Expression Evaluation: Evaluating the Evaluator: How RPN Streamlines Complex Calculations

1. The Basics

Expression evaluation is a fundamental concept in computer science and mathematics, serving as the cornerstone for computations in various programming languages and mathematical computations. It involves the process of determining the value of a mathematical expression written in a certain notation. This process is not just about understanding the numbers and operators involved but also about appreciating the underlying principles that govern the order of operations and the methods used to simplify and solve expressions.

From a programmer's perspective, expression evaluation is crucial for interpreting and executing code efficiently. In contrast, a mathematician might view expression evaluation as a structured approach to solving problems. Regardless of the viewpoint, the basics of expression evaluation remain the same: recognizing the types of expressions, understanding precedence and associativity of operators, and applying the correct algorithm to evaluate the expression.

Here's an in-depth look at the basics of expression evaluation:

1. Types of Expressions:

- Arithmetic expressions: Involve operators like addition (+), subtraction (-), multiplication (*), and division (/).

- Logical expressions: Use logical operators such as AND, OR, and NOT to return a boolean value.

- Relational expressions: Compare values using relational operators like greater than (>), less than (<), and equals (==).

2. Operator Precedence and Associativity:

- Operators have a predefined order of precedence that determines the order in which operations are performed.

- Associativity defines the order in which operators of the same precedence level are evaluated.

3. Evaluation Methods:

- Infix notation: The operator is placed between operands (e.g., A + B).

- Prefix notation (Polish notation): The operator precedes the operands (e.g., +AB).

- Postfix notation (Reverse Polish Notation - RPN): The operator follows the operands (e.g., AB+).

4. Reverse Polish Notation (RPN):

- RPN simplifies complex calculations by eliminating the need for parentheses.

- It is processed using a stack data structure, pushing operands onto the stack and applying operators to the operands that are popped from the stack.

Example: Consider the infix expression `3 + 4 2 / (1 - 5)`. In RPN, this expression is written as `3 4 2 1 5 - / +`. To evaluate:

- Push 3 onto the stack.

- Push 4 onto the stack.

- Push 2 onto the stack, multiply by 4 (top of the stack), and push the result (8) back onto the stack.

- Push 1 onto the stack.

- Push 5 onto the stack, subtract from 1 (top of the stack), and push the result (-4) back onto the stack.

- Divide 8 (top of the stack) by -4 and push the result (-2) back onto the stack.

- Add 3 (bottom of the stack) to -2 (top of the stack) to get the final result of 1.

Understanding the basics of expression evaluation is essential for anyone involved in programming or mathematics, as it lays the groundwork for more advanced topics and applications. Whether you're a student learning to code or a seasoned developer working on complex algorithms, mastering these fundamentals is key to success.

2. The Reverse Polish Notation

Reverse Polish Notation (RPN) is a mathematical notation wherein every operator follows all of its operands. It is also known as postfix notation and is distinct from the more common infix notation, which places the operator between the operands. RPN's unique structure eliminates the need for parentheses that are required in infix notation to dictate the order of operations. This makes RPN particularly advantageous for computer and calculator parsing because it reflects the actual steps taken to perform calculations. The notation has been widely adopted in calculators, particularly those used in the fields of engineering and computer science, due to its efficiency and speed.

Insights from Different Perspectives:

1. Computational Efficiency: From a computational standpoint, RPN is highly efficient for stack-based machines or algorithms. Since operations are performed immediately after their second operand is received, there is no need to retain operator precedence rules or parenthesis.

2. Human Readability: While RPN can be less intuitive for those accustomed to traditional infix notation, it offers clarity in complex calculations. Once familiar with the notation, users can often calculate expressions more quickly than with infix notation.

3. Historical Usage: RPN gained prominence with the introduction of Hewlett-Packard's scientific calculators. It was a key feature that set these calculators apart and showcased the practical application of RPN in performing quick and error-free calculations.

In-Depth Information:

1. The Stack: The core concept of RPN is the stack, where values are pushed onto the stack, and operators pop values from the stack to perform operations. For example, the expression $$ 3 \, 4 + $$ in RPN is equivalent to $$ 3 + 4 $$ in infix notation. Here, '3' and '4' are pushed onto the stack, and '+' pops them off to perform the addition, pushing the result back onto the stack.

