Formal Theories for Linear Algebra

We introduce two-sorted theories in the style of [CN10] for the complexity classes \oplusL 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.Comment: This is a revised journal version of the paper "Formal Theories for Linear Algebra" (Computer Science Logic) for the journal Logical Methods in Computer Scienc

Circuit Evaluation for Finite Semirings

The circuit evaluation problem for finite semirings is considered, where semirings are not assumed to have an additive or multiplicative identity. The following dichotomy is shown: If a finite semiring R (i) has a solvable multiplicative semigroup and (ii) does not contain a subsemiring with an additive identity 0 and a multiplicative identity 1 != 0, then its circuit evaluation problem is in the complexity class DET (which is contained in NC^2). In all other cases, the circuit evaluation problem is P-complete

On matrix rank function over bounded arithmetics

In arXiv:1811.04313, a definition of determinant is formalized in the bounded arithmetic $VNC^{2}$. Following the presentation of [Gathen, 1993], we can formalize a definition of matrix rank in the same bounded arithmetic. In this article, we define a bounded arithmetic $LAPPD$, and show that $LAPPD$ seems to be a natural arithmetic theory formalizing the treatment of rank function following Mulmuley's algorithm. Furthermore, we give a formalization of rank in $VNC^{2}$ by interpreting $LAPPD$ by $VNC^{2}$.Comment: 60 pages, section 4 added for readability, typos modified, acknowledgement revised, results unchanged, no figur

Formal Theories for Linear Algebra

We introduce two-sorted theories in the style of [CN10] for the complexity classes \oplusL 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

Formal Theories for Linear Algebra

Abstract. We introduce two-sorted theories in the style of [CN10] 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.