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Workshop on Admissible Rules and Unification

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Workshop on Admissible Rules and Unification Algebra and Coalgebra meet Proof Theory Utrecht University, April — , organised by Nick Bezhanishvili · Jeroen Goudsmit · Rosalie Iemhoff Support by the Netherlands Organisation for Scientific Research under grant .. is gratefully acknowledged. ii Program Take care to note that the entrance to Dri is through Dri . ursday April Dri room : — : Daniele Mundici page : — : Break : — : Tadeusz Litak page : — : Alexander Kurz page : — : Lunch : — : George Metcalfe page : — : Marcello Bonsangue page : — : Break : — : Lutz Schröder page : — : Marta Bílková page : — : Mehrnoosh Sadrzadeh page iii Friday April Dri room : — : Grigori Mints page : — : Roman Kuznets page : — : Break : — : Emil Jeřábek page : — : Giuseppe Greco page : — : Lunch : — : Nick Galatos page : — : Rostislav Horčík page : — : Break : — : Wojciech Dzik page : — : Silvio Ghilardi page : — Dinner Saturday April Dri room : — : Mai Gehrke page : — : Break : — : Sumit Sourabh page : — : Lunch Dri room : — : Clemens Kupke page : — : Albert Visser page iv Table of Contents Ni Galatos page Hyper-Residuated Frames Mai Gehrke page Duality and Recognition Silvio Ghilardi page Step Algebras, Step Frames and Beyond George Metcalfe page Gödel Modal Logics Grigori Mints page Kripke Models, Proof Search and Cut-elimination for LJ Daniele Mundici page e Differential Semantics of Łukasiewicz Syntactic Consequence Albert Visser page Degrees of Interpretability of Finitely Axiomatized Sequential eories v Marta Bílková page A Coalgebraic Logic for Preordered Coalgebras Marcello Bonsangue page Polynomials and Signal Flow Graphs: A Completeness Result Wojcie Dzik page Consequence Relations Extending Modal Logic S4:3 Giuseppe Greco page Dynamic Epistemic Logic Displayed Rostislav Horčík page Universal eory of Residuated Distributive Laice-Ordered Groupoids and its Complexity Emil Jeřábek page Logics with Directed Unification Clemens Kupke page Well-founded Semantics for Description Logics Alexander Kurz page Positive Coalgebraic Logic Roman Kuznets page Craig Interpolation, Proof-eoretically via Nested Sequents vi Tadeusz Litak page Mehrnoosh Sadrzadeh page Lutz Sröder page Lightweight Coalgebraic Description Logics Sumit Sourabh page Sahlqvist Preservation for Topological Modal Fixed-Point Logic vii viii Hyper-Residuated Frames Nick Galatos University of Denver A big portion of the third level of the substuctural hierarchy can be handled proof- theoretically via hypersequents. e main tool for a semantic proof of cut elimi- nation at that level has been that of a hyper-residuated frame. is is the hyper- sequent analogue of residuated frames used for sequent calculi based on FL. I will explain the mechanics and uses of hyper-residuated frames and I will discuss some of the successes and even more of the challenges that recent research has had in these directions. is will include mention of joint research with various collabo- rators. Duality and Recognition Mai Gehrke CNRS and University Paris Diderot - Paris e fact that one can associate a finite monoid with universal properties to each language recognised by an automaton is central to the solution of many practi- cal and theoretical problems in automata theory. It is particularly useful, via the advanced theory initiated by Eilenberg and Reiterman, in separating various com- plexity classes and, in some cases it leads to decidability of such classes. In joint work with Serge Grigorieff and Jean-Eric Pin, we have shown that this theory may be seen as a special case of Stone-Priestley duality extended to a duality between bounded distributive laices with additional operations and the appropriate spaces equipped with Kripke style relations. is is a duality which also plays a fundamen- tal role in semantics. In this talk I will give a general introduction to the extended duality and explain what this has to do with the connection between regular lan- guages and monoids. Step Algebras, Step Frames and Beyond Silvio Ghilardi University of Milan We review and compare existing constructions of finitely generated free algebras in modal logic focusing on step-by-step methods. We discuss the notions of step alge- bras and step frames arising from these investigations, as well as the role played by finite duality. Finally, we report ongoing joint work (with Nick Bezhanishvili) con- cerning the use of the above conceptual framework in proof-theoretic applications, where step semantics can be used to achieve bounds in proof search space. Gödel Modal Logics George Metcalfe Bern University ere exist in the literature a diverse range of motivations and methods for defin- ing and investigating many-valued modal logics. One popular and quite natural proposal