Some Lower Bound Results for Set-Multilinear Arithmetic Computations

In this paper, we study the structure of set-multilinear arithmetic circuits and set-multilinear branching programs with the aim of showing lower bound results. We define some natural restrictions of these models for which we are able to show lower bound results. Some of our results extend existing lower bounds, while others are new and raise open questions. More specifically, our main results are the following: (1) We observe that set-multilinear arithmetic circuits can be transformed into shallow set-multilinear circuits efficiently, similar to depth reduction results of [VSBR83,RY08] for more general commutative circuits. As a consequence, we note that polynomial size set-multilinear circuits have quasi-polynomial size set-multilinear branching programs. We show that \emph{narrow} set-multilinear ABPs (with a restricted number of set types) computing the Permanent polynomial $\mathrm{PER}_n$ require $2^{n^{\Omega(1)}}$ size. A similar result for general set-multilinear ABPs appears difficult as it would imply that the Permanent requires superpolynomial size set-multilinear circuits. It would also imply that the noncommutative Permanent requires superpolynomial size noncommutative arithmetic circuits. (2) Indeed, we also show that set-multilinear branching programs are exponentially more powerful than \emph{interval} multilinear circuits (where the index sets for each gate is restricted to be an interval w.r.t.\ some ordering), assuming the sum-of-squares conjecture. This further underlines the power of set-multilinear branching programs. (3) Finally, we consider set-multilinear circuits with restrictions on the number of proof trees of monomials computed by it, and prove exponential lower bounds results. This raises some new lower bound questions

Detecting and resolving spatial ambiguity in text using named entity extraction and self learning fuzzy logic techniques

Information extraction identifies useful and relevant text in a document and converts unstructured text into a form that can be loaded into a database table. Named entity extraction is a main task in the process of information extraction and is a classification problem in which words are assigned to one or more semantic classes or to a default non-entity class. A word which can belong to one or more classes and which has a level of uncertainty in it can be best handled by a self learning Fuzzy Logic Technique. This paper proposes a method for detecting the presence of spatial uncertainty in the text and dealing with spatial ambiguity using named entity extraction techniques coupled with self learning fuzzy logic techniquesComment: National Conference on Recent Trends in Data Mining and Distributed Systems September 201

A Fuzzy Logic based Method for Efficient Retrieval of Vague and Uncertain Spatial Expressions in Text Exploiting the Granulation of the Spatial Event Queries

The arrangement of things in n-dimensional space is specified as Spatial. Spatial data consists of values that denote the location and shape of objects and areas on the earths surface. Spatial information includes facts such as location of features, the relationship of geographic features and measurements of geographic features. The spatial cognition is a primal area of study in various other fields such as Robotics, Psychology, Geosciences, Geography, Political Sciences, Geographic Economy, Environmental, Mining and Petroleum Engineering, Natural Resources, Epidemiology, Demography etc., Any text document which contains physical location specifications such as place names, geographic coordinates, landmarks, country names etc., are supposed to contain the spatial information. The spatial information may also be represented using vague or fuzzy descriptions involving linguistic terms such as near to, far from, to the east of, very close. Given a query involving events, the aim of this ongoing research work is to extract the relevant information from multiple text documents, resolve the uncertainty and vagueness and translate them in to locations in a map. The input to the system would be a text Corpus and a Spatial Query event. The output of the system is a map showing the most possible, disambiguated location of the event queried. The author proposes Fuzzy Logic Techniques for resolving the uncertainty in the spatial expressions.Comment: National Conference on Future Computing,0975 8887,IJCA,February201

A Study on Application of Spatial Data Mining Techniques for Rural Progress

This paper focuses on the application of Spatial Data mining Techniques to efficiently manage the challenges faced by peripheral rural areas in analyzing and predicting market scenario and better manage their economy. Spatial data mining is the task of unfolding the implicit knowledge hidden in the spatial databases. The spatial Databases contain both spatial and non-spatial attributes of the areas under study. Finding implicit regularities, rules or patterns hidden in spatial databases is an important task, e.g. for geo-marketing, traffic control or environmental studies. In this paper the focus is on the effective use of Spatial Data Mining Techniques in the field of Economic Geography constrained to the rural areasComment: International Conference on Innovative Computing, information and communication technology ICICT09; souvenir pp no 6

Computational Model to Generate Case-Inflected Forms of Masculine Nouns for Word Search in Sanskrit E-Text

The problem of word search in Sanskrit is inseparable from complexities that include those caused by euphonic conjunctions and case-inflections. The case-inflectional forms of a noun normally number 24 owing to the fact that in Sanskrit there are eight cases and three numbers-singular, dual and plural. The traditional method of generating these inflectional forms is rather elaborate owing to the fact that there are differences in the forms generated between even very similar words and there are subtle nuances involved. Further, it would be a cumbersome exercise to generate and search for 24 forms of a word during a word search in a large text, using the currently available case-inflectional form generators. This study presents a new approach to generating case-inflectional forms that is simpler to compute. Further, an optimized model that is sufficient for generating only those word forms that are required in a word search and is more than 80% efficient compared to the complete case-inflectional forms generator, is presented in this study for the first time

A Binary Schema and Computational Algorithms to Process Vowel-based Euphonic Conjunctions for Word Searches

Comprehensively searching for words in Sanskrit E-text is a non-trivial problem because words could change their forms in different contexts. One such context is sandhi or euphonic conjunctions, which cause a word to change owing to the presence of adjacent letters or words. The change wrought by these possible conjunctions can be so significant in Sanskrit that a simple search for the word in its given form alone can significantly reduce the success level of the search. This work presents a representational schema that represents letters in a binary format and reduces Paninian rules of euphonic conjunctions to simple bit set-unset operations. The work presents an efficient algorithm to process vowel-based sandhis using this schema. It further presents another algorithm that uses the sandhi processor to generate the possible transformed word forms of a given word to use in a comprehensive word search

