Processing Succinct Matrices and Vectors

We study the complexity of algorithmic problems for matrices that are represented by multi-terminal decision diagrams (MTDD). These are a variant of ordered decision diagrams, where the terminal nodes are labeled with arbitrary elements of a semiring (instead of 0 and 1). A simple example shows that the product of two MTDD-represented matrices cannot be represented by an MTDD of polynomial size. To overcome this deficiency, we extended MTDDs to MTDD_+ by allowing componentwise symbolic addition of variables (of the same dimension) in rules. It is shown that accessing an entry, equality checking, matrix multiplication, and other basic matrix operations can be solved in polynomial time for MTDD_+-represented matrices. On the other hand, testing whether the determinant of a MTDD-represented matrix vanishes PSPACE$-complete, and the same problem is NP-complete for MTDD_+-represented diagonal matrices. Computing a specific entry in a product of MTDD-represented matrices is #P-complete.Comment: An extended abstract of this paper will appear in the Proceedings of CSR 201

SISTEM PENDUKUNG KEPUTUSAN PENENTUAN PENERIMA BEASISWA DENGAN METODE SIMPLE ADDITIVE WEIGHTING DI SMPN 19 TANGERANG

In determining scholarship recipients in schools manually, errors often occur causing inefficient management of scholarship data in terms of time and the absence of clear criteria for how students can obtain scholarships. To anticipate that mistakes do not occur, a Decision Support System is needed. One method that can be used is the Simple Additive Weighting (SAW) method. The data collection method used was the observation method to the school and the interview method with the related teachers regarding the scholarship acceptance. In this research, system development is carried out using the Waterfall method. And for system testing used Black-box Testing. The system design uses the Unified Model Language (UML), which includes usecase diagrams, activity diagrams, sequence diagrams and class diagrams. Then the program implementation uses the PHP programming language and MySQL database. This application will also display the value, criteria, alternatives and then a ranking of the scholarship recipient determination process. With this decision support system, the school will get the results of who is entitled to receive the scholarship. Keywords: Decision Support System, Scholarship, Simple Additive Weighting (SAW) Method, Unified Model Language (UML)

Tensor Networks or Decision Diagrams? Guidelines for Classical Quantum Circuit Simulation

Classically simulating quantum circuits is crucial when developing or testing quantum algorithms. Due to the underlying exponential complexity, efficient data structures are key for performing such simulations. To this end, tensor networks and decision diagrams have independently been developed with differing perspectives, terminologies, and backgrounds in mind. Although this left designers with two complementary data structures for quantum circuit simulation, thus far it remains unclear which one is the better choice for a given use case. In this work, we (1) consider how these techniques approach classical quantum circuit simulation, and (2) examine their (dis)similarities with regard to their most applicable abstraction level, the desired simulation output, the impact of the computation order, and the ease of distributing the workload. As a result, we provide guidelines for when to better use tensor networks and when to better use decision diagrams in classical quantum circuit simulation.Comment: 7 pages, 4 figures, comments welcom

Binary Decision Diagrams: from Tree Compaction to Sampling

Any Boolean function corresponds with a complete full binary decision tree. This tree can in turn be represented in a maximally compact form as a direct acyclic graph where common subtrees are factored and shared, keeping only one copy of each unique subtree. This yields the celebrated and widely used structure called reduced ordered binary decision diagram (ROBDD). We propose to revisit the classical compaction process to give a new way of enumerating ROBDDs of a given size without considering fully expanded trees and the compaction step. Our method also provides an unranking procedure for the set of ROBDDs. As a by-product we get a random uniform and exhaustive sampler for ROBDDs for a given number of variables and size

Circuit Testing Based on Fuzzy Sampling with BDD Bases

Fuzzy testing of integrated circuits is an established technique. Current approaches generate an approximately uniform random sample from a translation of the circuit to Boolean logic. These approaches have serious scalability issues, which become more pressing with the ever-increasing size of circuits. We propose using a base of binary decision diagrams to sample the translations as a soft computing approach. Uniformity is guaranteed by design and scalability is greatly improved. We test our approach against five other state-of-the-art tools and find our tool to outperform all of them, both in terms of performance and scalability

The Complexity of Reasoning with FODD and GFODD

Recent work introduced Generalized First Order Decision Diagrams (GFODD) as a knowledge representation that is useful in mechanizing decision theoretic planning in relational domains. GFODDs generalize function-free first order logic and include numerical values and numerical generalizations of existential and universal quantification. Previous work presented heuristic inference algorithms for GFODDs and implemented these heuristics in systems for decision theoretic planning. In this paper, we study the complexity of the computational problems addressed by such implementations. In particular, we study the evaluation problem, the satisfiability problem, and the equivalence problem for GFODDs under the assumption that the size of the intended model is given with the problem, a restriction that guarantees decidability. Our results provide a complete characterization placing these problems within the polynomial hierarchy. The same characterization applies to the corresponding restriction of problems in first order logic, giving an interesting new avenue for efficient inference when the number of objects is bounded. Our results show that for $\Sigma_k$ formulas, and for corresponding GFODDs, evaluation and satisfiability are $\Sigma_k^p$ complete, and equivalence is $\Pi_{k+1}^p$ complete. For $\Pi_k$ formulas evaluation is $\Pi_k^p$ complete, satisfiability is one level higher and is $\Sigma_{k+1}^p$ complete, and equivalence is $\Pi_{k+1}^p$ complete.Comment: A short version of this paper appears in AAAI 2014. Version 2 includes a reorganization and some expanded proof

Extension to UML-B Notation and Toolset

The UML-B notation has been created as an attempt to combine the success and ease of use of UML, with the verification and rigorous development capabilities of formal methods. However, the notation currently only supports a basic diagram set. To address this we have, in this project, designed and implemented a set of extensions to the UML-B notation that provide a much fuller software engineering experience, critically making UML-B more appealing to industry partners. These extensions comprise five new diagram types, which are aimed at supplying a broader range of design capabilities, such as conceptual Use-Case design and future integration with the ProB animator tool