Team Automata and Ordinal Register Machines Ryan Bissell-Siders University of Helsinki 2009 January 12 New Worlds of Computation Orl´ eans 1 / 27
team automata and infinitary computation team automata (definition) team automata (definition). team automata and ordinal register team automata and infinitary machines team automata and ordinal Turing computation machines computations on Turing Machines and team automata relationship to other hypercomputation systems questions 2 / 27
team automata (definition) team automata and ✔ Only change state when they collide or split; infinitary computation team automata ✔ only collide in pairs. (definition) team automata (definition). ✔ the pair of new states is a function of the old states. team automata and ordinal register machines team automata and ✔ can split; this always occurs before any other collision. ordinal Turing machines ✔ move in model M from a ∈ M at time s to computations on Turing Machines and team automata ✘ time s + 1 and the unique b ∈ M such that relationship to other hypercomputation M | = φ ( a, b ) , or systems questions ✘ times t > s and locations b ∈ M such that { ( b, t ) : M | = φ (( a, s ) , ( b, t )) } is a bijection between times and locations. 3 / 27
team automata (definition). Teams of automata exploring a graph and marking interesting team automata and infinitary locations are studied by, for instance, (P. Flocchini, D. Ilcinkas, N. computation team automata Santoro, Ping Pong in Dangerous Graphs: Optimal Black Hole (definition) team automata Search with Pure Tokens , 5218, LNCS, Springer, 227-241 (2008)). (definition). team automata and ordinal register machines team automata and ordinal Turing machines computations on Turing Machines and team automata relationship to other hypercomputation systems questions 4 / 27
team automata and ordinal register machines An Ordinal Register Machine stores ordinals α 0 . . . α 19 in its team automata and infinitary registers and runs for ordinal time. computation team automata A universal ORM program exists, which reads an input n as a code (definition) team automata and runs it. We simulate this universal ORM with a team of (definition). team automata and automata A 0 . . . A 19 , because the ORM always passes over limit ordinal register machines times with its registers in fixed state (either increasing linearly or team automata and ordinal Turing stationary). machines For instance, to compute ⌊ α 0 / 5 ⌋ , we set α 1 to 0 and let it grow computations on Turing Machines with speed 5 until it collides with α 0 ; the last value it takes will be and team automata relationship to other the greatest multiple of 5 below α 0 , and the elapsed time is hypercomputation systems ⌊ α 0 / 5 ⌋ . questions (Bissell-Siders and Koepke, 2006 CiE) 5 / 27
team automata and ordinal Turing machines Given a team A 0 . . . A n of automata moving on the ordinals, a team automata and infinitary Turing machine with the ordinals as its tape and with a finite set computation team automata of heads H 0 . . . H n simulates the team of automata by computing (definition) team automata the movement of A n and keeping H n there; if A n divides and (definition). team automata and leaves a marker, H n markes the Turing tape. ordinal register machines The behavior of a Turing machien with the ordinals as its tape can team automata and ordinal Turing be modeled by an Ordinal Reigster Machine. machines (Bissell-Siders and Koepke, 2008 AML) computations on Turing Machines and team automata relationship to other hypercomputation systems questions 6 / 27
team automata and infinitary computation computations on Turing Machines and team automata A Turing Machine rolling right A Turing Machine computations on Turing Machines and rolling right. A Turing Machine team automata rolling right.. A Turing Machine rolling right... Team Automata write an ordinal in R Team Automata write an ordinal in R . Team Automata write an ordinal in R .. Team Automata hit a wall Team Automata hit a wall. team automata simulate a Turing machine in R . relationship to other hypercomputation 7 / 27 systems questions
