First and follow
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This document discusses how to compute the nullable, first, and follow sets for grammar symbols in order to build a predictive parser. It provides the base cases and inductive steps for computing each set iteratively. An example grammar is used to demonstrate computing the sets and reaching a fixed point after multiple rounds of induction.
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2.3 Operations that preserve convexity & 2.4 Generalized inequalities
凸最適化のBoyd本の輪読会の発表資料です。
[References]
Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004.
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Introduction:
[Start with a brief introduction about yourself, including your profession or main area of expertise.]
Background:
[Discuss your background, education, and any relevant experiences that have shaped your journey.]
Accomplishments:
[Highlight notable achievements, awards, or significant projects you've been involved in.]
Expertise:
[Detail your areas of expertise, skills, or specific knowledge that sets you apart in your field.]
Passions and Interests:
[Share your passions, hobbies, or interests outside of your professional life, adding depth to your personality.]
Vision or Mission:
[If applicable, articulate your vision, mission, or goals in your chosen field or in life in general.]
Closing Statement:
[End with a closing statement that summarizes your essence or leaves a lasting impression.]
Feel free to customize each section with your own personal details and experiences. If you need further assistance or have specific points you'd like to include, feel free to let me know!
April 7, 2014
1) The document outlines a class agenda that includes reviewing quadratics and new solving methods in preparation for a Friday test. It also notes a third quarter exam on Friday covering quadratics, factoring, exponents, and polynomials.
2) The class will begin with warm-up problems reviewing graphing quadratics and finding axes of symmetry. Steps for graphing the equation y + 6x = x^2 + 9 are provided.
3) Additional examples are given for graphing a quadratic function modeling the height of a basketball after being shot, finding the maximum height and time to reach it by using the vertex of the parabola.
Vectors and 3 d
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Ch01
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- 1. Notes on First and Follow Written by Ashok Kumar Pachauri ITM Aligarh
- 2. Computing Nullable Sets • Non-terminal X is Nullable only if the following constraints are satisfied (computed using iterative analysis) – base case: • if (X := ) then X is Nullable – inductive case: • if (X := ABC...) and A, B, C, ... are all Nullable then X is Nullable
- 3. Computing First Sets • First(X) is computed iteratively – base case: • if T is a terminal symbol then First (T) = {T} – inductive case: • if X is a non-terminal and (X:= ABC...) then – First (X) = First (X) U First (ABC...) where First(ABC...) = F1 U F2 U F3 U ... and » F1 = First (A) » F2 = First (B), if A is Nullable » F3 = First (C), if A is Nullable & B is Nullable » ...
- 4. Computing Follow Sets • Follow(X) is computed iteratively – base case: • initially, we assume nothing in particular follows X – (Follow (X) is initially { }) – inductive case: • if (Y := s1 X s2) for any strings s1, s2 then – Follow (X) = First (s2) U Follow (X) • if (Y := s1 X s2) for any strings s1, s2 then – Follow (X) = Follow(Y) U Follow (X), if s2 is Nullable
- 5. building a predictive parser Z ::= X Y Z Z ::= d Y ::= c Y ::= X ::= a X ::= b Y e nullable first follow Z Y X
- 6. building a predictive parser Z ::= X Y Z Z ::= d Y ::= c Y ::= X ::= a X ::= b Y e nullable first follow Z no Y yes X no base case
- 7. building a predictive parser Z ::= X Y Z Z ::= d Y ::= c Y ::= X ::= a X ::= b Y e nullable first follow Z no Y yes X no after one round of induction, we realize we have reached a fixed point
- 8. building a predictive parser Z ::= X Y Z Z ::= d Y ::= c Y ::= X ::= a X ::= b Y e nullable first follow Z no d Y yes c X no a,b base case (note, first of Y should include empty string)
- 9. building a predictive parser Z ::= X Y Z Z ::= d Y ::= c Y ::= X ::= a X ::= b Y e nullable first follow Z no d,a,b Y yes c X no a,b after one round of induction, no fixed point
- 10. building a predictive parser Z ::= X Y Z Z ::= d Y ::= c Y ::= X ::= a X ::= b Y e nullable first follow Z no d,a,b Y yes c X no a,b after two rounds of induction, no more changes ==> fixed point
- 11. building a predictive parser Z ::= X Y Z Z ::= d Y ::= c Y ::= X ::= a X ::= b Y e nullable first follow Z no d,a,b { } Y yes c { } X no a,b { } base case
- 12. building a predictive parser Z ::= X Y Z Z ::= d Y ::= c Y ::= X ::= a X ::= b Y e nullable first follow Z no d,a,b { } Y yes c e,d,a,b X no a,b c,d,a,b after one round of induction, no fixed point
- 13. building a predictive parser Z ::= X Y Z Z ::= d Y ::= c Y ::= X ::= a X ::= b Y e nullable first follow Z no d,a,b { } Y yes c e,d,a,b X no a,b c,d,a,b after two rounds of induction, fixed point (but notice, computing Follow(X) before Follow (Y) would have required 3rd round)