Descriptive Complexity of Jonni Virtema Deterministic - PowerPoint PPT Presentation

Descriptive Complexity of Deterministic Polylogarithmic Time Descriptive Complexity of Jonni Virtema Deterministic Polylogarithmic Time Descriptive Complexity Polylogarithmic Time Jonni Virtema Index Logic Fixed points Hasselt University, Belgium Syntax on IL jonni.virtema@gmail.com Semantics of IL Joint work with Flavio Ferrarotti, Sen´ en Gonz´ alez, Results Jos´ e Mar´ ıa Turull Torres, and Jan Van den Bussche. Open Question WoLLIC 2019 – July 4th 2019 1 of 14

Descriptive Descriptive Complexity Complexity of Deterministic Polylogarithmic Time Jonni Virtema Descriptive Complexity ◮ Offers a machine independent description of complexity classes: Polylogarithmic ◮ Time/Space used by a machine to decide a problem Time ⇒ richness of the logical language needed to describe the problem. Index Logic Fixed points ◮ Complexity classes can/could be then separated by separating logics. Syntax on IL ◮ Many characterisations are known: Semantics of IL ◮ Fagin’s Theorem 1973: Existential second-order logic characterises NP. Results Open Question 2 of 14

Descriptive Descriptive Complexity Complexity of Deterministic Polylogarithmic Time Jonni Virtema ◮ Offers a machine independent description of complexity classes: ◮ Time/Space used by a machine to decide a problem Descriptive Complexity ⇒ richness of the logical language needed to describe the problem. Polylogarithmic Time ◮ Complexity classes can/could be then separated by separating logics. Index Logic ◮ Many characterisations are known: Fixed points ◮ Fagin’s Theorem 1973: Existential second-order logic characterises NP. Syntax on IL Semantics of IL ”A graph is three colourable” = Results � ∃ R ∃ B ∃ G ”each node is labeled by exactly one colour” Open Question � ∧ ”adjacent nodes are always coloured with distinct colours” 2 of 14

Descriptive Descriptive Complexity Complexity of Deterministic Polylogarithmic Time Jonni Virtema ◮ Offers a machine independent description of complexity classes: Descriptive ◮ Time/Space used by a machine to decide a problem Complexity ⇒ richness of the logical language needed to describe the problem. Polylogarithmic Time ◮ Complexity classes can/could be then separated by separating logics. Index Logic ◮ Many characterisations are known: Fixed points ◮ Fagin’s Theorem 1973: Existential second-order logic characterises NP. Syntax on IL ◮ ESO polylog characterises NPolylogTime . Semantics of IL ◮ Second-order logic characterises the polynomial time hierarchy. Results ◮ Least fixed point logic LFP characterises P on ordered structures. Open Question ◮ ... ◮ Major open problem: Does there exist a logic for P? 2 of 14

Descriptive Sublinear Complexity Classes and Random Access Machines Complexity of Deterministic Polylogarithmic Time ◮ In sublinear time the whole the input cannot be read. Jonni Virtema ◮ Turing machines with sequental access to the input does not suffice. Descriptive Complexity ◮ Instead random access model is used (cf. random access memory RAM) Polylogarithmic ◮ Random access machine model: Time Index Logic . . . Input Tape (read only) 1 0 0 1 1 0 B Fixed points . . . Address Tape 1 0 Syntax on IL B B B B B Semantics of IL . . . k Work Tapes 0 1 1 0 1 B B Results Open Question ◮ Finite control of the machine as for TM. k ∈ N DTIME [log k n ] ◮ PolylogTIME = � 3 of 14

Descriptive Sublinear Complexity Classes and Random Access Machines Complexity of Deterministic Polylogarithmic Time ◮ In sublinear time the whole the input cannot be read. Jonni Virtema ◮ Turing machines with sequental access to the input does not suffice. Descriptive Complexity ◮ Instead random access model is used (cf. random access memory RAM) Polylogarithmic ◮ Random access machine model: Time Index Logic . . . Input Tape (read only) 1 0 0 1 1 0 B Fixed points . . . Address Tape 1 0 Syntax on IL B B B B B Semantics of IL . . . k Work Tapes 0 1 1 0 1 B B Results Open Question ◮ Finite control of the machine as for TM. k ∈ N DTIME [log k n ] ◮ PolylogTIME = � 3 of 14

