Daniele Mundici

Averaging the truth-value in Łukasiewicz logic

Changs MV algebras are the algebras of the infinite-valued sentential calculus of Łukasiewicz. We introduce finitely additive measures (called states) on MV algebras with the intent of capturing the notion of ‘average degree of truth’ of a proposition. Since Boolean algebras coincide with idempotent MV algebras, states yield a generalization of finitely additive measures. Since MV algebras stand to Boolean algebras as AFC*-algebras stand to commutative AFC*-algebras, states are naturally related to noncommutativeC*-algebraic measures.

Satisfiability in many-valued sentential logic is NP-complete

Abstract Cooks NP-completeness theorem is extended to all many-valued sentential logics, including the infinite-valued calculus of Łukasiewicz.

Advanced Łukasiewicz calculus and MV-algebras

Preface.- Chapter 1. Prologue: de Finetti coherence criterion and Lukasiewicz logic.- Chapter 2. Rational polyhedra, Interpolation, Amalgamation.- Chapter 3. The Galois connection (Mod, Th) in L 21.- Chapter 4. The spectral and the maximal spectral space.- Chapter 5. De Concini-Procesi theorem and Schauder bases.- Chapter 6. Bases and finitely presented MV-algebras.- Chapter 7. The free product of MV-algebras.- The construction of free products.- Chapter 8. Direct limits, confluence and multisets.- Chapter 9. Tensors.- Chapter 10. States and the Kroupa-Panti Theorem.- Chapter 11. The MV-algebraic Loomis-Sikorski theorem.- Chapter 12. The MV-algebraic Stone-von Neumann theorem.- Chapter 13. Recurrence, probability, measure.- Chapter 14. Measuring polyhedra and averaging truth-values.- Chapter 15. A Renyi conditional in Lukasiewicz logic.- Chapter 16. The Lebesgue state and the completion of FREEn.- Chapter 17. Finitely generated projective MV-algebras.- Chapter 18. E ective procedures for L and MV-algebras.- Chapter 19. A first-order Lukasiewicz logic with [0, 1]-identity.- Chapter 20. Applications, further reading, selected problems.- Chapter 21. Background results.- Special Bibliography. References. Index.

Ulam's searching game with lies

Abstract We determine the minimal number of yes-no queries sufficient to find an unknown integer between 1 and 2 m if at most two of the answers may be erroneous.

The Lebesgue state of a unital abelian lattice-ordered group

Abstract Various combinatorial equivalents are given to the Lebesgue state in archimedean lattice-ordered groups with order-unit. The proofs use piecewise linear functions on polyhedra with rational vertices.

Least adaptive optimal search with unreliable tests

We consider the basic problem of searching for an unknown m-bit number by asking the minimum possible number of yes-no questions, when up to a finite number e of the answers may be erroneous. In case the (i+1)th question is adaptively asked after receiving the answer to the ith question, the problem was posed by Ulam and R&enyi and is strictly related to Berlekamps theory of error correcting communication with noiseless feedback. Conversely, in the fully non-adaptive model when all questions are asked before knowing any answer, the problem amounts to finding a shortest e-error correcting code. Let qe(m) be the smallest integer q satisfying Berlekamps bound i=0e()2qm. Then at least qe(m) questions are necessary, in the adaptive, as well as in the non-adaptive model. In the fully adaptive case, optimal searching strategies using exactly qe(m) questions always exist up to finitely many exceptional ms. At the opposite non-adaptive case, searching strategies with exactly qe(m) questions or equivalently, e-error correcting codes with 2m codewords of length qe(m)---are rather the exception, already for e=2, and are generally not known to exist for e>2. In this paper, for each e>1 and all sufficiently large m, we exhibit searching strategies that use a first batch of m non-adaptive questions and then, only depending on the answers to these m questions, a second batch of qe(m)m non-adaptive questions. These strategies are automatically optimal. Since even in the fully adaptive case, qe(m)1 questions do not suffice to find the unknown number, and qe(m) questions generally do not suffice in the non-adaptive case, the results of our paper provide e, fault tolerant searching strategies with minimum adaptiveness and minimum number of tests.

Structures and Norms in Science

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Free products in the category of abelian l-groups with strong unit

Abstract We examine the structure of free products in the category A 1 of abelian l-groups with strong unit. We give several examples of free product computations, with particular regard to l-groups in A 1 corresponding, via the functor K0, to AF C∗-algebras considered by Glimm, Bratteli, Elliott, Pimsner, Voiculescu, Effros, and Others.

Resolution and model building in the infinite-valued calculus of Lukasiewicz

Abstract We discuss resolution and its complexity in the infinite-valued sentential calculus of L ukasiewicz, with special emphasis on model building algorithms for satisfiable sets of clauses. We prove that resolution and model building are polynomially tractable in the fragments given by Horn clauses and by Krom clauses, i.e., clauses with at most two literals. Generalizing the pre-existing literature on resolution in infinite-valued logic, by a positive literal we mean a negationless formula in one variable, built only from the connectives ⊕, ⊙, ν, Λ. We prove that the expressive power of our literals encompasses all monotone McNaughton functions of one variable.

Optimal comparison strategies in Ulam's searching game with two errors

Abstract Suppose x is an n-bit integer. By a comparison question we mean a question of the form “does x satisfy either condition a ⩽x ⩽b or c ⩽x ⩽d?”. We describe strategies to find x using the smallest possible number q(n) of comparison questions, and allowing up to two of the answers to be erroneous. As proved in this self-contained paper, with the exception of n = 2, q(n) is the smallest number q satisfying Berlekamps inequality 2 q ⩾2 n q 2 + q + 1 . This result would disappear if we only allowed questions of the form “does x satisfy the condition a⩽x⩽b?”. Since no strategy can find the unknown x ∈ {0,1,…,2n −1} with less than q(n) questions, our result provides extremely simple optimal searching strategies for Ulams game with two lies—the game of Twenty Questions where up to two of the answers may be erroneous.