Critical thinking and mathematical proofs are two learning outcomes of focus in our Discrete Structures course. In this paper, we study how these two learning outcomes affect each other. Our study is based on assessment data from two editions of our Discrete Structures course, with mathematical proofs being taught before critical thinking in the first edition and taught after critical thinking in the second edition. Results of our data show that students in the first edition performed significantly better in understanding proofs and better in critical thinking than students in the second edition did. However, the difference between the mean scores on understanding proofs of the two editions is not statistically significant. These results suggest that teaching/learning mathematical proofs may help students understand critical thinking better, while teaching/learning critical thinking may not help students understand proofs

Quantum computers can break the RSA, El Gamal, and elliptic curve public-key cryptosystems, as they can efficiently factor integers and extract discrete logarithms. The power of such quantum attacks lies in quantum Fourier sampling, an algorithmic paradigm based on generating and measuring coset states. In this article we extend previous negative results of quantum Fourier sampling for Graph Isomorphism, which corresponds to hidden subgroups of order two (over S n ), to several cases corresponding to larger hidden subgroups. For one case, we strengthen some results of Kempe, Pyber, and Shalev on the Hidden Subgroup Problem over the symmetric group. In another case, we show the failure of quantum Fourier sampling on the Hidden Subgroup Problem over the general linear group GL 2 (F q ). The most important case corresponds to Code Equivalence, the problem of determining whether two given linear codes are equivalent to each other up to a permutation of the coordinates. Our results suggest that for many codes of interest--including generalized Reed Solomon codes, alternant codes, and Reed-Muller codes--solving these instances of Code Equivalence via Fourier sampling appears to be out of reach of current families of quantum algorithms.

Coverage of discrete structures is necessary for any computer science curriculum. More broadly, teaching skill in critical thinking is essential to providing a well-rounded college education. We describe our experience with incorporating a 'critical thinking module' (CTM) into a discrete structures (DS) course.

Many studies in heuristic search suggest that the accuracy of the heuristic used has a positive impact on improving the performance of the search. In another direction, historical research perceives that the performance of heuristic search algorithms, such as A* and IDA*, can be improved by requiring the heuristics to be consistent – a property satisfied by any perfect heuristic. However, a few recent studies show that inconsistent heuristics can also be used to achieve a large improvement in these heuristic search algorithms. These results raise a natural question: which property of heuristics, accuracy or consistency/inconsistency, should we focus on when building heuristics? In this article, we investigate the relationship between the inconsistency and the accuracy of heuristics with A* search. Our analytical result reveals a correlation between these two properties. We then run experiments on the domain for the Knapsack problem with a family of practical heuristics. Our empirical results show that in many cases, the more accurate heuristics also have higher level of inconsistency and result in fewer node expansions by A*.

This paper investigates the achievable rate and ergodic capacity of a non-orthogonal amplify-and-forward (NAF) half-duplex multi-relay network where multiple relays exploit channel state information (CSI) to cooperate with a pair of source and destination. In the first step, for a given input covariance matrix at the source, we derive an optimal power allocation scheme among the relays via optimal instantaneous power amplification coefficients to maximize the achievable rate. Given the nature of broadcasting and receiving collisions in NAF, the considered problem in this step is non-convex. To overcome this drawback, we propose a novel method by evaluating the achievable rate in different sub-domains of the vector channels. It is then demonstrated that the globally optimal solution can be derived in closed-form. In the next step, we establish the ergodic channel capacity by jointly optimizing the input covariance matrix at the source and the power allocation among the relays. We show that this is a bi-level non-convex problem and solve it using Tammer decomposition method. This approach allows us to transform the original optimization problem into an equivalent master problem and a set of sub-problems having closed-form solutions derived in the first step. The channel capacity is then obtained using an iterative water-filling-based algorithm. Finally, we analyze the capacity-achieving input covariance matrix at the source in high and low signal-to-noise ratio (SNR) regimes. At sufficiently high SNRs, it is shown that the transmit power at the source should be equally distributed in all broadcasting and cooperative phases. On the other hand, in low SNR regions, the source should spend all its power in the broadcasting phase associated with a relay having the strongest cascaded source-relay and relay-destination channels.

