Induction, Transitive Closure and Cycles Liron Cohen, Cornell - PowerPoint PPT Presentation

Induction, Transitive Closure and Cycles Liron Cohen, Cornell University, Reuben Rowe, University of Kent ASL North American Annual Meeting, 2018

Verification Database Complexity Applications MKM of Logic in CS Type Theory Knowledge Reasoning Model Checking Halpern, Harper, Immerman, Kolaitis, Vardi, Vianu. On the unusual effectiveness of logic in computer science, 2001

Inductive arguments on programs Verification Database Complexity Applications MKM of Logic in CS Type Theory Knowledge Reasoning Model Checking Halpern, Harper, Immerman, Kolaitis, Vardi, Vianu. On the unusual effectiveness of logic in computer science, 2001

Inductive arguments on programs Expressive Query languages (WITH RECURSIVE) Verification SQL3, IBM DB2, Datalog Database Complexity Applications MKM of Logic in CS Type Theory Knowledge Reasoning Model Checking Halpern, Harper, Immerman, Kolaitis, Vardi, Vianu. On the unusual effectiveness of logic in computer science, 2001

Inductive arguments on programs Expressive Query languages (WITH RECURSIVE) Verification SQL3, IBM DB2, Datalog Database Characterization of complexity classes Complexity Applications MKM of Logic in CS Type Theory Knowledge Reasoning Model Checking Halpern, Harper, Immerman, Kolaitis, Vardi, Vianu. On the unusual effectiveness of logic in computer science, 2001

Inductive arguments on programs Expressive Query languages (WITH RECURSIVE) Verification SQL3, IBM DB2, Datalog Database Characterization of complexity classes Complexity Applications MKM of Logic in CS Type Theory Knowledge Inductive definition of Reasoning type judgments Model Checking Halpern, Harper, Immerman, Kolaitis, Vardi, Vianu. On the unusual effectiveness of logic in computer science, 2001

Inductive arguments on programs Expressive Query languages (WITH RECURSIVE) Verification SQL3, IBM DB2, Datalog Database Characterization of complexity classes Complexity Applications MKM of Logic in CS Type Theory Knowledge Inductive definition of Reasoning type judgments Model Checking Reachability properties Halpern, Harper, Immerman, Kolaitis, Vardi, Vianu. On the unusual effectiveness of logic in computer science, 2001

Inductive arguments on programs Expressive Query languages (WITH RECURSIVE) Verification SQL3, IBM DB2, Datalog Database Characterization of complexity classes Complexity Applications MKM of Logic in CS Type Theory Knowledge Inductive definition of Reasoning type judgments Model Checking Common knowledge, defined inductively Reachability properties Halpern, Harper, Immerman, Kolaitis, Vardi, Vianu. On the unusual effectiveness of logic in computer science, 2001

Inductive arguments on programs Expressive Query languages (WITH RECURSIVE) Verification SQL3, IBM DB2, Datalog Database Characterization of complexity classes Complexity Natural numbers Applications MKM of Logic in CS Type Theory Knowledge Inductive definition of Reasoning type judgments Model Checking Common knowledge, defined inductively Reachability properties Halpern, Harper, Immerman, Kolaitis, Vardi, Vianu. On the unusual effectiveness of logic in computer science, 2001

Inductive arguments on programs Expressive Query languages (WITH RECURSIVE) Verification SQL3, IBM DB2, Datalog Database Characterization of complexity classes Complexity Natural numbers Applications MKM of Logic in CS Type Theory What Logic? Knowledge Inductive definition of Reasoning type judgments Model Checking Common knowledge, defined inductively Reachability properties Halpern, Harper, Immerman, Kolaitis, Vardi, Vianu. On the unusual effectiveness of logic in computer science, 2001

What Logic? FOL SOL

What Logic? FOL SOL No inductive machinery

What Logic? FOL SOL No inductive machinery Overkill

What Logic? FOL SOL No inductive machinery Overkill natural, effective extensions of FOL that allow inductive definitions

What Logic? FOL SOL No inductive machinery Overkill natural, effective extensions of FOL that allow inductive definitions Transitive Closure Logic

Transitive Closure Logic Transitive Closure Logic = FOL + a transitive closure operator.

Transitive Closure Logic Transitive Closure Logic = FOL + a transitive closure operator. The transitive closure R ∗ of binary relation R is defined by: R ∗ = R ( n ) � where R (0) = Id , R ( n +1) = R ( n ) ◦ R .

Transitive Closure Logic Transitive Closure Logic = FOL + a transitive closure operator. The transitive closure R ∗ of binary relation R is defined by: R ∗ = R ( n ) � where R (0) = Id , R ( n +1) = R ( n ) ◦ R . Alternatively, R ∗ = Id ∪ � { S | R ∪ S ◦ R ⊆ S } (Least fixed point of the composition operator)

Why Transitive Closure Logic? • The concept of the transitive closure is truly basic.

