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1. Kirchhoff-Lukasiewicz Machines
I ve been working on analog computing since 1990 when I invented, and my students and I built, the first Lukasiewicz logic Arrays.
Kirchhoff-Lukasiewicz Machines
I've been working on analog computing since 1990 when I invented, and my students and I built, the first Lukasiewicz Logic Arrays. This is an image of the first VLSI LLA, which is still used as a comparison benchmark for other designs. This is a schematic of the compacted "railroad-tie" design of the LLA I use today. It generates a complete set of piecewise-linear functions that are digitally reconfigurable, and would otherwise fill four MOSIS TinyChips, if implemented as an H-tree LLA. I have spent the past year working more on robotics there's a large NSF award and students to consider so my research into KLMs has been slowed. However, I spent my sabbatical in 1996 doing research on tools and methods to design and use these machines that are so trivially easy to build. Here's a screen shot of the KLM visual layout editor and simulator I am programming for the Macintosh. Layout is wholly iconic, so one can move one step upward from a transistor-level layout tool like MAGIC. This is similar to using a schematic layout editor in digital. No such CAD tool yet exists for KLMs, which is why I am writing one! Of course, this means I'm using a digital computer to design analog computers...I find that amusing. This is the schematic layout of the first KLM. It has a conductive sheet surrounded by a ring diode to restrict current flow into and out of the sheet to the contacts only. Two compact LLAs are available to shape outputs from the sheet. The sheet and LLA connections are made at the output pins of the VLSI circuit, which also permits the sheet and LLAs on the prototype to be tested independently.

2. An Approach To Fuziness In The Setting Of Lukasiewicz Logic,
The paper seeks to understand the meaning of fuzziness. Its aim is to show how in some cases fuzziness comes from the indistinguishability between a fuzzy

3. Lukasiewicz S Logic And Prime Numbers Introduction And Contents
For the first time in the world literature this monograph shows a direct relation between logic and prime numbers. Although manyvalued Lukasiewicz s logics

4. IngentaConnect On Normal Forms In Lukasiewicz Logic
On normal forms in ukasiewicz logic. Authors Di Nola, A.1; Lettieri, A.2. Source Archive for Mathematical logic, Volume 43, Number 6, August 2004 , pp.
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5. On An Infinite-valued Lukasiewicz Logic That Preserves Degrees Of Truth
On an infinitevalued Lukasiewicz logic that preserves degrees of truth.
On an infinite-valued Lukasiewicz logic that preserves degrees of truth
Angel J. Gil Universitat Pompeu Fabra, Barcelona Infinite-valued Lukasiewicz can be defined from a class of matrices constituted by Wajsberg algebras (also called MV-algebras) where the set of designated values is an arbitrary implicative filter. This logic is protoalgebraic, algebraizable but not selfextensional. It does not satisfy the Deduction-Detachment Theorem, nor the Graded Deduction Theorem, but it satisfies the Local Deduction-Detachment Theorem. In this paper we will study a new logic, determined also by Wajsberg algebras, but focusing on the order relation instead of the implication. This new logic is an example of a ``logic that preserves degrees of truth'', and has the following properties: it can be defined from the class of matrices constituted by Wajsberg algebras where the set of designated values is an arbitrary lattice filter, it is not protoalgebraic, not algebraizable and selfextensional. It does not satisfy the Deduction-Detachment Theorem, nor the Local Deduction-Detachment Theorem, but it does satisfy the Graded Deduction Theorem. Since the new logic is selfextensional and has conjuntion, it has a Gentzen system that is both fully adequate for it and algebraizable, having the same algebraic counterpart as the logic. Wegive a a sequent calculus of the Gentzen system corresponding to this logic, we study its properties and models, and we determine several relationships between the Gentzen system and the infinite valued Lukasiewicz logic.

