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In the fields related to school mathematics there is some acitivity on proving (or disproving) deducibility/decidability for some classes of school identities. In particular,

1) In logic they considered not long ago the base identities problem (this term is the translation from Russian, I am not sure that it is correct). The problem was the following. Let $N$ be the set of positive integers, and $\mathcal K$ a class of all functions from $N^k$ into $N$ ($k$ runs over $N$) which can be represented as compositions of usual algebraic operations $x+y$, $x\cdot y$ and $x^y$. Let us call a base of identities in $\mathcal K$ a set $B$ of identities for functions in ${\mathcal K}$, such that any identity for functions in $\mathcal K$ can be deduced from $B$. The question was, does there exist a finite base of identities for ${\mathcal K}$? This question appeared when A.Wilkie gave a counterexample for the Tarski high school algebra problem (where a list of identities was suggested by Tarski, and the question was whether this list is a base). In 1980-es R.Gurevich proved that there is no finite base of identities, so the problem of base identities is solved in negative. At the same time, as far as I understand, R.Gurevich proved that instead of finite base of identities, there exists a recursive base of identities, and as far as I understand this is an example of what logicians call decidability.

2) In computer algebra there is the so-called Richardson theorem, which states that if $\mathcal R$ is a class of expressions generated by

-- the rational numbers and the two real numbers $\pi$ and $ln 2$,

-- the variable $x$,

-- the operations of addition, multiplication, and composition, and

-- the sine, exponential, and absolute value functions,

then for $F\in {\mathcal R}$ the predicate $F=0$ is recursively undecidable.

My question is whether these two fields are related to each other? Is decidability for Richardson the same as decidability for logicians? If yes, then which exactly logical system does Richardson mean?

I am not a specialist here, I am interested in this because I write a textbook on mathematical analysis (I am sorry, this happens sometimes with mathematicians), and when describing elementary functions I faced a problem analogous to the base identities problem above, but the difference is that the list of operations (and elementary functions) is wider (for example, both $x-y$ and $x^y$ are included), and as a corollary the arising functions are defined not everywhere on $R$ (one can look at the details at page 197 in the draft of the first volume of my textbook -- unfortunately, it is in Russian).

This is strange, but I can't find anyone who could explain me this. I asked this question in sci.math.research some time ago, but the problem of overcoming the Kevin Buzzard resistance turned out to be undecidable for me there. So I would be much obliged to MO if my question will hang here for some time so that, perhaps, some specialitsts in logic could clarify me something.

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You formulation of Richardson's Theorem is wrong since the set $E$ of constant functions given by rational numbers is a counterexample of your claim. After reading Wikipedia page it seems clear to me that you have forgotten to write down some extra hypothesis on the set $E$. –  boumol May 11 '12 at 13:50
It is the same notion but note that in Richardson's theorem you need also the function sin(x) etc., see Wikipedia: en.m.wikipedia.org/wiki/Richardson's_theorem#section_1 –  Bjørn Kjos-Hanssen May 11 '12 at 13:53
@boumol, @Bjørn Kjos-Hanssen: Yes, excuse me, I have just corrected the formulation. –  Sergei Akbarov May 11 '12 at 15:34
I recommend you ask Mark Sapir a very specific question regarding your interest. I think there is an issue that you are trying to state but have not yet. My belief is that Mark's Russian is very good and his knowledge of decidability more than sufficient to resolve your stated and unstated issues. If you are able to pose a well formed question to him, you may get a very good answer very quickly. Gerhard "Ask Me About System Design" Paseman, 2012.05.11 –  Gerhard Paseman May 12 '12 at 0:19
@Gerhard: If I were a specialist, I could of course formulate a question in such a way that it would be much easier for another specialist to give an answer, since there would not be necessity to explain elementary things to non-specialists. But in this case there would not be a necessity for me to ask elementry questions, since, being a specialist, I could understand everything independently, without other specialists. –  Sergei Akbarov May 12 '12 at 8:03
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3 Answers

I am not a professional logician, but I have studied mathematical logic, and in past work I used the rough notion (as have many before me) that if you can write a Pascal program to decide correctly the yes or no answer to a problem given the finite set of parameters as input, then the problem or issue is decidable. Otherwise it isn't. Taken at this level, I see both uses of decidability as the same. In one, there is a finite specification which can be used to test whether an identity is in the one set, in the other there is no such program to test whether an equation/identity is in the other set.

(There are technical arguments to be made as to which machine model, complexity, degree of undecidability if one looks at e.g. Turing equivalent degrees, and so on. I am setting aside all these complexities and ways to distinguish the two uses of decidability, since they seem to me irrelevant to the basic intent of your question.)

I can see both problems as problems of clone theory. Again roughly, the first problem talks about whether there are a finite number of relations in the generators in addition to the general relations for a clone that can be used to describe the collection of equivalence classes of terms (there are not, but there is a recursive set of such relations). The second talks about whether the set of terms in the clone equivalent to the term 0 is describable by a computer program; according to Richardson, it is not. There are other ways to recast the problems to see some similarities and highlight the differences; it depends on just what you want to see.

EDIT: Another view of many issues of decidability is this one, borrowed and simplified from one used in complexity studies in computer science. If you have a decision or labelling problem, where you have a set S of instances and for each instance you want to say "yes, instance I has property P" or "no, I does not have P", you take a somewhat Platonist viewpoint and say " I will group those instance which have P into this subset R", and then you end up with two sets, S and a proper subset R. Then you shift to a constructivist mode and ask "Is there a way I can tell quickly, or even mechanically, when a member of S is also a member of R or not?" Then you switch to programmer/computer scientist mode and say "Let's see if I can either a) write a program to determine if an instance is a member of R, or b) translate the domain to one where I can encode the halting problem, so that determining membership in R solves the halting problem" . If the set R is recursive inside S, then a) is possible in theory, but may be difficult or impossible in practice, depending on the complexity of the set R. If the set R is not recursive in S, then b) may or may not be possible, but is usually the first step one tries.

