Transitive Closure Logic Infinitary and Cyclic Proof Systems 1 - PowerPoint PPT Presentation

Transitive Closure Logic Infinitary and Cyclic Proof Systems 1 School of Computing, University of Kent, Canterbury, UK 2 Dept of Computer Science, Cornell University, Ithaca, NY, USA Reuben N. S. Rowe 1 Liron Cohen 2 PARIS Workshop @ FLoC, Sunday 8 th July 2018, Oxford, UK

whose intended meaning is an infinite disjunction s x w 1 y w 1 x t y w 1 w 2 s x w 1 y w 1 x w 2 y w 2 x t y Transitive Closure ( TC ) Logic extends FOL with formulas: s x t y w 1 t s • s and t are terms 1 • ( RTC x , y φ )( s , t ) • φ is a formula • x and y are distinct variables (which become bound in φ )

Transitive Closure ( TC ) Logic extends FOL with formulas: • s and t are terms 1 • ( RTC x , y φ )( s , t ) • φ is a formula • x and y are distinct variables (which become bound in φ ) whose intended meaning is an infinite disjunction s = t ∨ φ [ s / x , t / y ] ∨ ( ∃ w 1 . φ [ s / x , w 1 / y ] ∧ φ [ w 1 / x , t / y ]) ∨ ( ∃ w 1 , w 2 . φ [ s / x , w 1 / y ] ∧ φ [ w 1 / x , w 2 / y ] ∧ φ [ w 2 / x , t / y ]) ∨ . . .

a i y The formal semantics: 1 v t v s a n 1 a n a 2 a 1 a 0 n for all i a i • M is a (standard) first-order model with domain D M v x a n v t a 0 D v s a n a 0 • v is a valuation of terms in M : 2 M , v | = ( RTC x , y φ )( s , t )

a i y The formal semantics: v t v s a n a 2 a 1 a 0 n for all i 1 a i • M is a (standard) first-order model with domain D M v x a n v t a 0 v s • v is a valuation of terms in M : 2 M , v | = ( RTC x , y φ )( s , t ) ⇔ ∃ a 0 , . . . , a n ∈ D . . . a n − 1

a i y The formal semantics: • M is a (standard) first-order model with domain D a n a 2 a 1 a 0 n for all i 1 a i M v x • v is a valuation of terms in M : 2 M , v | = ( RTC x , y φ )( s , t ) ⇔ ∃ a 0 , . . . , a n ∈ D . v ( s ) = a 0 ∧ v ( t ) = a n . . . v ( s ) v ( t ) a n − 1

The formal semantics: a 0 a n a 2 • M is a (standard) first-order model with domain D a 1 2 • v is a valuation of terms in M : M , v | = ( RTC x , y φ )( s , t ) ⇔ ∃ a 0 , . . . , a n ∈ D . v ( s ) = a 0 ∧ v ( t ) = a n ∧ M , v [ x := a i , y := a i + 1 ] | = φ for all i < n ϕ ϕ ϕ ϕ v ( s ) v ( t ) a n − 1

Why ‘Transitive Closure’ logic? ‘denotes’ the reflexive, transitive closure of M v x y v t v s s t RTC x y M v : RTC x y • Consider the binary relation induced by • b a y M v x a b M v x y (wrt. x and y ): 3

Why ‘Transitive Closure’ logic? • RTC x y ‘denotes’ the reflexive, transitive closure of : M v RTC x y s t v s v t x y M v 3 • Consider the binary relation induced by φ (wrt. x and y ): � φ ( x , y ) � M , v = { ( a , b ) | M , v [ x := a , y := b ] | = φ }

Why ‘Transitive Closure’ logic? 3 • Consider the binary relation induced by φ (wrt. x and y ): � φ ( x , y ) � M , v = { ( a , b ) | M , v [ x := a , y := b ] | = φ } • ( RTC x , y φ ) ‘denotes’ the reflexive, transitive closure of φ : = ( RTC x , y φ )( s , t ) ⇔ ( v ( s ) , v ( t )) ∈ ( � φ ( x , y ) � M , v ) ∗ M , v |

