Introduction
One goal with the explanations offered in this essay on the subject of the P vs. NP problem is to examine the current debate on the P vs NP problem , recent attempts at solving PNP and explaining the philosophical dimensions of the problem , including an analysis of Deolalikar´s Proof.
There has been a lot of speculation about the P vs. NP problem in various Academies around the world , and undeniably the P NP Problem has generated an enormous quantity of data concerning the 3000 sections or parts which this problem consists of.
Those who are already familiar with the P vs NP Problem at the University of Iceland will find enjoy to know that P NP is discussed by more Icelanders than has been previously considered , in light of the fact that generally the overall Icelandic Population is not much concerned with Mathematical Philosophy.
Rather than attempting to present further parts of the Puzzle over PNP , I have decided instead to present a Philosophical Essay concerning the Philosophical Dimensions by which numerous formulae and examples which have been concluded to be P or NP compatible are frequently discussed within the limits of the PNP debate. In light of the fact that this debate is still going on it is only natural and logical for members of the Icelandic Academic Community and beyound to express some enthusiasm concerning PNP.
Generally when articles about PNP are written , it is commonly asked in those articles what PNP actually is and what it stands for , which in itself is considered to be one of the most fundamentally important and challenging Philosophical Problems in modern times.
To ignore the status that PNP holds therefore as a Philosophical Problem , would be a grave mistake.
There are numerous reasons for why.
One reason being that despite there have been tendencies to keep mathematical calculations and formulae separate within the limits of most debates on subjects of Philosophical Research going on in Icelandic Society today , it is virtually undeniable that sooner or later there will be an increase in cases where serious attempts are made to explain mathematical problems from a Philosophical Perspective with an emphasis on the usefulness of formulae to explain the nature of each such problem.
Understandably mathematics has often been avoided as a subject in the Icelandic Philosophical Community on the ground that some members seem to consider mathematics to be somehow separate from modern Philosophy and that the limits by which modern Philosophy presents itself should be strictly explained within language alone , rather than through much use or even overuse of numbers.
But in fact , language and numbers are linked with each other inseparably , so it is therefore impossible to succeed in separating numbers from language in a Philosophical Debate , such as one on PvsNP.
This can be shown by attempting to present a Philosophical Problem which depends on the use of numbers to explain it´s concept without introducing the numerical symbolism , which will probably result in scrambling.
The first part of this article deals with introducing and explaining P vs NP , and the Second Part goes into current attempts to explain PNP in a modern context and recent attempts to resolve PNP will be explored
My current view of the P / NP problem is , that it can be solved by accepting that it is possible that a particular problem presented as being NP compatible may in fact turn out to be P compatible. If this is true it means that some problems which have until so far been considered to be NP compatible , are actually P compatible , which I believe can be considered as indicating that it is not possible to prove positively that P is NP per ce , or that NP is P per ce , since so many of the NP compatible problems we know about today , have not been able to offer us to our knowledge with a final solution as to whether P can be considered NP per ce.Instead it appears to be the case that only in certain isolated cases may we have a problem which had been considered NP compatible which turns out to be P compatible. In that case , we´d be dealing with a P compatible problem.
P vs. NP in the early 21st Century
Essentially PNP asks whether any problem can be quickly verified by a computer can also be solved by a computer.
The simple answer to this question is obviously no. Some problems quickly verified by a computer can also be solved by a computer , but it can take a long time to solve a problem verified by a computer which has not been verified quickly.
At first glance this problem seems rather simple , but as it´s analysis clearly demonstrates it becomes less and less apparent that that is so.
Significant progress has been made in trying to explain P vs. NP in modern times , resulting in P vs. NP being considered the single most important Philosophical Problem for Computer Science today.
Is P vs. NP however a Philosophical Problem?
A simple answer would be yes , simply because of how the problem presents itself as one of the 7 Millenium Prize Problems.
Famously it was Stephen Cook who introduced the problem in his now famous text , ´the Complexity of Theorem Solving Procedures´in 1971.
After that the world of Computer Science and Philosophy has never remained the same , simply because of the significance this problem has had as one of the most fundamentally difficult and challenging problems facing science and philosophy today.
