sfakehw12dot1

# sfakehw12dot1 - 6.045J/18.400J Automata Computability and...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 6.045J/18.400J: Automata, Computability and Complexity Nancy Lynch Homework 12.1 (Fake) Due: Never Elena Grigorescu Readings: Sipser, Chapter 8 (the whole chapter). Problem 1 : Solutions borrowed from Susan Hohenberger (2004.) (Sipser Exercise 8.1) Show that for any function f : N → N , where f ( n ) ≥ n , the space complexity class SPACE( f ( n )) is the same whether you define the class by using the single-tape TM model or the two tape read-only TM model. Solution 1 : First, we can simulate a SPACE( f ( n )), for f ( n ) ≥ n , single-tape TM on a SPACE( f ( n )) two tape read-only TM as follows. Step one: scan the read-only tape, copying its contents to the work tape. Step two: simulate the remainder of the computation, treating the read/write work-tape as the input tape of a “single-tape” TM. We can copy the contents of the counter, using only log n space to keep track of our position in the input. We can write all of the input on our work-tape, since f ( n ) ≥ n . Secondly, we can simulate a SPACE( f ( n )), for f ( n ) ≥ n , two tape read-only TM on a SPACE( f ( n )) single-tape TM by refraining from writing over the first n input symbols, using the remainder of the tape for our work area. Since this only adds n amount of space used, and we were already using at least n space, this increases our space by at most a constant. Problem 2 : (Sipser Problem 8.10) The Japanese game go-moku is played by two players, “X” and “O”, on a 19 × 19 grid. Players take turns placing markers, and the first player to achieve 5 of his/her markers consecutively in a row, column, or diagonal, is the winner. Consider this game generalized to an n × n board. Let GM = {( P )| P is a position in generalized go-moku, where player “X” has a winning strategy } . By a position we mean a board with markers placed on it, such as may occur in the middle of a play of the game. Show that GM ∈ PSPACE. Solution 2 : (borrowed from Spring 1999) First let’s assume that P is written as a grid of X,O and empty, so the length of the input is O ( n 2 ). Let’s define a recursive algorithm to solve GM(P), which accepts if there is a winning strategy for player X starting at position P: GM(P) (1) For all spaces i in position P without markers on them, (potential X moves) 1. Put an X marker on space i , thus changing the position to P’. If there are now 5 X’s in a row, accept , this is obviously a good move. If the board is now full, and no one has won, reject . 2. Otherwise, for all spaces j in position P’ without markers on them, (potential O moves) (a) Put an O on space j , thus changing the position to P”. If there are now 5 O’s in a row, or the board is full and no one has won, loop to the next i (goto step (1)); putting an X on i is obviously a bad move....
View Full Document

## This note was uploaded on 08/12/2009 for the course CS 430 taught by Professor Nancylynch during the Spring '07 term at New Mexico Junior College.

### Page1 / 5

sfakehw12dot1 - 6.045J/18.400J Automata Computability and...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online