is, very roughly speaking, to interpret formulas in a fixed algebra at each world of a standard or many-valued Kripke frame. In the laer case, such an ap- proach also provides a framework for defining many-valued description logics. e main goal of this talk will be to explain and explore this methodology in the seing of (the many-valued and intermediate) Gödel logic. Issues of axiomatization, proof theory, decidability, and complexity will be addressed, as will relationships to other approaches. A new decidability result will also be presented for the S5 Gödel logic corresponding to the one-variable fragment of first-order Gödel logic. Kripke Models, Proof Search and Cut-elimination for LJ Grigori Mints Stanford University Existing Schüe-style completeness proofs for intuitionistic predicate logic with respect to Kripke models provide cut-elimination only for some semantic tableau formulations. Beth models extend this to multiple-succedent Gentzen calculus, but simple translation back to familiar one-succedent Gentzen calculus LJ introduces cuts. We present a short (non-effective) proof of completeness for Kripke models and cut-elimination for LJ. e Differential Semantics of Łukasiewicz Syntactic Consequence Daniele Mundici University of Florence Boolean logic L2 handles f0; 1g-valued observables. us, e.g., Cook’s celebrated formula CT;x;n represents a f0; 1g-observable whose output is iff a nondetermin- istic machine T accepts an input x within n steps. Since most observables in real life are continuous, one may use [0; 1]-valued logics to handle [0; 1]-valued observables as L2 does for f0; 1g- observables. Since every [0; 1]-measurement has an error, and small errors should have small effects on composite observables, one may naturally restrict aention to continuous observables, and to continuous connectives. Since there cannot be functional completeness in [0; 1]-logic, it is natural to list a few desiderata, at least for a connective, denoted !, suitable for a formulation of syn- tactic consequence via modus ponens. eorem: Up to inessential isomorphisms, there is a unique [0; 1]-valued continuous function ! defined on [0; 1]2 and satis- fying the two conditions: x ! (y ! z) = y ! (x ! z) and x ! y = 1 iff x ≤ y. Such unique map is Łukasiewicz implication x ! y = min(1; 1 − x + y). Many people contributed to this result, including Trillas, Valverde, Smets, Magrez, Fodor, Roubens, Bakzynski. Tautologies in the resulting logic L1 are now defined seman- tically in the usual way, and are axiomatized by ŁLukasiewicz’s four axioms (Chang completeness theorem). One then stipulates that formula F “syntactically” follows from a set P of premises if it is obtainable from P and the tautologies via modus ponens. is is syntactic consequence in L1. Traditionally, one may also say that F is a “semantic” consequence of P if every valuation V satisfying all formulas in P also satisfies F ; however, this semantic consequence is not finitary: F may seman- tically follow from P without semantically following from any finite subset of P . Of course, syntactic consequence is finitary. One of the best services L1 can offer to continuously valued logics is a refinement of the notion of semantic consequence that Bolzano and Tarski devised for f0; 1g-logic. One first notes that, for every n L1-formula F (X1; :::; Xn), truth-functionality yields a map fF : [0; 1] ?[0; 1] by the stipulation V (F ) = fF (v), where v is the restriction of the valuation V to fX1; :::; Xng. Since ! is piecewise linear then so is fF , and hence fF has all direc- tional derivatives. en, as recently proved by the author, syntactic consequence in L1 coincides with the “stable” refinement of semantic consequence given by the following stipulation: F stably follows from P iff every valuation satisfying P also satisfies F , and whenever a directional derivative of fG vanishes (for each G in P ), then so does the corresponding directional derivative of fF , and so on for higher-order differentials. References [] Roberto L.O. Cignoli, Itala M.L. D’Oaviano, and Daniele Mundici. Algebraic foundations of many-valued reasoning. Vol. Trends in Logic. Kluwer Aca- demic Publishers, . : ---. [] Daniele Mundici. Advanced Łukasiewicz calculus and MV-algebras. Vol. Trends in Logic. Springer, . [] Daniele Mundici. e differential semantics of Lukasiewicz syntactic consequence. Ed. by Franco Montagna. Trends in Logic. Volume in honor of Petr Hàjek, to appear. Springer. Degrees of Interpretability of Finitely Axiomatized Sequential eories Albert Visser Utrecht University Finitely axiomatized sequential theories are something like a natural kind of theo- ries. ey share a lot of salient and important properties. Moreover, many familiar theories belong to this kind. Examples of finitely axiomatized sequential theories − 1 are the basic
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