Why Transitive Closure logic? • It has an intuitive, easy-to-understand semantics • It turns out to be surprisingly expressive Theorem (Avron ’03) All finitely inductively defined relations are definable in TC . 4 • It is a minimal extension of FOL

Why Transitive Closure logic? • It has an intuitive, easy-to-understand semantics • It turns out to be surprisingly expressive Theorem (Avron ’03) A. Avron, Transitive Closure and the Mechanization of Mathematics , 2003. * as defined in: S. Feferman, Finitary Inductively Presented Logics , 1989 † with signatures containing a pairing function 4 • It is a minimal extension of FOL All finitely inductively defined relations * are definable in TC . †

RTC v w n 1 n 2 v n 1 n 2 s n 1 s n 2 s n 1 0 s z 0 s z y s s 0 s s 0 v x s Example: Arithmetic s x Nat x s 0 y s 0 s y s s 0 s s y 0 s y y x x y s x 0 x s x natural numbers in TC : • The following axioms categorically characterise the z x 0 y w z ” y “ x 5 • Take a signature Σ = { 0 , s } + equality and pairing Nat ( x ) ≡ ( RTC v , w s v = w )( 0 , x )

RTC v w n 1 n 2 v n 1 n 2 s n 1 s n 2 s z 0 s z y Example: Arithmetic x y s x s s 0 s s y s 0 s y 0 y s s 0 s 0 0 x Nat x y x s y 0 w “ x y z ” 5 0 y z x • The following axioms categorically characterise the natural numbers in TC : x s x • Take a signature Σ = { 0 , s } + equality and pairing Nat ( x ) ≡ ( RTC v , w s v = w )( 0 , x ) s · = · s · = · s · = · s · = · s n - 1 0 v ( x )

s n 1 0 s z 0 s z y Example: Arithmetic s 0 s s 0 s s y s 0 s y 0 y s s s s v x s s 0 0 x Nat x y x s y x y s x 0 x s x natural numbers in TC : • The following axioms categorically characterise the 5 • Take a signature Σ = { 0 , s } + equality and pairing Nat ( x ) ≡ ( RTC v , w s v = w )( 0 , x ) “ x = y + z ” ≡ ( RTC v , w ∃ n 1 , n 2 . v = ⟨ n 1 , n 2 ⟩ ∧ w = ⟨ s n 1 , s n 2 ⟩ )( ⟨ 0 , y ⟩ , ⟨ z , x ⟩ )

s n 1 0 s z 0 s z y Example: Arithmetic s 0 s s 0 s s y s 0 s y s s s s v x s s 0 0 x Nat x y x s y x y s x 0 x s x natural numbers in TC : • The following axioms categorically characterise the 5 • Take a signature Σ = { 0 , s } + equality and pairing Nat ( x ) ≡ ( RTC v , w s v = w )( 0 , x ) “ x = y + z ” ≡ ( RTC v , w ∃ n 1 , n 2 . v = ⟨ n 1 , n 2 ⟩ ∧ w = ⟨ s n 1 , s n 2 ⟩ )( ⟨ 0 , y ⟩ , ⟨ z , x ⟩ ) ⟨ 0 , y ⟩

s n 1 0 s z 0 s z y Example: Arithmetic x Nat x s s 0 s s y s s s s v x s s 0 s 0 0 y x s y x y s x 0 x s x natural numbers in TC : • The following axioms categorically characterise the 5 • Take a signature Σ = { 0 , s } + equality and pairing Nat ( x ) ≡ ( RTC v , w s v = w )( 0 , x ) “ x = y + z ” ≡ ( RTC v , w ∃ n 1 , n 2 . v = ⟨ n 1 , n 2 ⟩ ∧ w = ⟨ s n 1 , s n 2 ⟩ )( ⟨ 0 , y ⟩ , ⟨ z , x ⟩ ) ⟨ 0 , y ⟩ ⟨ s 0 , s y ⟩

