The Quine—McCluskey algorithm QMCalso known as the method of prime implicantsis a method used for minimization of Boolean functions that was developed by Willard V. Quine in [1] [2] and extended by Edward J. McCluskey in Samson and Burton E. Mills in [10] [6] and by Raymond J. Nelson in Abrahams and John G. Nordahl [12] as well as Albert A. Mullin and Wayne G. Kellner [13] [14] [15] [16] proposed a decimal variant of the method. The Quine—McCluskey algorithm is functionally identical to Karnaugh mappingbut the tabular form makes it more efficient for use in computer algorithms, and it also gives a deterministic way to check that the minimal form of a Boolean function has been reached.

It is sometimes referred to as the tabulation method. Although more practical than Karnaugh mapping when dealing with more than four variables, the Quine—McCluskey algorithm also has a limited range of use since the problem it solves is NP-complete.

For a function of n variables the number of prime implicants can be as large as 3 n ln ne. Functions with a large number of variables have to be minimized with potentially non-optimal heuristic methods, of which the Espresso heuristic logic minimizer was the de facto standard in Step two of the algorithm amounts to solving the set cover problem ; [26] NP-hard instances of this problem may occur in this algorithm step.

We encode all of this information by writing. One can easily form the canonical sum of products expression from this table, simply by summing the minterms leaving out don't-care terms where the function evaluates to one:. So to optimize, all minterms that evaluate to one are first placed in a minterm table. Don't-care terms are also added into this table names in parenthesesso they can be combined with minterms:.

At this point, one can start combining minterms with other minterms.Some of the tables have many columns and millions of rows. When the model is being processed, all the memory of the server is consumed. Is there a way to reduce the size of the model so I can keep the entire model in memory? This means all data is compressed using the columnstore technology and then loaded into memory.

Due to the compression, the size of the data is typically much smaller in memory than it is on disk. However, models can grow too large to fit into memory. Keep in mind, processing a Tabular model can be very memory intensive. The new data has to be loaded into memory — uncompressed while it is being encoded — and the old data might be kept around as well for transaction purposes.

If you have multiple tables being processed at once, a lot of data is being put into RAM. You can avoid this by having a different processing strategy than just doing a Process Full. You can for example do a Process Clear first to remove existing data out of memory and then a Process Default to process the model again.

You can use combine this with the techniques described in this tip to further trim down on your memory usage. On my machine, I removed some blob columns to save space and I put a clustered columnstore index on the table. The following query will be used to extract the data from the database:.

You might adjust this number to your environment. As you can see in the query, I added two extra columns to showcase two typical scenarios you might find in a Tabular model:. Both columns will compress very badly and will take up a large portion of the model size. If you are using the compatibility level for SSAS Tabularyou need to modify the registry so you can use the legacy data sources in Visual Studio. There you can specify SQL statements, while in the modern get-data experience you can only create M queries.

In the wizard, choose Microsoft SQL Server as the data source type, specify the connection information to the StackOverflow database, specify the impersonation information I choose the service account, which means you need to give the account read permissions on the StackOverflow database and then choose how you want to import the data:. Before we start reducing the size of our Tabular model, we first have to figure out how big it is.

There are a couple of methods to get the size; some are more accurate than others. In earlier versions of SSAS Tabular, the easiest method is to right-click on a Tabular database and go to its properties. Another option is to go to the data folders and check the size of the model on disk the model is saved here when it is unloaded from memory. For the StackOverflow database, we find a size of about 2. Again, this is a very rough estimate.

Luckily, a free-to-use tool has been built by sqlbi. The VertiPaq Analyzer add an extra tab to the ribbon of the Excel workbook where you can also find more information:. To get the information from your model, you need to go to the PowerPivot model, where you click on Existing Connections in the ribbon. Once the connection is modified, you need to refresh the workbook so it can retrieve all the data.

You can do this in the PowerPivot model itself, in the Vertipaq Analyzer tab or in the Analyze tab when you have selected a pivot table. Here we can find the following information:. The model takes up 4. The high cardinality columns take up the most space.Table of Contents.

