Formalization of the Resolution Calculus for First-Order Logic - PowerPoint PPT Presentation

Formalization of FO resolution definition applicable C 1 C 2 L 1 L 2 σ ⟷ 
 C 1 ≠ {} ∧ C 2 ≠ {} ∧ L 1 ≠ {} ∧ L 2 ≠ {} 
 ∧ vars C 1 ∩ vars C 2 = {} 
 ∧ L 1 ⊆ C 1 ∧ L 2 ⊆ C 2 
 C )" ∧ mgu σ (L 1 ∪ L 2 definition resolution C 1 C 2 L 1 L 2 σ = ((C 1 - L 1 ) ∪ (C 2 - L 2 )) ⋅ σ DTU Compute, Technical University of Denmark 17

Formalization of FO resolution definition applicable C 1 C 2 L 1 L 2 σ ⟷ 
 C 1 ≠ {} ∧ C 2 ≠ {} ∧ L 1 ≠ {} ∧ L 2 ≠ {} 
 ∧ vars C 1 ∩ vars C 2 = {} 
 ∧ L 1 ⊆ C 1 ∧ L 2 ⊆ C 2 
 C )" ∧ mgu σ (L 1 ∪ L 2 definition resolution C 1 C 2 L 1 L 2 σ = ((C 1 - L 1 ) ∪ (C 2 - L 2 )) ⋅ σ inductive resolution_step 
 :: fterm clause set ⇒ fterm clause set ⇒ bool where 
 resolution_rule: 
 C 1 ∈ Cs ⟹ C 2 ∈ Cs ⟹ applicable C 1 C 2 L 1 L 2 σ ⟹ 
 resolution_step Cs (Cs ∪ {resolution C 1 C 2 L 1 L 2 σ }) 
 | standardize_apart: 
 C ∈ Cs ⟹ var_renaming_of C C' ⟹ resolution_step Cs (Cs ∪ {C'}) DTU Compute, Technical University of Denmark 17

Formalization of FO resolution definition applicable C 1 C 2 L 1 L 2 σ ⟷ 
 C 1 ≠ {} ∧ C 2 ≠ {} ∧ L 1 ≠ {} ∧ L 2 ≠ {} 
 ∧ vars C 1 ∩ vars C 2 = {} 
 ∧ L 1 ⊆ C 1 ∧ L 2 ⊆ C 2 
 C )" ∧ mgu σ (L 1 ∪ L 2 definition resolution C 1 C 2 L 1 L 2 σ = ((C 1 - L 1 ) ∪ (C 2 - L 2 )) ⋅ σ inductive resolution_step 
 :: fterm clause set ⇒ fterm clause set ⇒ bool where 
 resolution_rule: 
 C 1 ∈ Cs ⟹ C 2 ∈ Cs ⟹ applicable C 1 C 2 L 1 L 2 σ ⟹ 
 resolution_step Cs (Cs ∪ {resolution C 1 C 2 L 1 L 2 σ }) 
 | standardize_apart: 
 C ∈ Cs ⟹ var_renaming_of C C' ⟹ resolution_step Cs (Cs ∪ {C'}) definition resolution_deriv = rtranclp resolution_step DTU Compute, Technical University of Denmark 17

Refutational completeness DTU Compute, Technical University of Denmark 18

Refutational completeness Refutational completeness: 
 If C is unsatisfiable then the calculus can derive a contradiction DTU Compute, Technical University of Denmark 18

Refutational completeness Refutational completeness: 
 If C is unsatisfiable then the calculus can derive a contradiction unsatisfiable C ⟹ (C ⊢ {}) DTU Compute, Technical University of Denmark 18

Semantic tree DTU Compute, Technical University of Denmark 19

Semantic tree Enumeration of ground terms: p, q, r(c), … DTU Compute, Technical University of Denmark 19

Semantic tree Enumeration of ground terms: p, q, r(c), … DTU Compute, Technical University of Denmark 19


 
 
 
 
 Semantic tree Enumeration of ground terms: p, q, r(c), … Semantic trees are decision trees assigning True and False to the ground atoms. DTU Compute, Technical University of Denmark 19


 
 
 
 
 Semantic tree Enumeration of ground terms: p, q, r(c), … Semantic trees are decision trees assigning True and False to the ground atoms. Node on depth i makes decision for atom i . DTU Compute, Technical University of Denmark 19