6. Atlas: On An Arithmetic In A Set Theory Within Lukasiewicz Logic By Shunsuke Yat
Recursion contradicts to induction within Lukasiewicz logic. Accepted to Many Valued logic and Cognition Trends in logic V Conference in July 2007.
August 5-9, 2007
St Anne's College, University of Oxford
Oxford, England Organizers
Mai Gehrke and Hilary Priestley View Abstracts
Conference Homepage
On an arithmetic in a set theory within ukasiewicz logic
Shunsuke Yatabe
Kobe University
On an arithmetic in a set theory within ukasiewicz logic
Shunsuke Yatabe
A significance of the set theory with the comprehension principle is to allow a general form of the recursive definition []: For any formula j j (x, ..., z)] within CFLew , i.e. we can define a set z by using a parameter z itself. This allows us to represent any partial recursive function on w Let H is a set theory with the comprehension principle within ukasiewicz infinite-valued predicate logic with its standard semantics. It has been conjectured that H is enough strong to develop an arithmetic because the recursive definition on w can be used in place of mathematical induction: We review about this. The arithmetic in H is somehow similar to one in "non-standard models" of PA . For example, we can prove an

7. Area-efficient Implication Circuits For Very Dense Lukasiewicz Logic Arrays - US
Two Lukasiewicz logic arrays (.English Pound.LAs) are proposed that use areaefficient implementations of the one-diode and three-transistor implication
United States Patent 5770966
Area-efficient implication circuits for very dense lukasiewicz logic arrays
US Patent Issued on June 23, 1998
No. 783196 filed on 1997-01-15
Current US Class
Combining of plural signals Stabilized (e.g., compensated, regulated, maintained, etc.) Using field-effect transistor
Field of Search
MOSFET (i.e., metal-oxide semiconductor field-effect transistor) Current mode logic (CML) Diode THRESHOLD (E.G., MAJORITY, MINORITY, OR WEIGHTED INPUTS, ETC.) With field-effect transistor Having at least two cross-coupling paths Logarithmic Combining of plural signals Stabilized (e.g., compensated, regulated, maintained, etc.) Using field-effect transistor Nonlinear amplifying circuit Negative resistance type
Attorney, Agent or Firm
US Patent References
    Switched capacitor precision current source
    Issued on: February 15, 1983
    Inventor: Olesin, et al.
    Apparatus for high speed analysis of two-dimensional images
    Issued on: November 17, 1987

8. Lukasiewicz Logic: From Proof Systems To Logic Programming -- Metcalfe Et Al. 13
We present logic programming style goaldirected proof methods for L We then provide an algorithm for fuzzy logic programming in Rational Pavelka
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Original Articles
ukasiewicz Logic: From Proof Systems To Logic Programming
George Metcalfe Nicola Olivetti and Dov Gabbay Institute of Discrete Mathematics and Geometry, Technical University Vienna, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria. E-mail: Department of Computer Science, University of Turin, Corso Svizzera 185, 10149 Turin, Italy. Email: Department of Computer Science, King's College London, Strand, London WC2R 2LS, UK. Email: We present logic programming style "goal-directed" proof methods for ukasiewicz logic that both have a logical interpretation

9. Fuzzy Logic Of Lukasiewicz Logic
Fuzzy logic of Lukasiewicz logic a clarification. Source, Fuzzy Sets and Systems archive Volume 95 , Issue 3 (May 1998) table of contents. Pages 369 379

10. Area-efficient Implication Circuits For Very Dense Lukasiewicz Logic Arrays - Pa
are the basis for analog array processors called Lukasiewicz logic arrays (. Applications for Lukasiewicz logic arrays include fuzzy controllers,
Login or Create Free Account Search Go to Advanced Search Home Search Patents Data Services ... Help Title: Area-efficient implication circuits for very dense lukasiewicz logic arrays Document Type and Number: United States Patent 5770966 Link to this page: Abstract: A one-diode circuit for negated implication (.about.➝) is derived from a 12-transistor Lukasiewicz implication circuit (➝). The derivation also yields an adjustable three-transistor implication circuit with maximum error less than 1% of full scale. Two Lukasiewicz logic arrays (.English Pound.LAs) are proposed that use area-efficient implementations of the one-diode and three-transistor implication circuits. The very dense diode-tower .English Pound.LA contains 36,000 implications in an area that previously held 92 implications; the three-transistor .English Pound.LA contains 1,990 implications. Both .English Pound.LAs double the number of inputs per pin on the IC package. Very dense .English Pound.LAs make .English Pound.LA-based fuzzy controllers and neural networks practical. As an example, an .English Pound.LA retina that detects edges in 15 nanoseconds is described. Inventors: Mills, Jonathan W. (Bloomington, IN)

