Annals of Pure and Applied Logic最新文献

We provide a general and syntactically defined family of sequent calculi, called semi-analytic, to formalize the informal notion of a “nice” sequent calculus. We show that any sufficiently strong (multimodal) substructural logic with a semi-analytic sequent calculus enjoys the Craig Interpolation Property, CIP. As a positive application, our theorem provides a uniform and modular method to prove the CIP for several multimodal substructural logics, including many fragments and variants of linear logic. More interestingly, on the negative side, it employs the lack of the CIP in almost all substructural, superintuitionistic and modal logics to provide a formal proof for the well-known intuition that almost all logics do not have a “nice” sequent calculus. More precisely, we show that many substructural logics including $U{L}^{-}$, $\mathrm{MTL}$, $R$, Ł_n (for $n⩾3$), G_n (for $n⩾4$), and almost all extensions of $\mathrm{IMTL}$, $Ł$, $\mathrm{BL}$, $R{M}^e$, $\mathrm{IPC}$, $\mathrm{S4}$, and $\mathrm{Grz}$ (except for at most 1, 1, 3, 8, 7, 37, and 6 of them, respectively) do not have a semi-analytic calculus.

We classify intermediate logics according to their unification types. There are exactly two minimal intermediate logics with hereditary finitary unification: the least logic with hereditary unitary unification and the least logic with hereditary projective proximity (a notion close to projective approximation of Ghilardi [17], [18]), see Figure 4. They are locally tabular and are union splittings in the lattice Ext INT. There are exactly four maximal intermediate logics with nullary unification (see Figure 21) and they are tabular. Any intermediate logic with neither hereditary unitary unification nor with hereditary projective proximity is included in one of the four logics. There are logics with finitary/unitary (but not hereditary finitary) unification scattered among the majority of those with nullary unification, see Figure 23. Our main tools are the characterization of locally tabular logics with finitary (or unitary) unification, by their Kripke models [12], [13] and splittings.

We study a version of the Vitali covering theorem, which we call $\text{WHBU}$ and which is a direct weakening of the Heine-Borel theorem for uncountable coverings, called $\text{HBU}$. We show that $\text{WHBU}$ is central to measure theory by deriving it from various central approximation results related to Littlewood's three principles. A natural question is then how hard it is to prove $\text{WHBU}$ (in the sense of Kohlenbach's higher-order Reverse Mathematics), and how hard it is to compute the objects claimed to exist by $\text{WHBU}$ (in the sense of Kleene's computation schemes S1-S9). The answer to both questions is ‘extremely hard’, as follows: on one hand, in terms of the usual scale of (conventional) comprehension axioms, $\text{WHBU}$ is only provable using Kleene's ${\exists }^3$, which implies full second-order arithmetic. On the other hand, realisers (aka witnessing functionals) for $\text{WHBU}$, so-called Λ-functionals, are computable from Kleene's ${\exists }^3$, but not from weaker comprehension functionals. Despite this hardness, we show that $\text{WHBU}$, and certain Λ-functionals, behave much better than $\text{HBU}$ and the associated class of realisers, called Θ-functionals. In particular, we identify a specific Λ-functional called ${\Lambda }_{\text{S}}$ which adds no computational power to the Suslin functional, in contrast to Θ-functionals. Finally, we introduce a hierarchy involving Θ-functionals and $\text{HBU}$.

Here we give a complete list of the groups definable in Presburger arithmetic up to a finite index subgroup.

Given a weakly compact cardinal κ, we give an axiomatization of intuitionistic first-order logic over $L_{{\kappa }^{+},\kappa }$ and prove it is sound and complete with respect to Kripke models. As a consequence we get the disjunction and existence properties for that logic. This generalizes the work of Nadel in [8] for intuitionistic logic over $L_{{\omega }_1,\omega }$. When κ is a regular cardinal such that ${\kappa }^{<\kappa }=\kappa$, we deduce, by an easy modification of the proof, a complete axiomatization of intuitionistic first-order logic over $L_{{\kappa }^{+},\kappa ,\kappa }$, the language with disjunctions of at most κ formulas, conjunctions of less than κ formulas and quantification on less than κ many variables. In particular, this applies to any regular cardinal under the Generalized Continuum Hypothesis.

