Standard Completeness I: Proof Theoretic Approach Agata Ciabattoni Vienna University of Technology (TU Vienna) Joint work with P . Baldi, N. Galatos, G. Metcalfe, L. Spendier, K. Terui
Standard Completeness Completeness of axiomatic systems with respect to algebras whose lattice reduct is the real unit interval [ 0 , 1 ] .
Standard Completeness Completeness of axiomatic systems with respect to algebras whose lattice reduct is the real unit interval [ 0 , 1 ] . Why is the topic relevant for this workshop?
Standard Completeness Completeness of axiomatic systems with respect to algebras whose lattice reduct is the real unit interval [ 0 , 1 ] . Why is the topic relevant for this workshop? Why is the topic relevant? (Hajek 1998) Formalizations of Fuzzy Logic
Uninorm (based logics) Conjunction and implication are interpreted by a particular uninorm/t-norm (or a class of) and its residuum. A uninorm is a function ∗ : [ 0 , 1 ] 2 → [ 0 , 1 ] such that for all x , y , z ∈ [ 0 , 1 ] : x ∗ y = y ∗ x (Commutativity) ( x ∗ y ) ∗ z = x ∗ ( y ∗ z ) (Associativity) x ≤ y implies x ∗ z ≤ y ∗ z (Monotonicity) e ∈ [ 0 , 1 ] e ∗ x = x (Identity) The residuum of ∗ is a function ⇒ ∗ : [ 0 , 1 ] 2 → [ 0 , 1 ] where x ⇒ ∗ y = max { z | x ∗ z ≤ y } . A t-norm is a uninorm in which e = 1.
Some standard complete logics v : Propositions → [ 0 , 1 ] G¨ odel logic v ( A ∧ B ) = min { v ( A ) , v ( B ) } v ( A ∨ B ) = max { v ( A ) , v ( B ) } v ( A → B ) = 1 if v ( A ) ≤ v ( B ) , and v ( B ) otherwise v ( ⊥ ) = 0 UL Uninorm logic (Metcalfe, Montagna 2007) v ( A ⊙ B ) = v ( A ) ∗ v ( B ) , ∗ left continous uninorm v ( A ∨ B ) = max { v ( A ) , v ( B ) } v ( A → B ) = v ( A ) ⇒ ∗ v ( B ) v ( ⊥ ) = 0 MTL Monoidal T-norm logic (Godo, Esteva 2001) ∗ left continous t-norm
(Uninorm-based) Logics often described by adding axioms to already known logics. Example UL = FLe with (( α → β ) ∧ t ) ∨ (( β → α ) ∧ t ) (linearity) MTL = UL with weakening/integrality G¨ odel logic = MTL with contraction α → α ⊙ α SUL = UL with α → α ⊙ α and mingle α ⊙ α → α WMTL = MTL with ¬ ( α ⊙ β ) ∨ ( α ∧ β → α ⊙ β ) ....
(Uninorm-based) Logics are often described by adding axioms to already known logics. Question Given a logic L obtained by extending UL with α ⊙ α → α (mingle)? α n − 1 → α n ( n -contraction)? ¬ ( α ⊙ β ) n ∨ (( α ∧ β ) n − 1 → ( α ⊙ β ) n ) ? .... Is L standard complete? ( is it a formalization of Fuzzy Logic? )
(Uninorm-based) Logics are often described by adding axioms to already known logics. Question Given a logic L obtained by extending UL with α ⊙ α → α (mingle)? α n − 1 → α n ( n -contraction)? ¬ ( α ⊙ β ) n ∨ (( α ∧ β ) n − 1 → ( α ⊙ β ) n ) ? .... Is L standard complete? ( is it a formalization of Fuzzy Logic? ) Many papers written for individual logics!