Many studies in heuristic search suggest that the accuracy of the heuristic used has a positive impact on improving the performance of the search. In another direction, historical research perceives that the performance of heuristic search algorithms, such as A* and IDA*, can be improved by requiring the heuristics to be consistent -- a property satisfied by any perfect heuristic. However, a few recent studies show that inconsistent heuristics can also be used to achieve a large improvement in these heuristic search algorithms. These results leave us a natural question: which property of heuristics, accuracy or consistency/inconsistency, should we focus on when building heuristics? While there are studies on the heuristic accuracy with the assumption of consistency, no studies on both the inconsistency and the accuracy of heuristics are known to our knowledge. In this study, we investigate the relationship between the inconsistency and the accuracy of heuristics with A* search. Our analytical result reveals a correlation between these two properties. We then run experiments on the domain for the Knapsack problem with a family of practical heuristics. Our empirical results show that in many cases, the more accurate heuristics also have higher level of inconsistency and result in fewer node expansions by A*.

We study the behavior of the We then experimentally explore to what extent our rigorous upper bounds predict the behavior of

The Code Equivalence problem is that of determining whether two given linear codes are equivalent to each other up to a permutation of the coordinates. This problem has a direct reduction to a nonabelian hidden subgroup problem (HSP), suggesting a possible quantum algorithm analogous to Shor's algorithms for factoring or discrete log. However, we recently showed that in many cases of interest---including Goppa codes---solving this case of the HSP requires rich, entangled measurements. Thus, solving these cases of Code Equivalence via Fourier sampling appears to be out of reach of current families of quantum algorithms. Code equivalence is directly related to the security of McEliece-type cryptosystems in the case where the private code is known to the adversary. However, for many codes the support splitting algorithm of Sendrier provides a classical attack in this case. We revisit the claims of our previous article in the light of these classical attacks, and discuss the particular case of the Sidelnikov cryptosystem, which is based on Reed-Muller codes.

Quantum computers can break the RSA, El Gamal, and elliptic curve public-key cryptosystems, as they can efficiently factor integers and extract discrete logarithms. This motivates the development of post-quantum cryptosystems: classical cryptosystems that can be implemented with today's computers, that will remain secure even in the presence of quantum attacks. In this article we show that the McEliece cryptosystem over rational Goppa codes and the Niederreiter cryptosystem over classical Goppa codes resist precisely the attacks to which the RSA and El Gamal cryptosystems are vulnerable-namely, those based on generating and measuring coset states. This eliminates the approach of strong Fourier sampling on which almost all known exponential speedups by quantum algorithms are based. Specifically, we show that the natural case of the Hidden Subgroup Problem to which McEliece-type cryptosystems reduce cannot be solved by strong Fourier sampling, or by any measurement of a coset state. To do this, we extend recent negative results on quantum algorithms for Graph Isomorphism to subgroups of the automorphism groups of linear codes. This gives the first rigorous results on the security of the McEliece-type cryptosystems in the face of quantum adversaries, strengthening their candidacy for post-quantum cryptography. We also strengthen some results of Kempe, Pyber, and Shalev on the Hidden Subgroup Problem in Sn.

Both A* search and local search are heuristic algorithms widely used for problem-solving in Artificial Intelligence and Combinatorial Optimization. Understanding their performance is a problem of both practical and theoretical importance. Our goal in this thesis is to fill significant gaps in the rigorous analyses of these two heuristic algorithms. In particular, we study the behavior of the classical A* search algorithm when coupled with a heuristic that provides estimates, accurate to within small multiplicative factors, of the optimal cost to reach a solution. We prove a general, essentially tight upper bound on the time complexity of A* search on trees that depends on both the accuracy of the heuristic and the distribution of solution objective values. A consequence of our rigorous analysis indicates that the effective branching factor of the search will be reduced as long as the heuristic is sufficiently accurate and the number of near-optimal solutions in the search tree is not too large. We go on to provide an upper bound for A* search on graphs and, in this context, establish a bound on running time determined by the spectrum of the graph. Finally, we apply our analysis of A* search for the partial Latin square problem, and compare the theoretically predicted running times with experimental data. These results demonstrate dramatic reduction in effective branching factor of A* when heuristic error is small. For local search, we focus on understanding its limitations, which can be reduced to studying the problem of finding a local minimum of a real-valued black-box function on a graph. In 1983, Aldous gave the first strong lower bound for any randomized algorithm to determine a local minimum on hypercubes. The next major step forward was not until 2004 when Aaronson, introducing a new method for query complexity bounds, both strengthened Aldous's lower bound and gave an analogous quantum lower bound. While these bounds are very strong, they are known only for narrow families of graphs (hypercubes and grids). We show how to generalize Aaronson's techniques in order to give randomized (and quantum) lower bounds for local search on vertex-transitive graphs, a significantly larger family of graphs. The techniques of our proof follow Aaronson's idea of proving classical results by quantum arguments.

