Effectiveness in RPL, with Applications to Continuous Logic

In this paper, we introduce a foundation for computable model theory of rational Pavelka logic (an extension of {\L}ukasiewicz logic) and continuous logic, and prove effective versions of some theorems in model theory. We show how to reduce continuous logic to rational Pavelka logic. We also define notions of computability and decidability of a model for logics with computable, but uncountable, set of truth values; show that provability degree of a formula w.r.t. a linear theory is computable, and use this to carry out an effective Henkin construction. Therefore, for any effectively given consistent linear theory in continuous logic, we effectively produce its decidable model. This is the best possible, since we show that the computable model theory of continuous logic is an extension of computable model theory of classical logic. We conclude with noting that the unique separable model of a separably categorical and computably axiomatizable theory (such as that of a probability space or an $L^p$ Banach lattice) is decidable

Uniformity and Nonuniformity in Proof Complexity

This thesis is dedicated to the study of the relations between uniform and nonuniform proof complexity and computational complexity. Nonuniform proof complexity studies the lengths of proofs in various propositional proof systems such as Frege . Uniform proof complexity studies the provability strength of bounded arithmetic theories which use only concepts computable in specific computational complexity classes, e.g. the two-sorted bounded arithmetic theory VNC1 uses only concepts computable in NC1. We are interested in transferring concepts, tools, and results from computational complexity to proof complexity. We introduce the notion of proof complexity class which corresponds to the notion of computational complexity class. We show the possibility of developing a systematic framework for studying proof complexity classes associated with computational complexity classes. The framework is based on soundness statements for proof complexity classes and evaluation problems for circuit complexity classes. The soundness statements are universal for proof complexity classes as circuit evaluation problems are complete for computational complexity classes. We introduce the notion of io-typed theories to design theories corresponding to computational complexity classes which are not closed under composition. We use io-types to control the composition of provably total functions of theories. We design a new class of theories n^ε-ioV∞ (εPh.D

Theories for Subexponential-size Bounded-depth Frege Proofs ∗

This paper is a contribution to our understanding of the relationship between uniform and nonuniform proof complexity. The latter studies the lengths of proofs in various propositional proof systems such as Frege and bounded-depth Frege systems, and the former studies the strength of the corresponding logical theories such as VNC 1 and V 0 in [7]. A superpolynomial lower bound on the length of proofs in a propositional proof system for a family of tautologies expressing a result like the pigeonhole principle implies that the result is not provable in the theory associated with the propositional proof system. We define a new class of bounded arithmetic theories nε-ioV ∞ for ε < 1 and show that they correspond to complexity classes AltTime(O(1), O(nε)), uniform classes of subexponential-size bounded-depth circuits DepthSize(O(1), 2O(nε)). To accomplish this we introduce the novel idea of using types to control the amount of composition in our bounded arithmetic theories. This allows our theories to capture complexity classes that have weaker closure properties and are not closed under composition. We show that the proofs of ΣB 0-theorems in our theories translate to subexponential-size bounded-depth Frege proofs

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status: publishe