Collegium Logicum

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Collegium Logicum

Dialectica 48 2 : , In this talk we investigate the properties of medial as rewriting rule independently from logic. We present a graph theoretical criterion for checking whether there exists a medial rewriting path between two formulas. Finally, we return to logic and apply our criterion for giving a combinatorial proof for a decomposition theorem, i. Terui : How and when axioms can be transformed into good structural rules joint work with A.

Ciabattoni and N.

Galatos The class of substructural logics consists of axiomatic extensions of FL full Lambek calculus, or the multiplicative-additive fragment of intuitionistic noncommutative linear logic. When an axiom is added to FL, it may happen that it can be "structuralized," in the sense that it can be transformed into an equivalent set of structural rules. Furthermore, it may also happen that suc, when suitably structuralized, admits cut-elimination.

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In this talk, we identify a natural class of axioms which can be well structuralized over FL. We also give a subclass of axioms that admit cut-elimination, and give a necessary and sufficient condition under which an axiom of the former type belongs to the latter. Our condition can be stated both in terms of an acylclicity property and a conservative extension property. Terwijn : Embeddings into the Medvedev lattice The Medvedev lattice is a structure from computability theory that is interesting for various reasons. It was originally introduced for its connections with constructive logic, but it is also interesting in other respects, for example in connection with computation on the reals, algorithmic randomness, or as a generalization of the Turing degrees.

We study embeddability of lattices, algebras, and semilattices into the Medvedev lattice M. Sorbi showed that the countable dense Boolean algebra is embeddable into M. We show that this result is optimal: if a Boolean algebra is embeddable into M then it must be countable. On the other hand, if one drops the requirement that meets are preserved then much larger structures are embeddable. For example the large Boolean algebra of all subsets of the reals is embeddable as an upper semilattice into M.

As for the closely related Muchnik lattice, we show that the previous large Boolean algebra is embeddable into it as an algebra. Weller : Implementing CERES: tools for proof analysis This talk deals with the implementation of tools necessary to analyse formalized mathematical proofs using cut-elimination. Three tools are presented: Handy LK, a compiler that transforms a proof specified in a higher level language into a formal LK proof; ProofTool, a graphical tool for viewing and editing formal proofs; and CERES, an implementation of cut-elimination by resolution.

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