Questions tagged [bounded-arithmetic]
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25 questions
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Can Witnessing theorems lead to code extraction from proofs? (bounded arithmetics)
In bounded arithmetic (or bounded reverse mathematics) we study formal systems so weak that the structures of their proofs correspond to some known complexity classes such as PTIME, LOGSPACE or AC0.
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Reference request: Proof theory in $W_1^1$
Buss defined $V_2^1$ as a second-order bounded arithmetic corresponding to $\mathsf{PSPACE}$.
Later, Skelley introduced $W_1^1$, a third-order bounded arithmetic of $\mathsf{PSPACE}$.
Since the ...
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4
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Does this hierarchy of fragments of $I \Sigma_1$ collapse?
Does anyone know whether the following hierarchy of fragments of
$\mathrm{I} \Sigma_1$ (or rather
$\mathrm{I} \Pi_1$) collapses or not?
Let $\Sigma^b_n$ denote formulas in the language of arithmetic ...
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Induction on open formulas vs. Induction on $\Pi_1$ formulas
There are infinitely many extension to Robinson's $Q$ arithmetic many of which are defined by adding an axiom schema of induction for particular set of formulas.
I am confused about the theory $\text{...
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Finite axiomatizability and $\mathrm{PA^{top}}$
Is $\mathrm{PA^{top}}$ finitely axiomatizable? If not, does it have a finitely axiomatizable extension (allowing new predicates but not new variable types) that has arbitrarily large finite models?
$\...
- 8,668
6
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What is the consistency strength of this theory?
Language: first-order logic
Primitives: $=, S, \in $ (the first denotes identity, the second denotes “is a successor of”, and the third denotes membership relation).
Axioms: those of identity ...
- 13.2k
6
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1
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338
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Provability in $S^1_2$
What are some examples of natural true statements of the form $∀n φ(n)$ ($φ$ is a polynomial time computation/test) that are unprovable in $S^1_2$?
Examples may be unconditional or dependent on ...
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Does Bounded Arithmetic, $I\Delta_0$, prove the Recursion Theorem?
Compare my question Does Robinson Arithmetic already entail the Recursion Theorem. I would find it desirable and interesting if $I\Delta_0$ could be extended with a comprehension principle CP stating ...
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6
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Logical complexity of hard functions conjectures
Let $\phi_1$ and $\phi_2$ be the following statements:
$\phi_1:$ There is a function $f:\{0,1\}^*\to\{0,1\}$ computable in $E$ that has circuit complexity $2^{\Omega(n)}$.
$\phi_2:$ There is a ...
- 1,709
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Collapsing the Intuitionistic Bounded Arithmetics Hierarchy
Let $iT$ be the intuitionistic first order theory with non-logical axioms of classical first order theory $T$.
Theorem1. If $\mathsf{T^i_2}\vdash \mathsf{T_2}$, then $\mathsf{T^i_2}$ proves that the ...
- 1,709
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On subtheories of $\mathsf{T_2+EXP}$
By famous KPT theorem, if $\mathsf{T^i_2}\vdash \mathsf{T^j_2}$ for some $0<i<j$, then the polynomial hierarchy collapses. So it seems this bounded arithmetic hierarchy does not collapse. Also, ...
- 1,709
2
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Bounded Arithmetic and Counting
Let $\mathcal{L}=\{0,S,+,\cdot,=,<,X,R,S\}$ be the language of arithmetic with three additional predicate symbols $X(v)$, $R(v,u)$ and $S(v,u)$.
Let $\phi(x),\psi(x,y)$ and $\eta(x,y)$ be formulas ...
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2
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$P=NP$ and provability of family of propositional formulas
Let $\mathcal{L}'=\{+,\cdot,0,S,=,<,||,\#,R\}$ be the lanugage of bounded arithmetic with a $k$-ary relation $R$.
For every bounded sentence $\phi({\bf\bar{n}})$ in $\mathcal{L}'$ define ...
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Collapsing the Linear Time Hierarchy and finite axiomatizability of bounded arithmetic
It is well known that if ${\bf T_2}$ (or $I\Delta_0+\Omega_1$) is finitely axiomatizable, then the Polynomial Hierarchy collapses.
Q. Is there any similar relation between $I\Delta_0$ and Linear ...
- 1,709
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Equational theory for resolution proof system
Is there any equational theory $T$ like $PV$ with following properties:
If $T\vdash f=g$ for terms $f$ and $g$, translation of $f=g$ to propositional formulas has polynomial resolution proof.(like $PV$...
- 1,709
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Bounded Arithmetic vs Complexity Theory
In this post, when I talk about bounded arithmetic theories,
I mean the theories of arithmetic according to "Logical Foundations of Proof Complexity", which capture the complexity classes between $AC^...
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Weak Bounded Arithmetics
Let $\Sigma^b_i$ and $\Pi^b_i$ formulas be bounded formulas defined by Buss in language of $L_b$. $PIND(\phi(x))$ is the formula:
$$\phi(0)\land \forall x(\phi(\left \lfloor \frac{x}{2} \right \...
- 1,709
12
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Intutionistic Robinson Arithmetic
By Friedman translation $HA$ and $PA$ prove the same $\Pi_2$ formulas.
Is it true for Intutionistic Robinson arithmetic(Robinson axioms with intutionistic logic) and classic Robinson arithmetic?
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- 1,709
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models of $I\exists^+_1$
$\phi$ is a $\exists^+_1$ formula iff it is in language of arithmetic and does not have $\forall$,$\neg$ and $\rightarrow$, therefore $I\exists^+_1$ is theory of $Q$+induction axioms for $\exists^+_1$ ...
- 1,709
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Does every model of $I\Delta_0$ has an end extension to a model of $I\Delta_0+\Omega_1$?
Does every model of $I\Delta_0$ has an end extension to a model of $I\Delta_0+\Omega_1$?
In End extensions of models of linearly bounded arithmetic paper the author said this problem is open. I want ...
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Kripke models of $HA$
Let $K$ be a kripke model and $k$ be one of its node, then $\mathcal{M}_k$ is classical structure of $k$.
What is the strongest theory of arithmetic like $T$ such that for
every kripke model $K\...
- 1,709
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3
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434
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End Extension models of $I\Delta_0$
Recently I'm thinking about question below, but I can not prove or disprove it.
Is it true that for every model $M\models I\Delta_0$ there exists a
model $M'\models PA$ such that $M'$ is end ...
- 1,709
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3
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Model-theoretic accounts of feasibility in bounded arithmetic and related systems
Various weak theories of arithmetic have been partially motivated by a concern with numbers (or functions/proofs) that are feasible. This concern is sometimes connected to an interest in strictly ...
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775
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Unboundedness of primes in bounded arithmetic
Wilkie's well known question asks whether $I\Delta_{0}$ proves the unboundedness of primes. We know that by adding a sentence to $I\Delta_{0}$ which says "the exponential function is total", it is ...
- 1,413
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Is there exponentiation in "sufficiently large" models of $I\Delta_{0}$?
Let $L_{E}$ be the language of discretely ordered rings together with an extra predicate symbol $E$. The system $A$ consists of the axioms of $I\Delta_{0}$ (basic arithmetic plus induction for bounded ...
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