Let $\mu <\kappa <\lambda$ be three infinite cardinals, the first two being regular. We show that if there is no inner model with large cardinals, $u(\kappa ,\lambda )$ is regular, where $u(\kappa ,\lambda )$ denotes the least size of a cofinal subset in $\left(P_{\kappa }\right(\lambda ),\subseteq )$, and $\mathrm{cf}\left(\lambda \right)\ne \mu$, then (a) the μ-club filters on $P_{\kappa }\left(\lambda \right)$ and $P_{\kappa }\left(u\right(\kappa ,\lambda \left)\right)$ are isomorphic, and (b) the ideal dual to the μ-club filter on $P_{\kappa }\left(\lambda \right)$ (and hence the restriction of the nonstationary ideal on $P_{\kappa }\left(\lambda \right)$ to sets of uniform cofinality μ) is not $I_{\kappa ,\lambda }$-$b_{u(\kappa ,\lambda )}$-saturated.

Semi-algebraic proof systems such as sum-of-squares ($\mathrm{SoS}$) have attracted a lot of attention due to their relation to approximation algorithms: constant degree semi-algebraic proofs lead to conjecturally optimal polynomial-time approximation algorithms for important $\mathrm{NP}$-hard optimization problems. Motivated by the need to allow a more streamlined and uniform framework for working with $\mathrm{SoS}$ proofs than the restrictive propositional level, we initiate a systematic first-order logical investigation into the kinds of reasoning possible in algebraic and semi-algebraic proof systems. Specifically, we develop first-order theories that capture in a precise manner constant degree algebraic and semi-algebraic proof systems: every statement of a certain form that is provable in our theories translates into a family of constant degree polynomial calculus or $\mathrm{SoS}$ refutations, respectively; and using a reflection principle, the converse also holds.

This places algebraic and semi-algebraic proof systems in the established framework of bounded arithmetic, while providing theories corresponding to systems that vary quite substantially from the usual propositional-logic ones.

We give examples of how our semi-algebraic theory proves statements such as the pigeonhole principle, we provide a separation between algebraic and semi-algebraic theories, and we describe initial attempts to go beyond these theories by introducing extensions that use the inequality symbol, identifying along the way which extensions lead outside the scope of constant degree $\mathrm{SoS}$. Moreover, we prove new results for propositional proofs, and specifically extend Berkholz's dynamic-by-static simulation of polynomial calculus (PC) by $\mathrm{SoS}$ to PC with the radical rule.

Inspired by Owings's problem, we investigate whether, for a given an Abelian group G and cardinal numbers $\kappa ,\theta$, every colouring $c:G⟶\theta$ yields a subset $X\subseteq G$ with $\left|X\right|=\kappa$ such that $X+X$ is monochromatic. (Owings's problem asks this for $G=Z$, $\theta =2$ and $\kappa ={\aleph }_0$; this is known to be false for the same G and κ but $\theta =3$.) We completely settle the question for κ and θ both finite (by obtaining sufficient and necessary conditions for a positive answer) and for κ and θ both infinite (with a negative answer). Also, in the case where θ is infinite but κ is finite, we obtain some sufficient conditions for a negative answer as well as an example with a positive answer.

We prove that the standard computable presentation of the space $C[0,1]$ of continuous real-valued functions on the unit interval is computably and punctually (primitively recursively) universal. From the perspective of modern computability theory, this settles a problem raised by Sierpiński in the 1940s.

We prove that the original Urysohn's construction of the universal separable Polish space $U$ is punctually universal. We also show that effectively compact, punctual Stone spaces are punctually homeomorphically embeddable into Cantor space $2^{\omega }$; note that we do not require effective compactness be primitive recursive. We also prove that effective compactness cannot be dropped from the premises by constructing a counterexample.

As consequence of the VC theorem, any pseudo-finite measure over an NIP ultraproduct is generically stable. We demonstrate a converse of this theorem and prove that any finitely approximable measure over an ultraproduct is itself pseudo-finite (even without the NIP assumption). We also analyze the connection between the Morley product and the pseudo-finite product. In particular, we show that if μ is definable and both μ and ν are pseudo-finite, then the Morley product of μ and ν agrees with the pseudo-finite product of μ and ν. Using this observation, we construct generically stable idempotent measures on pseudo-finite NIP groups.