Sum of Product is the abbreviated form of SOP. Sum of product form is a form of expression in Boolean algebra in which different product terms of inputs are being summed together. To understand better about SOP, we need to know about min term. Minterm means the term that is true for a minimum number of combination of inputs. That is true for only one combination of inputs. Since AND gate also gives True only when all of its inputs are true so we can say min terms are AND of input combinations like in the table given below.

Each combination has a min terms denoted by small m and its decimal combination number written in subscript. Each of these minterms will be only true for the specific input combination. This is the standard form of Sum of Product. It is formed by O Ring the minterms of the function for which the output is true. It is just a fancy name. For example, a functions truth table is given below.

For this function the canonical SOP expression is. By expanding the summation we get. Now putting min terms in the expression. Canonical form contains all inputs either complemented or non-complemented in its product terms. As the name suggests, this form is the non-standardized form of SOP expressions. The product terms are not the min terms but they are simplified. This expression is still in Sum of Product form but it is non-canonical or non-standardized form.

This form is the most simplified SOP expression of a function. It is also a form of non-canonical form. Minimal SOP form is preferred because it uses the minimum number of gates and input lines.

Its K-map is given below. According to the K-map, the output expression will be. Every SOP expression has somewhat same designing i.

### Quine–McCluskey algorithm

The number of inputs and the number of AND gates depend upon the expression one is implementing. Example of designs of canonical and minimal SOP expression for a function is given below. Conversion from minimal or any sort of non-canonical form to canonical form is very simple.

Example of conversion for the above function in minimal SOP form is given below. Minimal SOP form. Now, this expression is in canonical form. For example, the function given above is in canonical SOP form. The remaining terms of this function are maxterms for which output is false.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Artificial Intelligence Stack Exchange is a question and answer site for people interested in conceptual questions about life and challenges in a world where "cognitive" functions can be mimicked in purely digital environment.

It only takes a minute to sign up. While learning RL, I came across some problems where the Q-matrix that I need to make is very very large. I am not sure if it is ever practical. Then I research and came to this conclusion that using the tabular method is not the only way, in fact, it is a very less powerful tool as compared to other methods such as deep RL methods.

Am I correct in this understanding that with the increasing complexity of problems, tabular RL methods are getting obsolete? Individual problems don't get any more complex, but the scope of solvable environments increases due to research and discovery of better or more apt methods.

Using deep RL methods with large neural nets can be a lot less efficient for solving simple problems. So tabular methods still have their place there. So tabular approaches are often preferred if they are appropriate. Many interesting, cutting edge problems that are relevant to AI, such as autonomous robots acting in the real world, do not fit the tabular approach.

In that sense, the approach is "obsolete" in that it no longer provides challenging research topics for practical AI there are still unanswered theoretical questions, such as proof of convergence for Monte Carlo control. It is still worth understanding tabular value-based methods in detail, because they form the foundations of the more complex deep learning methods.

In some sense they represent ideal solutions that deep RL tries to approximate, and the design of tabular solutions can be the inspiration for changes and adjustments to neural-network methods. Sign up to join this community.

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The best answers are voted up and rise to the top. Are tabular reinforcement learning methods obsolete or getting obsolete? Ask Question. Asked 2 months ago. Active 2 months ago. Viewed 93 times. Kashan Kashan 8 8 bronze badges.Solution: To solve the given problem, we follow the steps given below: Step 1 : The first step in the QM method is to separate the minterms into specific groups, as shown in Table 2. These groups are formed on the basis of the number of 1 s in their binary form.

## Quine McCluskey (Tabular) Method Example

For example, the binary number has no 1 in it and hence forms the first group. Binary numbers, etc. This process is continued to form groups with two 1 s, three 1 s, etc.