11. On Lukasiewicz Logic With Truth-constants - Mathfuzzlog
the expansion of Lukasiewicz logic with a countable set of truthconstants \mathcal{C} , have been recently proved for the case when the algebra of truth
On Lukasiewicz logic with truth-constants
From Mathfuzzlog
Jump to: navigation search Authors: Roberto Cignoli Francesc Esteva Llu­s Godo Title of the chapter: On Lukasiewicz logic with truth-constants Title of the book: Theoretical Advances and Applications of Fuzzy Logic and Soft Computing Editor(s): O. Castillo
Pages: Publisher: Springer-Verlag City: Year:
Canonical completeness results for L , the expansion of Lukasiewicz logic with a countable set of truth-constants , have been recently proved for the case when the algebra of truth constants is a subalgebra of the rational interval . The case when was left as an open problem. In this paper we solve positively this open problem by showing that L is strongly canonical complete for finite theories for any countable subalgebra of the standard Lukasiewicz chain L Retrieved from " Categories Book chapters Publications ... Publications by Llu­s Godo Views Personal tools Navigation Search Toolbox

12. [math/0508445] Invariant Measures In Free MV-algebras
Invariant measures in Lukasiewicz logic. Authors Giovanni Panti Comments 10 pages, 3 figures Subjclass logic; Dynamical Systems MSC-class 03B50; 37A05 math
Search or Article-id Help Advanced search All papers Titles Authors Abstracts Full text Help pages
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Mathematics > Logic
Title: Invariant measures in free MV-algebras
Authors: Giovanni Panti (Submitted on 24 Aug 2005 ( ), last revised 20 Apr 2007 (this version, v2)) Abstract: Comments: 12 pages, 4 figures. Title changed, motivational section rewritten, mathematics unchanged. To appear in Communications in Algebra Subjects: Logic (math.LO) ; Dynamical Systems (math.DS) MSC classes: Cite as: arXiv:math/0508445v2 [math.LO]
Submission history
From: Giovanni Panti [ view email
Wed, 24 Aug 2005 13:30:10 GMT (14kb)
Fri, 20 Apr 2007 09:29:13 GMT (16kb)
Which authors of this paper are endorsers?
Link back to: arXiv form interface contact

13. Analog Computation
The chip on the right is a Lukasiewicz logic Array that was patented by Indiana University. Although it is built from transistors, it computes continuous
Indiana University April 1998 Volume XXI Number 2
by Leigh Hedger
Each new decade sees a new wave of innovative technology. Experts in the 1950s thought that the world's computing needs could be supplied by half a dozen computers. The 1990s saw digital processors become so inexpensive that a car could use dozens of computers to control it. Today just one Porsche contains ninety-two processors. According to Paul Saffo, director of the Institute for the Future, changes just as dramatic will be caused by massive arrays of sensors that will allow computers to control our environment. In Saffo's vision of the future, we will have smart rooms that clean themselves with a carpet of billions of tiny computer-controlled hairs. Airplanes will have wings with micromachine actuators that will automatically adjust to handle turbulence. These smart machines will require computers that can process vast amounts of sensor data rapidly, far faster than today's digital computers. To make Saffo's vision a reality may well require a rebirth of technology that many consider obsolete in this era of digital computersanalog computers.
Johnathan Mills
, Associate Professor of Computer Science, Indiana University Bloomington

14. DC MetaData For: Algebras Of Lukasiewicz's Logic And Their Semiring Reducts
it makes full sense to consider manyvalued automata and many-valued formal languages interpreted in Lukasiewicz logic.
A. Di Nola, B. Gerla
Algebras of Lukasiewicz's Logic and their Semiring Reducts

Preprint series:
ESI preprints
Abstract In this paper we shall establish some links between the
The relationship of these algebraic structures gives a hint
on how to construct linear algebra starting from MV-algebras.
Indeed here the role of sum and product is played respectively
by a lattice operation and by an arithmetical operation.
In this way, following the tradition of semirings,
it makes full sense to consider "many-valued automata"
and "many-valued formal languages" interpreted in Lukasiewicz logic.