2. No Ambiguity: RPN expressions do not require parentheses because the order of operations is inherently clear. This removes ambiguity in expressions like $$ 3 \, 4 \, 5 + \times $$, which would require parentheses in infix notation to clarify the order of operations.

3. Error Reduction: The sequential nature of RPN reduces the likelihood of errors in complex calculations. Each step is processed and completed before moving on to the next, minimizing the cognitive load and potential for mistakes.

Examples to Highlight Ideas:

- Simple Arithmetic: For the expression $$ 5 \, 1 \, 2 + 4 \times + 3 - $$, the calculation proceeds as follows:

1. Push 5 onto the stack.

2. Push 1 onto the stack.

3. Push 2 onto the stack.

4. Add 1 and 2, push the result (3) onto the stack.

5. Push 4 onto the stack.

6. Multiply 3 (the result of 1+2) by 4, push the result (12) onto the stack.

7. Add 5 and 12, push the result (17) onto the stack.

8. Push 3 onto the stack.

9. Subtract 3 from 17, resulting in 14.

- Logical Operations: In programming, RPN can be used for logical operations such as $$ true \, false \, OR \, true \, AND $$. This would be evaluated as:

1. Push 'true' onto the stack.

2. Push 'false' onto the stack.

3. Perform OR operation, push 'true' (result of true OR false) onto the stack.

4. Push 'true' onto the stack.

5. Perform AND operation, resulting in 'true' (since true AND true is true).

RPN's utility in various domains, from calculators to computer algorithms, demonstrates its versatility and enduring relevance. Its ability to streamline complex calculations and reduce errors makes it a valuable tool for anyone working with intricate mathematical expressions.

3. The History of RPN and Its Mathematical Significance

Reverse Polish Notation (RPN), also known as postfix notation, is a mathematical notation in which every operator follows all of its operands. It is a key concept in the field of computer science, particularly in the realm of expression evaluation. Unlike traditional infix notation, which can be complex and require rules for operator precedence and parentheses to enforce order, RPN simplifies the process by eliminating the need for these conventions. The history of RPN is intertwined with the evolution of computational methods and the quest for efficiency in calculation.

The inception of RPN can be traced back to the early 20th century, but it was polish mathematician Jan Łukasiewicz who formalized the notation in the 1920s, hence the name 'Polish' notation. However, it wasn't until the 1950s and the work of Australian philosopher and computer scientist Charles Hamblin that RPN gained prominence. Hamblin proposed a computer design based on a stack, where operands would be pushed onto the stack, and operators would pop operands off, perform the calculation, and push the result back onto the stack. This method proved to be a significant advancement in the efficiency of expression evaluation.

From a mathematical standpoint, RPN is significant because it reflects the actual order in which operations are performed when evaluating an expression. This directness allows for easier manipulation of complex expressions and can be particularly advantageous in fields such as engineering and computer programming. Let's delve deeper into the historical development and mathematical significance of RPN through a detailed exploration:

1. Early Development and Theoretical Foundations

- Jan Łukasiewicz invents Polish notation in the 1920s.

- Charles Hamblin develops a stack-based computer architecture in the 1950s, leveraging RPN.

2. Adoption in Technology

- The first handheld calculator to use RPN was the HP-35, released in 1972 by Hewlett-Packard.

- RPN becomes a standard feature in many scientific calculators, favored for its speed and efficiency.

3. Mathematical Advantages

- Eliminates the need for parentheses, reducing errors in manual calculations.

- Reflects the sequential processing of operations, aligning with computational methods.

4. Impact on Computer Science

- Facilitates the creation of compilers and interpreters by simplifying the parsing process.

- Enhances the understanding of algorithms that manage data in a last-in, first-out (LIFO) manner.

5. Examples and Applications

- In an RPN calculator, to calculate ( (3 + 4) \times 5 ), one would input ( 3 \ 4 + 5 \times ), which directly corresponds to the order of operations.

- Programming languages like Forth and PostScript are based on RPN, showcasing its utility in real-world applications.

RPN's history is not just a tale of a notational system but a narrative of the relentless pursuit of simplicity and efficiency in mathematical operations. Its significance lies in its ability to streamline complex calculations, a principle that resonates with the core of computational thinking. As we continue to advance in technology and mathematics, RPN remains a testament to the elegance of logical structure in expression evaluation. It is a powerful tool that, once mastered, offers a clear and concise framework for tackling even the most intricate of mathematical challenges.