We shall make use of Table 2. By comparison of the minterms in our example with the tabulation given in Table 2. It may be noted that the members in each group differ in one-bit position only from the members of the adjacent group below or above that group. This rule follows for all the subsequent groups in the first as well as other charts. It may also be noted that the groups are named as PQRetc. Once we are familiar with the procedure of groupings, this designation is not required as will be clear from the examples to follow.

**Quine McCluskey Simplification with Don't Cares (Tabular Method) with k-map verification.**

Now, the first column of the first chart shows the names of the groups as PQRSetc. Below this, in Group Qwe have numbers 1, 2, and 8 as its members.

Similarly, every member of Q is different from every member of R in only one position. Thus, we conclude that the groups PQR, etc. It may also be noted that this is not true for distant groups. Hence grouping and pairing is not possible among distant groups. Step 2 : In this, as stated above, we first form the groups, as shown in Chart 1. From this chart, we prepare Chart 2, which shows the members of each group that can be paired together, these pairings being indicated by tick marksas shown in Chart 2.

For this, we inspect adjacent groups and collect members that differ from one another by a power of 2.

Hence, they can be paired together to form the members of Chart 3. The principle behind these pairings is that each of these pairs can be combined to eliminate one bit, as shown: Table 2.

As stated, in the pairing shown above, we find that variable D is eliminated. Illustration 2: As in Illustration 1, we find that decimal numbers 0 and 8 can be combined to eliminate Ashown below. Thus we find that appropriate pairing eliminates the variables appearing in the complemented and uncomplemented forms. In this way, one pairing operation between adjacent groups results in logic reduction by one bit.

As stated previously, it may be noted that pairing must be done strictly between adjacent groups, i. Pairing cannot be done between P and RP and Setc.

The results of the pairing are tabulated, as shown in Chart 3. This is shown in Column 2 of Chart 3. Column 3 shows the difference between the decimal numbers in each pair; it must be remembered that this difference must be exactly a power of 2, as indicated in Column 4. Usually, Columns 3 and 4 need not be shown in the charts; these can be easily avoided in practical solutions using the QM method. Here, they are shown only for illustration.

It also may be noted that to find the difference between two numbers, decimal numbers are easier to remember and handle than binary numbers. For example, it is easy to subtract mentally 7 from 12 than subtracting from For example, a pair from group P 1 can be combined with a pair from the adjacent group Q 1 if they differ by a power of 2.

Thus, in Chart 3, the pair 0, 1 of Group P 1 can be combined with the pair 2, 3 from Group Q 1as they differ by the same power of 2 i.In previous chapter, we discussed K-map method, which is a convenient method for minimizing Boolean functions up to 5 variables. But, it is difficult to simplify the Boolean functions having more than 5 variables by using this method.

Quine-McClukey tabular method is a tabular method based on the concept of prime implicants. This tabular method is useful to get the prime implicants by repeatedly using the following Boolean identity. If there is a change in only one-bit position, then take the pair of those two min terms. It consists of set of rows and columns.

Prime implicants can be placed in row wise and min terms can be placed in column wise. If the min term is covered only by one prime implicant, then it is essential prime implicant. Those essential prime implicants will be part of the simplified Boolean function. Repeat step 5 for Reduced prime implicant table. Stop this process when all min terms of given Boolean function are over.

The given Boolean function is in sum of min terms form. The given min terms are 2, 6, 8, 9, 10, 11, 14 and The ascending order of these min terms based on the number of ones present in their binary equivalent is 2, 8, 6, 9, 10, 11, 14 and The following table shows these min terms and their equivalent binary representations. The given min terms are arranged into 4 groups based on the number of ones present in their binary equivalents.

The following table shows the possible merging of min terms from adjacent groups. The min terms, which are differed in only one-bit position from adjacent groups are merged. In this case, there are three groups and each group contains combinations of two min terms. The following table shows the possible merging of min term pairs from adjacent groups.

The successive groups of min term pairs, which are differed in only one-bit position are merged. In this case, there are two groups and each group contains combinations of four min terms. Here, these combinations of 4 min terms are available in two rows. So, we can remove the repeated rows. The reduced table after removing the redundant rows is shown below.

