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# How to calculate transitive relationships [JAVA]

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Posted · Original PosterOP

If I have IF A THEN B and IF B THEN C I can form the relationship IF A THEN C by the property of transitivity. How might one go about this efficiently in Java? I am thinking have two arrays and one contains the IF variable while the other array is the THEN variable. The index in the array tells us what relationship it is. So index 0 in both arrays is A -> B and index 1 B -> C. I could go one index at a time and search the IF array for values containing the THEN variable and continue that loop as the association could be many relations deep, (A -> B) (B -> C) (C -> E) (E -> F) so A -> F. Any ideas on how to make this the fastest way possible? Thanks

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Just keeping this here as a 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i see you are learning descrete math. Nice.

Well, the thing is, it has nothing to do with java. You are really learning them to prepare you for the theory of computation(another CS course standard in all CS curriculum) which is when you start studying finite state automata, none deterministic automata, and then finally the Turing machine and how you express regular language with it. This stuffs is more relevant to a computer scientist and mathematicians than a programmer.

I will say though, graph theories also fall under discrete math and these will be very important for studying algorithm analysis which will be something that's going to be relevant for you as a coder. Idk if you are gonna be learning these in entry level CS course however since for many schools, these are taught by the math department instead and are usually math designated courses so in the same category as calculus.

Sudo make me a sandwich

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One efficient way to express such a relationship would be a map. A map has the advantage that it is pretty quick in terms of lookup.

```Map<String, String> relationshipMap = new HashMap<>();
relationshipMap.put("A", "B");
relationshipMap.put("B", "C");```

Questions:

• Are all of your relationships 1:1, or could there also be something like A->(B,C)?
• What's the expected input and output of that program?

My understanding is that you would e.g. get "A" as input and you want to follow the relationship to the furthest element in it? If so, you would use the input as the key to look up the value it refers to. If a value exists, use that as a key for another lookup. Keep looking until you no longer get a result. The last value you got is your result.

Remember to quote or @mention others, so they are notified of your reply

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Posted · Original PosterOP
6 minutes ago, wasab said:

i see you are learning descrete math. Nice.

Well, the thing, it has nothing to do with java. You are really learning them to prepare you for the theory of computation(another CS course standard in all CS curriculum) which is when you start studying finite state automata, none deterministic automata, and then finally the Turing machine and how you express regular language with it. This stuffs is more relevant to a computer scientist and mathematicians than a programmer.

I will say though, graph theories also fall under discrete math and these will be very important for studying algorithm analysis which will be something that's going to be relevant for you as a coder. Idk if you are gonna be learning these in entry level CS course however since for many schools, these are taught by the math department instead and are usually math designated courses so in the same category as calculus.

Your pretty much right except this is for a database class that uses discrete math and I have a separate discrete math class entirely. We went over graph theory just recently and also learned about spanning trees with things like Prim's Algorithm

5 minutes ago, Eigenvektor said:

One efficient way to express such a relationship would be a map. A map has the advantage that it is pretty quick in terms of lookup.

```
Map<String, String> relationshipMap = new HashMap<>();
relationshipMap.put("A", "B");
relationshipMap.put("B", "C");```

Questions:

• Are all of your relationships 1:1, or could there also be something like A->(B,C)?
• What's the expected input and output of that program?

My understanding is that you would e.g. get "A" as input and you want to follow the relationship to the furthest element in it? If so, you would use the input as the key to look up the value it refers to. If a value exists, use that as a key for another lookup. Keep looking until you no longer get a result. The last value you got is your result.

The relationships can be greater than 1:1. (W,V) -> (B,A).

Expected output is every possible rule you can generate with the given data. So for 9 base relations you will be able to generate 8 new relationships.