Semantic tree A path represents a partial (Herbrand) interpretation. E.g. {p ↦ T , q ↦ F , r(c) ↦ F } DTU Compute, Technical University of Denmark 20

Formalized enumeration DTU Compute, Technical University of Denmark 21

Formalized enumeration definition nat_from_hatom :: hterm atom ⇒ nat where 
 nat_from_hatom ≡ (SOME f. bij f) DTU Compute, Technical University of Denmark 21

Formalized enumeration definition nat_from_hatom :: hterm atom ⇒ nat where 
 nat_from_hatom ≡ (SOME f. bij f) instantiation hterm :: countable begin 
 instance by countable_datatype 
 end DTU Compute, Technical University of Denmark 21

Formalized enumeration definition nat_from_hatom :: hterm atom ⇒ nat where 
 nat_from_hatom ≡ (SOME f. bij f) instantiation hterm :: countable begin 
 instance by countable_datatype 
 end lemma infinite_hatoms: infinite (UNIV :: 't atom set) 
 <proof> DTU Compute, Technical University of Denmark 21

Formalized enumeration definition nat_from_hatom :: hterm atom ⇒ nat where 
 nat_from_hatom ≡ (SOME f. bij f) instantiation hterm :: countable begin 
 instance by countable_datatype 
 end lemma infinite_hatoms: infinite (UNIV :: 't atom set) 
 <proof> lemma nat_from_hatom_bij: bij nat_from_hatom 
 proof - 
 have countable (UNIV :: hterm atom set) by simp 
 moreover 
 have infinite (UNIV :: hterm atom set) using infinite_hatoms by auto 
 ultimately 
 obtain x where bij (x :: hterm atom ⇒ nat) using countableE_infinite by blast 
 then show ?thesis using … someI by metis 
 qed DTU Compute, Technical University of Denmark 21

Formalized enumeration definition nat_from_hatom :: hterm atom ⇒ nat where 
 nat_from_hatom ≡ (SOME f. bij f) instantiation hterm :: countable begin 
 instance by countable_datatype 
 end lemma infinite_hatoms: infinite (UNIV :: 't atom set) 
 <proof> lemma nat_from_hatom_bij: bij nat_from_hatom 
 proof - 
 have countable (UNIV :: hterm atom set) by simp 
 moreover 
 have infinite (UNIV :: hterm atom set) using infinite_hatoms by auto 
 ultimately 
 obtain x where bij (x :: hterm atom ⇒ nat) using countableE_infinite by blast 
 then show ?thesis using … someI by metis 
 qed DTU Compute, Technical University of Denmark 21

Formalized semantic trees DTU Compute, Technical University of Denmark 22

Formalized semantic trees Finite trees: datatype tree = 
 Leaf 
 | Branching tree tree DTU Compute, Technical University of Denmark 22

Formalized semantic trees Finite trees: datatype tree = 
 Leaf 
 | Branching tree tree Paths: type_synonym path = bool list DTU Compute, Technical University of Denmark 22

Formalized semantic trees Finite trees: datatype tree = 
 Leaf 
 | Branching tree tree Paths: type_synonym path = bool list Possibly infinite trees: type_synonym inftree = path set abbreviation wf_tree :: path set ⇒ bool where 
 wf_tree T ≡ ( ∀ ds d. (ds @ d) ∈ T ⟶ ds ∈ T) DTU Compute, Technical University of Denmark 22

Falsification by partial interpretation DTU Compute, Technical University of Denmark 23

Falsification by partial interpretation Falsification of ground clause: 
 {p ↦ T , q ↦ F , r(c) ↦ T } falsifies {q,¬r(c)} DTU Compute, Technical University of Denmark 23

Falsification by partial interpretation Falsification of ground clause: 
 {p ↦ T , q ↦ F , r(c) ↦ T } falsifies {q,¬r(c)} abbreviation falsifies g :: path ⇒ fterm clause ⇒ bool where 
 falsifies g G C ≡ ground C ∧ ( ∀ l ∈ C. falsifies G l) DTU Compute, Technical University of Denmark 23

Falsification by partial interpretation Falsification of ground clause: 
 {p ↦ T , q ↦ F , r(c) ↦ T } falsifies {q,¬r(c)} abbreviation falsifies g :: path ⇒ fterm clause ⇒ bool where 
 falsifies g G C ≡ ground C ∧ ( ∀ l ∈ C. falsifies G l) Falsification of FO clause: 
 {p ↦ T , q ↦ F , r(c) ↦ T } falsifies {q,¬r( x )} DTU Compute, Technical University of Denmark 23

