Model BNF Formally in mathematical terms
$begingroup$
Usually, a language is formally defined as some subset $L$ of the Kleene-closure $Sigma^*$ of some "alphabet" (i.e. finite set) $Sigma$, where by Kleene closure we mean the infinite union $$Sigma^* = bigcup_{k=0}^infty Sigma^k$$ where $Sigma^0 := {epsilon}$, and $Sigma^k$ for $kgeqslant 1$ denotes the Cartesian product of sets. In view of this definition, it is more convenient to model strings of length $k$ (i.e. $k$-tuples) here as functions $fcolon{1,dots,k}to Sigma$ so that $(a_1, dots, a_n)$ becomes $(f(1), dots, f(n))$ as opposed to the usual definition with nesting of pairs (this avoids discomfort about tuples of length zero or one, so that $epsilon$ is just $fcolonvarnothingtovarnothing$, and also tuples of length one are distinguished structurally as functions and cannot be confused with members of the alphabet; i.e. $Sigma^1 neq Sigma$).
Coming from a mathematical background to the subject of formal languages, I am used to the comfortable notion that the machinery underlying everything is built on the ZF(C) axioms. So far, these definitions are satisfactory and nicely describe the idea of languages in a mathematical framework.
But then when Backus–Naur form is used, the mathematical rigour starts slacking a bit, I can't seem to find any books/resources which formally define how BNF's construct languages as sets. I'd appreciate some guidance as to how BNFs can be studied in a formal setting which continues in the spirit of the definitions above.
computer-science context-free-grammar
edited Dec 27 '18 at 13:24
Asaf Karagila♦
306k33438769
asked Dec 27 '18 at 1:29
Luke CollinsLuke Collins
754419
$endgroup$
add a comment |
$begingroup$
Usually, a language is formally defined as some subset $L$ of the Kleene-closure $Sigma^*$ of some "alphabet" (i.e. finite set) $Sigma$, where by Kleene closure we mean the infinite union $$Sigma^* = bigcup_{k=0}^infty Sigma^k$$ where $Sigma^0 := {epsilon}$, and $Sigma^k$ for $kgeqslant 1$ denotes the Cartesian product of sets. In view of this definition, it is more convenient to model strings of length $k$ (i.e. $k$-tuples) here as functions $fcolon{1,dots,k}to Sigma$ so that $(a_1, dots, a_n)$ becomes $(f(1), dots, f(n))$ as opposed to the usual definition with nesting of pairs (this avoids discomfort about tuples of length zero or one, so that $epsilon$ is just $fcolonvarnothingtovarnothing$, and also tuples of length one are distinguished structurally as functions and cannot be confused with members of the alphabet; i.e. $Sigma^1 neq Sigma$).
Coming from a mathematical background to the subject of formal languages, I am used to the comfortable notion that the machinery underlying everything is built on the ZF(C) axioms. So far, these definitions are satisfactory and nicely describe the idea of languages in a mathematical framework.
But then when Backus–Naur form is used, the mathematical rigour starts slacking a bit, I can't seem to find any books/resources which formally define how BNF's construct languages as sets. I'd appreciate some guidance as to how BNFs can be studied in a formal setting which continues in the spirit of the definitions above.
computer-science context-free-grammar
edited Dec 27 '18 at 13:24
Asaf Karagila♦
306k33438769
asked Dec 27 '18 at 1:29
Luke CollinsLuke Collins
754419
$endgroup$
$begingroup$
I'm not sure what is your confusion.
$endgroup$
– Asaf Karagila♦
Dec 27 '18 at 8:17
$begingroup$
@AsafKaragila What is BNF? How does it formally define languages? I want a description of BNF's which highlights the underlying set structure.
$endgroup$
– Luke Collins
Dec 27 '18 at 13:06
$begingroup$
you already mentioned the main feature, i.e. the language is a subset of all letter combinations taken from the alphabet. I.e. good way to handle this is via $f : Sigma^*rightarrow 2$ functions which are characteristic function of the subset.