Descriptive Sublinear Complexity Classes and Random Access Machines Complexity of Deterministic Polylogarithmic Time ◮ In sublinear time the whole the input cannot be read. Jonni Virtema ◮ Turing machines with sequental access to the input does not suffice. Descriptive Complexity ◮ Instead random access model is used (cf. random access memory RAM) Polylogarithmic ◮ Random access machine model: Time Index Logic . . . Input Tape (read only) 1 0 0 1 1 0 B Fixed points . . . Address Tape 1 0 Syntax on IL B B B B B Semantics of IL . . . k Work Tapes 0 1 1 0 1 B B Results Open Question ◮ Finite control of the machine as for TM. k ∈ N DTIME [log k n ] ◮ PolylogTIME = � 3 of 14

Descriptive Example computation in deterministic polylogarithmic time Complexity of Deterministic Polylogarithmic Time Jonni Virtema Descriptive Complexity Polylogarithmic ◮ Calculate the length n of the input. Time . . . Index Logic 1 0 0 Input Tape (read only) B B B B Fixed points . . . Index Tape 0 B B B B B B Syntax on IL Semantics of IL Results Open Question 4 of 14

Descriptive Example computation in deterministic polylogarithmic time Complexity of Deterministic Polylogarithmic Time Jonni Virtema Descriptive Complexity Polylogarithmic ◮ Calculate the length n of the input. Time . . . Index Logic 1 0 0 Input Tape (read only) B B B B Fixed points . . . Index Tape 1 0 0 B B B B Syntax on IL Semantics of IL Results Open Question 4 of 14

Descriptive Example computation in deterministic polylogarithmic time Complexity of Deterministic Polylogarithmic Time Jonni Virtema Descriptive Complexity Polylogarithmic ◮ Calculate the length n of the input. Time . . . Index Logic 1 0 0 Input Tape (read only) B B B B Fixed points . . . Index Tape 1 0 B B B B B Syntax on IL Semantics of IL Results Open Question 4 of 14

Descriptive Example computation in deterministic polylogarithmic time Complexity of Deterministic Polylogarithmic Time Jonni Virtema Descriptive Complexity Polylogarithmic ◮ Calculate the length n of the input. Time . . . Index Logic 1 0 0 Input Tape (read only) B B B B Fixed points . . . Index Tape 1 1 B B B B B Syntax on IL Semantics of IL Results Open Question 4 of 14

Descriptive Example computation in deterministic polylogarithmic time Complexity of Deterministic Polylogarithmic Time Jonni Virtema Descriptive Complexity ◮ Calculate the length n of the input. Polylogarithmic . . . Time Input Tape (read only) 1 0 0 B B B B Index Logic . . . Index Tape 1 0 B B B B B Fixed points Syntax on IL ◮ The index tape has n − 1 as binary. Semantics of IL ◮ Any polynomial time numerical property of n (in binary) can be computed. Results Open Question 4 of 14

Descriptive Structures as inputs to the Turing machine Complexity of Deterministic Polylogarithmic Time Jonni Virtema ◮ Finite ordered structures with domain { 0 , . . . , n } and finite vocabularies. Descriptive Complexity ◮ Structures are encoded as strings as usual in descriptive complexity. Polylogarithmic ◮ Relation R A of arity k is encoded as a binary string of length | A | k , where 1 Time in a given position indicates that the corresponding tuple is in the relation. Index Logic ◮ Constant number c A is encoded as a binary string of length ⌈ log n ⌉ . Fixed points Syntax on IL ◮ k -ary functions are viewed as ⌈ log n ⌉ -many k -ary relations, where the i -th Semantics of IL relation indicates whether the i -th bit is 1. Results ◮ DTIME [log k ˆ n ] = DTIME [log k n ], where ˆ n is the length of the encoding Open Question and n the domain size. 5 of 14

Descriptive Structures as inputs to the Turing machine Complexity of Deterministic Polylogarithmic Time Jonni Virtema ◮ Finite ordered structures with domain { 0 , . . . , n } and finite vocabularies. Descriptive Complexity ◮ Structures are encoded as strings as usual in descriptive complexity. Polylogarithmic ◮ Relation R A of arity k is encoded as a binary string of length | A | k , where 1 Time in a given position indicates that the corresponding tuple is in the relation. Index Logic ◮ Constant number c A is encoded as a binary string of length ⌈ log n ⌉ . Fixed points Syntax on IL ◮ k -ary functions are viewed as ⌈ log n ⌉ -many k -ary relations, where the i -th Semantics of IL relation indicates whether the i -th bit is 1. Results ◮ DTIME [log k ˆ n ] = DTIME [log k n ], where ˆ n is the length of the encoding Open Question and n the domain size. 5 of 14