Falsification by partial interpretation Falsification of ground clause: 
 {p ↦ T , q ↦ F , r(c) ↦ T } falsifies {q,¬r(c)} abbreviation falsifies g :: path ⇒ fterm clause ⇒ bool where 
 falsifies g G C ≡ ground C ∧ ( ∀ l ∈ C. falsifies G l) Falsification of FO clause: 
 {p ↦ T , q ↦ F , r(c) ↦ T } falsifies {q,¬r( x )} abbreviation falsifies :: path ⇒ fterm clause ⇒ bool where 
 falsifies G C ≡ ( ∃ C'. instance_of C' C ∧ falsifies g G C') DTU Compute, Technical University of Denmark 23


 Closed semantic tree Definition of closed semantic tree: 
 All branches falsify a ground instance of a clause in Cs DTU Compute, Technical University of Denmark 24


 Closed semantic tree Definition of closed semantic tree: 
 Definition of closed semantic tree: 
 All branches falsify a ground instance of a clause in Cs All branches falsify a ground instance of a clause in Cs 
 Cs = { {¬q,¬p}, {r( x )}, {¬p,q,¬r( y )}, {p} } p ↦ T p ↦ F q ↦ T q ↦ F r(c) ↦ T r(c) ↦ F DTU Compute, Technical University of Denmark 24


 Closed semantic tree Definition of closed semantic tree: 
 Definition of closed semantic tree: 
 Definition of closed semantic tree: 
 All branches falsify a ground instance of a clause in Cs All branches falsify a ground instance of a clause in Cs 
 All branches falsify a ground instance of a clause in Cs 
 Cs = { {¬q,¬p}, {r( x )}, {¬p,q,¬r( y )}, {p} } Cs = { {¬q,¬p}, {r( x )}, {¬p,q,¬r( y )}, {p} } p ↦ T p ↦ T p ↦ F p ↦ F {p ↦ T , q ↦ T } q ↦ T q ↦ T q ↦ F q ↦ F falsifies {¬q,¬p} r(c) ↦ T r(c) ↦ T r(c) ↦ F r(c) ↦ F DTU Compute, Technical University of Denmark 24


 Closed semantic tree Definition of closed semantic tree: 
 Definition of closed semantic tree: 
 Definition of closed semantic tree: 
 Definition of closed semantic tree: 
 All branches falsify a ground instance of a clause in Cs 
 All branches falsify a ground instance of a clause in Cs All branches falsify a ground instance of a clause in Cs 
 All branches falsify a ground instance of a clause in Cs 
 Cs = { {¬q,¬p}, {r( x )}, {¬p,q,¬r( y )}, {p} } Cs = { {¬q,¬p}, {r( x )}, {¬p,q,¬r( y )}, {p} } Cs = { {¬q,¬p}, {r( x )}, {¬p,q,¬r( y )}, {p} } p ↦ T p ↦ T p ↦ T p ↦ F p ↦ F p ↦ F {p ↦ T , q ↦ F , r(c) ↦ T } falsifies {p ↦ T , q ↦ T } q ↦ T q ↦ T q ↦ T q ↦ F q ↦ F q ↦ F {¬p,q,¬r(c)} falsifies ground instance of {¬q,¬p} r(c) ↦ T r(c) ↦ T r(c) ↦ T r(c) ↦ F r(c) ↦ F r(c) ↦ F {¬p,q,¬r( y )} DTU Compute, Technical University of Denmark 24


