p) ∃s@i ((∃s1¬@s1 ¬p ∨ ∀s2¬@s2 p) ∧ @s¬p) ∃s(@i (∃s1@s1 ¬¬p ∨ ∀s2@s2 ¬p) ∧ @s¬p) ∃s((@i ∃s1@s1 p ∨ @i ∀s2@s2 ¬p) ∧ @s¬p) ∃s((∃s1@i @s1 p ∨ ∀s2@i @s2 ¬p) ∧ @s¬p) ∃s((∃s1@s1 p ∨ ∀s2@s2 ¬p) ∧ @s¬p) ∃s∃s1∀s2((@s1 p ∨ @s2 ¬p) ∧ @s¬p) ψ2 = ∀s2((@c2 p ∨ @s2 ¬p) ∧ @c1 ¬p) 32/37
55. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 3 Let ϕ3 = @i (∀s∃t@s♦t) ϕ3 = @i (∀s∃t@s♦t) 33/37
56. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 3 (Step 2) Retification Let ϕ3 = @i (∀s∃t@s♦t) ϕ3 = @i (∀s∃t@s♦t) ϕ3 = @i (∀s∃t@s♦t) 33/37
57. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 3 (Step 2) Existencial Closure Let ϕ3 = @i (∀s∃t@s♦t) ϕ3 = @i (∀s∃t@s♦t) ϕ3 = @i (∀s∃t@s♦t) ψ3 = @i (∀s∃t@s♦t) 33/37
58. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 3 (Step 3) Prenex Conjutive Form Let ϕ3 = @i (∀s∃t@s♦t) ϕ3 = @i (∀s∃t@s♦t) ϕ3 = @i (∀s∃t@s♦t) ψ3 = @i (∀s∃t@s♦t) ∀s∃t(@i @s♦t) ∀s∃t(@s♦t) 33/37
59. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 3 (Step 3) Prenex Conjutive Form Let ϕ3 = @i (∀s∃t@s♦t) ϕ3 = @i (∀s∃t@s♦t) ϕ3 = @i (∀s∃t@s♦t) ψ3 = @i (∀s∃t@s♦t) ∀s∃t(@i @s♦t) ∀s∃t(@s♦t) ∀s∃t(@s♦t) 33/37
60. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic Example 3 (Step 4) Skolemization Let ϕ3 = @i (∀s∃t@s♦t) ϕ3 = @i (∀s∃t@s♦t) ϕ3 = @i (∀s∃t@s♦t) ψ3 = @i (∀s∃t@s♦t) ∀s∃t(@i @s♦t) ∀s∃t(@s♦t) ∀s∃t(@s♦t) ψ3 = ∀s(@s♦f (s)) 33/37
61. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic (Second) Herbrand-like Herbrand-like Let L and Σ be, respectively, a hybrid and an algebraic similary type, and let Φ ⊆ Form@,∀(L, Term(Σ, WVar)). Then Φ is unsatisfiable iff some finite set Φ∗ of ground instances of Skolem forms of Φ ∪ Eq(L) is propositionally unsatisfiable. 34/37
62. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic (Second) Herbrand-like Herbrand-like Let L and Σ be, respectively, a hybrid and an algebraic similary type, and let Φ ⊆ Form@,∀(L, Term(Σ, WVar)). Then Φ is unsatisfiable iff some finite set Φ∗ of ground instances of Skolem forms of Φ ∪ Eq(L) is propositionally unsatisfiable. Strategy of the proof: A set of formulas Φ is unsatisfiable iff the set Ψ of its Skolem forms its unsatisfiable. ⇒ If Φ is unsatisfiable then some finite set Φ∗ of ground instances of Skolem forms of Φ ∪ Eq(L) is propositionally unsatisfiable. ⇐ If some finite set Φ∗ of ground instances of Skolem forms of Φ ∪ Eq(L) is propositionally unsatisfiable then Φ is unsatisfiable 34/37
63. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic (Second) Herbrand-like ⇐ 35/37
64. Summary Herbrand Context Hybrid Logic Hybrid Logic Quantified Conclusion Logic (Second) Herbrand-like ⇒ 36/37
65. Conclusion Conclusion