Understanding the effect of a heuristic on the time complexity of the well-known A * search algorithm has been a subject of a large research body in AI. Among those researches is Korf et al. that the effect of a heuristic is to reduce the effective depth rather than effective branching factor. Since this conclusion conflicts with both previous analysis and recent analysis by other authors, we try to investigate the causes of such contradictions and finally find out that Korf et al.'s analysis has many serious flaws from which their main conclusion was drawn.

Knowledge validation, as part of knowledge base verification and validation, is a critical process in knowledge engineering. The ultimate goal of this process is to make the knowledge base satisfy all test cases given by human experts. This is further complicated by factors such as uncertainty and incompleteness. Our paper covers theoretical results in knowledge validation for Bayesian Knowledge Bases (BKBs), a probabilistic model extended from Bayesian Networks for representing knowledge in uncertain domains. First, we study the consistency of test case sets by identifying the necessary and sufficient conditions for a test case set such that there exists a knowledge base satisfying all of its test cases. Second, we analyze the thrashing problem which is the interminable oscillation of the knowledge base's state when validating by parameter refinement. We propose an approach to validating BKBs that effectively eliminates thrashing under certain conditions of the original knowledge base and the test case set.

We study the problem of local search on a graph. Given a real-valued black-box functionf on the graph's vertices, this is the problem of determining a local minimum of f--a vertex v for which f(v) is no more than f evaluated at any of v's neighbors. In1983, Aldous gave the first strong lower bounds for the problem, showing that anyrandomized algorithm requires Ω(2n/2-o(n)) queries to determine a local minimum onthe n-dimensional hypercube. The next major step forward was not until 2004 whenAaronson, introducing a new method for query complexity bounds, both strengthened thislower bound to Ω(2n/2/n2) and gave an analogous lower bound on the quantum querycomplexity. While these bounds are very strong, they are known only for narrow familiesof graphs (hypercubes and grids). We show how to generalize Aaronson's techniques inorder to give randomized (and quantum) lower bounds on the query complexity of localsearch for the family of vertex-transitive graphs. In particular, we show that for anyvertex-transitive graph G of N vertices and diameter d, the randomized and quantumquery complexities for local search on G are Ω (√N/dlogN) and (4√N / √dlogN),respectively.

We study the behavior of the classical A* search algorithm when coupled with a heuristic that provides estimates, accurate to within a small multiplicative factor, of the distance to a solution. We prove general upper bounds on the complexity of A* search, for both admissible and unconstrained heuristic functions, that depend only on the distribution of solution objective values. We go on to provide nearly matching lower bounds that are attained even by non-adversarially chosen solution sets induced by a simple stochastic model.

Knowledge validation is a vital phase in knowledge engineering which is ultimately aimed at correcting the knowledge-base such that when inferenced over, it satisfy all test cases specified by the expert users. Naturally, such a goal is unachievable if there is some contradiction in the given set of test cases. In this work, we analyze this property of test case sets in knowledge validation for knowledge bases that are modelled in terms of Bayesian knowledge bases (BKBs) in order to determine the necessary and sufficient condition for a test case set to be contradiction-free, i.e., "Does there exist a knowledge base satisfying all test cases in the set?" We show that the complexity of deciding if a test case set meets that condition is NP-compete. As such, we also present some special cases in which it is tractable to make this determination.