 Closed semantic tree Definition of closed semantic tree: 
 Definition of closed semantic tree: 
 Definition of closed semantic tree: 
 Definition of closed semantic tree: 
 Definition of closed semantic tree: 
 All branches falsify a ground instance of a clause in Cs 
 All branches falsify a ground instance of a clause in Cs 
 All branches falsify a ground instance of a clause in Cs 
 All branches falsify a ground instance of a clause in Cs 
 All branches falsify a ground instance of a clause in Cs Cs = { {¬q,¬p}, {r( x )}, {¬p,q,¬r( y )}, {p} } Cs = { {¬q,¬p}, {r( x )}, {¬p,q,¬r( y )}, {p} } Cs = { {¬q,¬p}, {r( x )}, {¬p,q,¬r( y )}, {p} } Cs = { {¬q,¬p}, {r( x )}, {¬p,q,¬r( y )}, {p} } p ↦ T p ↦ T p ↦ T p ↦ T p ↦ F p ↦ F p ↦ F p ↦ F {p ↦ T , q ↦ F , r(c) ↦ T } {p ↦ T , q ↦ F , r(c) ↦ T } falsifies falsifies {p ↦ T , q ↦ T } q ↦ T q ↦ T q ↦ T q ↦ T q ↦ F q ↦ F q ↦ F q ↦ F {¬p,q,¬r(c)} {r(c)} falsifies ground instance of ground instance of {¬q,¬p} r(c) ↦ T r(c) ↦ T r(c) ↦ T r(c) ↦ T r(c) ↦ F r(c) ↦ F r(c) ↦ F r(c) ↦ F {¬p,q,¬r( y )} {r( x )} DTU Compute, Technical University of Denmark 24


 Closed semantic tree Definition of closed semantic tree: 
 Definition of closed semantic tree: 
 Definition of closed semantic tree: 
 Definition of closed semantic tree: 
 Definition of closed semantic tree: 
 Definition of closed semantic tree: 
 All branches falsify a ground instance of a clause in Cs 
 All branches falsify a ground instance of a clause in Cs 
 All branches falsify a ground instance of a clause in Cs 
 All branches falsify a ground instance of a clause in Cs 
 All branches falsify a ground instance of a clause in Cs All branches falsify a ground instance of a clause in Cs 
 Cs = { {¬q,¬p}, {r( x )}, {¬p,q,¬r( y )}, {p} } Cs = { {¬q,¬p}, {r( x )}, {¬p,q,¬r( y )}, {p} } Cs = { {¬q,¬p}, {r( x )}, {¬p,q,¬r( y )}, {p} } Cs = { {¬q,¬p}, {r( x )}, {¬p,q,¬r( y )}, {p} } Cs = { {¬q,¬p}, {r( x )}, {¬p,q,¬r( y )}, {p} } p ↦ T p ↦ F p ↦ T p ↦ T p ↦ T p ↦ T p ↦ F p ↦ F p ↦ F p ↦ F {p ↦ T , q ↦ F , r(c) ↦ T } {p ↦ T , q ↦ F , r(c) ↦ T } {p ↦ F } falsifies falsifies {p ↦ T , q ↦ T } q ↦ T q ↦ T q ↦ T q ↦ T q ↦ T q ↦ F q ↦ F q ↦ F q ↦ F q ↦ F falsifies {¬p,q,¬r(c)} {r(c)} falsifies {p} ground instance of ground instance of {¬q,¬p} r(c) ↦ T r(c) ↦ F r(c) ↦ T r(c) ↦ T r(c) ↦ T r(c) ↦ T r(c) ↦ F r(c) ↦ F r(c) ↦ F r(c) ↦ F {¬p,q,¬r( y )} {r( x )} DTU Compute, Technical University of Denmark 24

Completeness proof 1. Herbrand’s theorem: 
 Any unsatisfiable set of clauses has a finite closed semantic tree. 2. {} is derivable from any set of clauses with a closed semantic tree. The proof follows Chang & Lee (1973). DTU Compute, Technical University of Denmark 25

Completeness proof ↳ 1. Herbrand’s theorem 2. Deriving {} Herbrand’s theorem: Any unsatisfiable set of clauses Cs has a finite closed semantic tree. Proof: Let T be a full infinite semantic tree. 
 Consider any infinite p path in T . 
 p is an interpretation and thus falsifies Cs . 
 A (finite) prefix also falsifies Cs . 
 Let T’ be a copy of T with all paths replaced with finite falsifying prefixes. 
 T’ is finite by König’s lemma. DTU Compute, Technical University of Denmark 26

Completeness proof ↳ 1. Herbrand’s theorem 2. Deriving {} Herbrand’s theorem: Any unsatisfiable set of clauses Cs has a finite closed semantic tree. p is an interpretation? Proof: A path is a list of bools. An interpretation is a Let T be a full infinite semantic tree. 
 fun_sym ⇒ 'u list ⇒ 'u Consider any infinite p path in T . 
 and a p is an interpretation and thus falsifies Cs . 
 pred_sym ⇒ 'u list ⇒ bool A (finite) prefix also falsifies Cs . 
 Let T’ be a copy of T with all paths replaced with finite falsifying prefixes. 
 T’ is finite by König’s lemma. DTU Compute, Technical University of Denmark 26

