Nonuniform Reductions and NP-Completeness

Nonuniformity is a central concept in computational complexity with powerful connections to circuit complexity and randomness. Nonuniform reductions have been used to study the isomorphism conjecture for NP and completeness for larger complexity classes. We study the power of nonuniform reductions for NP0completeness, obtaining both separations and upper bounds for nonuniform completeness vs uniform complessness in NP. Under various hypotheses, we obtain the following separations: 1. There is a set complete for NP under nonuniform many-one reductions, but not under uniform many-one reductions. This is true even with a single bit of nonuniform advice. 2. There is a set complete for NP under nonuniform many-one reductions with polynomial-size advice, but not under uniform Turing reductions. That is, polynomial nonuniformity is stronger than a polynomial number of queries. 3. For any fixed polynomial p(n), there is a set complete for NP under uniform 2-truth-table reductions, but not under nonuniform many-one reductions that use p(n) advice. That is, giving a uniform reduction a second query makes it more powerful than a nonuniform reduction with fixed polynomial advice. 4. There is a set complete for NP under nonuniform many-one reductions with polynomial ad- vice, but not under nonuniform many-one reductions with logarithmic advice. This hierarchy theorem also holds for other reducibilities, such as truth-table and Turing. We also consider uniform upper bounds on nonuniform completeness. Hirahara (2015) showed that unconditionally every set that is complete for NP under nonuniform truth-table reductions that use logarithmic advice is also uniformly Turing-complete. We show that under a derandomization hypothesis, the same statement for truth-table reductions and truth-table completeness also holds

Autoreducibility of NP-Complete Sets

We study the polynomial-time autoreducibility of NP-complete sets and obtain separations under strong hypotheses for NP. Assuming there is a p-generic set in NP, we show the following: - For every $k \geq 2$, there is a $k$-T-complete set for NP that is $k$-T autoreducible, but is not $k$-tt autoreducible or $(k-1)$-T autoreducible. - For every $k \geq 3$, there is a $k$-tt-complete set for NP that is $k$-tt autoreducible, but is not $(k-1)$-tt autoreducible or $(k-2)$-T autoreducible. - There is a tt-complete set for NP that is tt-autoreducible, but is not btt-autoreducible. Under the stronger assumption that there is a p-generic set in NP $\cap$ coNP, we show: - For every $k \geq 2$, there is a $k$-tt-complete set for NP that is $k$-tt autoreducible, but is not $(k-1)$-T autoreducible. Our proofs are based on constructions from separating NP-completeness notions. For example, the construction of a 2-T-complete set for NP that is not 2-tt-complete also separates 2-T-autoreducibility from 2-tt-autoreducibility

An effectiveness of high order thinking skills (HOTS) self- instructional manual for students’ assignment achievement

High Order Thinking Skills (HOTS) is an important aspect of teaching and learning. An individual’s thinking can affect learning ability, learning speed and effectiveness of learning. The weakness in implementing HOTS is one of the reasons for a student not being creative in solving all the problems that arise. The purpose of this study was to evaluate the effectiveness of HOTS’s Self-Instructional Manual (SIM) in teaching and learning for assignment achievement among polytechnic students. This study uses a quantitative approach and the Quasi Experimental design - “Pretest-Posttest Non-equivalent Comparison Group Design” consisting of one Treatment Group (TG) and one Control Group (CG) involving 78 students at Polytechnic of Sultan Abdul Halim Muad'zam Shah. The assignment evaluation rubric was modified to assess the level of students’ assignment achievement. The findings showed that most of the students for TG and CG achieved good result in the individual pre-assignment. Additionally, there is a significant difference mean scores of individual post assignment between TG and CG. Students for TG achieved excellent result in the individual post assignments. However, students for CG only achieved good result in the individual post assignments. This means that HOTS Self-Instructional Manual has a significant impact on the student assignments achievement. Therefore, we encourage all students of higher education institutions to use Self-Instructional Manual in teaching and learning in order to score better achievement especially in the course work

Teaching Assistantship from Department of Computer Science

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The John P. Ellbogen Outstanding Graduate Assistant Teaching Award

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