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Abstract In this paper we calibrate the strength of the soundness of a set theory KP ω + (Π 1 -Collection) with the assumption that 'there exists an uncountable regular ordinal' in terms of the existence of ordinals. [ABSTRACT FROM AUTHOR]

We describe a new quantifier elimination algorithm for real closed fields based on Thom encoding and sign determination. The complexity of this algorithm is elementary recursive and its proof of correctness is completely algebraic. In particular, the notion of connected components of semialgebraic sets is not used. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we show that the intuitionistic theory for finitely many iterations of strictly positive operators is a conservative extension of Heyting arithmetic. The proof is inspired by the quick cut-elimination due to G. Mints. This technique is also applied to fragments of Heyting arithmetic. [Copyright &y& Elsevier]

Abstract: We first look at an existing infinitary sequent system for common knowledge for which there is no known syntactic cut-elimination procedure and also no known non-trivial bound on the proof-depth. We then present another infinitary sequent system based on nested sequents that are essentially trees and with inference rules that apply deeply inside these trees. Thus we call this system “deep” while we call the former system “shallow”. In contrast to the shallow system, the deep system allows one to give a straightforward syntactic cut-elimination procedure. Since both systems can be embedded into each other, this also yields a syntactic cut-elimination procedure for the shallow system. For both systems we thus obtain an upper bound of on the depth of proofs, where is the Veblen function. [Copyright &y& Elsevier]

Abstract: This paper proves the strong normalization of classical natural deduction with disjunction and permutative conversions, by using CPS-translation and augmentations. Using them, this paper also proves the strong normalization of classical natural deduction with general elimination rules for implication and conjunction, and their permutative conversions. This paper also proves that natural deduction can be embedded into natural deduction with general elimination rules, strictly preserving proof normalization. [Copyright &y& Elsevier]

Abstract: A simple cut elimination proof for arithmetic with the epsilon symbol is used to establish the termination of a modified epsilon substitution process. This opens a possibility of extension to much stronger systems. [Copyright &y& Elsevier]

Abstract: We introduce a system of simply typed lambda terms (with fixed point combinators) and show that a rather comprehensive class of (co-)recursion equations on streams or non-wellfounded trees can be solved in our system. Moreover certain conditions are presented which guarantee that the defined functionals are primitive recursive. As a major example we give a co-recursive treatment of Mints’ continuous cut-elimination operator. [Copyright &y& Elsevier]

Abstract: We give geometric axioms for existentially closed Hasse fields. We prove a quantifier elimination result for existentially closed -truncated Hasse fields and characterize them as reducts of existentially closed Hasse fields. [Copyright &y& Elsevier]

We introduce the new notion of additive “multibox” for linear logic proof-nets. Thanks to this notion, we define a cut-elimination procedure which associates with every proof-net of multiplicative and additive linear logic a unique cut-free one. [Copyright &y& Elsevier]

Abstract: We develop a general algebraic and proof-theoretic study of substructural logics that may lack associativity, along with other structural rules. Our study extends existing work on (associative) substructural logics over the full Lambek Calculus (see, for example, Ono (2003) , Galatos and Ono (2006) , Galatos et al. (2007) ). We present a Gentzen-style sequent system that lacks the structural rules of contraction, weakening, exchange and associativity, and can be considered a non-associative formulation of . Moreover, we introduce an equivalent Hilbert-style system and show that the logic associated with and is algebraizable, with the variety of residuated lattice-ordered groupoids with unit serving as its equivalent algebraic semantics. Overcoming technical complications arising from the lack of associativity, we introduce a generalized version of a logical matrix and apply the method of quasicompletions to obtain an algebra and a quasiembedding from the matrix to the algebra. By applying the general result to specific cases, we obtain important logical and algebraic properties, including the cut elimination of and various extensions, the strong separation of , and the finite generation of the variety of residuated lattice-ordered groupoids with unit. [Copyright &y& Elsevier]

Abstract: The larger project broached here is to look at the generally sentence “if is well-ordered then is well-ordered”, where is a standard proof-theoretic function from ordinals to ordinals. It has turned out that a statement of this form is often equivalent to the existence of countable coded -models for a particular theory whose consistency can be proved by means of a cut elimination theorem in infinitary logic which crucially involves the function . To illustrate this theme, we prove in this paper that the statement “if is well-ordered then is well-ordered” is equivalent to . This was first proved by Marcone and Montalban [Alberto Marcone, Antonio Montalbán, The epsilon function for computability theorists, draft, 2007] using recursion-theoretic and combinatorial methods. The proof given here is principally proof-theoretic, the main techniques being Schütte’s method of proof search (deduction chains) [Kurt Schütte, Proof Theory, Springer-Verlag, Berlin, Heidelberg, 1977] and cut elimination for a (small) fragment of . [Copyright &y& Elsevier]

Abstract: We give inequational and equational axioms for semirings with a fixed-point operator and formally develop a fragment of the theory of context-free languages. In particular, we show that Greibach’s normal form theorem depends only on a few equational properties of least pre-fixed points in semirings, and eliminations of chain and deletion rules depend on their inequational properties (and the idempotence of addition). It follows that these normal form theorems also hold in non-continuous semirings having enough fixed points. [Copyright &y& Elsevier]

Abstract: For some modal fixed point logics, there are deductive systems that enjoy syntactic cut-elimination. An early example is the system in Pliuskevicius (1991) for LTL. More recent examples are the systems by the authors of this paper for the logic of common knowledge (Brünnler and Studer, 2009) and by Hill and Poggiolesi (2010) for PDL, which are based on a form of deep inference. These logics can be seen as fragments of the modal mu-calculus. Here we are interested in how far this approach can be pushed in general. To this end, we introduce a nested sequent system with syntactic cut-elimination which is incomplete for the modal mu-calculus, but complete with respect to a restricted language that allows only fixed points of a certain form. This restricted language includes the logic of common knowledge and PDL. There is also a traditional sequent system for the modal mu-calculus by Jäger et al. (2008) , without a syntactic cut-elimination procedure. We embed that system into ours and vice versa, thus establishing cut-elimination also for the shallow system, when only the restricted language is considered. [Copyright &y& Elsevier]

Abstract: Starting from Girard’s seminal paper on light linear logic (LLL), a number of works investigated systems derived from linear logic to capture polynomial time computation within the computation-as-cut-elimination paradigm. The original syntax of LLL is too complicated, mainly because one has to deal with sequents which do not just consist of formulas but also of ‘blocks’ of formulas. We circumvent the complications of ‘blocks’ by introducing a new modality which is exclusively in charge of ‘additive blocks’. One of the most interesting features of this purely multiplicative is the possibility of the second-order encodings of additive connectives. The resulting system (with the traditional syntax), called Easy-LLL, is still powerful to represent any deterministic polynomial time computations in purely logical terms. Unlike the original LLL, Easy-LLL admits polynomial time strong normalization together with the Church–Rosser property, namely, cut elimination terminates in a unique way in polytime by any choice of cut reduction strategies. [Copyright &y& Elsevier]

Abstract: A sequent calculus for Odintsov’s Hilbert-style axiomatization of a logic related to the trilattice of generalized truth values is introduced. The completeness theorem w.r.t. a simple semantics for is proved using Maehara’s decomposition method that simultaneously derives the cut-elimination theorem for . A first-order extension of and its semantics are also introduced. The completeness and cut-elimination theorems for are proved using Schütte’s method. [Copyright &y& Elsevier]