Standard Completeness: algebraic approach Given a logic L : 1 Identify the algebraic semantics of L ( L -algebras) 2 Show completeness of L w.r.t. linear, countable L -algebras 3 (Rational completeness): Find an embedding into linear, dense countable L -algebras 4 Dedekind-Mac Neille style completion (embedding into L -algebras with lattice reduct [ 0 , 1 ] )
Standard Completeness: algebraic approach Given a logic L : 1 Identify the algebraic semantics of L ( L -algebras) 2 Show completeness of L w.r.t. linear, countable L -algebras 3 (Rational completeness): Find an embedding into linear, dense countable L -algebras 4 Dedekind-Mac Neille style completion (embedding into L -algebras with lattice reduct [ 0 , 1 ] ) Step 3: problematic (mainly ( ∗ ) ad hoc solutions) ( ∗ ) see Paolo’s talk!
Standard Completeness: proof theoretic approach (Metcalfe, Montagna JSL 2007) Given a logic L : Add Takeuti and Titani’s density rule ( p eigenvariable) ( α → p ) ∨ ( p → β ) ∨ γ ( density ) ( α → β ) ∨ γ (= L + ( density ) is rational complete) Show that density produces no new theorems (Rational completeness) Dedekind-Mac Neille style completion
Our result Uniform (and automated) proofs of standard completeness for large classes of axiomatic extensions of UL How?
Our result Uniform (and automated) proofs of standard completeness for large classes of axiomatic extensions of UL How? (Step 1) Defining suitable calculi for axiomatic extensions of UL (Step 2) General conditions for the elimination of the density rule from these calculi (Step 3) Dedekind-Mac Neille style completion
Our result Uniform (and automated) proofs of standard completeness for large classes of axiomatic extensions of UL How? (Step 1) Defining suitable calculi for axiomatic extensions of UL (Step 2) General conditions for the elimination of the density rule from these calculi (Step 3) Dedekind-Mac Neille style completion (Avron JSL ’89) Hypersequents: Γ 1 ⇒ Π 1 | . . . | Γ n ⇒ Π n where for all i = 1 , . . . n , Γ i ⇒ Π i is an ordinary sequent
Our base calculus: FLe ( tr ) ( fl ) ( init ) α ⇒ α ⇒ t f ⇒ ( ⊤ ) ( ⊥ ) Γ ⇒ ⊤ Γ , ⊥ ⇒ ∆ Γ ⇒ α α, ∆ ⇒ Π Γ ⇒ Π Γ ⇒ ( tl ) ( fr ) ( cut ) t , Γ ⇒ Π Γ ⇒ f Γ , ∆ ⇒ Π Γ ⇒ α Γ ⇒ β α i , Γ ⇒ Π Γ ⇒ α i ( ∧ r ) ( ∧ l ) ( ∨ r ) Γ ⇒ α ∧ β α 1 ∧ α 2 , Γ ⇒ Π Γ ⇒ α 1 ∨ α 2 α, Γ ⇒ Π β, Γ ⇒ Π Γ ⇒ α β, ∆ ⇒ Π α, Γ ⇒ β ( ∨ l ) ( → l ) ( → r ) α ∨ β, Γ ⇒ Π Γ , α → β, ∆ ⇒ Π Γ ⇒ α → β Γ ⇒ α ∆ ⇒ β α, β, Γ ⇒ Π ( ⊙ r ) ( ⊙ l ) Γ , ∆ ⇒ α ⊙ β α ⊙ β, Γ ⇒ Π
Calculi for axiomatic extensions of FLe E.g. UL = FLe + (( α → β ) ∧ t ) ∨ (( β → α ) ∧ t ) (linearity) Cut elimination is not preserved when axioms are added (Idea) Axioms are transformed into ‘good’ structural rules in the ‘appropriate’ formalism
