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Chapter 8: Q15P (page 331)

Consider the following two-person version of the language that was described in Problem 7.28. Each player starts with an ordered stack of puzzle cards. The players take turns placing the cards in order in the box and may choose which side faces up. Player I wins if all hole positions are blocked in the final stack, and Player II wins if some hole position remains unblocked. To prove that the problem of determining which player has a winning strategy for a given starting configuration of the cards is PSPACE-Complete.

Short Answer

Expert verified

The PUZZLE problem is already in NP and NP-HARD, it is proved that the PUZZLE problem is PSPACE-complete.

Step by step solution

01

Introduction

The two-person version of the language PUZZLE. Each player takes turns to play with the cards until the player finally wins. The problem of determining which player has a winning strategy for a given starting configuration of the cards is PSPACE-complete.

02

Prove that the given problem is in NP.

The players take turns using the sorted cards. Each of the cards used in the problem should have a pair of records, as proven by the nondeterministic Turning Machine. The utilised cards have already been assigned to the pair of records in the PUZZLE challenge. As a result, the PUZZLE issue is an NP-complete problem.

03

Prove that the given problem is in NP-HARD.

The 3CNF recipe components are mapped to the card service in multinomial time. All of the cards in the Cset, as well as the extra card E, have been put together. Only when the formula receives a wonderful contention will the PUZZLE problem come to an agreement. Otherwise, the PUZZLE issue will not be able to organise anything. In multinomial time, the 3CNF is degraded to the PUZZLE . As a result, the PUZZLE issue is NP-HARD to solution.

04

Final answer

Therefore, it is proved that the PUZZLE problem is PSPACE-complete.

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Most popular questions from this chapter

Show that 2SAT is NL-complete.

Let B be the language of properly nested parentheses and brackets. For example,is in B but []is not. Show that B is in L.

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The cat-and-mouse game is played by two players, "Cat" and "Mouse," on an arbitrary undirected graph. At a given point, each player occupies a node of the graph. The players take turns moving to a node adjacent to the one that they currently occupy. A special node of the graph is called "Hole." Cat wins if the two players ever occupy the same node. Mouse wins if it reaches the Hole before the preceding happens. The game is a draw if a situation repeats (i.e., the two players simultaneously occupy positions that they simultaneously occupied previously, and it is the same player's turn to move).
HAPPY-CAT={<G,c,m,h>G,c,m,hAre respectively a graph and positions of the Cat, Mouse, and Hole, such that Cat has a winning strategy if Cat moves first}.
Show thatHAPPY-CAT is in P. (Hint: The solution is not complicated and doesn't depend on subtle details in the way the game is defined. Consider the entire game tree. It is exponentially big, but you can search it in polynomial time.)

Show that TQBF restricted to formulas where the part following the quantifiers is in conjunctive normal form is still PSPACE-complete.
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