ƆԀ S₱▓Ɇ▓cs: i7 6ʇɥפᴉƎ00K (4.4ghz), Asus DeLuxe X99A II, GT҉X҉1҉0҉8҉0 Zotac Amp ExTrꍟꎭe),Si6F4Gb D🅾🅼🅸🅽🅰🆃🅾r PlatinUm, EVGA G2 Sǝʌǝᘉ5ᙣᙍᖇᓎᙎᗅᖶt, Phanteks Enthoo Primo, 3TB WD Black, 500gb 850 Evo, H100iGeeTeeX, Windows 10, K70 R̸̢̡̭͍͕̱̭̟̩̀̀̃́̃͒̈́̈́͑̑́̆͘͜ͅG̶̦̬͊́B̸͈̝̖͗̈́, G502, HyperX Cloud 2s, Asus MX34. פN∩SW∀S 960 EVO

Just keeping this here as a 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13 minutes ago, BuckGup said:

The relationships can be greater than 1:1. (W,V) -> (B,A).

Expected output is every possible rule you can generate with the given data. So for 9 base relations you will be able to generate 8 new relationships.

In that case you could use a map that contains a list. Btw, a map is also known as an associative array, or a dictionary. Alternatively you could use a class that can be used to express relationships. You essentially have a data structure that represents a "tree" and, given a node in that tree, you need to find all leaves that are reachable from that node.

```Map<String, List<String>> relationshipMap = new HashMap<>();
relationshipMap.put("A", Arrays.asList("B", "C"));
relationshipMap.put("B", Arrays.asList("D", "E"));
relationshipMap.put("C", Arrays.asList("D", "F"));```

Your lookup will be a bit more complicated, because for each key in your input you'll have to determine all possible outputs, which are then the inputs for your next lookup, Whenever an input no longer has anything associated with it, it's a leaf element.

I'd probably look into creating a proper representation of a tree, though. Something like this:

```public class Node {
private final String name;
private final List<Node> children = new ArrayList<>();
}```

Remember to quote or @mention others, so they are notified of your reply

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14 minutes ago, BuckGup said:

Your pretty much right except this is for a database class that uses discrete math and I have a separate discrete math class entirely. We went over graph theory just recently and also learned about spanning trees with things like Prim's Algorithm.

I do not recall ever seeing a database course that will teach you discrete math. I mean they will go over some general algorithms that DBMS use to minimize disk I/Os but nothing more advance than that. You will encounter graphs like binary trees and spanning trees in data structure course but I wouldn't count that as learning graph theories. It is like learning to fly an airplane vs learning the laws of aerodynamics. I guess if you are actually learning to apply algorithms that operate on these structures, that would be more akin to doing actual math.

Most schools would want their mathematics professors to actually teach the nitty gritty details of the graph theories. That is just how it is in most places.

Sudo make me a sandwich

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6 hours ago, BuckGup said:

If I have IF A THEN B and IF B THEN C I can form the relationship IF A THEN C by the property of transitivity. How might one go about this efficiently in Java? I am thinking have two arrays and one contains the IF variable while the other array is the THEN variable.

This can be thought of as a string replacement system. Since you mention that you can have rules of the form AB -> C, or more generally, <input0 ... inputn> := <output0 ... ouputn> you are looking for Context Sensitive systems. Are you sure you're looking for a context sensitive system, or in other words, is the tuple (A, B) really two tokens or is it the single token AB, which is unique from either A or B?

In your case, you have a production rule like A -> B. A is known as the predecessor, and B is known as the successor. You then have a production rule B -> C. There is no production rule for C, so C is known as a terminal character (trivially, this is the Identity property, or simply C -> C). As long as you only have one production rule per token, or per tuple of tokens in the case of context-sensitive systems, then your system is deterministic. If you have multiple rules, like A -> B and A -> C, then you have a non-deterministic system.

What you can do is put all of your rules in some <key, value> pair collection, like a hash map or a dictionary. Then, you can iterate through the characters in some input string, known as the axiom, transforming each character by looking it's rule up in the collection. Do this for one replacement per character or tuple of characters, per iteration. When the entire string is terminal characters you are finished, or in this case, your axiom is "proven".

For more information on this method of doing things, you can look into L-Systems, Semi-Thue Systems, or String Replacement Systems. I gather that you are just starting out in discrete mathematics. You should then be warned that you are beginning to dabble in Formal Languages, which while I find the subject very challenging and thus very rewarding, you may have some difficulties in fully understanding what is going on without first completing the set theory portion of your class: For some reason people like to express the ideas behind Language Theory with formal mathematics instead of practical programs, even though one of the basic tenants of Formal languages is that any given language can be defined by the automaton (in our case, the program) that accepts it.

"Ultimately, saying that you don’t care about privacy because you have nothing to hide is no different from saying you don’t care about freedom of speech because you have nothing to say." ~Verax

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Posted · Original PosterOP
5 hours ago, straight_stewie said:

This can be thought of as a string replacement system. Since you mention that you can have rules of the form AB -> C, or more generally, <input0 ... inputn> := <output0 ... ouputn> you are looking for Context Sensitive systems. Are you sure you're looking for a context sensitive system, or in other words, is the tuple (A, B) really two tokens or is it the single token AB, which is unique from either A or B?

In your case, you have a production rule like A -> B. A is known as the predecessor, and B is known as the successor. You then have a production rule B -> C. There is no production rule for C, so C is known as a terminal character (trivially, this is the Identity property, or simply C -> C). As long as you only have one production rule per token, or per tuple of tokens in the case of context-sensitive systems, then your system is deterministic. If you have multiple rules, like A -> B and A -> C, then you have a non-deterministic system.

What you can do is put all of your rules in some <key, value> pair collection, like a hash map or a dictionary. Then, you can iterate through the characters in some input string, known as the axiom, transforming each character by looking it's rule up in the collection. Do this for one replacement per character or tuple of characters, per iteration. When the entire string is terminal characters you are finished, or in this case, your axiom is "proven".

For more information on this method of doing things, you can look into L-Systems, Semi-Thue Systems, or String Replacement Systems. I gather that you are just starting out in discrete mathematics. You should then be warned that you are beginning to dabble in Formal Languages, which while I find the subject very challenging and thus very rewarding, you may have some difficulties in fully understanding what is going on without first completing the set theory portion of your class: For some reason people like to express the ideas behind Language Theory with formal mathematics instead of practical programs, even though one of the basic tenants of Formal languages is that any given language can be defined by the automaton (in our case, the program) that accepts it.

Very good info. I am looking for just one token that consists of a string. So it could be A or Apple. So I don't have to worry about finding relations in something like A  -> B & C turning to A -> C and A -> B. As well as reflexive properties being implied

ƆԀ S₱▓Ɇ▓cs: i7 6ʇɥפᴉƎ00K (4.4ghz), Asus DeLuxe X99A II, GT҉X҉1҉0҉8҉0 Zotac Amp ExTrꍟꎭe),Si6F4Gb D🅾🅼🅸🅽🅰🆃🅾r PlatinUm, EVGA G2 Sǝʌǝᘉ5ᙣᙍᖇᓎᙎᗅᖶt, Phanteks Enthoo Primo, 3TB WD Black, 500gb 850 Evo, H100iGeeTeeX, Windows 10, K70 R̸̢̡̭͍͕̱̭̟̩̀̀̃́̃͒̈́̈́͑̑́̆͘͜ͅG̶̦̬͊́B̸͈̝̖͗̈́, G502, HyperX Cloud 2s, Asus MX34. פN∩SW∀S 960 EVO

Just keeping this here as a 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8 hours ago, straight_stewie said:

For some reason people like to express the ideas behind Language Theory with formal mathematics instead of practical programs, even though one of the basic tenants of Formal languages is that any given language can be defined by the automaton (in our case, the program) that accepts it.

That's because you can express formal langauge without an automaton. Heck, English language I am writing to express my ideas is a formal langauge.

It is broad field that goes beyond just it's applicability in computer science. Formal langauge has been study by linguistics and mathmaticians alike and it has predated the invention of pushdown automatas or a Turing machine just like computer science is a disciplain that has predated the invention of modern day computers. As a result, I would treat them as such by disassociating them from any given programming language when learning about them.

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5 hours ago, wasab said:

That's because you can express formal langauge without an automaton

That is correct. The word "can" in this context usually implies that it's not the only way. Often, it implies that it's not the most common way.

5 hours ago, wasab said:

Formal langauge ... has predated the invention of pushdown automatas or a Turing machine

This is false. In fact, there was no definition for the term "Formal Language", nor widespread idea of thinking about language as transformational generative rules (which is what formal languages are) until Noam Chomsky's 1957 book "syntactic structures".

Turing Automata were discovered by Alan Turing in 1936, a full 20 years before Chomsky's book.

In Chomsky's own book, he comes up with pushdown automata specifically to describe a machine that can accept any context-free grammar. Which is ironic, considering that what you are arguing against is my proposition that it would be oft beneficial to talk about formal languages from the perspective of someone writing a program that understands them, which is what the creator of formal languages did himself.

"Ultimately, saying that you don’t care about privacy because you have nothing to hide is no different from saying you don’t care about freedom of speech because you have nothing to say." ~Verax

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32 minutes ago, straight_stewie said:

Which is ironic, considering that what you are arguing against is my proposition that it would be oft beneficial to talk about formal languages from the perspective of someone writing a program that understands them, which is what the creator of formal languages did himself.

Nah. It is better to talk about them in terms of sets and other mathmatical constructs then to say it is something understood by automatas. If formal langauge theory doesn't predate automata, the notion that languages are constructed from letters of an alphabets and follow certain rules certainly do predate automata.

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28 minutes ago, wasab said:

the notion that languages are constructed from letters of an alphabets and follow certain rules certainly do predate automata.

Well I would hope so. That's the most general definition of any form of text based communication... But we call that "linguistics", not "formal languages"...

The term "Formal Language" is very well defined: A language consisting of strings of characters concatenated together from an alphabet and that follows a set of production rules, which are the so-called "formal grammar" of that language.

For clarity sake, a production-rule is of the form we discussed above: A := BA, B := A, for example.

28 minutes ago, wasab said:

Nah. It is better to talk about them in terms of sets and other mathmatical constructs then to say it is something understood by automatas

With all due respect, that's not for you to decide. That's for the people studying the subject to decide for themselves. In this case, the OP specifically asked how to implement such a thing. Therefore, only discussing it in terms of the pure mathematics is absolutely useless, hence:

1 hour ago, straight_stewie said:

my proposition that it would be oft beneficial to talk about formal languages from the perspective of someone writing a program that understands them

"Ultimately, saying that you don’t care about privacy because you have nothing to hide is no different from saying you don’t care about freedom of speech because you have nothing to say." ~Verax

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1 hour ago, straight_stewie said:

That's for the people studying the subject to decide for themselves. In this case, the OP specifically asked how to implement such a thing.

You can draw a state diagram with pen and paper that models an automaton interpreting a certain langauge. it isnt require that you actually do this through code.

Natural langauge theory is still math, it just happens to lie in the intersection between computer science and descret math. I find the best way to learn it is in the same way you should learn physics, in the langauge of math.

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Posted · Original PosterOP
19 hours ago, straight_stewie said:

This can be thought of as a string replacement system. Since you mention that you can have rules of the form AB -> C, or more generally, <input0 ... inputn> := <output0 ... ouputn> you are looking for Context Sensitive systems. Are you sure you're looking for a context sensitive system, or in other words, is the tuple (A, B) really two tokens or is it the single token AB, which is unique from either A or B?

In your case, you have a production rule like A -> B. A is known as the predecessor, and B is known as the successor. You then have a production rule B -> C. There is no production rule for C, so C is known as a terminal character (trivially, this is the Identity property, or simply C -> C). As long as you only have one production rule per token, or per tuple of tokens in the case of context-sensitive systems, then your system is deterministic. If you have multiple rules, like A -> B and A -> C, then you have a non-deterministic system.

What you can do is put all of your rules in some <key, value> pair collection, like a hash map or a dictionary. Then, you can iterate through the characters in some input string, known as the axiom, transforming each character by looking it's rule up in the collection. Do this for one replacement per character or tuple of characters, per iteration. When the entire string is terminal characters you are finished, or in this case, your axiom is "proven".

For more information on this method of doing things, you can look into L-Systems, Semi-Thue Systems, or String Replacement Systems. I gather that you are just starting out in discrete mathematics. You should then be warned that you are beginning to dabble in Formal Languages, which while I find the subject very challenging and thus very rewarding, you may have some difficulties in fully understanding what is going on without first completing the set theory portion of your class: For some reason people like to express the ideas behind Language Theory with formal mathematics instead of practical programs, even though one of the basic tenants of Formal languages is that any given language can be defined by the automaton (in our case, the program) that accepts it.

So I have managed to get two arrays that look like this from the database

```[1A, 2B, 3C, 4A, 5E, 5F, 6A, 6B, 7H, 8H, 9J]
[1B, 2C, 3D, 4E, 4F, 5G, 6H, 7I, 7J, 8K, 9L]```

The number corresponds with the relationship it is in. So 1A means A -> B and 4A means A -> E & F.  I'm wondering if maybe I should have a linked list as I want to check if the next element contains the same number as if it doesn't I want to stop and not search them all. I also am thinking of writing a loop inside this that traces the relations by hopping from one array to the other checking if it exists anywhere else. This sounds costly as it would have to check every element in the opposite array every time it found itself. So if I go A to B then I would check the opposite array for B. If I find B then I check the other array for B that doesn't contain the same leading number as me. You mentioned a hashmap earlier. Could I use the number as the key then the letter as the value? This would provide faster lookup and eliminate the issue of trying to eliminate duplicate searches.

ƆԀ S₱▓Ɇ▓cs: i7 6ʇɥפᴉƎ00K (4.4ghz), Asus DeLuxe X99A II, GT҉X҉1҉0҉8҉0 Zotac Amp ExTrꍟꎭe),Si6F4Gb D🅾🅼🅸🅽🅰🆃🅾r PlatinUm, EVGA G2 Sǝʌǝᘉ5ᙣᙍᖇᓎᙎᗅᖶt, Phanteks Enthoo Primo, 3TB WD Black, 500gb 850 Evo, H100iGeeTeeX, Windows 10, K70 R̸̢̡̭͍͕̱̭̟̩̀̀̃́̃͒̈́̈́͑̑́̆͘͜ͅG̶̦̬͊́B̸͈̝̖͗̈́, G502, HyperX Cloud 2s, Asus MX34. פN∩SW∀S 960 EVO

Just keeping this here as a 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7 hours ago, wasab said:

You can draw a state diagram with pen and paper that models an automaton interpreting a certain langauge.

That's still writing a program...

3 hours ago, BuckGup said:

The number corresponds with the relationship it is in. So 1A means A -> B and 4A means A -> E & F.  I'm wondering if maybe I should have a linked list as I want to check if the next element contains the same number as if it doesn't I want to stop and not search them all. I also am thinking of writing a loop inside this that traces the relations by hopping from one array to the other checking if it exists anywhere else. This sounds costly as it would have to check every element in the opposite array every time it found itself. So if I go A to B then I would check the opposite array for B. If I find B then I check the other array for B that doesn't contain the same leading number as me. You mentioned a hashmap earlier. Could I use the number as the key then the letter as the value? This would provide faster lookup and eliminate the issue of trying to eliminate duplicate searches.

Consider that 1A and 4A are actually unique tokens. We know this because they have different successors. 1A and 4A will never generate the same output. Then consider that 4A has two separate successors, but you always want both of them, therefore 4E and 4F are really the same token, 4E4F.

This being the case, it would be better to redefine your inputs in terms of a single character (this is just easier to tokenize if you ever read them from a file). For example, 1A is just A, and 4A is just B.

Then, we can do something simple like use the predecessor as the key, and the successor string as the value, like this (not in any particular language):

```Dictionary<string, string> productionRules = new Dictionary<string, string>();

string input = "AB";
string output = "";

foreach (char c in input)
output += productionRules.GetValue((string)c);

Print(output);

// This will print "BEF"```

After getting this down, what you are really trying to do is say something like:

• If A then B
• If B then C
• If C then D
• Therefore, A then D.

