Using our second grammar for arithmetic expressions, draw the two possible parse trees for the e

Using our second grammar for arithmetic expressions, draw...

Key Concepts

Context-Free Grammar (CFG)

A context-free grammar is a set of recursive rewriting rules (or productions) used to generate patterns of strings. In the study of formal languages and programming languages, CFGs are used to describe the syntax of languages in a concise way, making them a key tool for parsing and compiler design.

Parse Trees

A parse tree is a tree representation that depicts the syntactic structure of a string based on a grammar. Each node of the tree represents a grammar symbol, and the branches represent the application of production rules. Parse trees provide a visual way to understand how a particular string is derived from the start symbol of a grammar.

Grammatical Ambiguity

Grammatical ambiguity occurs when a single string can be generated by a grammar in more than one way, resulting in multiple distinct parse trees. This is an important concept in the study of grammars because ambiguous grammars complicate parsing, as the intended structure of a program or expression might not be clear without additional rules or context.

Operator Associativity

Operator associativity refers to the rule that determines how operators of the same precedence are grouped in the absence of parentheses. In arithmetic expressions, associativity affects the structure of the parse tree by guiding the order in which operations are performed, which is particularly significant in ambiguous grammars where multiple interpretations exist.

Transcript

00:04 We're given arithmetic expressions written in prefix notation, and we're asked to draw the ordered rooted tree corresponding to each of these expressions, and then we're asked to write each expression using infix notation.

00:23 So in part a, we're given the expression and prefix notation, addition, multiplication, addition, subtraction, 53214.

00:41 So to form the order of rooted tree, we're going to start with the addition as the root.

01:10 And then one of the children is multiplication.

01:15 And one of the children of multiplication is addition.

01:26 And one of the children of the addition is subtraction.

01:32 And then we have the children of subtraction are five and three.

01:37 And then we have the children of addition, our subtraction, and two.

01:47 And then we have the children of multiplication are addition and one.

01:55 And finally, we have that the children of addition are multiplication and four.

02:02 So this is the binary tree.

02:08 And using this tree, you can write the expression using infix notation.

02:12 So the way i like to do this is i go around the tree and counterclockwise.

02:15 Clockwise order.

02:17 First time i encounter a symbol that's not a leaf, i write a left parentheses.

02:23 The second time i encounter that symbol, i write the symbol, and the third time i encounter that symbol, i write a right parentheses.

02:32 So i have first i encounter addition as the root.

02:39 So i have a left parenthesis, then i encounter multiplication first time, left parentheses, then i encounter this edition the first time, another left parentheses.

02:49 Then i encounter subtraction for the first time, one more with left parentheses, then a five, which is a leaf.

03:04 So i'll rate a five since this is a leaf.

03:09 I encounter the subtraction for the second time, i write the subtraction.

03:12 Then i encounter three, a leaf.

03:14 Then i encounter subtraction for the third time, right parentheses.

03:19 Then i encounter addition for the second time, addition, then the leaf two, then addition for the third time, right parentheses, then multiplication for the second time, multiplication, then a one, a leaf, then multiplication for the third time, right parentheses.

03:40 And here i'm going to have to break this into two lines.

03:42 Then addition for the second time, addition, then the leaf four, and then the root addition for the third time, write parentheses.

03:54 And this concludes the infix notation for this expression.

04:03 So we have that, and we also have the tree.

04:09 Now, in part b, we are given the expression in prefix notation, exponentiation, addition to three, subtraction 5 -1.

04:33 So to construct the tree, i'm going to start with exponentiation as the root, which has children addition.

04:45 Well, it's one of the children.

04:46 And then we see addition has the children two and three, which are leaves.

04:54 And then we see that exponentiation also has the child in addition to addition of subtraction.

05:26 And subtraction has the children 5 and 1, which are leaves...

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