Completeness proof ↳ 1. Herbrand’s theorem 2. Deriving {} Herbrand’s theorem: Any unsatisfiable set of clauses Cs has a finite closed semantic tree. p is an interpretation? Proof: A path is a list of bools. An interpretation is a Let T be a full infinite semantic tree. 
 fun_sym ⇒ 'u list ⇒ 'u Consider any infinite p path in T . 
 and a p is an interpretation and thus falsifies Cs . 
 pred_sym ⇒ 'u list ⇒ bool A (finite) prefix also falsifies Cs . 
 Let T’ be a copy of T with all paths replaced with Yes, we can make a finite falsifying prefixes. 
 conversion function T’ is finite by König’s lemma. extend. DTU Compute, Technical University of Denmark 26

Completeness proof ↳ 1. Herbrand’s theorem 2. Deriving {} Herbrand’s theorem: Any unsatisfiable set of clauses Cs has a finite closed semantic tree. Proof: Let T be a full infinite semantic tree. 
 Consider any infinite p path in T . 
 p is an interpretation and thus falsifies Cs . 
 Does it? A (finite) prefix also falsifies Cs . 
 Let T’ be a copy of T with all paths replaced with finite falsifying prefixes. 
 T’ is finite by König’s lemma. DTU Compute, Technical University of Denmark 26

Completeness proof ↳ 1. Herbrand’s theorem 2. Deriving {} If an infinite path falsifies a set of clauses, then so does a finite prefix. FO clause set Cs falsified by Interpretation extend p ⟹ Cs falsified by Partial prefix of p interpretation DTU Compute, Technical University of Denmark 27

Completeness proof ↳ 1. Herbrand’s theorem 2. Deriving {} If an infinite path falsifies a set of clauses, then so does a finite prefix. FO clause set Ground clause set Cs falsified by Interpretation extend p ⟹ Cs falsified by Partial prefix of p interpretation DTU Compute, Technical University of Denmark 27

Completeness proof ↳ 1. Herbrand’s theorem 2. Deriving {} If an infinite path falsifies a set of clauses, then so does a finite prefix. FO clause set Ground clause set Cs ʹ falsified by Cs falsified by Interpretation extend p extend p ⟹ Cs falsified by Partial prefix of p interpretation DTU Compute, Technical University of Denmark 27

Completeness proof ↳ 1. Herbrand’s theorem 2. Deriving {} If an infinite path falsifies a set of clauses, then so does a finite prefix. FO clause set Ground clause set ⟹ Cs ʹ falsified by Cs falsified by Interpretation extend p extend p ⟹ Cs falsified by Partial prefix of p interpretation DTU Compute, Technical University of Denmark 27

Completeness proof ↳ 1. Herbrand’s theorem 2. Deriving {} If an infinite path falsifies a set of clauses, then so does a finite prefix. FO clause set Ground clause set ⟹ Cs ʹ falsified by Cs falsified by Interpretation extend p extend p ⟹ Cs ʹ falsified by Cs falsified by Partial prefix of p prefix of p interpretation DTU Compute, Technical University of Denmark 27

Completeness proof ↳ 1. Herbrand’s theorem 2. Deriving {} If an infinite path falsifies a set of clauses, then so does a finite prefix. FO clause set Ground clause set ⟹ Cs ʹ falsified by Cs falsified by Interpretation extend p extend p ⟹ ⟹ Cs ʹ falsified by Cs falsified by Partial prefix of p prefix of p interpretation DTU Compute, Technical University of Denmark 27

Completeness proof ↳ 1. Herbrand’s theorem 2. Deriving {} If an infinite path falsifies a set of clauses, then so does a finite prefix. FO clause set Ground clause set Cs ʹ falsified by Cs falsified by Interpretation ⟹ extend p extend p ⟹ ⟹ Cs ʹ falsified by Cs falsified by Partial ⟹ prefix of p prefix of p interpretation DTU Compute, Technical University of Denmark 27

Completeness proof ↳ 1. Herbrand’s theorem 2. Deriving {} DTU Compute, Technical University of Denmark 28

