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Computer Theory, NP completness

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  1. 1. Computability Anand Janakiraman UMN, Twin Cities Hilbert’s dream : Euclid & Formal Reasoning Euclid successfully axiomized geometry. Axioms are statements that are assumed to be true Uncomputability (see box on formal reasoning ) Theorems are proved/deduced using rules of inference Hilbert dreamt to determine a complete axiom set assuming axioms are true. There are some functions which cannot be computed by for mathematics. The Axiom set would be Rules of inference: An example is Modus Ponens any turing machines. The table shows the value computed by the i th machine on the j th input . Machine 1 produces the ouput [0110....] No machine can compute the Finite: without this one could take as one’s axioms the set of all complement of the diagonal [1000....] true propositions. Sound: if all provable theorems are true Formal Reasoning was introduced by Euclid in his book “Elements” which describes Geometry in an axiomatic Complete: the system is able to prove all true theorems manner.The method of formal reasoning came to be known as Aristotlean school of thought. Decidable: if there is a mechanical procedure for determining whether or not an arbitrary theorem is provable. Other attempts to axiomize mathematics were “Peano’s arithmetic” and Elliptical and Hyperbolic geometry which were done by relaxing Euclid’s fifth postulate Godel Godel proved that for any consistent axioms F there is a true statement of first order number theory that is not provable or disprovable by F. ( i.e., a true statement that can be made using 0, 1, plus, times, Mechanical computation is limited. Turing machines can for every, there exists, AND, OR, NOT, parentheses, and compute all that can be computed. The number of turing variables that refer to natural numbers. ) machines is enumerable, whereas the number of functions The proof is on the lines of liar's paradox ( "I am lying" ). is not. Thus there are some functions that are not computable. An example of such a problem is halting Godel constructs a statement similar to S: problem. "This theorem is not provable in number theory". if S is false, then S is provable ( this leads to a contradiction . Is S provable or not provable) . Thus we are forced to assume S is true and arithmetic itself cannot prove it Thus we cannot obtain a system that is complete (since there are Aristotle Euclid unproven true statements). It may seem that we could obtain a complete axiomization Halting Problem by simply taking all true stmts as axioms. But one requirement is that these axioms should be recognizable by mechanical method. As Turing subsequently showed Turing showed that the halting problem is uncomputable. that the true statements about natural numbers cannot be mechanically recognized. Turing Turing showed there no is a mechanical procedure for determining whether or not an arbitrary theorem is provable. Mechanical Procedure Hilbert In order to formalize the notion of mechanical procedure , Turing introduced a simplified model of computer (the person who computes ) " assume computation is carried on one- dimensional paper ie a tape divided into squares.... The behavior of the computer is determined by the symbols he is observing and the state of mind at that moment" Decision Problem A function is computable if any turing machine computes it. The Turing Machine is an abstract, mathematical model that Turing proved that the decision problem is uncomputable from describes what can and cannot be computed. the uncomputability of halting problem. The halting problem (Machine M halts on tape T) can be expressed as logical formula. If there were a procedure for the x,y, provability of arbitrary propositions (the decision problem) , then y,z, there would be one for halting problem. The fact that halting Finite state brain problem is uncomputable means that there is no procedure for x,y, Finite alphabet of determining the provability of arbitrary theorem. Thus shattering symbols Infinite supply of Hilbert’s dream. notebooks Godel Turing x