s n 1 0 s z 0 s z y Example: Arithmetic x Nat x s s s s v x s s 0 s 0 0 y x s y x y s x 0 x s x natural numbers in TC : • The following axioms categorically characterise the 5 • Take a signature Σ = { 0 , s } + equality and pairing Nat ( x ) ≡ ( RTC v , w s v = w )( 0 , x ) “ x = y + z ” ≡ ( RTC v , w ∃ n 1 , n 2 . v = ⟨ n 1 , n 2 ⟩ ∧ w = ⟨ s n 1 , s n 2 ⟩ )( ⟨ 0 , y ⟩ , ⟨ z , x ⟩ ) ⟨ 0 , y ⟩ ⟨ s 0 , s y ⟩ ⟨ s s 0 , s s y ⟩

s n 1 0 Example: Arithmetic y s s s s v x s s 0 s 0 x Nat x 0 x s y x y s x 0 x s x natural numbers in TC : • The following axioms categorically characterise the 5 • Take a signature Σ = { 0 , s } + equality and pairing Nat ( x ) ≡ ( RTC v , w s v = w )( 0 , x ) “ x = y + z ” ≡ ( RTC v , w ∃ n 1 , n 2 . v = ⟨ n 1 , n 2 ⟩ ∧ w = ⟨ s n 1 , s n 2 ⟩ )( ⟨ 0 , y ⟩ , ⟨ z , x ⟩ ) ⟨ 0 , y ⟩ ⟨ s 0 , s y ⟩ ⟨ s s 0 , s s y ⟩ ⟨ s z 0 , s z y ⟩

s n 1 0 s z 0 s z y Example: Arithmetic s s 0 s s y s 0 s y 0 y s s s s v x s s 0 s 0 0 natural numbers in TC : • The following axioms categorically characterise the 5 • Take a signature Σ = { 0 , s } + equality and pairing Nat ( x ) ≡ ( RTC v , w s v = w )( 0 , x ) “ x = y + z ” ≡ ( RTC v , w ∃ n 1 , n 2 . v = ⟨ n 1 , n 2 ⟩ ∧ w = ⟨ s n 1 , s n 2 ⟩ )( ⟨ 0 , y ⟩ , ⟨ z , x ⟩ ) ∀ x . s x ̸ = 0 ∀ x , y . s ( x ) = s ( y ) → x = y ∀ x . Nat ( x )

Applications e.g. SQL3, IBM DB2, Datalog J. Halpern Et Al, On the Unusual Effectiveness of Logic in Computer Science , 2001 defined inductively Common knowledge, Reachability properties of type judgments Inductive definition complexity classes Characterization of (WITH RECURSIVE) Expressive query languages, of Logic in CS data in programs Loops/inductive Databases Verification Complexity Type Theory Checking Model Reasoning Knowledge 6

Applications e.g. SQL3, IBM DB2, Datalog J. Halpern Et Al, On the Unusual Effectiveness of Logic in Computer Science , 2001 defined inductively Common knowledge, Reachability properties of type judgments Inductive definition complexity classes Characterization of (WITH RECURSIVE) Expressive query languages, of Logic in CS data in programs Loops/inductive Databases Verification Complexity Type Theory Checking Model Reasoning Knowledge 6

FOL SOL TC Weak SOL -logic Cardinality logic FOL + Henkin Quantifiers FOM FOL + ML Ind. Defs “Everything should be made as simple as possible but not simpler” —Albert Einsten 7

FOL SOL TC Weak SOL -logic Cardinality logic FOL + Henkin Quantifiers FOM FOL + ML Ind. Defs “Everything should be made as simple as possible but not simpler” —Albert Einsten 7

FOL SOL TC Weak SOL Cardinality logic FOL + Henkin Quantifiers FOM FOL + ML Ind. Defs “Everything should be made as simple as possible but not simpler” —Albert Einsten 7 ω -logic

FOL SOL TC Weak SOL Cardinality logic FOL + Henkin Quantifiers FOL + ML Ind. Defs “Everything should be made as simple as possible but not simpler” —Albert Einsten 7 ω -logic FOM µ

where, for binary relations R and S , we define The transitive closure 8 R + = ∪ R i , where R 0 = R i ≥ 0 R i + 1 = R i ◦ R ( i ≥ 0 ) is a particular kind of fixed point: R + = µ X . Ψ R ( X ) Ψ R ( S ) = R ∪ ( R ◦ S )