How does one show R recursive in S or not? One takes an encoding, which is an injective and computable map from S into the natural numbers (or computably functional equivalent), and then sees if the image of R under this map is a recursive subset of the natural numbers. So this and the previous paragraph are a long winded way of saying that most issues of decidability involve coding the problem up in a way as to move the question into the realm of subsets of natural numbers, and using recursion theory or diagonalization or something to determine the status of the image set. For me, I picture the set of identities or the set of terms as a set of numbers, each number colored with label or term or identity it represents, and I picture the subset with property P as a subset of integers which may or may not be a recursive subset. The set of identities satisfied by the real numbers with exponentiation , addition and multiplication is a set which has a logically equivalent, recursive, and non finite subset. The set of terms in the Richardson theorem which are equivalent to 0 is a nonrecursive subset of the set of all terms used in the context of the theorem. END EDIT

Gerhard "Ask Me About System Design" Paseman, 2012.05.11

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Gerhard, are you saying that these two questions are not connected with each other? –  Sergei Akbarov May 11 '12 at 17:11
By the way, there are two ways to interpret "base" in this context: as in a basis or generating set of identities, and as in basic or simple as in high school knowledge is basic knowledge needed to help form knowledge coming from graduate studies. Gerhard "Ask Me About System Design" Paseman, 2012.05.11 –  Gerhard Paseman May 11 '12 at 17:15
I think there is some commonality, but I do not think they are very closely related. In a broad study of decidability I would look at both problems, but I would not take the specific techniques of one problem and try to apply them to the other. For one, they have different outcomes: one has a Pascal program to describe it, the other does not. Gerhard "Ask Me About System Design" Paseman, 2012.05.11 –  Gerhard Paseman May 11 '12 at 17:18
I thought, I gave an accurate definition for base... –  Sergei Akbarov May 11 '12 at 17:19
Which one has a Pascal program to describe it? –  Sergei Akbarov May 11 '12 at 17:22
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Note: I'm not actually familiar with either problem that you ask about, so I'm going by your description.

Recursive base of identities means there is a computer program P such that given an identity I, running P will tell you in a finite amount of time whether I is in the base or not. P is called a "decision procedure".

Richardson problem being undecidable means something like: given an arbitrary program (Turing machine) P, you can encode the halting problem for P an an expression in $\mathcal R$. That is you can write down a formula that is identically zero if and only if P halts. Since the halting problem is undecidable, there is no decision procedure for telling if such a formula in $\mathcal R$ is identically zero. That's sort of like Hilbert's tenth problem, where you can encode an arbitary program P as a set of diophantine equations, that has a solution iff P halts. Again since the halting problem is undecidable, there is no algorithm to tell whether an arbitrary diophantine system has a solution.

I think the absolute value function being available in $\mathcal R$ may have something to do with the undecidability. In symbolic algebra, the Risch algorithm is a finite procedure for telling whether a given expression made from elementary functions and composition has a closed-form indefinite integral. But I seem to remember that if you add the absolute value function, the problem becomes undecidable.

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I need somebody to talk about this. When reading texts on this topic I see a lot of unclear places. For example, according to definition - en.wikipedia.org/wiki/Recursive_set - recursive set must lie in the set of positive integers. When Richardson says that the set of true identities is not recursive, this means that the set of all identities is enumerated, I suppose... So there must be a standard procedure for numeration formulas, is it? What is this procedure? Or I don't understand something? –  Sergei Akbarov May 12 '12 at 20:35
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Sergei, this is a reply to your comment asking about enumerating formulas in $\mathcal R$. Sorry to post it as a separate answer but I no longer have the browser cookie to post it as a followup comment.

You don't need a particular standardized enumeration, but just some computable mapping between formulas and natural numbers so that each formula gets a unique number. Such a numbering scheme is traditionally called a "Gödel numbering" and the numbers are called "Gödel numbers" because the idea was (I think) introduced in Gödel's landmark paper (1931) about the incompleteness theorem.

A simple Gödel numbering scheme (similar to the one Gödel used) is like this: say the formulas are written in an "alphabet" whose "letters" are $\{\sigma_1,\sigma_2,\ldots\}$. Treat those as natural numbers the obvious way (i.e. $\sigma_k\mapsto k$). So a formula F might be written as $(F_1,F_2,\ldots F_n)$ where the $F_i$ are natural numbers. Then let

$$N_F=2^{F_1}\cdot 3^{F_2} \cdot 5^{F_3} \cdots p_n^{F_n}$$

where $p_i$ is the $i$'th prime number. That is the Gödel number for F (under this particular scheme). It's pretty easy to see how to convert a formula to a number and back. Some numbers won't correspond to valid formulas so treat them as identically zero, for example.

Maybe you should read an introductory book on logic, if you want more clarity about this stuff. There are some other threads suggesting them.

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Thank you, now I recall that in the university our lecturer told something similar. But I suppose, it's too difficult to communicate each other in this way. If you have time and forbearance, you can contact me by e-mail, it is indicated at the website that I gave at my page here in Mathoverflow. Or you can just edit your answer and put there a name of a book about all this. –  Sergei Akbarov May 12 '12 at 22:12
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