4. A Detailed Analysis

When delving into the world of expression evaluation, one quickly encounters two predominant notations: Reverse Polish Notation (RPN) and Infix Notation. These systems represent the syntax for writing expressions that include operators and operands. While Infix Notation is the conventional form most people are accustomed to, RPN presents a compelling alternative, particularly in computational contexts. This analysis will explore the nuances of each notation, providing insights from the perspectives of readability, computational efficiency, and historical usage. We'll dissect the mechanics of each system, using examples to illuminate their practical implications, and ultimately, we'll understand why RPN, despite its less intuitive appearance, can often be the superior choice for processing complex calculations.

1. Readability and Cognitive Load: Infix Notation, where operators are placed between operands (e.g., $$ A + B $$), is universally taught and recognized. It aligns with natural language patterns, making it more immediately understandable. For example, the expression $$ (3 + 4) \times 5 $$ is clear in its order of operations. However, RPN, which places operators after their operands (e.g., $$ AB+ $$), reduces cognitive load during manual parsing as it eliminates the need for parentheses. The previous example in RPN would be $$ 3 4 + 5 \times $$, which is evaluated left to right without ambiguity.

2. Computational Efficiency: From a computational standpoint, RPN shines. It aligns closely with stack-based algorithms, allowing for straightforward implementation in programming. Computers can evaluate RPN expressions efficiently by pushing operands onto a stack and popping them off to apply operators. For instance, evaluating the RPN expression $$ 5 1 2 + 4 \times + 3 - $$ is a direct process: push 5, push 1, push 2, add (resulting in 3), push 4, multiply (resulting in 12), add to 5 (resulting in 17), and finally subtract 3 to get 14.

3. Error Minimization: RPN reduces the potential for errors in calculation. Infix expressions can become complex and prone to mistakes, especially with nested parentheses. RPN's sequential operation application mitigates this risk. Consider the infix expression $$ ((15 / (7 - (1 + 1))) \times 3) - (2 + (1 + 1)) $$ versus its RPN counterpart $$ 15 7 1 1 + - / 3 \times 2 1 1 + + - $$. The latter is less likely to result in misplaced parentheses or operator precedence errors.

4. Historical and Practical Usage: Historically, RPN gained prominence with Hewlett-Packard's calculators, which favored the notation for its efficient use of memory and keystrokes. In modern times, programming languages like Lisp and Forth use variations of RPN due to its simplicity and speed. It's also the preferred method for many computer algebra systems.

While Infix Notation is more aligned with human language patterns and intuition, RPN offers significant advantages in computational contexts. Its alignment with stack-based processing, reduction of errors, and historical significance in computing tools make it a powerful method for expression evaluation. As we continue to push the boundaries of complex calculations, particularly in the realm of computer science, RPN's streamlined approach may become increasingly relevant.

5. The Advantages of Using RPN in Complex Calculations

Reverse Polish Notation (RPN), also known as postfix notation, is a mathematical notation wherein every operator follows all of its operands. It is a robust method of expression evaluation that is particularly advantageous in complex calculations. Unlike traditional infix notation, which can be cluttered with parentheses and require the application of the order of operations, RPN provides a streamlined approach that can significantly simplify the process of evaluating expressions, especially when dealing with a stack-based data structure.

From the perspective of computational efficiency, RPN shines due to its inherent ability to eliminate the need for parentheses. This not only reduces the cognitive load for humans when parsing and evaluating expressions but also simplifies the computational parsing process. In a stack-based computing environment, RPN expressions are evaluated from left to right, with each operator acting on the two most recent operands. This leads to a natural and efficient calculation flow that is less prone to errors.

Advantages of Using RPN in Complex Calculations:

1. Clarity in Evaluation: RPN expressions are unambiguous. Since each operator follows its operands, there's no need to consider operator precedence or parentheses, making it clear which operations are performed first.

2. Reduced Cognitive Load: When dealing with complex calculations, RPN reduces the mental effort required to keep track of the order of operations and parentheses, allowing the focus to remain on the actual problem-solving process.

