Over several holiday and train journeys, I’ve been slowly reading and digesting this book by Blum, Cucker, Shub, and Smale. It’s one of a number of books I’ve read that are definitively not in my research field – hence holiday reading – and yet touches it tangentially enough to get me excited. I’ve been interested in computation defined in terms of real numbers ever since my own PhD thesis back in 1998-2001, which looked at ways to take a specific class of computation defined on the reals and synthesise hardware circuits operating in very efficient fixed-point arithmetic. While my own work has often involved specifying computation on the reals, including recently exploiting the differences between real and finite precision computation for circuits, this book takes a fundamental look at computation over the reals themselves. In particular, it looks at what happens to notions of computability and complexity if you consider real addition, multiplication, division, and subtraction as basic operators in a computation model: what is computable, what resulting complexity bounds can be proved? Ever since my first exposure to algebraic geometry, especially real algebraic geometry, I see it everywhere – and have even made some small forays [1,2] into harnessing its power for hardware design. It’s a delight to see computability and algebraic geometry coming together so neatly in this text.

Part I: Basic Development

The authors define a computational model that is very natural for analysis: a computation consists of a control-flow graph consisting of input nodes, computation nodes (polynomial or rational maps), branch nodes (checking for equality or inequality, depending on whether the ring / field being computed over is ordered), and output nodes. For the remainder of this review, I will consider only computation over the reals, though some of the results are more general. Theorems naturally flow from this setting. My favourites: the halting set of a machine is a countable union of semialgebraic sets; there are “small” systems of polynomial inequalities, of low degree, describing behaviour of a machine up to T time steps. As a by product, the first practical uses of this theory are illustrated, e.g. a proof that the Mandelbrot set is not decidable over {\mathbb R}. The interplay between the uncountable (program state) and the countable (program counter state) leads to some intricate and enjoyable analysis.

Chapter 3 generalises classical results for universal Turing machines, typically expressed in terms of unbounded tapes with binary storage at each location, to the case where an element of an arbitrary ring is stored at each location (imagine a real number stored at each location). Additional shift operations are introduced to deal with moving along the tape. It is shown that operation over unbounded sequences allows for uniformity of algorithm across dimensionality but does not add additional computational power for fixed dimensionality. In finite time we can only explore a finite portion of the tape, so we still retain the semi-algebraic (or countable union of semi-algebraic) sets for the halting sets derived in Chapter 2.

Chapter 4 extends the usual complexity-theoretic classes P and EXP to computation over a ring. Computation over {\mathbb Z}_2 corresponds to the classical case. There are no great surprises here for a reader who has read a basic text in computational complexity together with Chapters 1-3 of this book. The authors round off the chapter by using this new machinery to cast results from Goedel and Tarski from an interesting perspective. For example, Goedel becomes there exist definable undecidable sets over {\mathbb Z}, while Tarski becomes every definable set over {\mathbb R} is decidable over {\mathbb R}. Questions of definability are then put off until the very end of this book – and this review.

Chapter 5 shows how classical NP, NP-complete and NP-hard definitions carry forward into computation over a general ring. Then, specialising the results to particular rings and particular costs of elementary operations (unit? dependent on size of numbers?), the NP-completeness of various key problems are shown. In particular, it is shown that the classical 3-SAT NP-completeness result can be obtained by working over the ring {\mathbb Z}_2. The chapter ends with an interesting discussion of the P ?= NP question over various rings and with various costs. It’s interesting, for example, that the problem of determining whether there is a zero of a degree-4 polynomial is NP-complete over {\mathbb R} while that of a degree 3 is in P; analogous to the situation with 3-SAT versus 2-SAT in the classical setting.

Chapter 6 moves on to look specifically at machines over {\mathbb Z} with bit cost, i.e. the bigger the number, the more it costs to operate on it. It is shown that these machines are polynomially equivalent to those operating over {\mathbb Z}_2 with unit cost, i.e. anything you can write in terms of algebraic operations over {\mathbb Z} you can also write under the classical Turing model of computation, and vice versa, with only polynomial blow up in execution steps. This is not too surprising, and ultimately derives from the standard possibility of writing an integer as its binary expansion. The chapter then takes a detour via Farkas’s Lemma to show that linear programming feasibility (LPF) is in NP, a rather intricate result I first saw in Schrijver’s encyclopaedic book when working with my former PhD student Qiang Liu on discrete linear algebra for hardware memory optimization. It is noted that LPF’s status depends strongly on the choice of computation model and ring: in the (equivalent) classic setting of integers with bit cost it is NP-complete, over rationals with bit cost it is in P, while its status over {\mathbb R} is apparently open.

