Outline Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion SGGS Theorem Proving: an Exposition 1 Maria Paola Bonacina Dipartimento di Informatica Universit` a degli Studi di Verona Verona, Italy, EU 4th Workshop on Practical Aspects of Automated Reasoning (PAAR), 23 July 2014 (Subsuming the talk at the Annual Meeting of the IFIP WG 1.6 on Term Rewriting, 13 July 2014) Satellite of the 7th Int. Joint Conf. on Automated Reasoning (IJCAR) 6th Federated Logic Conf. (FLoC), Vienna Summer of Logic (Extended version based also on talks at MPI Saarbr¨ ucken, June 2014, and U. Koblenz-Landau, Sept. 2014) 1 Joint work with David A. Plaisted Maria Paola Bonacina SGGS Theorem Proving: an Exposition
Outline Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion Maria Paola Bonacina SGGS Theorem Proving: an Exposition
Outline Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion Motivation A first-order theorem-proving method simultaneously ◮ DPLL-style model based ◮ Proof confluent ◮ Semantically guided ◮ Goal sensitive Maria Paola Bonacina SGGS Theorem Proving: an Exposition
Outline Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion DPLL-style model based ◮ Derivation state includes candidate (partial) model ◮ Inference and search (for model) guide each other (e.g., CDCL in DPLL) ◮ Inference as model transformation Maria Paola Bonacina SGGS Theorem Proving: an Exposition
Outline Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion Proof confluent ◮ Confluence: diamond property: ւ ց ⇒ ց ւ ◮ Proof confluence: Committing to an inference never prevents proof ◮ No backtracking Maria Paola Bonacina SGGS Theorem Proving: an Exposition
Outline Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion Semantically guided ◮ Semantic guidance by a given initial interpretation I ◮ In theory and manual examples: e.g., based on sign ◮ In practice: problems and knowledge bases may come with it ◮ SGGS: semantic guidance and model-based style connected; I as starting point and default Maria Paola Bonacina SGGS Theorem Proving: an Exposition
Outline Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion Goal sensitive ◮ Notion of goal: = ? ϕ ◮ H | ◮ H ∪ {¬ ϕ } ⊢ ? ⊥ ◮ H ∪ {¬ ϕ } ❀ S set of clauses ◮ S = T ⊎ iSOS where H ❀ T , {¬ ϕ } ❀ iSOS ◮ S = T ⊎ iSOS , iSOS input set of support ◮ Alternatively: S = T ⊎ iSOS with T consistent, iSOS = S \ T ◮ Generate only clauses connected with iSOS Maria Paola Bonacina SGGS Theorem Proving: an Exposition
Outline Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion Motivation summary ◮ A first-order reasoning method with all these properties?! ◮ Yes!!! SGGS Semantically Guided Goal Sensitive reasoning Maria Paola Bonacina SGGS Theorem Proving: an Exposition
Outline Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion Model Representation Model representation from PL to FOL: ◮ DPLL: Trail of literals L 1 , . . . , L n ◮ SGGS: ◮ Initial interpretation I ◮ Sequence of constrained clauses with selected literals Γ = A 1 ✄ C 1 [ L 1 ] , . . . , A n ✄ C n [ L n ] ◮ That modify I Maria Paola Bonacina SGGS Theorem Proving: an Exposition
Outline Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion Example I: unit clauses ◮ I : all negative ◮ Sequence Γ: P ( a , x ) , P ( b , y ) , ¬ P ( z , z ) , P ( u , v ) ◮ Interpretation I [Γ]: I [Γ] | = P ( a , t ) for all ground terms t I [Γ] | = P ( b , t ) for all ground terms t I [Γ] �| = P ( t , t ) for t other than a and b I [Γ] | = P ( u , v ) for all distinct ground terms u and v Maria Paola Bonacina SGGS Theorem Proving: an Exposition
Outline Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion Example II: non-unit clauses, selected literals ◮ I : all negative ◮ Sequence Γ: [ P ( x )] , ¬ P ( f ( y )) ∨ [ Q ( y )] , ¬ P ( f ( z )) ∨ ¬ Q ( g ( z )) ∨ [ R ( f ( z ) , g ( z ))] ◮ Interpretation I [Γ]: I [Γ] | = P ( x ) I [Γ] | = Q ( y ) I [Γ] | = R ( f ( z ) , g ( z )) I [Γ] �| = L for all other positive literals L Maria Paola Bonacina SGGS Theorem Proving: an Exposition