Further merging of the combinations of min terms from adjacent groups is not possible, since they are differed in more than one-bit position. There are three rows in the above table. So, each row will give one prime implicant. The prime implicants are placed in row wise and min terms are placed in column wise. So, it is an essential prime implicant.

This will be part of simplified Boolean function. Now, remove this prime implicant row and the corresponding min term columns. The reduced prime implicant table is shown below. The min term 15 is covered only by one prime implicant WY. In this example problem, we got three prime implicants and all the three are essential. Therefore, the simplified Boolean function is.

Previous Page. Next Page.This article provides insight into the Karnaugh map K-map Boolean algebraic simplification technique via a few examples. It also includes a brief note on the advantages and the disadvantages of K-maps. Digital electronics deals with the discrete-valued digital signals. In general, any electronic system based on the digital logic uses binary notation zeros and ones to represent the states of the variables involved in it. Thus, Boolean algebraic simplification is an integral part of the design and analysis of a digital electronic system.

Although Boolean algebraic laws and DeMorgan's theorems can be used to achieve the objective, the process becomes tedious and error-prone as the number of variables involved increases.

This necessitates the use of a suitable, relatively-simple simplification technique like that of Karnaugh map K-mapintroduced by Maurice Karnaugh in The K-map method of solving the logical expressions is referred to as the graphical technique of simplifying Boolean expressions. K-maps are also referred to as 2D truth tables as each K-map is nothing but a different format of representing the values present in a one-dimensional truth table.

Further, each cell within a K-map has a definite place-value which is obtained by using an encoding technique known as Gray code. The specialty of this code is the fact that the adjacent code values differ only by a single bit. This means that each K-map cell can be addressed using a unique Gray Code-Word. These concepts are further emphasized by a typical celled K-map shown in Figure 1, which can be used to simplify a logical expression comprising of 4-variables A, B, C and D mentioned at its top-left corner.

Here the rows and the columns of the K-map are labeled using 2-bit Gray code, shown in the figure, which assigns a definite address for each of its cells. We'll start with a given problem statement. Design a digital system whose output is defined as logically low if the 4-bit input binary number is a multiple of 3; otherwise, the output will be logically high. The output is defined if and only if the input binary number is greater than 2. The first step in designing any digital system is to have a clear idea of the variables involved in the process, along with their state-values.

Further, depending on the problem statement, we have to arrive at the number of output variables and their values for each and every combination of the input literals, which can be conveniently represented in the form of a truth table.

Note that, in addition to the input and output columns, the truth table also has a column which gives the decimal equivalent of the input binary combination, which makes it easy for us to arrive at the minterm or maxterm expansion for the given problem.

Thus for the given example. However, sometimes the logical expression which is to be simplified might be directly given in terms of SOP or POS forms. In this case, the requirement for the truth table can be overlooked provided that we express the given expression in its canonical form, from which the corresponding minterms or maxterms can be obtained.

From Step 1, we know the number of input variables involved in the logical expression from which size of the K-map required will be decided. Further, we also know the number of such K-maps required to design the desired system as the number of output variables would also be known definitely.

The procedure is to be repeated for every single output variable. K-map simplification can also be referred to as the "simplification by grouping" technique as it solely relies on the formation of clusters. This can be done by expressing the bits which are common amongst the Gray code-words which represent the cells contained within the considered group. Another way to describe the process of obtaining the simplified logical expression for a group is to eliminate the variable s for which the corresponding bits appear within the group as both 0 and 1.

Finally, all these group-wise logical expressions need to be combined appropriately to form the simplified Boolean equation for the output variable. The same procedure must be repeated for every output variable of the given problem. Having obtained the simplified logical expression, we can decide on the type and the number of gates required to realize the expected logic for every output bit, which further results in the complete design of the desired system.

The digital system designed to realize the full adder in terms of sum and carry outputs in POS form is shown by Figure An alternative for the inputs.