This means that [A-C] are non terminal characters, and D is a terminal character. So what we can do is define our grammar such that if the chain of logic follows, the output string will be made entirely of terminal characters, in this case D.

This yields the string replacement system:

```axiom: A

rules:
A -> B
B -> C
C -> D```

Then we can iterate over this system until one of two cases becomes true:

• The output string is entirely terminal characters. In this case this means that there is a path between A and D.
• We timeout. This doesn't necessarily mean that there isn't a path between A and D, it just means that the system didn't halt in a reasonable amount of time and we gave up.

That looks like this:

```Dictionary<string, string> ProductionRules = GetProductionRules();

string axiom = GetAxiom();
string doubleBuffer = "";

while ((!OnlyTerminals(axiom)) && (!OnlyTerminals(DoubleBuffer)))
{
doubleBuffer = "";

foreach (char c in axiom)
doubleBuffer += ProductionRules.GetValue(c);

axiom = doubleBuffer;
}

Print(axiom);

// This will print the string D```

Bear in mind that this type of system only proves that there is a path of transitive relationships between A and D. It does not decide whether the conditions (such as if A then B) actually hold true.

"Ultimately, saying that you don’t care about privacy because you have nothing to hide is no different from saying you don’t care about freedom of speech because you have nothing to say." ~Verax

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On 12/12/2019 at 8:46 AM, BuckGup said:

So I have managed to get two arrays that look like this from the database

```
[1A, 2B, 3C, 4A, 5E, 5F, 6A, 6B, 7H, 8H, 9J]
[1B, 2C, 3D, 4E, 4F, 5G, 6H, 7I, 7J, 8K, 9L]```

The number corresponds with the relationship it is in. So 1A means A -> B and 4A means A -> E & F.  I'm wondering if maybe I should have a linked list as I want to check if the next element contains the same number as if it doesn't I want to stop and not search them all. I also am thinking of writing a loop inside this that traces the relations by hopping from one array to the other checking if it exists anywhere else. This sounds costly as it would have to check every element in the opposite array every time it found itself. So if I go A to B then I would check the opposite array for B. If I find B then I check the other array for B that doesn't contain the same leading number as me. You mentioned a hashmap earlier. Could I use the number as the key then the letter as the value? This would provide faster lookup and eliminate the issue of trying to eliminate duplicate searches.

I'd definitely go for a hash map or, as I said, go for a tree directly. Technically a hash map is a tree (red-black tree). Using an explicit tree structure has the advantage that it is more clear what you're doing and you can design them in a way that prevents cycles. If your hash map contains something like A -> B -> A your resolver that tries to find the last node in your relationship will have a fit.

Here's an example of a Node class that prevents you from creating cycles:

```public class Node {

private final String name;

private final Node parent;
private final List<Node> children = new ArrayList<>();

public Node(final String name) {
this.name = name;
this.parent = null;
}

private Node(final String name, final Node parent) {
this.name = name;
this.parent = parent;
}

public Node addChild(final String name) {
final Node child = new Node(name, this);
return child;
}

public String getName() {
return name;
}

public Node getParent() {
return parent;
}

public List<Node> getChildren() {
return Collections.unmodifiableList(children);
}

public boolean hasChildren() {
return !children.isEmpty();
}

@NonNull
@Override
public String toString() {
return name;
}
}```

Here's an example how you can follow a relationship to find all leave nodes that are reachable from a given node:

```@Test
public void test() {
final Node root = new Node("A");

// The node for which you want to find all leaves
final Node start = root;
final List<Node> leaves = new ArrayList<>();

// Code to find leaves
final Stack<Node> stack = new Stack<>();
stack.push(start);

while (!stack.isEmpty()) {
final Node current = stack.pop();

if (!current.hasChildren()) {
} else {
current.getChildren().forEach(stack::push);
}
}

// Output
System.out.format("%s -> %s", start, leaves);
}```

This will give you the output "A -> [D, C]".

In terms of database I'd probably create a table like this:

```Id | Name | ParentId |
---+------+----------+
1  | "A"  | null     |
2  | "B"  | 1        |
3  | "C"  | 2        |
4  | "D"  | 2        |```

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