3. Efficiency in Computation: Computers can evaluate RPN expressions more efficiently. The stack-based nature of RPN means that as soon as an operator is encountered, the calculation can be performed immediately using the operands at the top of the stack.

4. Ease of Implementation: Implementing an RPN calculator is straightforward in programming languages that support stack data structures. The algorithm is simple: push operands onto the stack, and when an operator is encountered, pop the required number of operands, perform the operation, and push the result back onto the stack.

5. Error Reduction: The directness of RPN minimizes the chance of making syntactical errors in expression formulation, which can be common in complex infix expressions with many nested parentheses.

6. History of Success: RPN has a proven track record in various fields, including computer science, engineering, and finance, where HP calculators utilizing RPN have been the tools of choice for professionals for decades.

Examples Highlighting the Advantages:

Consider the infix expression ( (3 + 4) \times (5 - 2) ). In traditional evaluation, one must first calculate the expressions within the parentheses. In RPN, this expression becomes ( 3 \ 4 + 5 \ 2 - \times ), which is evaluated as follows:

- Push 3 onto the stack.

- Push 4 onto the stack.

- Encounter '+', pop 3 and 4, add them (resulting in 7), push the result onto the stack.

- Push 5 onto the stack.

- Push 2 onto the stack.

- Encounter '-', pop 5 and 2, subtract them (resulting in 3), push the result onto the stack.

- Encounter '×', pop 7 and 3, multiply them (resulting in 21), push the result onto the stack.

The final result is 21, achieved without the need for parentheses or concern for operator precedence. This example illustrates the clarity and efficiency of RPN, particularly in complex calculations where such advantages are amplified. The simplicity of the RPN evaluation process makes it a powerful tool in the arsenal of anyone dealing with intricate mathematical expressions.

6. Converting Infix to RPN

Converting an expression from infix notation to Reverse Polish Notation (RPN), also known as postfix notation, is a critical step in expression evaluation that offers a myriad of benefits. This conversion simplifies complex calculations by eliminating the need for parentheses and the rules of operator precedence, allowing for a more straightforward evaluation process. The RPN format is particularly advantageous in computer algorithms and calculators because it aligns with the stack-based nature of computation, leading to efficient and rapid evaluations. The process of conversion, while systematic, requires a keen understanding of the intricacies involved in the parsing of expressions. It's a fascinating interplay between algorithmic precision and the inherent logic of mathematical operations. By adopting different perspectives, such as that of a computer scientist optimizing for performance or a mathematician ensuring the accuracy of the expression, we can appreciate the elegance and utility of RPN in various contexts.

Here is a detailed, step-by-step guide to converting an infix expression to RPN:

1. Understand the Components: An infix expression is composed of operands (numbers or variables) and operators (+, -, *, /) that denote the operations to be performed. Parentheses are used to override the usual precedence of operators.

2. Initialize a Stack: The stack will temporarily hold operators and parentheses during the conversion process.

3. Process the Expression Left to Right:

- Operands: When an operand is encountered, it is immediately added to the output queue.

- Operators: When an operator is encountered, the algorithm must decide whether to stack it or output it based on the precedence of the current operator relative to the operator on top of the stack.

- Parentheses: If a left parenthesis is encountered, it is pushed onto the stack. A right parenthesis causes the stack to be popped onto the output queue until a left parenthesis is encountered, which is then discarded.

4. Apply the shunting-Yard algorithm: This algorithm, developed by Edsger Dijkstra, is used to handle the operator precedence and associativity:

- If the stack is empty or contains a left parenthesis on top, push the current operator onto the stack.

- If the current operator has higher precedence than the top of the stack, push it on the stack.

- If the precedence is lower or equal, pop the stack to the output queue and then push the current operator onto the stack.

5. Empty the Stack: After the entire expression has been processed, any remaining operators on the stack are popped to the output queue.

Example: Let's convert the infix expression `3 + 4 * 2 / ( 1 - 5 )` to RPN:

- Start with an empty stack and output queue.

- Process `3`: Output `3`.

- Process `+`: Stack `+`.

- Process `4`: Output `4`.

- Process ``: Since `` has higher precedence than `+`, stack `*`.

- Process `2`: Output `2`.

- Process `/`: Since `/` and `` have the same precedence, pop `` to output, then stack `/`.

- Process `(`: Stack `(`.

- Process `1`: Output `1`.

- Process `-`: Stack `-`.