Completeness proof 1. Herbrand’s theorem 1. Herbrand’s theorem ↳ 2. Deriving {} 2. Deriving {} DTU Compute, Technical University of Denmark 28

Completeness proof 1. Herbrand’s theorem 1. Herbrand’s theorem ↳ 2. Deriving {} 2. Deriving {} closed semantic tree for Cs q ↦ F r(c) ↦ T r(c) ↦ F DTU Compute, Technical University of Denmark 28

Completeness proof 1. Herbrand’s theorem 1. Herbrand’s theorem ↳ 2. Deriving {} 2. Deriving {} closed semantic tree for Cs q ↦ F r(c) ↦ T r(c) ↦ F falsifies C 1 C 2 DTU Compute, Technical University of Denmark 28

Completeness proof 1. Herbrand’s theorem 1. Herbrand’s theorem ↳ 2. Deriving {} 2. Deriving {} closed semantic tree for Cs q ↦ F r(c) ↦ T r(c) ↦ F falsifies C 1 C 2 C DTU Compute, Technical University of Denmark 28

Completeness proof 1. Herbrand’s theorem 1. Herbrand’s theorem ↳ 2. Deriving {} 2. Deriving {} closed semantic tree for Cs q ↦ F r(c) ↦ T r(c) ↦ F falsifies C 1 C 2 C DTU Compute, Technical University of Denmark 28

Completeness proof 1. Herbrand’s theorem 1. Herbrand’s theorem ↳ 2. Deriving {} 2. Deriving {} closed semantic tree for Cs ⋃ { C } q ↦ F r(c) ↦ T r(c) ↦ F falsifies C 1 C 2 C DTU Compute, Technical University of Denmark 28

Completeness proof 1. Herbrand’s theorem 1. Herbrand’s theorem ↳ 2. Deriving {} 2. Deriving {} closed semantic tree for Cs ⋃ { C } q ↦ F DTU Compute, Technical University of Denmark 28

Completeness proof 1. Herbrand’s theorem 1. Herbrand’s theorem ↳ 2. Deriving {} 2. Deriving {} closed semantic tree for Cs ⋃ { C } q ↦ F DTU Compute, Technical University of Denmark 28

Completeness proof 1. Herbrand’s theorem 1. Herbrand’s theorem ↳ 2. Deriving {} 2. Deriving {} closed semantic tree for Cs ⋃ { C } q ↦ F DTU Compute, Technical University of Denmark 28

Completeness proof 1. Herbrand’s theorem ↳ 2. Deriving {} Eventually the empty tree is closed for our Cs . Then we have derived {} . DTU Compute, Technical University of Denmark 29

Completeness proof 1. Herbrand’s theorem 1. Herbrand’s theorem ↳ 2. Deriving {} 2. Deriving {} closed semantic tree for Cs q ↦ F r(c) ↦ T r(c) ↦ F falsifies C 1 C 2 C DTU Compute, Technical University of Denmark 30

Completeness proof 1. Herbrand’s theorem 1. Herbrand’s theorem ↳ 2. Deriving {} 2. Deriving {} closed semantic tree for Cs q ↦ F r(c) ↦ T r(c) ↦ F falsifies C 1 C 2 DTU Compute, Technical University of Denmark 30

Completeness proof 1. Herbrand’s theorem 1. Herbrand’s theorem ↳ 2. Deriving {} 2. Deriving {} closed semantic tree for Cs q ↦ F r(c) ↦ T r(c) ↦ F falsifies C 1 C 2 instance of C 1 ʹ C 2 ʹ DTU Compute, Technical University of Denmark 30

Completeness proof 1. Herbrand’s theorem 1. Herbrand’s theorem ↳ 2. Deriving {} 2. Deriving {} closed semantic tree for Cs q ↦ F r(c) ↦ T r(c) ↦ F falsifies C 1 C 2 instance of C 1 ʹ C 2 ʹ C ʹ DTU Compute, Technical University of Denmark 30

Completeness proof 1. Herbrand’s theorem 1. Herbrand’s theorem ↳ 2. Deriving {} 2. Deriving {} closed semantic tree for Cs q ↦ F r(c) ↦ T r(c) ↦ F falsifies falsifies C 1 C 2 instance of C 1 ʹ C 2 ʹ by arguments about enumeration C ʹ DTU Compute, Technical University of Denmark 30