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Digital theory, logic, and two-state control
Published in Raymond F. Gardner, Introduction to Plant Automation and Controls, 2020
Schematic diagrams are useful when ganging several Boolean-logic operations having many inputs to ultimately yield a single binary output. However, the schematic diagrams alone do not readily indicate the output state without careful tracking of each combination of input values. Truth tables are used in conjunction with schematic diagrams to relate each input-signal combination to its final true or false output. Sometimes, the logic-gate arrangements can be simplified to provide the same output from the same inputs. Three techniques for simplifying the logic gates are Boolean algebra, Karnaugh mapping, and computer methods. Table 4.1 summarizes the rules of Boolean algebra for simplifying more complicated functions. Some examples of Boolean algebra and truth tables are provided.
Digital Image Processing Preliminaries
Published in Qin Zhang, Roger Skjetne, Sea Ice Image Processing with MATLAB®, 2018
The logical operations are derived from Boolean algebra, which is a mathematical approach to describe propositions whose outcome would be either TRUE or FALSE. The logical operations consist of three basic operations: NOT, OR, and AND. The terms NOT, OR, and AND are commonly used to denote complementation, union, and intersection, respectively. The NOT operation simply inverts the input value, that is, the output is FALSE if the input is TRUE, and it sets to TRUE if the input is FALSE. The OR operation produces the output TRUE if either one of the inputs is TRUE, and FALSE if and only if all the inputs are FALSE. The AND operation produces the output TRUE if and only if all inputs are TRUE, and FALSE otherwise. Any other logic operator, such as NAND, NOR, and XOR, etc., can be implemented by using only these three operators.
Binary arithmetic, sets, logic and Boolean algebra
Published in H. G. Davies, G. A. Hicks, Mathematics for scientific and technical students, 2014
For any Boolean expression a truth table can be set up which lists all possible combinations of true and false values for the function. This is shown for two Boolean functions Z =ab and Z =a+ b in Tables 15.3 and 15.4.
Of gaps, gluts, and God's ability to change the past
Published in Journal of Applied Non-Classical Logics, 2022
Let us first look at the logic of gaps. This logic interprets ‘0.5’ as a truth-value gap of being neither true nor false. That is, any sentence that has it as a truth value – e.g. sentences about the future or vague sentences – is in the gap between truths and falsities.
Indefinite abductive explanations
Published in Journal of Applied Non-Classical Logics, 2019
Luciano Caroprese, Ester Zumpano
The set of ground instances of an atom a (resp. literal l, rule r, program ), denoted by (resp. , , ) is built by replacing variables with constants in all possible ways. An interpretation is a set of facts. The truth value of ground atoms, literals and rules with respect to an interpretation M is as follows: , , and , where A is an atom, are literals and true>false. An interpretation M is a model for a program , if all rules in are true w.r.t. M. A model M of a program P is said to be minimal if there is no model N of P such that . We denote the set of minimal models of a program with . The semantics of a positive program is given by its unique minimal model which can be computed by applying the immediate consequence operator until the fixpoint is reached (). The semantics of a program with negation is given by the set of its stable models, denoted as . An interpretation M is a stable model of if M is the unique minimal model of the positive program , where is obtained from by: (i) removing all rules r such that there exists a negative literal in the body of r and A is in M and (ii) removing all negative literals from the remaining rules (Gelfond & Lifschitz, 1988). It is well known that stable models are minimal models (i.e. ).
Rapid assessment of geo-hydrological hazards in Antananarivo (Madagascar) historical centre for damage prevention
Published in Geomatics, Natural Hazards and Risk, 2019
Andrea Ciampalini, William Frodella, Claudio Margottini, Nicola Casagli
The Analamanga hill shows a peculiar hydrographic setting characterized by two distinct networks: (i) the ‘natural’ hydrographic network, made of those creeks formed as a consequence of water erosion; (ii) the ‘anthropic’ hydrographic network formed by all the flow-paths represented by small streets and small artificial channels. The ‘natural’ hydrographic network was mapped during the field survey performed in October 2017. When possible, each identified creek was described in detail considering the presence of man-made structures, vegetation cover, presence/absence of running water, presence of soils or cover in the related watershed. Field surveys were carried out along the entire accessible hydrographic network of the study area. The area was systematically inspected along all drainage lines up to the catchment divide. The goal of the field survey is the identification of the main water courses where rainfall usually accumulates. The complexity of the urbanization of the slopes of the study area did not allow to accurately map all the flow-paths. In this case, the ‘anthropic’ hydrographic network was automatically mapped using the DEM and the LIDAR products in GIS environment. The use of a DEM aims at identifying all the possible flow-paths from every point to the main water courses (Tarboton et al. 1991). This step is very important in hilly areas like Antananarivo where the shallow drainage network and the sewer system are completely lacking. The procedure followed to extract the hydrographic network from both the DEMs was the following: anomalous DEM sinks and peaks, representing discontinuities in the DEM surface have been filled (Planchon and Darboux 2002). Once that the imperfection has been removed we applied the Flow Direction algorithm. This step takes the filled DEM as input and measures the direction of flow out of each cell of the raster (Greenlee 1987; Jenson and Domingue 1988). The Flow Direction raster was used to obtain the Flow Accumulation, a raster allowing for the calculation of the accumulated flow (Tarboton et al. 1991; Jenson and Domingue 1988). A conditional tool was applied to control the output value for each cell, based on whether the cell value is evaluated as true or false in a specified conditional statement. This tool has been used to extract the stream network avoiding mapping very small lines produced by the noise of the LiDAR following the Strahler (1957) method (Tarboton et al. 1991). The raster represented by the Stream ordering can be transformed into a vector layer in order to be easily used in GIS environment. The transformation from a raster to a feature allowed to efficiently manage the stream network extracted from a LiDAR or a DEM. The last step concerns the identification of the hydrographic basins. Basins were identified by recognizing ridge lines between basins (divides) on the input flow direction raster, which is analyzed to find all sets of connected cells that belong to the same basin. The results are a raster of drainage basins that can be transformed into a shape file.