$endgroup$
– tp1
Dec 27 '18 at 14:22
$begingroup$
I think I'm after something along the lines of "A BNF is a set of production rules", where production rules are as described here.
$endgroup$
– Luke Collins
Dec 27 '18 at 14:26
add a comment |
$begingroup$
Usually, a language is formally defined as some subset $L$ of the Kleene-closure $Sigma^*$ of some "alphabet" (i.e. finite set) $Sigma$, where by Kleene closure we mean the infinite union $$Sigma^* = bigcup_{k=0}^infty Sigma^k$$ where $Sigma^0 := {epsilon}$, and $Sigma^k$ for $kgeqslant 1$ denotes the Cartesian product of sets. In view of this definition, it is more convenient to model strings of length $k$ (i.e. $k$-tuples) here as functions $fcolon{1,dots,k}to Sigma$ so that $(a_1, dots, a_n)$ becomes $(f(1), dots, f(n))$ as opposed to the usual definition with nesting of pairs (this avoids discomfort about tuples of length zero or one, so that $epsilon$ is just $fcolonvarnothingtovarnothing$, and also tuples of length one are distinguished structurally as functions and cannot be confused with members of the alphabet; i.e. $Sigma^1 neq Sigma$).
Coming from a mathematical background to the subject of formal languages, I am used to the comfortable notion that the machinery underlying everything is built on the ZF(C) axioms. So far, these definitions are satisfactory and nicely describe the idea of languages in a mathematical framework.
But then when Backus–Naur form is used, the mathematical rigour starts slacking a bit, I can't seem to find any books/resources which formally define how BNF's construct languages as sets. I'd appreciate some guidance as to how BNFs can be studied in a formal setting which continues in the spirit of the definitions above.
computer-science context-free-grammar
edited Dec 27 '18 at 13:24
Asaf Karagila♦
306k33438769
asked Dec 27 '18 at 1:29
Luke CollinsLuke Collins
754419
$endgroup$
Usually, a language is formally defined as some subset $L$ of the Kleene-closure $Sigma^*$ of some "alphabet" (i.e. finite set) $Sigma$, where by Kleene closure we mean the infinite union $$Sigma^* = bigcup_{k=0}^infty Sigma^k$$ where $Sigma^0 := {epsilon}$, and $Sigma^k$ for $kgeqslant 1$ denotes the Cartesian product of sets. In view of this definition, it is more convenient to model strings of length $k$ (i.e. $k$-tuples) here as functions $fcolon{1,dots,k}to Sigma$ so that $(a_1, dots, a_n)$ becomes $(f(1), dots, f(n))$ as opposed to the usual definition with nesting of pairs (this avoids discomfort about tuples of length zero or one, so that $epsilon$ is just $fcolonvarnothingtovarnothing$, and also tuples of length one are distinguished structurally as functions and cannot be confused with members of the alphabet; i.e. $Sigma^1 neq Sigma$).
Coming from a mathematical background to the subject of formal languages, I am used to the comfortable notion that the machinery underlying everything is built on the ZF(C) axioms. So far, these definitions are satisfactory and nicely describe the idea of languages in a mathematical framework.
But then when Backus–Naur form is used, the mathematical rigour starts slacking a bit, I can't seem to find any books/resources which formally define how BNF's construct languages as sets. I'd appreciate some guidance as to how BNFs can be studied in a formal setting which continues in the spirit of the definitions above.
computer-science context-free-grammar
computer-science context-free-grammar
edited Dec 27 '18 at 13:24
Asaf Karagila♦
306k33438769
asked Dec 27 '18 at 1:29
Luke CollinsLuke Collins
754419
edited Dec 27 '18 at 13:24
Asaf Karagila♦
306k33438769
asked Dec 27 '18 at 1:29
Luke CollinsLuke Collins
754419
edited Dec 27 '18 at 13:24
Asaf Karagila♦
306k33438769
edited Dec 27 '18 at 13:24
Asaf Karagila♦
306k33438769
edited Dec 27 '18 at 13:24
Asaf Karagila♦
306k33438769
306k33438769
asked Dec 27 '18 at 1:29
Luke CollinsLuke Collins
754419
asked Dec 27 '18 at 1:29
Luke CollinsLuke Collins
754419
asked Dec 27 '18 at 1:29
Luke CollinsLuke Collins
754419
754419
$begingroup$
I'm not sure what is your confusion.
