CMPSCI 741: Circuit Complexity Theory

David Mix Barrington

Fall, 2006

This is the home page for CMPSCI 741. CMPSCI 741 is an advanced graduate course in circuit complexity theory and will deal with circuit complexity classes, examples of algorithms in circuit models, and lower bounds. There are two texts, Vollmer's Introduction to Circuit Complexity (first edition, Springer Verlag) and the notes from the PCMI 2000 undergraduate complexity lectures by Alexis Maciel and I, available on the web at an address I gave in the first lecture (we don't want to link to these).

Instructor Contact Info: David Mix Barrington, 210 CMPSCI building, 545-4329, office hours MTuWThF 11-12. I generally answer my email fairly reliably.

The course will meet for two lecture meetings a week, TuTh 1:00-2:15 in room 140, Computer Science Building.

Announcements (5 October 2006):

• (17 Oct) Here's the summary of last week's lectures:
  • Lecture 10, 10 October: We discussed various versions of the class TC⁰, proving that the DLOGTIME-uniform version is equal to the descriptive complexity class FOM -- first-order logic with majority quantifiers and the BIT predicate. We briefly discussed the version of FOM without BIT (and allowing majority quantifiers to act only on single variables), and I suggested another possible final project topic, "The Descriptive Complexity Approach to LOGCFL" by Lautemann, McKenzie, Schwentick, and Vollmer (STACS 1999). This paper proves, among other things, that FOM without BIT cannot recognize the unary language of strings whose length is a perfect square. We developed integer arithmetic through multiplication of long integers in FOM, following PCMI lecture A6.
  • Lecture 11, 12 October: Following PCMI lecture A7, we proved a weak version of the Prime Number Theorem using Kolmogorov complexity, then applied this to show that the word problem for the free group on two generators is in LOGSPACE, and that threshold functions with poly-log thresholds are in FO. This also provides a warmup for the next lecture where we will use Chinese Remainder Representation to divide integers in FOM. That presentation will follow the paper by Hesse, Allender, and Barrington in JCSS 2002.
• (5 Oct) Hmm... I have gotten a little behind, haven't I? Here's the summaries of the last several lectures:
  • Lecture 6, 26 September: The inductive case of the FO = AC⁰ theorem, and the non-uniform base case. The base case, and the equivalence of FO + BIT and FO + (PLUS, TIMES) with each other, require a more complicated argument.
  • Lecture 7, 28 September: We reviewed and finished the argument from PCMI lecture A4, that FO + BIT and FO + (PLUS, TIMES) each have the capability to define a sequence mechanism, and that this mechanism can then define the BITSUM predicate we need to verify an alleged computation of a DLOGTIME Turing machine.
  • Lecture 8, 3 October: We reviewed the group theory presented in PCMI lecture A5, with particular attention to the group S₃ and what we can compute with programs over it. At the end we proved the non-uniform case of Barrington's Theorem, that programs over S₅ are equivalent to NC¹.
  • Lecture 9, 5 October: We proved the uniform case of Barrington's Theorem, that DLOGTIME-uniform programs over S₅ are equivalent to ALOGTIME and hence to DLOGTIME-ECL-uniform NC¹. We then showed that any programs over any non-solvable finite group are also equivalent to NC¹. The complement of this result is that programs over any solvable group can be computed in ACC⁰ (actually in CC⁰, once we convince ourselves that the latter class is closed under NC⁰ operations). I gave most of a proof of this, but got hung up on a technical point (embarassing, as the result is in my Ph.D. thesis). The reason is that I was trying to do too much:

    Suppose we have a group G with a normal subgroup H, where G/H is cyclic of order m and (by the IH) we assume that iterated products in H can be computed in CC⁰. I pick an element a such that the coset aH is a generator of G/H. Then, given an iterated product in G, I write each element as a power of a times an element of H. Then, as I said, I move all the a's to the right, so each element of h in the product is conjugated by some power of a. Although a^m is an element of H, it is not necessarily the identity of G, and conjugating by it is not necessarily the identity operation on H. But a^g, where g is the order of G, is the identity of G. Thus in order to find the effect of the conjugations, I only need to count the a's to the left of a given place modulo g, and I can do this with a mod-g counting gate. Thus these gates and the gates needed to multiply in H suffice to multiply in G.

  I also began talking about the class TC⁰, languages that can be computed by circuits of O(10 depth, polynomial size, unbounded fan-in, and gates for AND, OR, and MAJORITY. I asked you for homework to show that the following problems can be solved in TC⁰:

  1. Sorting n n-bit binary numbers
  2. Multiplying two n-bit binary numbers, which depends on adding together n n-bit numbers
  3. Recognizing the Dyck language, which we can think of as strings of left and right parentheses of two types, such that the parentheses are properly balanced. A context-free grammar for this language is:

            S --> aSb
            S --> cSd
            S --> SS
            S --> λ
     

     It may be easier to start with the single-type parenthesis language, where we drop the second rule of the grammar and have strings only over {a,b}.

  4. A language over {0,1} is symmetric if permuting the letters in the string preserves membership -- that is, membership depends only on the number of 0's and the number of 1's in the string. Ignoring uniformity issues, prove that any symmetric language is in TC⁰.

  I also made my first suggestion for a final project, on the paper of Buss, Cook, Gupta, and Ramachandran (SIAM J. Comp., 1990). This paper shows that the Boolean sentence value problem is in NC¹.

• (22 Sept) In Lecture 4 on Tuesday, we proved the Ruzzo Alternation/Circuit Theorem, that ACⁱ for i≥1 is captured by ATM's with log space and logⁱn alternations, that NCⁱ for i≥1 is captured by ATM's with log space and logⁱn time, and that AC⁰ is captured by ATM's with O(log n) time and O(1) alternations. In most cases we could define the uniform circuit classes by saying that the direct connection language must be in DLOGTIME, but for NC¹ we needed to say that the extended connection language is in DLOGTIME. This material is in Vollmer.

  In yesterday's Lecture 5 we stated a number of characterizations of NC¹ and AC⁰ and made a start on proving some of them (after reviewing the relationship between these two circuit classes, DLOGSPACE, and the regular languages. We defined finite monoids and gave the algebraic characterization of the regular languages (see PCMI lecture A1). Our next task will be to prove the BIS theorem, that AC⁰ = FO + BIT = ATM (log time, O(1) alternations), from Lecture A4.

  I also assigned two easy homework problems, and I've added a third one here:

  1. Using the Immerman-Szelepcsenyi Theorem (that NL = co-NL), prove that a language is in NL iff it is the language of an ATM with O(log n) space and O(1) alternations.
  2. Prove carefully that non-uniform PSIZE is equal to P/poly, the class of languages of poly-time Turing machines that are given n^O(1) bits of advice, that depends only on the length of the input.
  3. (new) A branching program is a labeled DAG where each node has out-degree 0 or 2, nodes of out-degree 0 are labeled "accept" or "reject", nodes of out-degree 2 are labeled by input variables, and the edges out of a node of out-degree 2 are labeled "0" and "1". Given a branching program with node labels taken from {1,...,n}, and an input of n bits, there is a path from the branching program's start node to some node of out-degree 0 defined by taking the edge out of each node corresponding to the value of the node's variable. A family of branching programs, one for each value of n, thus defines a language. Define the class POLYBP to be the set of languages of branching program sizes where the number of nodes in the program for length n is n^O(1). Give a characterization of non-uniform POLYBP. Define a reasonble definition of "uniform POLYBP", and give a characterization of this class in terms of Turing machines.
• (18 Sept) In Lecture 3, last Thursday, we reviewed alternating Turing machines and the Alternation Theorem. We concentrated on the result that AL = P, because tomorrow we'll do the Ruzzo Alternation/Circuit Theorem which shows that various subclasses of P correspond to logspace ATM's with various resource restrictions.
• (13 Sept) Yesterday we discussed the 2^n1/(d-1) size upper bound for depth-d unbounded fan-in AND-OR circuits for PARITY. We then showed that P is equal to both P-uniform and L-uniform PSIZE, as in Lecture B5. Finally, we showed that L is equal to L-uniform log-width polysize circuits. This last result forced us to revise our definition of uniformity for leveled circuits, so that the level number of a gate is available as part of the circuit definition. Next time we will work through the Ruzzi Circuit-Alternation Theorem, which is mentioned in Lecture B11 and proved in Lecture 24 of my Spring 2004 CMPSCI 601 notes.
• (7 Sept) We had our first lecture, in which I covered more or less the material of PCMI lecture B1, on basic machine and circuit complexity classes. I asked you to think about two problems. First, construct constant-depth unbounded fan-in circuits (of AND, OR, and NOT gates) for the parity language, using as few gates as you can. Second, review what you know about uniformity conditions on circuit families, that will allow you to prove that the classes P and "uniform PSIZE" are equal.

Last modified 18 October 2006