Part II: Some Geometry of Numerical Algorithms

This part of the book turns to look at specific algorithms such as Newton’s method and a homotopy method for nonlinear optimization. In each case, complexity results are given in terms of the number of basic real arithmetic operations required. Some fun analysis and quite a clear exposition of homotopy methods, which I’ve worked with before in the context of convex optimization, especially when using barrier methods for linear and convex quadratic programming, something I looked at quite carefully when trying to design FPGA-based computer architectures for this purpose.

The chapter on Bézout’s Theorem assumed a rather more thorough grounding in differential topology than I have, and I wasn’t entirely sure how it sat with the rest of the work, as the constructive / algorithmic content of the theorems was less apparent to me.

The final chapters of this part of the book took me back to more familiar territory – first introduced to me by the outstandingly clear encyclopaedic textbook by Nick Higham – the discussion of condition numbers for linear equations and the relationship to the loss of precision induced by solving such systems (roughly, the relative error in the right hand side of Ax = b gets amplified by the condition number of the matrix A). What was interesting here is the authors’ probabilistic analysis – looking at the expected condition number if the matrix elements are iid Gaussian variables, for example. I have skimmed another book also co-authored by Cucker – on this very topic, and I’m waiting to have the time to read it. Condition for polynomial problems is covered, leading to complexity results for logarithmic barrier interior point methods for linear equations over \mathbb{Q} (spoiler: it’s in \mathbb{P}).

Part III: Complexity Classes over the Reals

After Part II, which was only interesting to me in parts, I started to get excited again by Part III. Lower bounds on the computational complexity of various problem are derived here based on some very interesting ideas. The basic scheme is to bound the number of different connected components in a (semi-)algebraic set in terms of the degree of the polynomials and the number of variables involved. The number of different connected components can then be used to bound the number of steps an (algebraic) machine takes to decide membership of that set.

In Chapter 17, the framework is extended to probabilistic machines, and the standard definition of BPP is extended to computation over an arbitrary ring, giving BPP_{R}. The chapter then goes on to show that probabilistic machines over \mathbb{R} can be simulated by deterministic machines over \mathbb{R} without blowing up execution time, i.e. that P_\mathbb{R} = BPP_\mathbb{R}, whereas this is unknown for the classical setting. The essence of the argument makes use of a special real number (shown, non constructively, to exist), encoding all the different “coin flips” that a probabilistic program makes during its execution, which is then encoded as a machine constant in the simulating machine.

Parallelism – close to my heart – is covered in Chapter 18, through the introduction of complexity classes PL^k_\mathbb{R}, parallel polylogarithmic time, for sets that can be decided in time O(\log^k n) using a polynomial number of processors, and PAR_\mathbb{R}, parallel polynomial time, for sets that can be decided in polynomial time using an exponential number of processors (definitions following a SPMD computational model). Algorithms are derived for matrix inversion and product and a result is cited for parallel quantifier elimination over the reals.

Chapter 22 looks at the relative power of machines over \mathbb{Z}_2, i.e. the classical model, compared to real machines which take inputs restricted to be drawn from \{0,1\}. It’s fairly obvious that the latter setting can provide a lot of additional power, for example deciding membership of a set in \mathbb{Z}^\infty_2 can be done by encoding the characteristic function of the set as the digits of some real constant in the program. It is then shown that additive real machines (a restricted form of the machines discussed above, containing no multiplication or division) that additionally contain branching only on equality can solve exactly the same problems in polynomial time as in the classical model. This is unlike the case where branching can be over inequality, where some additional computational power is provided by this model.

Descriptive Complexity, Chapter 23 is interesting: the descriptive power of (a particular) first order logic extended with fixed point and maximisation operations and of existential second order logic are shown to correspond to a particular subclass of P_\mathbb{R} problems and EXP_\mathbb{R}, respectively. In the latter case, I think this essentially says that known results in the classical case carry forward to computation over the reals despite the move away from a finite ring. I am not familiar enough with computational complexity literature to comment on how the former result compares with the classical setting, and I didn’t feel that this point is was very well described in the book.

Conclusion

Overall, I found this book a little eclectic, especially Part II, reflective of the fact that it has clearly been put together drawing from a variety of different primary sources spanning several decades. This should not be seen as a criticism. The choice of material was wonderful for me – as holiday reading at least! – tangentially touching so many different topics I’ve seen during my career and drawing me in. In places, the development was over an arbitrary ring, in places over \mathbb{C} and in places over \mathbb{R}, and although for some sections it was absolutely clear why, for others it was less clear. Overall, though, I would very much recommend anyone who likes to keep a foot in both the continuous and discrete worlds to take a look at this book.

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