Tarski's influence on computer science

The influence of Alfred Tarski on computer science was indirect but significant in a number of directions and was in certain respects fundamental. Here surveyed is the work of Tarski on the decision procedure for algebra and geometry, the method of elimination of quantifiers, the semantics of formal languages, modeltheoretic preservation theorems, and algebraic logic; various connections of each with computer science are taken up

Dependent choice, properness, and generic absoluteness

We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to -preserving symmetric submodels of forcing extensions. Hence, not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in and. Our results confirm as a natural foundation for a significant portion of classical mathematics and provide support to the idea of this theory being also a natural foundation for a large part of set theory

Computation on abstract data types. The extensional approach, with an application to streams

AbstractIn this paper we specialize the notion of abstract computational procedure previously introduced for intensionally presented structures to those which are extensionally given. This is provided by a form of generalized recursion theory which uses schemata for explicit definition, conditional definition and least fixed point (LFP) recursion in functional of type level ⩽ 2 over any appropriate structure. It is applied here to the case of potentially infinite (and more general partial) streams as an abstract data type

Turing jumps through provability

Fixing some computably enumerable theory $T$, the Friedman-Goldfarb-Harrington (FGH) theorem says that over elementary arithmetic, each $\Sigma_1$ formula is equivalent to some formula of the form $\Box_T \varphi$ provided that $T$ is consistent. In this paper we give various generalizations of the FGH theorem. In particular, for $n>1$ we relate $\Sigma_{n}$ formulas to provability statements $[n]_T^{\sf True}\varphi$ which are a formalization of "provable in $T$ together with all true $\Sigma_{n+1}$ sentences". As a corollary we conclude that each $[n]_T^{\sf True}$ is $\Sigma_{n+1}$-complete. This observation yields us to consider a recursively defined hierarchy of provability predicates $[n+1]^\Box_T$ which look a lot like $[n+1]_T^{\sf True}$ except that where $[n+1]_T^{\sf True}$ calls upon the oracle of all true $\Sigma_{n+2}$ sentences, the $[n+1]^\Box_T$ recursively calls upon the oracle of all true sentences of the form $\langle n \rangle_T^\Box\phi$. As such we obtain a `syntax-light' characterization of $\Sigma_{n+1}$ definability whence of Turing jumps which is readily extended beyond the finite. Moreover, we observe that the corresponding provability predicates $[n+1]_T^\Box$ are well behaved in that together they provide a sound interpretation of the polymodal provability logic ${\sf GLP}_\omega$

Logics of Finite Hankel Rank

We discuss the Feferman-Vaught Theorem in the setting of abstract model theory for finite structures. We look at sum-like and product-like binary operations on finite structures and their Hankel matrices. We show the connection between Hankel matrices and the Feferman-Vaught Theorem. The largest logic known to satisfy a Feferman-Vaught Theorem for product-like operations is CFOL, first order logic with modular counting quantifiers. For sum-like operations it is CMSOL, the corresponding monadic second order logic. We discuss whether there are maximal logics satisfying Feferman-Vaught Theorems for finite structures.Comment: Appeared in YuriFest 2015, held in honor of Yuri Gurevich's 75th birthday. The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-23534-9_1

UNFOLDING FINITIST ARITHMETIC

The concept of the (full) unfolding \user1{{\cal U}}(S) of a schematic system $S$ is used to answer the following question: Which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted $S$? The program to determine \user1{{\cal U}}(S) for various systems $S$ of foundational significance was previously carried out for a system of nonfinitist arithmetic, $NFA$; it was shown that \user1{{\cal U}}(NFA) is proof-theoretically equivalent to predicative analysis. In the present paper we work out the unfolding notions for a basic schematic system of finitist arithmetic, $FA$, and for an extension of that by a form $BR$ of the so-called Bar Rule. It is shown that \user1{{\cal U}}(FA) and \user1{{\cal U}}(FA + BR) are proof-theoretically equivalent, respectively, to Primitive Recursive Arithmetic, $PRA$, and to Peano Arithmetic, $PA

Arithmetical conservation results

In this paper we present a proof of Goodman's Theorem, a classical result in the metamathematics of constructivism, which states that the addition of the axiom of choice to Heyting arithmetic in finite types does not increase the collection of provable arithmetical sentences. Our proof relies on several ideas from earlier proofs by other authors, but adds some new ones as well. In particular, we show how a recent paper by Jaap van Oosten can be used to simplify a key step in the proof. We have also included an interesting corollary for classical systems pointed out to us by Ulrich Kohlenbach

Logic and Methodology, Center Stage

The first international Congress for Logic, Methodology and Philosophy of Science was held at Stanford University in August of 1960. Occupying the vacuum created by the demise of the Unity of Science movement, it was the culminating event, on an international scale, of a long process of reorganization of communities of the philosophy of science and of logic that took place in the fifteen years following World War II—a process that involved many competing interests and personalities. Alfred Ta..

Global Greengrants Funds: Brazil Case Study

The Brazilian case study synthesises a decade of global greengrant funds' experience with small grants in the country. It describes what strategies have worked well, what kinds of impacts have been observed, as well as broad trends and challenges. The study seeks to illustrate how small grants have promoted social change and helped build the capacity of the Brazilian environmental movement over the past ten years