Related papers: RCF2: Evaluation and Consistency
This article discusses completeness of Boolean Algebra as First Order Theory in Goedel's meaning. If Theory is complete then any possible transformation is equivalent to some transformation using axioms, predicates etc. defined for this…
Incremental gradient and incremental proximal methods are a fundamental class of optimization algorithms used for solving finite sum problems, broadly studied in the literature. Yet, without strong convexity, their convergence guarantees…
In this paper we develop cyclic proof systems for the problem of inclusion between the least sets of models of mutually recursive predicates, when the ground constraints in the inductive definitions belong to the quantifier-free fragments…
Termination property of functions is an important issue in computability theory. In this paper, we show that repeated iterations of a function can induce an order amongst the elements of its domain set. Hasse diagram of the poset, thus…
Principal Component Analysis (PCA) has been widely used for dimensionality reduction and feature extraction. Robust PCA (RPCA), under different robust distance metrics, such as l1-norm and l2, p-norm, can deal with noise or outliers to some…
The reflection principle is the statement that if a sentence is provable then it is true. Reflection principles have been studied for first-order theories, but they also play an important role in propositional proof complexity. In this…
The complexity class $NP$ can be logically characterized both through existential second order logic $SO\exists$, as proven by Fagin, and through simulating a Turing machine via the satisfiability problem of propositional logic SAT, as…
An efficient entailment proof system is essential to compositional verification using separation logic. Unfortunately, existing decision procedures are either inexpressive or inefficient. For example, Smallfoot is an efficient procedure but…
Optimality Theory is a constraint-based theory of phonology which allows constraints to be violated. Consequently, implementing the theory presents problems for declarative constraint-based processing frameworks. On the basis of two…
We design a proof system for propositional classical logic that integrates two languages for Boolean functions: standard conjunction-disjunction-negation and binary decision trees. We give two reasons to do so. The first is…
Ordinal analysis induces a partition of $\Sigma^1_1$-definable and $\Pi^1_1$-sound theories whereby two theories are equivalent if they have the same proof-theoretic ordinal. We show that no equivalence relation $\equiv$ is finer than the…
Robust perception and reasoning require consistency across sensory modalities. Yet current multimodal models often violate this principle, yielding contradictory predictions for visual and textual representations of the same concept. Rather…
Stochastic compositional optimization arises in many important machine learning tasks such as value function evaluation in reinforcement learning and portfolio management. The objective function is the composition of two expectations of…
We investigate machine models similar to Turing machines that are augmented by the operations of a first-order structure $\mathcal{R}$, and we show that under weak conditions on $\mathcal{R}$, the complexity class $\text{NP}(\mathcal{R})$…
We prove the following two results. Theorem A: Let alpha be a limit ordinal. Suppose that 2^{|alpha|}<aleph_alpha and 2^{|alpha|^+}<aleph_{|alpha|^+}, whereas aleph_alpha^{|alpha|}>aleph_{|alpha|^+}. Then for all n< omega and for all…
We investigate infinitary wellfounded systems for linear logic with fixed points, with transfinite branching rules indexed by some closure ordinal $\alpha$ for fixed points. Our main result is that provability in the system for some…
We define a notion of model for the $\lambda$$\Pi$-calculus modulo theory and prove a soundness theorem. We then define a notion of super-consistency and prove that proof reduction terminates in the $\lambda$$\Pi$-calculus modulo any…
We present a logically principled foundation for systematizing, in a way that works with any computational effect and evaluation order, SMT constraint generation seen in refinement type systems for functional programming languages. By…
We study the recursion-theoretic complexity of Positive Almost-Sure Termination ($\mathsf{PAST}$) in an imperative programming language with rational variables, bounded nondeterministic choice, and discrete probabilistic choice. A program…
We study stochastic optimization from a joint continuous-discrete point of view. Starting from a second-order stochastic differential equation interpreted as a noisy accelerated gradient flow, we discretize the dynamics by a fully implicit…