Honest elementary degrees and degrees of relative provability without the cupping property

An element a of a lattice cups to an element b>ab>a if there is a c<bc<b such that a∪c=ba∪c=b. An element of a lattice has the cupping property if it cups to every element above it. We prove that there are non-zero honest elementary degrees that do not have the cupping property, which answers a question of Kristiansen, Schlage-Puchta, and Weiermann. In fact, we show that if b is a sufficiently large honest elementary degree, then b has the anti-cupping property, which means that there is an a with 0<Ea<Eb0<Ea<Eb that does not cup to b. For comparison, we also modify a result of Cai to show, in several versions of the degrees of relative provability that are closely related to the honest elementary degrees, that in fact all non-zero degrees have the anti-cupping property, not just sufficiently large degrees

Some results on PA-provably recursive functions

We provide some results which emerged from joint research carried out at CRM. The theorems are inspired by analogy with situations related to forcin

The prospects for mathematical logic in the twenty-first century

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.Comment: Association for Symbolic Logi

Towards a logical foundation of randomized computation

This dissertation investigates the relations between logic and TCS in the probabilistic setting. It is motivated by two main considerations. On the one hand, since their appearance in the 1960s-1970s, probabilistic models have become increasingly pervasive in several fast-growing areas of CS. On the other, the study and development of (deterministic) computational models has considerably benefitted from the mutual interchanges between logic and CS. Nevertheless, probabilistic computation was only marginally touched by such fruitful interactions. The goal of this thesis is precisely to (start) bring(ing) this gap, by developing logical systems corresponding to specific aspects of randomized computation and, therefore, by generalizing standard achievements to the probabilistic realm. To do so, our key ingredient is the introduction of new, measure-sensitive quantifiers associated with quantitative interpretations. The dissertation is tripartite. In the first part, we focus on the relation between logic and counting complexity classes. We show that, due to our classical counting propositional logic, it is possible to generalize to counting classes, the standard results by Cook and Meyer and Stockmeyer linking propositional logic and the polynomial hierarchy. Indeed, we show that the validity problem for counting-quantified formulae captures the corresponding level in Wagner's hierarchy. In the second part, we consider programming language theory. Type systems for randomized \lambda-calculi, also guaranteeing various forms of termination properties, were introduced in the last decades, but these are not "logically oriented" and no Curry-Howard correspondence is known for them. Following intuitions coming from counting logics, we define the first probabilistic version of the correspondence. Finally, we consider the relationship between arithmetic and computation. We present a quantitative extension of the language of arithmetic able to formalize basic results from probability theory. This language is also our starting point to define randomized bounded theories and, so, to generalize canonical results by Buss

Prolegomena to a Semantic Theory for Natural Languages Based on Recursive Artihmetic

In this dissertation, the possibility of employing a version of (primitive) recursive arith- metic to build the semantic representations of natural language sentences is explored. This idea derives from the fact that such a formal system differs under several respects from formalisms which have been traditionally employed in formal semantics, based on classical predicate logic. Specifically, in the case of recursive arithmetic, quantifiers are not primitive terms of the language, but they are defined as peculiar recursive functions; additionally, within it they cannot be defined in a way which corresponds to how they have traditionally been conceived, i.e. as “unbounded” quantifiers, whose domain is not necessarily finite. In recursive arithmetic, however, it is possible to convey something equivalent to general assertions, regarding any arbitrarily chosen individual, by using free variables; crucially, such variables do not establish relations of scope with other terms of the language, and their interpretation can to a large extent be assimilated to that of wide scope standard universal quantifiers. In the light of this, it is argued that several linguistic phenomena, attested in natural languages of different families, can be explained in an especially natural way by assuming that the lexical elements and syn- tactic structures involved are correlated with the presence of these free variables with generic value in the logical form of the sentence. In particular, generic indefinites, con- ditionals and habitual clauses are analyzed, in their interaction with the negation and, as for the first two, with quantified noun phrases; in connection with these aspects, the problem of the internal structure of negative indefinite is also addressed; finally, a pos- sible analysis of the Neg-Raising phenomenon in terms of generic variables is offered. Many of the proposals made here have already appeared in the literature, in Löbner (2000, 2013) and, moreover, in Goodstein (1951, 1957) and Hornstein (1984). Some apparent counterexamples to the theory outlined are explained by making appeal to an independently motivated treatment of embedded clauses. It is suggested that the analyzed phenomena, when collectively considered, confirm the validity of the initial project, letting one glimpse new potential scenarios for a fruitful exchange between the philosophy of mathematics and linguistic semantics

Complexities of Proof-Theoretical Reductions

The present thesis is a contribution to a project that is carried out by Michael Rathjen and Andreas Weiermann to give a general method to study the proof-complexity of Pi_2 sentences. This general method uses the generalised ordinal-analysis that was given by Buchholz, Rueede and Strahm as well as the generalised characterisation of provable-recursive functions of PA with axioms for transfinite induction that was given by Weiermann. The present thesis links these two methods by giving an explicit elementary bound, for the proof-complexity increment that occurs after the transition from the theory that was used by Rueede and Strahm, to PA with axioms for transfinite induction, which was analysed by Weiermann

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems