Talk:Presburger arithmetic

message: Some anon just rewrote most of the article, not too badly.  Please check that work. Thanks. --John Nagle 05:22, 13 July 2006 (UTC)
thread_title:  Rewrite by anon 

message: I do not understand why simplex has anything to do with Presburger. Presburger is for natural numbers, whereas simplex is for the reals. Simplex might be used inside an algorithm such as branch & bound to solve the ILP part, but this is not what is written in the page.
  
-- I added some references to explain this. As discussed above, for practical applications we look often at quantifier-free fragment as a special case, and this is typically solved in SMT solvers through a combination of a SAT solver and integer linear arithmetic solver. Integer linear arithmetic solvers use Simplex to solve a relaxation of the problem and then additional rules of integer linear programming to obtain completeness for integers. --vkuncak  — Preceding unsigned comment added by Vkuncak (talk • contribs) 11:19, 1 March 2015 (UTC)
thread_title: simplex 

message: Shouldn't there be some explicit mention of quantifiers in the induction axiom?
(P(0) AND (P(x)->P(x+1))->P(x)
say P is "is even." Now P(0)->P(1) is false so the axiom is vacuously true for x=0. 
P(0) AND P(1)->P(2) is true, but P(1) is false, hence the axiom fails.  I think one wants two variables. Thoughts?  MotherFunctor (talk) 17:17, 27 November 2007 (UTC)
replies: text: Yes, the third x should be a y or some other variable. — Carl (CBM · talk) 17:47, 27 November 2007 (UTC)
thread_title:  induction 

message: We should discuss the various decision procedures:
* Omega
* Wolper et al.'s methods using finite automata (for instance, the LIRA tool). David.Monniaux (talk) 07:24, 10 December 2007 (UTC)
thread_title:  Decision using automata etc. 

message: I removed phrases like 'invokes an object constant' in favor of what I feel to be more standard expressions; also -- equality: is it a nonlogical constant, therefore a predicate of the language of Presburger arithmetic (as the original revision had it) OR is the background logic 'first order logic with identity' in which case equality is NOT just another predicate.  I favor the latter. 
Zero sharp (talk) 21:16, 7 April 2008 (UTC)
thread_title:  Language 

message: "The Wikipedia reference is slightly incorrect. The upper bound grows exponentially by the size of the problem, so is infinitely large, not bounded as suggested by Wikipedia. It is, Oppen believes, the largest bounded problem thus shown" added by user User:Derekoppen Zero sharp (talk) 06:27, 1 June 2008 (UTC)
thread_title:  moved from article 

message: Could someone advise on the propriety of one putting references to one's
own work in a WP article (Derek C. Oppen as user 'Derekoppen' added
a reference to one of his own journal articles).  Thanks. Zero sharp (talk) 18:30, 2 June 2008 (UTC)
replies: text: See WP:COS and WP:COI. One should be cautious with such edits, but ultimately the standard is the same as if anybody else inserted the reference: if it's relevant, notable, and properly sourced, it's OK. In this particular case, I think the edit was entirely appropriate, except that it was somewhat misplaced (that's why I moved it and slightly reformulated). Notice also that the reference to Oppen and Nelson (1980) was added by someone else. — EJ (talk) 10:02, 3 June 2008 (UTC)
replies: text:  Understood, thanks very much. Zero sharp (talk) 14:52, 3 June 2008 (UTC)
thread_title:  self-reference 

message: In the Properties section, the following statement has scope ambiguity:
:Mojżesz Presburger proved Presburger arithmetic to be:
:...
:* decidable: There exists an algorithm which decides whether any given statement in Presburger arithmetic is true or false.
Does this mean 1 or 2?:
# For any statement S in Presburger arithmetic, there is some algorithm A such that (A decides whether S is true or false)
#*i.e., all P-statements are decidable
# There is an algorithm A such that for any statement S in Presburger arithmetic (A decides whether S is true or false)
#*i.e., P-arithmetic has some "master" decision algorithm
(Sorry for not using <math> notation, but I'm not a mathematician and not familiar with that part of Wikicode.) After writing them out that way, I think it's pretty clear that #1 is what's meant, but the English is ambiguous and should be cleaned up.
--Thnidu (talk) 14:42, 22 May 2009 (UTC)
replies: text: No, #2 is the correct meaning. #1 is nonsense, as it is trivially satisfied for any statement in any theory: one of the two algorithms "return YES" and "return NO" always works. I do not see any ambiguity in the formulation, it clearly starts with an existential quantifier, and in any case, there is a link to decidability (logic) which explains it in detail. You just have to read carefully. — Emil J. 15:20, 22 May 2009 (UTC)
thread_title:  Unclear assertion 

message: The article gives a nice example given for something that is provable in Presburger Arithmetic:
:for all x, there exists y : (y + y = x) ∨ (y + y + 1 = x). This states that every number is either even or odd.
But the ∨ symbol is inclusive-or. Don't we want exclusive-or here? If there's no standard exclusive-or symbol, it could be stated with a conjunction (∧).  — Preceding unsigned comment added by Iconjack (talk • contribs) 16:53, 24 November 2013 (UTC)
thread_title:  Exclusive OR 

message: Hi, axiom 1 says "¬(0 = x + 1)", which I take to mean that there are no negative numbers. Is this true, and out of interest, how would permitting them suddenly make thing undecidable? (edit: I mean really simply put as I'm no mathematician).
thanks 78.149.128.58 (talk) 11:11, 14 May 2016 (UTC)
replies: text: I guess, allowing negative numbers wouldn't make Presburger arithmetic undecidable, but a Bachelor's thesis might be needed to formally prove that. I guess, there are mainly historical reasons for Presburger arithmetic dealing with natural rather than integer numbers. In the 19th century, Peano gave his axioms for natural numbers (Peano axioms); in 1900, Hilbert posed the problem of proving their consistency (Peano axioms#Consistency), but no such proof was found; possibly this motivated Presburger to suggest in 1929 a simpler theory that is decidable (Presburger arithmetic#lead); in 1931, Gödel showed that Peano arithmetic cannot be proven consistent within itself (Peano axioms#Consistency). If I remember right, Gödel's proof doesn't use much features of Peano arithmetic beyond Presburger arithmetic, except for multiplication and prime numbers (he uses them to encode a sequence of symbols into a single number, thus enabling encoding of a logic formula into a number); so those are the features that make things undecidable. The reason why Peano used natural rather than integer numbers was probably that he intended to have an axiom system as simple as possible for the foundation of arithmetic. - Jochen Burghardt (talk) 17:25, 14 May 2016 (UTC)

text: Versions of Presburger arithmetic with or without negative numbers are bi-interpretable, and used quite interchangably in the literature, as they have virtually identical properties. (If you allow negative numbers, you have to include an explicit ordering predicate in the signature, lest the theory becomes a bit weaker.) But the official definition is with non-negative integers. The theory with negative integers is more precisely called the theory of \mathbb Z-groups.—Emil J. 12:11, 8 July 2023 (UTC)
thread_title:  negative numbers disallowed? 

message: The passage
> Since the axioms in the schema in (5) cannot be replaced by any finite number of axioms, Presburger arithmetic is not finitely axiomatizable in first-order logic.
follows the grammar of justification, but it seems to me that it's just a tautology. We're not defining the term finitely axiomatizable here. Why can't the schema be replaced by a finite number of axioms?   Luqui (talk) 06:20, 14 November 2019 (UTC)
replies: text: You are right. I'll change the sentence so that it reads like a tautology. However, I don't know the reason; so I can't do better than adding . - Jochen Burghardt (talk) 11:55, 14 November 2019 (UTC)
thread_title:  Finite axiomatization