$endgroup$
– Asaf Karagila♦
Dec 27 '18 at 8:17
$begingroup$
@AsafKaragila What is BNF? How does it formally define languages? I want a description of BNF's which highlights the underlying set structure.
$endgroup$
– Luke Collins
Dec 27 '18 at 13:06
$begingroup$
you already mentioned the main feature, i.e. the language is a subset of all letter combinations taken from the alphabet. I.e. good way to handle this is via $f : Sigma^*rightarrow 2$ functions which are characteristic function of the subset.
$endgroup$
– tp1
Dec 27 '18 at 14:22
$begingroup$
I think I'm after something along the lines of "A BNF is a set of production rules", where production rules are as described here.
$endgroup$
– Luke Collins
Dec 27 '18 at 14:26
add a comment |
$begingroup$
I'm not sure what is your confusion.
$endgroup$
– Asaf Karagila♦
Dec 27 '18 at 8:17
$begingroup$
@AsafKaragila What is BNF? How does it formally define languages? I want a description of BNF's which highlights the underlying set structure.
$endgroup$
– Luke Collins
Dec 27 '18 at 13:06
$begingroup$
you already mentioned the main feature, i.e. the language is a subset of all letter combinations taken from the alphabet. I.e. good way to handle this is via $f : Sigma^*rightarrow 2$ functions which are characteristic function of the subset.
$endgroup$
– tp1
Dec 27 '18 at 14:22
$begingroup$
I think I'm after something along the lines of "A BNF is a set of production rules", where production rules are as described here.
$endgroup$
– Luke Collins
Dec 27 '18 at 14:26
$begingroup$
I'm not sure what is your confusion.
$endgroup$
– Asaf Karagila♦
Dec 27 '18 at 8:17
$begingroup$
I'm not sure what is your confusion.
$endgroup$
– Asaf Karagila♦
Dec 27 '18 at 8:17
$begingroup$
@AsafKaragila What is BNF? How does it formally define languages? I want a description of BNF's which highlights the underlying set structure.
$endgroup$
– Luke Collins
Dec 27 '18 at 13:06
$begingroup$
@AsafKaragila What is BNF? How does it formally define languages? I want a description of BNF's which highlights the underlying set structure.
$endgroup$
– Luke Collins
Dec 27 '18 at 13:06
$begingroup$
you already mentioned the main feature, i.e. the language is a subset of all letter combinations taken from the alphabet. I.e. good way to handle this is via $f : Sigma^*rightarrow 2$ functions which are characteristic function of the subset.
$endgroup$
– tp1
Dec 27 '18 at 14:22
$begingroup$
you already mentioned the main feature, i.e. the language is a subset of all letter combinations taken from the alphabet. I.e. good way to handle this is via $f : Sigma^*rightarrow 2$ functions which are characteristic function of the subset.
$endgroup$
– tp1
Dec 27 '18 at 14:22
$begingroup$
I think I'm after something along the lines of "A BNF is a set of production rules", where production rules are as described here.
$endgroup$
– Luke Collins
Dec 27 '18 at 14:26
$begingroup$
I think I'm after something along the lines of "A BNF is a set of production rules", where production rules are as described here.
$endgroup$
– Luke Collins
Dec 27 '18 at 14:26
add a comment |
1 Answer
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$begingroup$
For simplicity, assume each non-terminal occurs exactly once as the left hand side of a BNF rule. (This just means that all the rules are joined together with alternation.) Let $N$ be the set of non-terminals and identify the terminals with the alphabet $Sigma$. Define a set function $g:mathcal P(Sigma^*)^Ntomathcal P(Sigma^*)^N$ by setting $g(S)(n)$ to the following interpretation of the right hand side of the rule for the non-terminal, $n$. Interpret each terminal (i.e. letter) as a singleton set of that letter. Interpret juxtaposition as language concatenation. Interpret alternation as union. Finally, interpret the non-terminal $m$ via as $S(m)$. Let $sin N$ be the starting non-terminal. The language recognized by the BNF is the least (pre-)fixed point of $g$ evaluated at $s$ where the ordering is the pointwise subset ordering.