Why Transitive Closure Logic? • The concept of the transitive closure is truly basic. • Being a ‘descendent of’ • The natural numbers • Well-formed formulas

Why Transitive Closure Logic? • The concept of the transitive closure is truly basic. • Being a ‘descendent of’ • The natural numbers • Well-formed formulas • A minimal extension.

Why Transitive Closure Logic? • The concept of the transitive closure is truly basic. • Being a ‘descendent of’ • The natural numbers • Well-formed formulas • A minimal extension. • A special case of a least fixed point.

Why Transitive Closure Logic? • The concept of the transitive closure is truly basic. • Being a ‘descendent of’ • The natural numbers • Well-formed formulas • A minimal extension. • A special case of a least fixed point. • Equivalent to other extensions of FOL , but the most convenient from a proof theoretical perspective.

Why Transitive Closure Logic? • The concept of the transitive closure is truly basic. • Being a ‘descendent of’ • The natural numbers • Well-formed formulas • A minimal extension. • A special case of a least fixed point. • Equivalent to other extensions of FOL , but the most convenient from a proof theoretical perspective. • Captures inductive principles in a uniform way.

Why Transitive Closure Logic? • The concept of the transitive closure is truly basic. • Being a ‘descendent of’ • The natural numbers • Well-formed formulas • A minimal extension. • A special case of a least fixed point. • Equivalent to other extensions of FOL , but the most convenient from a proof theoretical perspective. • Captures inductive principles in a uniform way. • Not parametrized by a set of inductive principles.

The Language The Language The language L TC is defined as L FOL , with the additional clause: • ( RTC x , y ϕ )( s , t ) is a formula, for ϕ a formula, x , y distinct variables, and s , t terms. ( x , y become bound in this formula.)

The Language The Language The language L TC is defined as L FOL , with the additional clause: • ( RTC x , y ϕ )( s , t ) is a formula, for ϕ a formula, x , y distinct variables, and s , t terms. ( x , y become bound in this formula.) Allows for: • Rich testing • Nested RTC

The Semantics The Intended Meaning of ( RTC x , y ϕ )( s , t ) s = t ∨ ϕ ( s , t ) ∨ ∃ w 1 .ϕ ( s , w 1 ) ∧ ϕ ( w 1 , t ) ∨ ∃ w 1 ∃ w 2 .ϕ ( s , w 1 ) ∧ ϕ ( w 1 , w 2 ) ∧ ϕ ( w 2 , t ) ∨ ...

The Semantics The Intended Meaning of ( RTC x , y ϕ )( s , t ) s = t ∨ ϕ ( s , t ) ∨ ∃ w 1 .ϕ ( s , w 1 ) ∧ ϕ ( w 1 , t ) ∨ ∃ w 1 ∃ w 2 .ϕ ( s , w 1 ) ∧ ϕ ( w 1 , w 2 ) ∧ ϕ ( w 2 , t ) ∨ ... Formal Definition Let M be a structure for L TC and v an assignment in M . M , v | = ( RTC x , y ϕ ) ( s , t ) iff there exist a 0 , ... a n ∈ D s.t. v [ s ] = a 0 ; v [ t ] = a n ; M , v [ x := a i , y := a i +1 ] | = ϕ for 0 ≤ i < n . s s ϕ ϕ ϕ ϕ t t a 0 a 1 a 2 a n − 1 a n

The Semantics The Intended Meaning of ( RTC x , y ϕ )( s , t ) s = t ∨ ϕ ( s , t ) ∨ ∃ w 1 .ϕ ( s , w 1 ) ∧ ϕ ( w 1 , t ) ∨ ∃ w 1 ∃ w 2 .ϕ ( s , w 1 ) ∧ ϕ ( w 1 , w 2 ) ∧ ϕ ( w 2 , t ) ∨ ... Formal Definition Let M be a structure for L TC and v an assignment in M . M , v | = ( RTC x , y ϕ ) ( s , t ) iff there exist a 0 , ... a n ∈ D s.t. v [ s ] = a 0 ; v [ t ] = a n ; M , v [ x := a i , y := a i +1 ] | = ϕ for 0 ≤ i < n . s s ϕ ϕ ϕ ϕ t t a 0 a 1 a 2 a n − 1 a n M , v | = ( RTC x , y ϕ ) ( s , t ) provided for every A ⊆ D , if v ( s ) ∈ A and ∀ a , b ∈ D : ( a ∈ A ∧ M , v [ x := a , y := b ] | = ϕ ) → b ∈ A , then v ( t ) ∈ A .

Expressive Power • The reflexive and the non-reflexive TC operators are equivalent (assuming equality).

Expressive Power • The reflexive and the non-reflexive TC operators are equivalent (assuming equality). Theorem [Avron, ’03] All recursive functions and relations are definable in L { 0 , s } TC

Expressive Power • The reflexive and the non-reflexive TC operators are equivalent (assuming equality). Theorem [Avron, ’03] All recursive functions and relations are definable in L { 0 , s } (with TC pairs)