Outline Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion What does a constrained clause represent? Its constrained ground instances (cgi’s) or ground instances satisfying the constraints Example: ◮ x �≡ y ✄ P ( x , y ) ◮ P ( a , b ) ∈ Gr ( x �≡ y ✄ P ( x , y )) ◮ P ( b , b ) �∈ Gr ( x �≡ y ✄ P ( x , y )) Maria Paola Bonacina SGGS Theorem Proving: an Exposition
Outline Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion Constraints ◮ Atomic constraint: true , x ≡ y , top ( t ) = f ◮ Constraint: atomic, ¬ , � , or � of constraints ◮ Standard form: � of x �≡ y , top ( x ) � = f Maria Paola Bonacina SGGS Theorem Proving: an Exposition
Outline Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion Literal selection ◮ Every literal in sequence is either I -true or I -false ◮ I -true: all cgi’s true in I ◮ I -false: all cgi’s false in I ◮ Literal tells truth value of all its cgi’s ◮ Prefer I -false literals for selection: If clause has I -false literals, one is selected ◮ I -true literal selected only if all literals I -true ( I -all-true clause) Maria Paola Bonacina SGGS Theorem Proving: an Exposition
Outline Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion SGGS clause sequence ◮ Initial interpretation I ◮ Sequence Γ = A 1 ✄ C 1 [ L 1 ] , . . . , A n ✄ C n [ L n ] ◮ Every literal is either I -true or I -false ◮ Literal L i in C i is selected ◮ If A i ✄ C i [ L i ] has I -false literals, one is selected Select I -false literals to modify I ◮ Empty sequence: ε Maria Paola Bonacina SGGS Theorem Proving: an Exposition
Outline Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion Interpretation I [Γ] represented by clause sequence Γ ◮ Partial interpretation I p (Γ | j ) for prefix Γ | j ◮ For each clause A j ✄ C j [ L j ] take its proper constrained ground instances (pcgi): ◮ Not satisfied by I p (Γ | j − 1 ) ◮ Satisfiable by adding the pcgi of L j ◮ I [Γ]: complete I p (Γ) by consulting I whenever I p (Γ) does not determine truth value ◮ I [Γ] is I modified to satisfy the pcgi’s of the selected literals Maria Paola Bonacina SGGS Theorem Proving: an Exposition
Outline Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion Example ◮ I : all negative ◮ Sequence Γ: [ P ( x )] , top ( y ) � = g ✄ [ Q ( y )] , z �≡ c ✄ [ Q ( g ( z ))] ◮ Interpretation I [Γ]: I [Γ] | = P ( x ) I [Γ] | = Q ( t ) for all ground terms t whose top symbol is not g I [Γ] | = Q ( g ( t )) for all ground terms t other than c I [Γ] �| = L for all other positive literals L Maria Paola Bonacina SGGS Theorem Proving: an Exposition
Outline Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion Induced partial interpretation I ◮ Defined inductively over length of clause sequence ◮ Each constrained clause in sequence may contribute ◮ Prefix of length j , 1 ≤ j ≤ n : Γ | j = A 1 ✄ C 1 [ L 1 ] , . . . , A j ✄ C j [ L j ] Maria Paola Bonacina SGGS Theorem Proving: an Exposition
Outline Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion Proper constrained ground instances ◮ A ✄ C [ L ] ◮ Interpretation J ◮ Proper constrained ground instance (pcgi) of A ✄ C [ L ] wrt J : constrained ground instance C ′ [ L ′ ]: ◮ Not satisfied: J ∩ C ′ [ L ′ ] = ∅ ◮ Satisfiable by adding L ′ : ¬ L ′ �∈ J Maria Paola Bonacina SGGS Theorem Proving: an Exposition
Outline Motivation: Why SGGS? Model representation Inferences Refutational Completeness Goal Sensitivity Discussion Induced partial interpretation II ◮ Initial interpretation I ◮ Sequence Γ = A 1 ✄ C 1 [ L 1 ] , . . . , A n ✄ C n [ L n ] ◮ Induced partial interpretation I p (Γ | j ): ◮ j = 0: empty sequence: empty interpretation ◮ j > 0: Take pcgi’s of A j ✄ C j [ L j ] wrt I p (Γ | j − 1 ) ◮ Take instances of L j in those pcgi’s ◮ Add them to I p (Γ | j − 1 ) to build I p (Γ | j ) ◮ Each clause adds the pcgi’s of its selected literal Maria Paola Bonacina SGGS Theorem Proving: an Exposition
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