A Turing Machine rolling right Given a Turing machine program P which can notice team automata and infinitary L -the left end of the tape, computation computations on we divide its tape into L = B . . . C, A . . . , where C and A are Turing Machines and team automata adjacent cells. We create a program with tape: A Turing Machine rolling right 0 . . . 0 A . . . B . . . C which pushes the interval ( B, C ) further to the A Turing Machine right whenever the interval ( A . . . ) needs to add something to its rolling right. A Turing Machine right, and which frequently erases the value of the tape at A and rolling right.. A Turing Machine adds it to the interveral ( B, C ) . rolling right... Team Automata write an ordinal in R Team Automata write an ordinal in R . Team Automata write an ordinal in R .. Team Automata hit a wall Team Automata hit a wall. team automata simulate a Turing machine in R . relationship to other hypercomputation 8 / 27 systems questions
A Turing Machine rolling right. We write states as sentences of sem-natural language. team automata and infinitary The program P is a set of tuples ( state, character, where to go, computation computations on what to write, what state to enter ) . Turing Machines and team automata The new program PR has, for each command A Turing Machine rolling right ( s 0 , χ 0 , d, χ 1 , s 1 ) ∈ P , (( s 0 , 0) , χ 0 , d, χ 1 , ( s 1 , 1)) ∈ PR . A Turing Machine ( s 0 , L, d, χ 1 , s 1 ) ∈ PR is replaced by ( s 0 , A, d, χ 1 , s 1 ) ∈ PR . rolling right. A Turing Machine The program PR has, additionally, the following two sequences of rolling right.. A Turing Machine commands rolling right... Team Automata Beginning in state ( s 0 , 1) , read the character χ 0 into memory. write an ordinal in R Team Automata Mark this location as D . find A , erase it, move right, read the write an ordinal in R . character as χ 1 , write A , move to C , print χ 1 , move right, print Team Automata write an ordinal in C , find D , print χ 0 , execute the rule ( s 0 , χ 0 , d, χ 1 , s 1 ) ∈ P – print R .. Team Automata hit χ 1 and move in direction d and enter state ( s 1 , 0) . a wall Team Automata hit a wall. team automata simulate a Turing machine in R . relationship to other hypercomputation 9 / 27 systems questions
A Turing Machine rolling right.. team automata and Beginning with state ( s 0 , 1) , read the character we see into infinitary memory as χ 0 , print F , move right, check whether we read B . If computation computations on not, move left, print χ 0 , and excute the sequence above. If we Turing Machines and team automata read B , print D , move to C , move right, print B . Move right. A Turing Machine rolling right Print E . (Find D , erase it, move right, read χ 1 . print D Move to A Turing Machine rolling right. E . Print χ 1 . Move right. Print E .) Repeat until the χ 1 we read is A Turing Machine C . Erase it. Find E . Print χ 1 . Move right. Print C . Find F . rolling right.. A Turing Machine print χ 0 and execute the sequence above. rolling right... Team Automata This program PR has the characteristic that it moves A every write an ordinal in R Team Automata time the program P makes one move. PR moves the interval write an ordinal in R . ( B, c ) whenever the program P desires to write on the n -th Team Automata write an ordinal in square, and | ( A, B ) | = n . R .. Team Automata hit a wall Team Automata hit a wall. team automata simulate a Turing machine in R . relationship to other hypercomputation 10 / 27 systems questions
A Turing Machine rolling right... If the Turing tape’s first ω -many cells have been written into team automata and infinitary (0 , 1) ⊂ R by a program whose curves φ are constant, then the computation computations on length of the tape’s elements ( n, ω ) diminishes exponentially with Turing Machines n (it is p n for some fraction n ). The time taken to execute and team automata A Turing Machine rolling right program PR is � n ∈ ω t n , where t n is the time taken to move A to A Turing Machine the right (and perhaps move ( B, C ) to the right) and execute the rolling right. A Turing Machine n -th command in P . This requires that the program traverse rolling right.. A Turing Machine subsets of the interval ( A, ω ) at most 2 n + 4 -many times (at most rolling right... Team Automata n elements in ( B, c ) to copy, which takes n passes left and right, write an ordinal in R Team Automata then back to the active cell, then to A , to C , and back to the write an ordinal in R . active cell). That would take time t n < p n (2 × n + 4) . Team Automata write an ordinal in R .. Team Automata hit a wall Team Automata hit a wall. team automata simulate a Turing machine in R . relationship to other hypercomputation 11 / 27 systems questions
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