One goal with the explanations offered in this essay on the subject of the P vs. NP problem is to examine the current debate on the P vs NP problem , recent attempts at solving PNP and explaining the philosophical dimensions of the problem , including an analysis of Deolalikar´s Proof.
There has been a lot of speculation about the P vs. NP problem in various Academies around the world , and undeniably the P NP Problem has generated an enormous quantity of data concerning the 3000 sections or parts which this problem consists of.
Those who are already familiar with the P vs NP Problem at the University of Iceland will find enjoy to know that P NP is discussed by more Icelanders than has been previously considered , in light of the fact that generally the overall Icelandic Population is not much concerned with Mathematical Philosophy.
Rather than attempting to present further parts of the Puzzle over PNP , I have decided instead to present a Philosophical Essay concerning the Philosophical Dimensions by which numerous formulae and examples which have been concluded to be P or NP compatible are frequently discussed within the limits of the PNP debate. In light of the fact that this debate is still going on it is only natural and logical for members of the Icelandic Academic Community and beyound to express some enthusiasm concerning PNP.
Generally when articles about PNP are written , it is commonly asked in those articles what PNP actually is and what it stands for , which in itself is considered to be one of the most fundamentally important and challenging Philosophical Problems in modern times.
To ignore the status that PNP holds therefore as a Philosophical Problem , would be a grave mistake.
There are numerous reasons for why.
One reason being that despite there have been tendencies to keep mathematical calculations and formulae separate within the limits of most debates on subjects of Philosophical Research going on in Icelandic Society today , it is virtually undeniable that sooner or later there will be an increase in cases where serious attempts are made to explain mathematical problems from a Philosophical Perspective with an emphasis on the usefulness of formulae to explain the nature of each such problem.
Understandably mathematics has often been avoided as a subject in the Icelandic Philosophical Community on the ground that some members seem to consider mathematics to be somehow separate from modern Philosophy and that the limits by which modern Philosophy presents itself should be strictly explained within language alone , rather than through much use or even overuse of numbers.
But in fact , language and numbers are linked with each other inseparably , so it is therefore impossible to succeed in separating numbers from language in a Philosophical Debate , such as one on PvsNP.
This can be shown by attempting to present a Philosophical Problem which depends on the use of numbers to explain it´s concept without introducing the numerical symbolism , which will probably result in scrambling.
The first part of this article deals with introducing and explaining P vs NP , and the Second Part goes into current attempts to explain PNP in a modern context and recent attempts to resolve PNP will be explored
My current view of the P / NP problem is , that it can be solved by accepting that it is possible that a particular problem presented as being NP compatible may in fact turn out to be P compatible. If this is true it means that some problems which have until so far been considered to be NP compatible , are actually P compatible , which I believe can be considered as indicating that it is not possible to prove positively that P is NP per ce , or that NP is P per ce , since so many of the NP compatible problems we know about today , have not been able to offer us to our knowledge with a final solution as to whether P can be considered NP per ce.Instead it appears to be the case that only in certain isolated cases may we have a problem which had been considered NP compatible which turns out to be P compatible. In that case , we´d be dealing with a P compatible problem.
P vs. NP in the early 21st Century
Essentially PNP asks whether any problem can be quickly verified by a computer can also be solved by a computer.
The simple answer to this question is obviously no. Some problems quickly verified by a computer can also be solved by a computer , but it can take a long time to solve a problem verified by a computer which has not been verified quickly.
At first glance this problem seems rather simple , but as it´s analysis clearly demonstrates it becomes less and less apparent that that is so.
Significant progress has been made in trying to explain P vs. NP in modern times , resulting in P vs. NP being considered the single most important Philosophical Problem for Computer Science today.
Is P vs. NP however a Philosophical Problem?
A simple answer would be yes , simply because of how the problem presents itself as one of the 7 Millenium Prize Problems.
Famously it was Stephen Cook who introduced the problem in his now famous text , ´the Complexity of Theorem Solving Procedures´in 1971.
After that the world of Computer Science and Philosophy has never remained the same , simply because of the significance this problem has had as one of the most fundamentally difficult and challenging problems facing science and philosophy today.
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