By Olaf Beyersdorff, Oliver Kutz (auth.), Nick Bezhanishvili, Valentin Goranko (eds.)

The eu summer time tuition in common sense, Language and knowledge (ESSLLI) is prepared each year via the organization for good judgment, Language and data (FoLLI) in numerous websites round Europe. the main target of ESSLLI is at the interface among linguistics, good judgment and computation. ESSLLI deals foundational, introductory and complicated classes, in addition to workshops, protecting a large choice of issues in the 3 components of curiosity: Language and Computation, Language and common sense, and good judgment and Computation. in the course of weeks, round 50 classes and 10 workshops are provided to the attendants, every one of 1.5 hours in step with day in the course of a 5 days week, with as much as seven parallel periods. ESSLLI additionally incorporates a pupil consultation (papers and posters by way of scholars in basic terms, 1.5 hour in line with day throughout the weeks) and 4 night lectures through senior scientists within the coated components. The 6 path notes have been conscientiously reviewed and chosen. The papers are equipped in topical sections on computational complexity, multi-agant platforms, average language processing, thoughts in video games and formal semantics.

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Additional info for Lectures on Logic and Computation: ESSLLI 2010 Copenhagen, Denmark, August 2010, ESSLLI 2011, Ljubljana, Slovenia, August 2011, Selected Lecture Notes

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Beyersdorff and O. Kutz As σ is consistent with ϕ there exists a model M, w of ϕ such that M, w |= Ai if and only if σ( Ai ) = 1. Let s¯ be the variables in π which do not appear in a modal context. For these variables we define an assignment ρ by setting ρ (s) = 1 if and only if M, w |= s. Let σ := ρ ∪ ρ . By Lemma 41 σ satisfies all formulae from the proof π. Therefore, in particular, σ (ϕ → ψ) = 1. On the other hand, by the choice of σ, we have σ(¬ ψ) = 1 and therefore also σ (¬ ψ) = 1. Also M, w is a model of ϕ and M, w is consistent with σ , implying σ (ϕ) = 1.

The following proposition provides a general method how to transform propositional tautologies into K-tautologies. Proposition 33. Let ϕ(¯ p, r¯) and ψ(¯ p, s¯) be propositional formulae which use common variables p¯ and let ϕ(¯ p, r¯) be monotone in p¯. If ϕ(¯ p, r¯) → ψ(¯ p, s¯) is a propositional tautology, then ϕ( p¯, r¯) → ψ(¯ p, s¯) is a K-tautology. Proof. e. p) (4) ϕ(¯ p, r¯) → θ(¯ and θ(¯ p) → ψ(¯ p, s¯) (5) are propositional tautologies. Substituting p¯ by p¯ in (4) we obtain the K-tautology ϕ( p¯, r¯) → θ( p¯) .

Pk , s¯) → C(p1 , . . , pk ) and C(p1 , . . , pk ) → ψ(¯ p, r¯) are propositional tautologies. Proof. The corollary follows from the previous theorem together with the following fact: if we start with a K-tautology θ and delete in θ all occurrences of , then we obtain a propositional tautology. 7 The Lower Bound Putting things together we obtain the lower bound for Frege systems in K which we already stated in the beginning of this section as Theorem 31: Theorem 46 (Hrubeˇ s [59,60]). Every K-Frege proof of the formulae √ Clique n n+1 ( p¯, r¯) → Ω(1) uses 2n steps.

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