Hypersequent Calculus for UL (UL = FLe + (( α → β ) ∧ t ) ∨ (( β → α ) ∧ t )) Hypersequent: Γ 1 ⇒ Π 1 | . . . | Γ n ⇒ Π n This calculus is obtained embedding sequents into hypersequents in FLe α, Γ ⇒ β Γ ⇒ α → β ( → , r ) i.e. G | α, Γ ⇒ β G | Γ ⇒ α → β ( → , r )
Hypersequent Calculus for UL (UL = FLe + (( α → β ) ∧ t ) ∨ (( β → α ) ∧ t )) Hypersequent: Γ 1 ⇒ Π 1 | . . . | Γ n ⇒ Π n This calculus is obtained embedding sequents into hypersequents in FLe adding suitable rules to manipulate the additional layer of structure. G | Γ ⇒ α | Γ ⇒ α G G | Γ ⇒ α ( ew ) ( ec ) G | Γ ⇒ α
Hypersequent Calculus for UL (UL = FLe + (( α → β ) ∧ t ) ∨ (( β → α ) ∧ t )) Hypersequent: Γ 1 ⇒ Π 1 | . . . | Γ n ⇒ Π n This calculus is obtained embedding sequents into hypersequents in FLe adding suitable rules to manipulate the additional layer of structure. G | Γ ⇒ α | Γ ⇒ α G G | Γ ⇒ α ( ew ) ( ec ) G | Γ ⇒ α G | Γ , Γ ′ ⇒ α G | Γ 1 , Γ ′ 1 ⇒ α ′ ( com ) G | Γ , Γ 1 ⇒ α | Γ ′ , Γ ′ 1 ⇒ α ′ (Avron 1991)
An example β ⇒ β α ⇒ α ( com ) α ⇒ β | β ⇒ α ( → , r ) α ⇒ β | ⇒ β → α ( → , r ) ⇒ t ⇒ α → β | ⇒ β → α ⇒ t 2x ( ∧ , r ) ⇒ ( α → β ) ∧ t | ⇒ ( β → α ) ∧ t ( ∨ i , r ) ⇒ ( α → β ) ∧ t | ⇒ (( α → β ) ∧ t ) ∨ (( β → α ) ∧ t ) ( ∨ i , r ) ⇒ (( α → β ) ∧ t ) ∨ (( β → α ) ∧ t ) | ⇒ (( α → β ) ∧ t ) ∨ (( β → α ) ∧ t ) ( EC ) ⇒ (( α → β ) ∧ t ) ∨ (( β → α ) ∧ t )
Our result Uniform (and automated) proofs of standard completeness for large classes of axiomatic extensions of UL (Step 1) Defining suitable calculi for axiomatic extensions of UL (Step 2) General conditions for the elimination of the density rule from these calculi (Step 3) Dedekind-Mac Neille style completion
Algorithmic introduction of analytic calculi I Definition (Classification; -, Galatos and Terui, LICS 2008) The classes P n , N n of positive and negative axioms/equations are: P 0 ::= N 0 ::= atomic formulas P n + 1 ::= N n | P n + 1 ∨ P n + 1 | P n + 1 ⊙ P n + 1 | t | ⊥ N n + 1 ::= P n | P n + 1 → N n + 1 | N n + 1 ∧ N n + 1 | f | ⊤
Examples Class Axiom Name N 2 α → t , ⊥ → α weakening α → α ⊙ α contraction α ⊙ α → α expansion α n → α m knotted axioms ¬ ( α ∧ ¬ α ) weak contraction P 2 α ∨ ¬ α excluded middle ( α → β ) ∨ ( β → α ) prelinearity P 3 ¬ α ∨ ¬¬ α weak excluded middle ¬ ( α ⊙ β ) ∨ ( α ∧ β → α ⊙ β ) (wnm) N 3 (( α → β ) → β ) → (( β → α ) → α ) L ukasiewicz axiom ( α ∧ β ) → α ⊙ ( α → β ) divisibility
Algorithmic introduction of analytic calculi II Theorem (AC, Galatos, Terui 2008) Algorithm to transform (almost all) axioms α up to the class N 2 into good structural rules in sequent calculus axioms α up to the class P 3 into good structural rules in hypersequent calculus
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