- Process `5`: Output `5`.

- Process `)`: Pop `-` to output, then discard `(`.

- Empty the stack: Pop `/` and `+` to output.

The final RPN expression is `3 4 2 * 1 5 - / +`.

By following these steps, one can systematically convert any infix expression to RPN, paving the way for efficient computation and evaluation. The beauty of this method lies in its algorithmic simplicity and the ease with which it can be implemented in programming languages to handle complex mathematical expressions.

7. Applications and Tools

Reverse Polish Notation (RPN), also known as postfix notation, is a mathematical notation wherein every operator follows all of its operands. It is a key concept in computer science, particularly in the realm of expression evaluation, as it allows for efficient and straightforward computation. Unlike traditional infix notation, which can require parentheses and operator precedence rules to be understood and applied correctly, RPN simplifies the process by eliminating the need for these complicating factors. This characteristic makes RPN an invaluable tool in modern computing, where speed and accuracy are paramount.

From the perspective of a compiler designer, RPN is a boon. It simplifies the parsing stage of compilation, allowing for a more straightforward conversion from source code to machine code. For software developers, especially those working with stack-based languages or designing calculators, RPN provides a clear and concise way to represent expressions that are easy to evaluate programmatically. Meanwhile, data scientists and analysts benefit from RPN's clarity when performing complex calculations that might otherwise be prone to errors in traditional notations.

Here are some in-depth insights into the applications and tools associated with RPN in modern computing:

1. Stack-Based Programming Languages: Languages like Forth and PostScript utilize RPN at their core. This approach allows for efficient memory usage and operation execution, as operators immediately act upon the operands that precede them.

2. Calculators: Many scientific calculators, including certain models from HP and Texas Instruments, employ RPN for its simplicity and speed in calculation. Users can input complex expressions without worrying about the order of operations or parentheses.

3. Algorithm Implementation: Algorithms that involve expression evaluation, such as Dijkstra's shunting-yard algorithm, often use RPN to convert infix expressions to postfix form before computation.

4. Graphical User Interfaces (GUIs): Tools like calculator apps or educational software often include an RPN mode to help users learn and apply the concept in a visual and interactive manner.

5. Educational Tools: RPN is used in teaching computer science and mathematics to illustrate the importance of expression evaluation order and to introduce students to stack operations.

6. Debugging Tools: Developers use RPN to debug complex expressions by breaking them down into simpler components, making it easier to isolate and fix errors.

To highlight the practicality of RPN, consider a simple arithmetic expression in infix notation: ( (3 + 4) \times 5 ). In RPN, this expression becomes ( 3 \ 4 + 5 \times ), which can be evaluated easily from left to right without the need for parentheses. This example underscores how RPN streamlines the calculation process, making it a preferred method in various computing applications. The adoption of RPN across different fields and tools underscores its enduring relevance and the efficiency it brings to modern computing practices.

8. Challenges and Limitations of RPN in Expression Evaluation

Reverse Polish Notation (RPN), also known as postfix notation, is a mathematical notation in which operators follow their operands. It has been widely adopted in computational systems due to its inherent advantage of eliminating the need for parentheses to define the order of operations. However, despite its streamlined approach and efficiency in many computational scenarios, RPN is not without its challenges and limitations, particularly when it comes to expression evaluation. From the perspective of a programmer, a mathematician, or an end-user, these challenges can manifest in various forms, impacting the usability and functionality of RPN in certain contexts.

1. Readability and Maintainability:

For individuals accustomed to the conventional infix notation, RPN can be less intuitive and harder to read. Consider the infix expression $$ (3 + 4) \times (5 - 2) $$, which in RPN becomes `3 4 + 5 2 - \times`. For complex expressions, this lack of intuitiveness can lead to difficulties in understanding and maintaining code or mathematical proofs.

2. Learning Curve:

The learning curve associated with RPN is steep. New users must invest time to become proficient in this notation, which can be a significant barrier to its adoption, especially when infix notation is deeply ingrained in educational systems.

3. Error Proneness:

While RPN reduces errors related to the order of operations, it introduces a different kind of error potential. Users must be vigilant about the sequence in which they input operands and operators. A misplaced operator or operand can lead to incorrect results, as seen in the difference between `3 4 5 + \times` and `3 4 + 5 \times`.