An Ontology for Comprehensive Tutoring of Euphonic Conjunctions of Sanskrit Grammar

Euphonic conjunctions (sandhis) form a very important aspect of Sanskrit morphology and phonology. The traditional and modern methods of studying about euphonic conjunctions in Sanskrit follow different methodologies. The former involves a rigorous study of the Paninian system embodied in Panini's Ashtadhyayi, while the latter usually involves the study of a few important sandhi rules with the use of examples. The former is not suitable for beginners, and the latter, not sufficient to gain a comprehensive understanding of the operation of sandhi rules. This is so since there are not only numerous sandhi rules and exceptions, but also complex precedence rules involved. The need for a new ontology for sandhi-tutoring was hence felt. This work presents a comprehensive ontology designed to enable a student-user to learn in stages all about euphonic conjunctions and the relevant aphorisms of Sanskrit grammar and to test and evaluate the progress of the student-user. The ontology forms the basis of a multimedia sandhi tutor that was given to different categories of users including Sanskrit scholars for extensive and rigorous testing

Computational Algorithms Based on the Paninian System to Process Euphonic Conjunctions for Word Searches

Searching for words in Sanskrit E-text is a problem that is accompanied by complexities introduced by features of Sanskrit such as euphonic conjunctions or sandhis. A word could occur in an E-text in a transformed form owing to the operation of rules of sandhi. Simple word search would not yield these transformed forms of the word. Further, there is no search engine in the literature that can comprehensively search for words in Sanskrit E-texts taking euphonic conjunctions into account. This work presents an optimal binary representational schema for letters of the Sanskrit alphabet along with algorithms to efficiently process the sandhi rules of Sanskrit grammar. The work further presents an algorithm that uses the sandhi processing algorithm to perform a comprehensive word search on E-text

Randomized Polynomial Time Identity Testing for Noncommutative Circuits

In this paper we show that the black-box polynomial identity testing for noncommutative polynomials $f\in\mathbb{F}\langle z_1,z_2,\cdots,z_n \rangle$ of degree $D$ and sparsity $t$, can be done in randomized \poly(n,\log t,\log D) time. As a consequence, if the black-box contains a circuit $C$ of size $s$ computing $f\in\mathbb{F}\langle z_1,z_2,\cdots,z_n \rangle$ which has at most $t$ non-zero monomials, then the identity testing can be done by a randomized algorithm with running time polynomial in $s$ and $n$ and $\log t$. This makes significant progress on a question that has been open for over ten years. The earlier result by Bogdanov and Wee [BW05], using the classical Amitsur-Levitski theorem, gives a randomized polynomial-time algorithm only for circuits of polynomially bounded syntactic degree. In our result, we place no restriction on the degree of the circuit. Our algorithm is based on automata-theoretic ideas introduced in [AMS08,AM08]. In those papers, the main idea was to construct deterministic finite automata that isolate a single monomial from the set of nonzero monomials of a polynomial $f$ in $\mathbb{F}\langle z_1,z_2,\cdots,z_n \rangle$. In the present paper, since we need to deal with exponential degree monomials, we carry out a different kind of monomial isolation using nondeterministic automata.Comment: As the number of monomials in a noncommutative polynomial which has anarithmetic circuit of size $s$ can actually be doubly exponential in $s$, our result does not imply a randomized polynomial-time identity test for all size s noncommutative circuits. The algorithm works only for noncommutative size s circuits which computes a polynomial with exp(s) many monomial

Noncommutative Valiant's Classes: Structure and Complete Problems

In this paper we explore the noncommutative analogues, $\mathrm{VP}_{nc}$ and $\mathrm{VNP}_{nc}$, of Valiant's algebraic complexity classes and show some striking connections to classical formal language theory. Our main results are the following: (1) We show that Dyck polynomials (defined from the Dyck languages of formal language theory) are complete for the class $\mathrm{VP}_{nc}$ under $\le_{abp}$ reductions. Likewise, it turns out that $\mathrm{PAL}$ (Palindrome polynomials defined from palindromes) are complete for the class $\mathrm{VSKEW}_{nc}$ (defined by polynomial-size skew circuits) under $\le_{abp}$ reductions. The proof of these results is by suitably adapting the classical Chomsky-Sch\"{u}tzenberger theorem showing that Dyck languages are the hardest CFLs. (2) Next, we consider the class $\mathrm{VNP}_{nc}$. It is known~\cite{HWY10a} that, assuming the sum-of-squares conjecture, the noncommutative polynomial $\sum_{w\in\{x_0,x_1\}^n}ww$ requires exponential size circuits. We unconditionally show that $\sum_{w\in\{x_0,x_1\}^n}ww$ is not $\mathrm{VNP}_{nc}$-complete under the projection reducibility. As a consequence, assuming the sum-of-squares conjecture, we exhibit a strictly infinite hierarchy of p-families under projections inside $\mathrm{VNP}_{nc}$ (analogous to Ladner's theorem~\cite{Ladner75}). In the final section we discuss some new $\mathrm{VNP}_{nc}$-complete problems under $\le_{abp}$-reductions. (3) Inside $\mathrm{VP}_{nc}$ too we show there is a strict hierarchy of p-families (based on the nesting depth of Dyck polynomials) under the $\le_{abp}$ reducibility