This is just the straightforward set-theoretic rendition of a collection of mutually inductively defined sets. Sure enough, if we use the BNF: $s ::= 0 mid 1 s$ with alphabet ${0,1}$, the least (pre-)fixed point property of the resulting language is essentially (the set-theoretic variant of) mathematical induction (of natural numbers). Call the resulting language $L$. The least pre-fixed point property states that for any set $S$ such that ${0}cup({1}frown S)subseteq S$ where $frown$ is language concatenation, $Lsubseteq S$.
answered Dec 29 '18 at 12:32
Derek ElkinsDerek Elkins
17.2k11437
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add a comment |
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$begingroup$
For simplicity, assume each non-terminal occurs exactly once as the left hand side of a BNF rule. (This just means that all the rules are joined together with alternation.) Let $N$ be the set of non-terminals and identify the terminals with the alphabet $Sigma$. Define a set function $g:mathcal P(Sigma^*)^Ntomathcal P(Sigma^*)^N$ by setting $g(S)(n)$ to the following interpretation of the right hand side of the rule for the non-terminal, $n$. Interpret each terminal (i.e. letter) as a singleton set of that letter. Interpret juxtaposition as language concatenation. Interpret alternation as union. Finally, interpret the non-terminal $m$ via as $S(m)$. Let $sin N$ be the starting non-terminal. The language recognized by the BNF is the least (pre-)fixed point of $g$ evaluated at $s$ where the ordering is the pointwise subset ordering.
This is just the straightforward set-theoretic rendition of a collection of mutually inductively defined sets. Sure enough, if we use the BNF: $s ::= 0 mid 1 s$ with alphabet ${0,1}$, the least (pre-)fixed point property of the resulting language is essentially (the set-theoretic variant of) mathematical induction (of natural numbers). Call the resulting language $L$. The least pre-fixed point property states that for any set $S$ such that ${0}cup({1}frown S)subseteq S$ where $frown$ is language concatenation, $Lsubseteq S$.
answered Dec 29 '18 at 12:32
Derek ElkinsDerek Elkins
17.2k11437
$endgroup$
add a comment |
$begingroup$
For simplicity, assume each non-terminal occurs exactly once as the left hand side of a BNF rule. (This just means that all the rules are joined together with alternation.) Let $N$ be the set of non-terminals and identify the terminals with the alphabet $Sigma$. Define a set function $g:mathcal P(Sigma^*)^Ntomathcal P(Sigma^*)^N$ by setting $g(S)(n)$ to the following interpretation of the right hand side of the rule for the non-terminal, $n$. Interpret each terminal (i.e. letter) as a singleton set of that letter. Interpret juxtaposition as language concatenation. Interpret alternation as union. Finally, interpret the non-terminal $m$ via as $S(m)$. Let $sin N$ be the starting non-terminal. The language recognized by the BNF is the least (pre-)fixed point of $g$ evaluated at $s$ where the ordering is the pointwise subset ordering.
This is just the straightforward set-theoretic rendition of a collection of mutually inductively defined sets. Sure enough, if we use the BNF: $s ::= 0 mid 1 s$ with alphabet ${0,1}$, the least (pre-)fixed point property of the resulting language is essentially (the set-theoretic variant of) mathematical induction (of natural numbers). Call the resulting language $L$. The least pre-fixed point property states that for any set $S$ such that ${0}cup({1}frown S)subseteq S$ where $frown$ is language concatenation, $Lsubseteq S$.
