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

Provably Total Functions of Arithmetic with Basic Terms

A new characterization of provably recursive functions of first-order arithmetic is described. Its main feature is using only terms consisting of 0, the successor S and variables in the quantifier rules, namely, universal elimination and existential introduction.Comment: In Proceedings DICE 2011, arXiv:1201.034

The depth of intuitionistic cut free proofs

Abstract We prove a quadratic upper bound for the depth of cut free proofs in propositional intuitionistic logic formalized with Gentzen's sequent calculus. We discuss bounds on the necessary number of reuses of left implication rules. We exhibit an example showing that this quadratic bound is optimal. As a corollary, this gives a new proof that propositional validity for intuitionistic logic is in PSPACE

Sharpened lower bounds for cut elimination

We present sharpened lower bounds on the size of cut free proofs for first-order logic. Prior lower bounds for eliminating cuts from a proof established superexponential lower bounds as a stack of exponentials, with the height of the stack proportional to the maximum depth d of the formulas in the original proof. Our new lower bounds remove the constant of proportionality, giving an exponential stack of height equal to d − O(1). The proof method is based on more efficiently expressing the Gentzen-Solovay cut formulas as low depth formulas

Strong isomorphism reductions in complexity theory

We give the first systematic study of strong isomorphism reductions, a notion of reduction more appropriate than polynomial time reduction when, for example, comparing the computational complexity of the isomorphim problem for different classes of structures. We show that the partial ordering of its degrees is quite rich. We analyze its relationship to a further type of reduction between classes of structures based on purely comparing for every n the number of nonisomorphic structures of cardinality at most n in both classes. Furthermore, in a more general setting we address the question of the existence of a maximal element in the partial ordering of the degrees

Bounded Arithmetic in Free Logic

One of the central open questions in bounded arithmetic is whether Buss' hierarchy of theories of bounded arithmetic collapses or not. In this paper, we reformulate Buss' theories using free logic and conjecture that such theories are easier to handle. To show this, we first prove that Buss' theories prove consistencies of induction-free fragments of our theories whose formulae have bounded complexity. Next, we prove that although our theories are based on an apparently weaker logic, we can interpret theories in Buss' hierarchy by our theories using a simple translation. Finally, we investigate finitistic G\"odel sentences in our systems in the hope of proving that a theory in a lower level of Buss' hierarchy cannot prove consistency of induction-free fragments of our theories whose formulae have higher complexity

AI Researchers, Video Games Are Your Friends!

If you are an artificial intelligence researcher, you should look to video games as ideal testbeds for the work you do. If you are a video game developer, you should look to AI for the technology that makes completely new types of games possible. This chapter lays out the case for both of these propositions. It asks the question "what can video games do for AI", and discusses how in particular general video game playing is the ideal testbed for artificial general intelligence research. It then asks the question "what can AI do for video games", and lays out a vision for what video games might look like if we had significantly more advanced AI at our disposal. The chapter is based on my keynote at IJCCI 2015, and is written in an attempt to be accessible to a broad audience.Comment: in Studies in Computational Intelligence Studies in Computational Intelligence, Volume 669 2017. Springe

A standard protocol to report discrete stage-structured demographic information

Stage-based demographic methods, such as matrix population models (MPMs), are powerful tools used to address a broad range of fundamental questions in ecology, evolutionary biology and conservation science. Accordingly, MPMs now exist for over 3000 species worldwide. These data are being digitised as an ongoing process and periodically released into two large open-access online repositories: the COMPADRE Plant Matrix Database and the COMADRE Animal Matrix Database. During the last decade, data archiving and curation of COMPADRE and COMADRE, and subsequent comparative research, have revealed pronounced variation in how MPMs are parameterized and reported. Here, we summarise current issues related to the parameterisation and reporting of MPMs that arise most frequently and outline how they affect MPM construction, analysis, and interpretation. To quantify variation in how MPMs are reported, we present results from a survey identifying key aspects of MPMs that are frequently unreported in manuscripts. We then screen COMPADRE and COMADRE to quantify how often key pieces of information are omitted from manuscripts using MPMs. Over 80% of surveyed researchers (n = 60) state a clear benefit to adopting more standardised methodologies for reporting MPMs. Furthermore, over 85% of the 300 MPMs assessed from COMPADRE and COMADRE omitted one or more elements that are key to their accurate interpretation. Based on these insights, we identify fundamental issues that can arise from MPM construction and communication and provide suggestions to improve clarity, reproducibility and future research utilising MPMs and their required metadata. To fortify reproducibility and empower researchers to take full advantage of their demographic data, we introduce a standardised protocol to present MPMs in publications. This standard is linked to www.compa dre-db.org, so that authors wishing to archive their MPMs can do so prior to submission of publications, following examples from other open-access repositories such as DRYAD, Figshare and Zenodo. Combining and standardising MPMs parameterized from populations around the globe and across the tree of life opens up powerful research opportunities in evolutionary biology, ecology and conservation research. However, this potential can only be fully realised by adopting standardised methods to ensure reproducibility

A standard protocol to report discrete stage-structured demographic information

Stage-based demographic methods, such as matrix population models (MPMs), are powerful tools used to address a broad range of fundamental questions in ecology, evolutionary biology and conservation science. Accordingly, MPMs now exist for over 3000 species worldwide. These data are being digitised as an ongoing process and periodically released into two large open-access online repositories: the COMPADRE Plant Matrix Database and the COMADRE Animal Matrix Database. During the last decade, data archiving and curation of COMPADRE and COMADRE, and subsequent comparative research, have revealed pronounced variation in how MPMs are parameterized and reported. Here, we summarise current issues related to the parameterisation and reporting of MPMs that arise most frequently and outline how they affect MPM construction, analysis, and interpretation. To quantify variation in how MPMs are reported, we present results from a survey identifying key aspects of MPMs that are frequently unreported in manuscripts. We then screen COMPADRE and COMADRE to quantify how often key pieces of information are omitted from manuscripts using MPMs. Over 80% of surveyed researchers (n = 60) state a clear benefit to adopting more standardised methodologies for reporting MPMs. Furthermore, over 85% of the 300 MPMs assessed from COMPADRE and COMADRE omitted one or more elements that are key to their accurate interpretation. Based on these insights, we identify fundamental issues that can arise from MPM construction and communication and provide suggestions to improve clarity, reproducibility and future research utilising MPMs and their required metadata. To fortify reproducibility and empower researchers to take full advantage of their demographic data, we introduce a standardised protocol to present MPMs in publications. This standard is linked to www.compadre-db.org, so that authors wishing to archive their MPMs can do so prior to submission of publications, following examples from other open-access repositories such as DRYAD, Figshare and Zenodo. Combining and standardising MPMs parameterized from populations around the globe and across the tree of life opens up powerful research opportunities in evolutionary biology, ecology and conservation research. However, this potential can only be fully realised by adopting standardised methods to ensure reproducibility