CERES for Propositional Proof Schemata Mikheil Rukhaia joint work - PowerPoint PPT Presentation

CERES for Propositional Proof Schemata Mikheil Rukhaia joint work with T. Dunchev, A. Leitsch and D. Weller Institute of Computer Languages, Vienna University of Technology. Workshop on Structural and Computational Proof Theory, Innsbruck, Austria. October 27, 2011.

Introduction Schematic LK Schematic Characteristic Clause Set Schematic Projections Ongoing and Future Work Introduction CERES for Proof Schemata M. Rukhaia Structural and Computational Proof Theory Oct 27, 2011 2 / 42

Introduction Schematic LK Schematic Characteristic Clause Set Schematic Projections Ongoing and Future Work Overview ◮ Schemata are very useful in mathematical proofs (avoids explicit use of the induction). ◮ Schemata are used on meta-level. ◮ Many problems can be expressed in propositional schema lan- guage, like: Circuit verification, Graph coloring, Pigeonhole principle, etc. CERES for Proof Schemata M. Rukhaia Structural and Computational Proof Theory Oct 27, 2011 3 / 42

Introduction Schematic LK Schematic Characteristic Clause Set Schematic Projections Ongoing and Future Work Propositional Schema Language ◮ Set of index variables is a set of variables over natural numbers. ◮ Linear arithmetic expression is as usual built on the signature 0 , s , + , − and on a set of index variables. ◮ Indexed proposition is an expression of the form p a , where a is a linear arithmetic expression. ◮ Propositional variable is an indexed proposition p a , where a ∈ N . CERES for Proof Schemata M. Rukhaia Structural and Computational Proof Theory Oct 27, 2011 4 / 42

Introduction Schematic LK Schematic Characteristic Clause Set Schematic Projections Ongoing and Future Work Syntax ◮ Formula schema is defined inductively: Indexed proposition is a formula schema. If φ 1 and φ 2 are formula schemata, then so are φ 1 ∨ φ 2 , φ 1 ∧ φ 2 and ¬ φ 1 . If φ is a formula schema, a , b are linear arithmetic expressions and i is an index variable, then � b i = a φ and � b i = a φ are formula schemata, called iterations. CERES for Proof Schemata M. Rukhaia Structural and Computational Proof Theory Oct 27, 2011 5 / 42

Introduction Schematic LK Schematic Characteristic Clause Set Schematic Projections Ongoing and Future Work Semantics ◮ Interpretation is a pair of functions, I = ( I , I p ) , s.t. I maps index variables to natural numbers and I p maps propositional variables to truth values. ◮ Truth value � φ � I of a formula schema φ in an interpretation I is defined inductively: � p a � I = I p ( p I ( a ) ) . � ¬ φ � I = T iff � φ � I = F . � φ 1 ∧ ( ∨ ) φ 2 � I = T iff � φ 1 � I = T and (or) � φ 2 � I = T . � � b � �� b � I = T iff for every (there is an) integer α s.t. φ i = a i = a I ( a ) ≤ α ≤ I ( b ) , � φ � I [ α/ i ] = T . CERES for Proof Schemata M. Rukhaia Structural and Computational Proof Theory Oct 27, 2011 6 / 42

Introduction Schematic LK Schematic Characteristic Clause Set Schematic Projections Ongoing and Future Work Cut-Elimination on Proof Schemata Aim: describe syntactically sequence of cut-free proofs ( χ n ) n ∈ N obtained by cut-elimination on proof sequences ( ϕ n ) n ∈ N . Cut-free proofs of schema typically are described in meta-language. Find object language to define sequence ( χ n ) n ∈ N . CERES for Proof Schemata M. Rukhaia Structural and Computational Proof Theory Oct 27, 2011 7 / 42

Introduction Schematic LK Schematic Characteristic Clause Set Schematic Projections Ongoing and Future Work Which cut-elimination method? ◮ Reductive cut-elimination. ◮ CERES. Efficient. Strong methods of redundancy-elimination. Atomic cut-normal form is constructed via parts of the original proof. CERES for Proof Schemata M. Rukhaia Structural and Computational Proof Theory Oct 27, 2011 8 / 42

Introduction Schematic LK Schematic Characteristic Clause Set Schematic Projections Ongoing and Future Work The CERES Method ◮ CERES is a cut-elimination method by resolution. ◮ Method consists of the following steps: Skolemization of the proof (if it is not already skolemized). 1 Computation of the characteristic clause set. 2 Refutation of the characteristic clause set. 3 Computation of the Projections and construction of the Atomic 4 Cut Normal Form. CERES for Proof Schemata M. Rukhaia Structural and Computational Proof Theory Oct 27, 2011 9 / 42

Introduction Schematic LK Schematic Characteristic Clause Set Schematic Projections Ongoing and Future Work Schematic LK CERES for Proof Schemata M. Rukhaia Structural and Computational Proof Theory Oct 27, 2011 10 / 42

Introduction Schematic LK Schematic Characteristic Clause Set Schematic Projections Ongoing and Future Work Basic Notions ◮ Sequent Schema is an expression of the form Γ ⊢ ∆ , where Γ and ∆ are multisets of formula schemata. ◮ Initial Sequent Schema is an expression of the form A ⊢ A , where A is an indexed proposition. ◮ Proof Link is a tuple ( ϕ, t ) , where ϕ is a proof name and t is a linear arithmetic expression. CERES for Proof Schemata M. Rukhaia Structural and Computational Proof Theory Oct 27, 2011 11 / 42

Introduction Schematic LK Schematic Characteristic Clause Set Schematic Projections Ongoing and Future Work Calculus LKS ◮ Axioms: initial sequent schemata or proof links. ◮ Rules: CERES for Proof Schemata M. Rukhaia Structural and Computational Proof Theory Oct 27, 2011 12 / 42

Introduction Schematic LK Schematic Characteristic Clause Set Schematic Projections Ongoing and Future Work Calculus LKS ◮ Axioms: initial sequent schemata or proof links. ◮ Rules: ∧ introduction: A , Γ ⊢ ∆ B , Γ ⊢ ∆ ∧ : l 1 ∧ : l 2 A ∧ B , Γ ⊢ ∆ A ∧ B , Γ ⊢ ∆ Γ ⊢ ∆ , A Π ⊢ Λ , B ∧ : r Γ , Π ⊢ ∆ , Λ , A ∧ B i = 0 A i ) ∧ A n + 1 ≡ � n + 1 Equivalences: A 0 ≡ � 0 i = 0 A i and ( � n i = 0 A i CERES for Proof Schemata M. Rukhaia Structural and Computational Proof Theory Oct 27, 2011 12 / 42

Introduction Schematic LK Schematic Characteristic Clause Set Schematic Projections Ongoing and Future Work Calculus LKS ◮ Axioms: initial sequent schemata or proof links. ◮ Rules: ∨ introduction: A , Γ ⊢ ∆ B , Π ⊢ Λ ∨ : l A ∨ B , Γ , Π ⊢ ∆ , Λ Γ ⊢ ∆ , A Γ ⊢ ∆ , B ∨ : r 1 ∨ : r 2 Γ ⊢ ∆ , A ∨ B Γ ⊢ ∆ , A ∨ B i = 0 A i ) ∨ A n + 1 ≡ � n + 1 Equivalences: A 0 ≡ � 0 i = 0 A i and ( � n i = 0 A i CERES for Proof Schemata M. Rukhaia Structural and Computational Proof Theory Oct 27, 2011 12 / 42

Introduction Schematic LK Schematic Characteristic Clause Set Schematic Projections Ongoing and Future Work Calculus LKS ◮ Axioms: initial sequent schemata or proof links. ◮ Rules: ¬ introduction: Γ ⊢ ∆ , A A , Γ ⊢ ∆ ¬ : r ¬ : l ¬ A , Γ ⊢ ∆ Γ ⊢ ∆ , ¬ A CERES for Proof Schemata M. Rukhaia Structural and Computational Proof Theory Oct 27, 2011 12 / 42

Introduction Schematic LK Schematic Characteristic Clause Set Schematic Projections Ongoing and Future Work Calculus LKS ◮ Axioms: initial sequent schemata or proof links. ◮ Rules: Weakening rules: Γ ⊢ ∆ Γ ⊢ ∆ w : r w : l A , Γ ⊢ ∆ Γ ⊢ ∆ , A CERES for Proof Schemata M. Rukhaia Structural and Computational Proof Theory Oct 27, 2011 12 / 42

Introduction Schematic LK Schematic Characteristic Clause Set Schematic Projections Ongoing and Future Work Calculus LKS ◮ Axioms: initial sequent schemata or proof links. ◮ Rules: Contraction rules: A , A , Γ ⊢ ∆ c : l Γ ⊢ ∆ , A , A c : r A , Γ ⊢ ∆ Γ ⊢ ∆ , A CERES for Proof Schemata M. Rukhaia Structural and Computational Proof Theory Oct 27, 2011 12 / 42

Introduction Schematic LK Schematic Characteristic Clause Set Schematic Projections Ongoing and Future Work Calculus LKS ◮ Axioms: initial sequent schemata or proof links. ◮ Rules: Cut rule: Γ ⊢ ∆ , A A , Π ⊢ Λ cut Γ , Π ⊢ ∆ , Λ CERES for Proof Schemata M. Rukhaia Structural and Computational Proof Theory Oct 27, 2011 12 / 42

Introduction Schematic LK Schematic Characteristic Clause Set Schematic Projections Ongoing and Future Work LKS -proof ◮ Derivation is a directed tree with nodes as sequences and edges as rules. ◮ LKS -proof of the sequence S is a derivation of S with axioms as leaf nodes. ◮ An LKS -proof is called ground if it does not contain free param- eters, index variables, or proof links. CERES for Proof Schemata M. Rukhaia Structural and Computational Proof Theory Oct 27, 2011 13 / 42

Introduction Schematic LK Schematic Characteristic Clause Set Schematic Projections Ongoing and Future Work Proof Schemata ◮ Proof schema ψ is a tuple of pairs � ( ψ 1 base , ψ 1 step ) , . . . , ( ψ m base , ψ m step ) � such that: ψ 1 ≺ ψ 2 ≺ · · · ≺ ψ m , base is a ground LKS -proof of S i { n ← 0 } , for i ∈ { 1 , . . . , m } , ψ i step is an LKS -proof of S i { n ← k + 1 } , where k is an index vari- ψ i able, and ψ i step contains proof links of the form (for i ≺ j ): ( ψ i , k ) ( ψ j , k j ) or S i { n ← k } S j � n ← k j � ◮ From now on m = 1. CERES for Proof Schemata M. Rukhaia Structural and Computational Proof Theory Oct 27, 2011 14 / 42