answered Dec 29 '18 at 12:32
Derek ElkinsDerek Elkins
17.2k11437
$endgroup$
add a comment |
$begingroup$
For simplicity, assume each non-terminal occurs exactly once as the left hand side of a BNF rule. (This just means that all the rules are joined together with alternation.) Let $N$ be the set of non-terminals and identify the terminals with the alphabet $Sigma$. Define a set function $g:mathcal P(Sigma^*)^Ntomathcal P(Sigma^*)^N$ by setting $g(S)(n)$ to the following interpretation of the right hand side of the rule for the non-terminal, $n$. Interpret each terminal (i.e. letter) as a singleton set of that letter. Interpret juxtaposition as language concatenation. Interpret alternation as union. Finally, interpret the non-terminal $m$ via as $S(m)$. Let $sin N$ be the starting non-terminal. The language recognized by the BNF is the least (pre-)fixed point of $g$ evaluated at $s$ where the ordering is the pointwise subset ordering.
This is just the straightforward set-theoretic rendition of a collection of mutually inductively defined sets. Sure enough, if we use the BNF: $s ::= 0 mid 1 s$ with alphabet ${0,1}$, the least (pre-)fixed point property of the resulting language is essentially (the set-theoretic variant of) mathematical induction (of natural numbers). Call the resulting language $L$. The least pre-fixed point property states that for any set $S$ such that ${0}cup({1}frown S)subseteq S$ where $frown$ is language concatenation, $Lsubseteq S$.
answered Dec 29 '18 at 12:32
Derek ElkinsDerek Elkins
17.2k11437
$endgroup$
For simplicity, assume each non-terminal occurs exactly once as the left hand side of a BNF rule. (This just means that all the rules are joined together with alternation.) Let $N$ be the set of non-terminals and identify the terminals with the alphabet $Sigma$. Define a set function $g:mathcal P(Sigma^*)^Ntomathcal P(Sigma^*)^N$ by setting $g(S)(n)$ to the following interpretation of the right hand side of the rule for the non-terminal, $n$. Interpret each terminal (i.e. letter) as a singleton set of that letter. Interpret juxtaposition as language concatenation. Interpret alternation as union. Finally, interpret the non-terminal $m$ via as $S(m)$. Let $sin N$ be the starting non-terminal. The language recognized by the BNF is the least (pre-)fixed point of $g$ evaluated at $s$ where the ordering is the pointwise subset ordering.
This is just the straightforward set-theoretic rendition of a collection of mutually inductively defined sets. Sure enough, if we use the BNF: $s ::= 0 mid 1 s$ with alphabet ${0,1}$, the least (pre-)fixed point property of the resulting language is essentially (the set-theoretic variant of) mathematical induction (of natural numbers). Call the resulting language $L$. The least pre-fixed point property states that for any set $S$ such that ${0}cup({1}frown S)subseteq S$ where $frown$ is language concatenation, $Lsubseteq S$.
answered Dec 29 '18 at 12:32
Derek ElkinsDerek Elkins
17.2k11437
answered Dec 29 '18 at 12:32
Derek ElkinsDerek Elkins
17.2k11437
answered Dec 29 '18 at 12:32
Derek ElkinsDerek Elkins
17.2k11437
answered Dec 29 '18 at 12:32
Derek ElkinsDerek Elkins
17.2k11437
17.2k11437
add a comment |
add a comment |
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$begingroup$
I'm not sure what is your confusion.
$endgroup$
– Asaf Karagila♦
Dec 27 '18 at 8:17
$begingroup$
@AsafKaragila What is BNF? How does it formally define languages? I want a description of BNF's which highlights the underlying set structure.
$endgroup$
– Luke Collins
Dec 27 '18 at 13:06
$begingroup$
you already mentioned the main feature, i.e. the language is a subset of all letter combinations taken from the alphabet. I.e. good way to handle this is via $f : Sigma^*rightarrow 2$ functions which are characteristic function of the subset.
$endgroup$
– tp1
Dec 27 '18 at 14:22
$begingroup$
I think I'm after something along the lines of "A BNF is a set of production rules", where production rules are as described here.
$endgroup$
– Luke Collins
Dec 27 '18 at 14:26