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I haven't found a good enough answer by googling. Here's what i know:

  1. TG's can have more than one initial state
  2. In TG's, Edges/transitions can be labelled with strings
  3. In TG's, it is not necessary to show transitions for all letters

Notes:

  • I am talking about plain TG's, not GTG's and DFA's only not NFA
  • Only thing i know about NFA's is that each a state can have multiple transitions for the same letter, so please keep that in mind if hte answer invloves NFA's
  • Links would be appreciated.
Awaisome
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    I’ve never heard of transition graphs. – Yuval Filmus Dec 02 '18 at 18:17
  • @YuvalFilmus , exactly, i can't find something definitive either. the internet seems to think TG is what i think FA is. so i am not sure how they are different, i am on a distant learning platform and low quality material, so they use these words interchangeably, only difference their notes (that's right, no textbooks) tell are listed above. they also always use FA where they mean a DFA. Take a look here and here – Awaisome Dec 03 '18 at 03:26
  • @YuvalFilmus would it be right to say that FA is a theoretical machine and it's diagram is what we call a TG. but then why did my notes specify differences between the two as i listed above. I think it's because in my notes DFA is referred to as FA. so a TG is the diagram of both a DFA and NFA, but since they referred to DFA as only FA and we haven't read about NFA yet, that's why they had to specify the differences, because in the notes FA only means DFA and not NFA. clear as mud, right? – Awaisome Dec 03 '18 at 03:38
  • @YuvalFilmus this site has the same notes my university provided me in pdf with better looking diagrams, i don't who copied whose. here's lecture 4 in my notes where they tell what an FA is (there's no index, CTRL+F for finite automaton) and here's lecture 7 where they talk about TGs. – Awaisome Dec 03 '18 at 03:42
  • Do they mean transition systems? – reinierpost May 02 '19 at 06:59

2 Answers2

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Your nomenclature is highly nonstandard. It seems that "finite automata" are what are usually known as DFAs, whereas "transition graphs" are a special case of GNFAs in which the only regular expressions allowed are words; alternatively, they are like $\epsilon$-NFAs, only transitions can be marked by arbitrary words (rather than just words of length at most 1).

There are two differences between your FAs and your TGs:

  1. FAs are deterministic: at each point in time, there is exactly one choice of which state to go to next. There is no such restriction on TGs. Moreover, whereas FAs have a unique initial state, TGs have an arbitrary number of initial states.

  2. Transitions in FAs are labeled by symbols. In contrast, transitions in TGs are labeled by words.

These are exactly the differences that you listed.

Yuval Filmus
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  • the notes my uni provided don't mention GNFAs but GTG or generalized TGs whose only difference from TGs is that the edges are labelled with regex. I am just starting the course so i hope it will be clear by the time.i finish the basics. i have asked a more practical question, no replies yet. kindly take a look – Awaisome Dec 03 '18 at 05:49
  • I contacted the faculty, their reply: _"In an FA, there must be a single out going edge for each input character at each state. Loop is considered as an outgoing edge.

    TG is more flexible than FA. The edges may be omitted in TG, substrings can be mentioned on edges as well as the NULL character. In non-deterministic FA (NFA), the outgoing edges may be missing for some alphabets or there may be present more than one edge for the same alphabet."_ This was all already in the notes. Doesn't help. I have downloaded proper notes.

     – Awaisome Dec 03 '18 at 05:53
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TL;DR version: Your initial statement was accurate - I'll add some more clarity (hopefully) for people who come across this post later on:

The formal definition (from Daniel I.A. Cohen):

A transition graph, abbreviated TG, is a collection of three things:

  1. A finite set of states, at least one of which is designated as the start state ( - ) and some (maybe none) of which are designated as final states ( + ) .
  2. An alphabet I of possible input letters from which input strings are formed.
  3. A finite set of transitions (edge labels) that show how to go from some states to some others, based on reading specified substrings of input letters (possibly even the null string lambda ).

I'll use TG to refer to transition graphs and use FA to refer to finite automata.

My notes on features of TGs:

  • A TG can have multiple start states, while a FA cannot
  • A TG can have lambda on an edge, while a FA cannot
  • A TG can have an entire string on an edge, while a FA cannot
  • A TG is non-deterministic; It can have multiple possible paths while following the input string. If even ONE of these paths leads to an accept state, the string is accepted.
  • A TG can crash because its transition functions do not constitute total functions, which means functions that provide output for the entire domain of input (in other words, in a FA, you NEED to consider all possible input at each state - with a TG you do not.)

It is also important to note that any TG can be converted to a FA. Part of the underlying theory of Kleene's theorem is that if a TG exists, OR an FA exists, or a regular language exists for a language, then they all have to exist.

Additional information:

Transition Graphs are a relaxing of the notion of a finite automaton. It's a bit confusing, because the diagrams of machines in general are often called state transition diagrams. But other authors (e.g., Lenz) refers to the diagrams (visuals) themselves as transition graphs.

However, the notion of a Transition Graph (TG) isn't "non-standard nomenclature" as has been suggested. It is probably less popular now, but this type of machine was originally discussed in Finite Automata and the Representation of Events by J. Myhill in 1957. It was made very popular in semi-modern times by Daniel I.A. Cohen's excellent (classic) book, Introduction to Computer Theory, specifically in chapter 6 (in the 2nd edition, 1997). Cohen was one of Alonzo Church's Ph.D. candidates, so I don't think this was just a matter of some no-named researcher trying to make a certain term or concept popular.

John Baugh PhD
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