On the relative strengths of fragments of collection

Let $\mathbf{M}$ be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, $\Delta_0$-separation and set foundation. This paper studies the relative strength of set theories obtained by adding fragments of the set-theoretic collection scheme to $\mathbf{M}$. We focus on two common parameterisations of collection: $\Pi_n$-collection, which is the usual collection scheme restricted to $\Pi_n$-formulae, and strong $\Pi_n$-collection, which is equivalent to $\Pi_n$-collection plus $\Sigma_{n+1}$-separation. The main result of this paper shows that for all $n \geq 1$, (1) $\mathbf{M}+\Pi_{n+1}\textrm{-collection}+\Sigma_{n+2}\textrm{-induction on } \omega$ proves the consistency of Zermelo Set Theory plus $\Pi_{n}$-collection, (2) the theory $\mathbf{M}+\Pi_{n+1}\textrm{-collection}$ is $\Pi_{n+3}$-conservative over the theory $\mathbf{M}+\textrm{strong }\Pi_n \textrm{-collection}$. It is also shown that (2) holds for $n=0$ when the Axiom of Choice is included in the base theory. The final section indicates how the proofs of (1) and (2) can be modified to obtain analogues of these results for theories obtained by adding fragments of collection to a base theory (Kripke-Platek Set Theory with Infinity and $V=L$) that does not include the powerset axiom.Comment: 22 page

Dependence Logic with Generalized Quantifiers: Axiomatizations

We prove two completeness results, one for the extension of dependence logic by a monotone generalized quantifier Q with weak interpretation, weak in the meaning that the interpretation of Q varies with the structures. The second result considers the extension of dependence logic where Q is interpreted as "there exists uncountable many." Both of the axiomatizations are shown to be sound and complete for FO(Q) consequences.Comment: 17 page

Capturing Hiproofs in HOL Light

Hierarchical proof trees (hiproofs for short) add structure to ordinary proof trees, by allowing portions of trees to be hierarchically nested. The additional structure can be used to abstract away from details, or to label particular portions to explain their purpose. In this paper we present two complementary methods for capturing hiproofs in HOL Light, along with a tool to produce web-based visualisations. The first method uses tactic recording, by modifying tactics to record their arguments and construct a hierarchical tree; this allows a tactic proof script to be modified. The second method uses proof recording, which extends the HOL Light kernel to record hierachical proof trees alongside theorems. This method is less invasive, but requires care to manage the size of the recorded objects. We have implemented both methods, resulting in two systems: Tactician and HipCam

Formalizing Mathematical Knowledge as a Biform Theory Graph: A Case Study

A biform theory is a combination of an axiomatic theory and an algorithmic theory that supports the integration of reasoning and computation. These are ideal for formalizing algorithms that manipulate mathematical expressions. A theory graph is a network of theories connected by meaning-preserving theory morphisms that map the formulas of one theory to the formulas of another theory. Theory graphs are in turn well suited for formalizing mathematical knowledge at the most convenient level of abstraction using the most convenient vocabulary. We are interested in the problem of whether a body of mathematical knowledge can be effectively formalized as a theory graph of biform theories. As a test case, we look at the graph of theories encoding natural number arithmetic. We used two different formalisms to do this, which we describe and compare. The first is realized in ${\rm CTT}_{\rm uqe}$, a version of Church's type theory with quotation and evaluation, and the second is realized in Agda, a dependently typed programming language.Comment: 43 pages; published without appendices in: H. Geuvers et al., eds, Intelligent Computer Mathematics (CICM 2017), Lecture Notes in Computer Science, Vol. 10383, pp. 9-24, Springer, 201

Rich Situated Attitudes

We outline a novel theory of natural language meaning, Rich Situated Semantics [RSS], on which the content of sentential utterances is semantically rich and informationally situated. In virtue of its situatedness, an utterance’s rich situated content varies with the informational situation of the cognitive agent interpreting the utterance. In virtue of its richness, this content contains information beyond the utterance’s lexically encoded information. The agent-dependence of rich situated content solves a number of problems in semantics and the philosophy of language (cf. [14, 20, 25]). In particular, since RSS varies the granularity of utterance contents with the interpreting agent’s informational situation, it solves the problem of finding suitably fine- or coarse-grained objects for the content of propositional attitudes. In virtue of this variation, a layman will reason with more propositions than an expert

Validation of Ultrahigh Dependability for Software-Based Systems

Modern society depends on computers for a number of critical tasks in which failure can have very high costs. As a consequence, high levels of dependability (reliability, safety, etc.) are required from such computers, including their software. Whenever a quantitative approach to risk is adopted, these requirements must be stated in quantitative terms, and a rigorous demonstration of their being attained is necessary. For software used in the most critical roles, such demonstrations are not usually supplied. The fact is that the dependability requirements often lie near the limit of the current state of the art, or beyond, in terms not only of the ability to satisfy them, but also, and more often, of the ability to demonstrate that they are satisfied in the individual operational products (validation). We discuss reasons why such demonstrations cannot usually be provided with the means available: reliability growth models, testing with stable reliability, structural dependability modelling, as well as more informal arguments based on good engineering practice. We state some rigorous arguments about the limits of what can be validated with each of such means. Combining evidence from these different sources would seem to raise the levels that can be validated; yet this improvement is not such as to solve the problem. It appears that engineering practice must take into account the fact that no solution exists, at present, for the validation of ultra-high dependability in systems relying on complex software