Formal Theories for Linear Algebra

Abstract

Abstract. We introduce two-sorted theories in the style of Cook and Nguyen for the complexity classes ⊕L and DET , whose complete problems include determinants over Z2 and Z, respectively. We then describe interpretations of Soltys' linear algebra theory LAp over arbitrary integral domains, into each of our new theories. The result shows equivalences of standard theorems of linear algebra over Z2 and Z can be proved in the corresponding theory, but leaves open the interesting question of whether the theorems themselves can be proved. IntroductionThis paper is a contribution to bounded reverse mathematics [Ngu08,CN10], that part of proof complexity concerned with determining the computational complexity of concepts needed to prove theorems of interest in computer science. We are specifically interested in theorems of linear algebra over finite fields and the integers. The relevant complexity classes for each case have been well-studied in the computational complexity literature. The classes are ⊕L and DET , associated with linear algebra over Z 2 and Z, respectively. We introduce formal theories V ⊕L and V #L for ⊕L and DET , each intended to capture reasoning in the corresponding class. Each theory allows induction over any relation in the associated complexity class, and the functions definable in each theory are exactly the functions in the class. In particular determinants and coefficients of the characteristic polynomial of a matrix can be defined.To study the question of which results from linear algebra can be proved in the theories we take advantage of Soltys's theory LAp [SK01, SC04] for formalizing linear algebra over an arbitrary field or integral domain. We present two interpretations of LAp: one into V ⊕L and one into V #L. Both interpretations translate theorems of LAp to theorems in the corresponding theory, but the meaning of the theorems differs in the two translations since the ring elements range over Z 2 in one and over Z in the other. From these interpretations and results in [SK01, SC04] we show that the theories prove some interesting properties of determinants, but leave open the question of whether the proofs of some basic theorems 1998 ACM Subject Classification: F.4.0.

Uniform, integral and efficient proofs for the determinant identities

“…Cook and Fontes [CF12] developed a bounded arithmetic theory V #L, corresponding to DET, where DET is the class of functions that can be computed by uniform families of polynomial-size constant-depth Boolean circuits with oracle access to the determinant over Z (where integer entries of matrices are presented in binary). In other words, DET is the AC 0 -closure of integer determinants.…”

“…In fact, within the NC := ∪ ∞ i=0 NC i hierarchy, which consists of all polynomial-size circuit families of poly-logarithmic depth, NC 2 is the weakest level known to compute the determinant (formally, the weakest circuit class computing integer determinants is the class DET that lies between NC 1 and NC 2 ; see below). Furthermore, the importance of linear algebra in bounded arithmetic and proof complexity has been identified in many works, and it has been conjectured that the determinant identities, and specifically the multiplicativity of the determinant function DET(A) · DET(B) = DET(AB), for two matrices A, B, can be proved in a formal theory that, loosely speaking, reasons with NC 2 concepts (Cook and Nguyen present this specific question in their monograph [CN10]; see also [CF12,BBP95,BP98,Sol01,SC04]). This conjecture is aligned with the intuition that basic properties of many constructions and functions of a given complexity class are provable in logical theories not using concepts beyond that class.…”

“…The weakest theory known to date to prove the determinant identities is PV which corresponds to polynomial-time reasoning; this was shown by Soltys and Cook [SC04] (cf. [CF12,Jeř05]). Quite recently, Hrubeš and Tzameret [HT15] showed that at least in the propositional case, the determinant identities expressing the multiplicativity of the determinant over GF (2) can be proved with polynomial-size propositional proofs operating with NC 2 -circuits (as well as with quasipolynomial size Frege proofs).…”

Formalizing Randomized Matching Algorithms

“…Since we need some basic linear algebra in this paper, we are interested in the twosorted theory V #L and its universal conservative extension V #L from [6]. Recall that #L is usually defined as the class of functions f such that for some nondeterministic logspace Turing machine M , f (x) is the number of accepting computations of M on input x.…”

“…Observe that VPV extends V #L since matrix powering can easily be carried out in polytime, and thus all theorems of V #L from [6,29] are also theorems of VPV. From results in [29] (see page 44 of [14] for a correction) it follows that VPV proves the CayleyHamilton Theorem, and hence the cofactor expansion formula and other usual properties of determinants of integer matrices.…”