On proofs about threshold circuits and counting hierarchies

Johannsen, Jan; Pollett, Chris

doi:10.1109/lics.1998.705678

Cited by 11 publications

(30 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [JP98], Johannsen and Pollett introduce a hierarchy {C 0 k } k≥1 of first-order theories, where C 0 k characterizes the class of functions computable by families of constant-depth threshold circuits of size bounded by τ k (n), where τ 1 (n) = O(n), τ k+1 (n) = 2 τ k (log n) . In particular, C 0 2 captures TC 0 .…”

Section: Theories For Tc 0 and Other Small Complexity Classesmentioning

confidence: 99%

See 1 more Smart Citation

Theories for TC0 and Other Small Complexity Classes

Nguyen¹,

Cook²

2006

Logical Methods in Computer Science

View full text Add to dashboard Cite

Abstract. We present a general method for introducing finitely axiomatizable "minimal" two-sorted theories for various subclasses of P (problems solvable in polynomial time). The two sorts are natural numbers and finite sets of natural numbers. The latter are essentially the finite binary strings, which provide a natural domain for defining the functions and sets in small complexity classes. We concentrate on the complexity class TC 0 , whose problems are defined by uniform polynomial-size families of bounded-depth Boolean circuits with majority gates. We present an elegant theory VTC 0 in which the provably-total functions are those associated with TC 0 , and then prove that VTC 0 is "isomorphic" to a differentlooking single-sorted theory introduced by Johannsen and Pollet. The most technical part of the isomorphism proof is defining binary number multiplication in terms a bit-counting function, and showing how to formalize the proofs of its algebraic properties.

show abstract

Section: Theories For Tc 0 and Other Small Complexity Classesmentioning

confidence: 99%

“…We also thank Christ Pollett for clarifying the proofs in [JP98], and Alan Skelley for helpful comments.…”

Section: Acknowledgmentmentioning

confidence: 99%

Theories for TC0 and Other Small Complexity Classes

Nguyen¹,

Cook²

2006

Logical Methods in Computer Science

View full text Add to dashboard Cite

show abstract

“…In [18], Johannsen and Pollett introduce a hierarchy ¼ ½ of first-order theories, where ¼ characterizes the class of functions computable by families of constant-depth threshold circuits of size bounded by ´Òµ, where ½´Ò µ Ç´Òµ ·½´Ò µ ¾ ´ÐÓ Òµ . In particular, ¼ ¾ captures Ì ¼ .…”

Section: First-order Theories For ì ¼mentioning

confidence: 99%

“…In [18], the first-order theories ¼ ·½ ( ½) have been shown RSUV isomorphic to the second-order theories ¼ . Thus, ¼ ½ can be seen as a theory for Ì ¼ .…”

Section: Second-order Theories For ì ¼mentioning

confidence: 99%

VTC/sup O/: a second-order theory for TC/sup 0/

Nguyen¹,

Cook²

2004

Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004.

View full text Add to dashboard Cite

We introduce a finitely axiomatizable second-order theory ÎÌ ¼ , which is associated with the class Ç-uniform Ì ¼ . It consists of the base theory Î ¼ for ¼ reasoning together with the axiom AEÍÅÇAE Ë, which states the existence of a "counting array" for any string : the th row of contains only the number of 1 bits up to (excluding) bit of . We introduce the notion of "strong ¡ ½ -definability" for relations in a theory, and use a recursive characterization of the Ì ¼ relations (rather than functions) to show that the Ì ¼ relations are strongly ¡ ½ -definable. It follows that the Ì ¼ functions are ¦ ½ -definable in ÎÌ ¼ .We prove a general witnessing theorem for second-order theories and conclude that the ¦ ½ theorems of ÎÌ ¼ are witnessed by Ì ¼ functions. We prove that ÎÌ ¼ is RSUV isomorphic to the first order theory ¡ ½ -Ê of Johannsen and Pollett (the "minimal theory for Ì ¼ "). ¡ ½ -Ê includes the ¡ ½ comprehension rule, and J and P ask whether there is an upper bound to the nesting depth required for this rule. We answer "yes", because ÎÌ ¼ , and therefore ¡ ½ -Ê, are finitely axiomatizable. Finally, we show that ¦ ¼ theorems of ÎÌ ¼ translate to families of tautologies which have polynomial-size constant-depth Ì ¼ -Frege proofs. We also show that ÈÀÈ is a ¦ ¼ theorem of ÎÌ ¼ . These together imply that the family of propositional tautologies associated with ÈÀÈ has polynomialsize constant-depth Ì ¼ -Frege proofs.

show abstract

“…More precisely, TCA 2 is ∀∃Σ 1,b ∞ -conservative over TCA. In turn, TCA is a reformulation of the theory D 0 2 of J. Johannsen and C. Pollett (see [7]). …”

Section: Introductionmentioning

confidence: 99%

Counting as integration in feasible analysis

Ferreira

2006

Mathematical Logic Qtrly

View full text Add to dashboard Cite

Suppose that it is possible to integrate real functions over a weak base theory related to polynomial time computability. Does it follow that we can count? The answer seems to be: obviously yes! We try to convince the reader that the severe restrictions on induction in feasible theories preclude a straightforward answer. Nevertheless, a more sophisticated reflection does indeed show that the answer is affirmative.

show abstract

On proofs about threshold circuits and counting hierarchies

Cited by 11 publications

References 15 publications

Theories for TC0 and Other Small Complexity Classes

Theories for TC0 and Other Small Complexity Classes

VTC/sup O/: a second-order theory for TC/sup 0/

Counting as integration in feasible analysis

Contact Info

Product

Resources

About