4. Stack Management:

RPN relies on stack data structures to evaluate expressions. This requires careful management of the stack, particularly in the case of nested expressions or when implementing RPN in programming. Stack overflow or underflow errors can occur if the stack is not managed correctly.

5. Limited Support for Functions:

In RPN, functions that take more than one argument can be awkward to represent. For example, a function like `min(a, b)` in infix notation does not translate as neatly into RPN, which might require additional conventions or syntax to handle such cases.

6. integration with Existing systems:

Many existing computational systems and software are built around infix notation. Integrating RPN into these systems can be challenging and may require significant refactoring or the development of translation layers, which can be resource-intensive.

7. user Interface design:

designing user interfaces that accommodate RPN can be challenging. For calculators, this often means a departure from the traditional layout, which can affect user adoption and satisfaction.

8. Algorithmic Complexity:

While RPN simplifies the parsing of expressions, it can complicate algorithms that rely on the structure of expressions, such as symbolic differentiation or integration. These operations often require a return to infix or tree representations, negating some of RPN's advantages.

While RPN offers a streamlined method for evaluating expressions, particularly in stack-based computing environments, it presents a set of challenges that can affect its applicability and user experience. These limitations must be carefully considered when choosing RPN as the notation of choice for expression evaluation, especially in educational and user-facing applications. The trade-offs between the efficiency gains and the potential for complexity and user friction are central to the ongoing discussion about the role of RPN in mathematical and computational contexts.

9. Trends and Innovations in Calculation Methods

Reverse Polish Notation (RPN), also known as postfix notation, is a method of notation for mathematical and logical expressions where the operator follows the operands. It is a key part of many computational systems due to its efficiency and the ease with which it can be implemented in computer algorithms. As we look towards the future, RPN is poised to evolve with the integration of new technologies and methodologies. This evolution is expected to address current limitations and expand its applicability in various fields, from education to advanced computing.

1. Integration with natural Language processing (NLP):

The future of RPN is likely to see a closer integration with NLP technologies. This will allow users to input expressions in a more natural and intuitive way, with the system converting these into RPN for efficient computation. For example, a user might say, "What is the square root of the sum of four and nine?" and the system would process this into the RPN expression `4 9 + √`.

2. Enhanced Educational Tools:

Educational platforms are beginning to incorporate RPN into their teaching methodologies. By using interactive tools that visualize the stack operations of RPN, students can gain a deeper understanding of the underlying processes involved in computation. Imagine a digital classroom where students can manipulate expressions in real-time, seeing the immediate effect of each operation on the stack.

3. Advanced error Detection and correction:

Error detection in RPN can be challenging due to its syntax. However, advancements in error-checking algorithms will make it easier to spot and correct mistakes. Future calculators might highlight potential errors in real-time, suggesting corrections before the calculation is even performed.

4. Expansion into New Programming Languages:

While rpn is already used in some programming languages, we can expect its principles to be adopted by more mainstream languages, particularly those focused on functional programming. This could lead to a new syntax that allows for more concise and readable code.

5. quantum Computing applications:

As quantum computing becomes more prevalent, RPN's stack-based operations could be adapted to work with quantum bits (qubits). This would involve rethinking the traditional stack to accommodate the probabilistic nature of quantum information.

6. collaboration with AI and Machine learning:

artificial intelligence and machine learning algorithms could utilize RPN to streamline complex calculations. By training AI models with RPN, these systems could perform large-scale computations more efficiently, potentially leading to breakthroughs in data analysis and modeling.

7. Cross-Disciplinary Uses:

RPN's application will extend beyond mathematics and computer science into fields like linguistics, psychology, and even art. For instance, RPN could be used to analyze sentence structures or to create generative art based on algorithmic patterns.

8. mobile and Wearable technology:

With the rise of mobile and wearable devices, RPN calculators will become more accessible and user-friendly. Future apps might use gesture-based controls to input expressions, making the process more interactive and engaging.

The future of RPN holds exciting prospects for innovation and expansion. Its unique characteristics make it a valuable tool for a wide range of applications, and as technology advances, so too will the capabilities and uses of RPN. The trends and innovations discussed here are just the beginning, and we can expect to see RPN continue to evolve in ways that enhance its efficiency, accessibility, and relevance in our increasingly digital world.

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