15. JSTOR Rational Pavelka Predicate Logic Is A Conservative
Rational Pavelka logic extends Lukasiewicz infinitely valued logic by adding truth It does not extend Lukasiewicz logic (Theorem 2), even for statements<669:RPPLIA>2.0.CO;2-D

16. Sibirskii Zhurnal Industrial'noi Matematiki
Lukasiewicz s logic as an architecture model of arithmetic We present a new view of the nature of Lukasiewicz s logic. UDC 510.644 Received 06.03.2003

17. George Metcalfe
Lukasiewicz logic From Proof Systems to logic Programming (with N. Olivetti and D. Gabbay). logic Journal of the IGPL 13, 561585, 2005.
Home Publications Short CV Activities ... Teaching
Proof Theory for Fuzzy Logics (with N. Olivetti and D. Gabbay). To appear for Research Studies Press.
Journal Articles
Herbrand's Theorem, Skolemization, and Proof Systems for First-Order Lukasiewicz Logic (with M. Baaz). Submitted.
Density Elimination (with A. Ciabattoni). Submitted.
Proof Theory for Admissible Rules (with R. Iemhoff). Submitted.
Substructural Fuzzy Logics (with F. Montagna). Journal of Symbolic Logic 72(3), pages 834-864, 2007.
Fuzzy Logics Based on [0,1)-Continuous Uninorms (with D. Gabbay). Archive for Mathematical Logic 46(6), pages 425-469, 2007.
Normal Forms for Fuzzy Logics: a Proof-Theoretic Approach (with P. Cintula). Archive for Mathematical Logic 46(5), pages 347-363, 2007.
Proof Calculi for Casari's Comparative Logics. Journal of Logic and Computation 16(4):405-422, 2006.
Sequent and Hypersequent Calculi for Abelian and Lukasiewicz logics (with N. Olivetti and D. Gabbay). ACM Transactions on Computational Logic 6(3): 578-613, 2005.

18. Prof. Jonathan Mills Leverhulme Research Professor Faculty Of
Lukasiewicz Insect The Role of ContinuousValued logic in a Mobile Robot s Lukasiewicz logic Arrays in Future Directions of Parallel Programming and
Prof. Jonathan Mills
Leverhulme Research Professor Faculty of Computing, Engineering and Mathematical Sciences University of the West of England, Bristol, UK
Associate Professor Department of Computer Science School of Informatics Indiana University, Bloomington, USA
Jonathan Mills' Analog Computing Page.

  • Applied for: Mills, J. November 2002. Extended Analog Computer. Issued: Mills, J. August 1997. U.S. Patent Interim Serial No. 08/783,196. Lukasiewicz Logic Arrays. Issued: Mills, J. August 1992. U.S. Patent 5,193,206 (GE-1906), Reduced Instruction Set Microprocessor.

  • Himebaugh, B. and J. Mills 2005. µEAC: USB Micro-Extended Analog Computer. Himebaugh, B. and J. Mills 2004. Internet-accessible Extended Analog Computer. Mills, J. 1995. PDIFFSHT: P-diffusion partial differential equation solver. Mills, J. 1995. NDIFFSHT: N-diffusion partial differential equation solver. Mills, J. 1995. The Foamputer: conductive foam extended analog computers. Heininger, A., A. Biswas and J. Mills. 1992. EYE2: VLSI LLA vertical edge detector. Heininger, A. and J. Mills. 1992. EYE1: VLSI photodetector array for LLA retina.

19. A Necessary And Sufficient Condition For Lukasiewicz Logic Functions
The literal, TSUM, min and max operations employed in multiplevalued logic design can be expressed in terms of the implication and the negation of
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20. [Abstract] Fuzzy DCG Syntactic Parser For Command Language Recognition Under Spe
Lukasiewicz developed the first Nvalued logic in the 1930’s. . The application of Lukasiewicz logic implements a kind of approximate reasoning,

21. DBLP: Daniele Mundici
15, Daniele Mundici Averaging the truthvalue in Lukasiewicz logic. Studia logica 55(1) 113-127 (1995). 1994. 14, Daniele Mundici A Constructive Proof of
Daniele Mundici
List of publications from the DBLP Bibliography Server FAQ Coauthor Index - Ask others: ACM DL Guide CiteSeer CSB ... Manuela Busaniche , Daniele Mundici: Geometry of Robinson consistency in Lukasiewicz logic. Ann. Pure Appl. Logic 147 EE Daniele Mundici: A Characterization of the free n -generated MV-algebra. Arch. Math. Log. 45 EE Daniele Mundici: Bookmaking over infinite-valued events. Int. J. Approx. Reasoning 43 EE Ferdinando Cicalese Christian Deppe ... Ferdinando Cicalese , Daniele Mundici, Ugo Vaccaro : Preface. Discrete Applied Mathematics 137 Silvio Ghilardi , Daniele Mundici: Foreword. Studia Logica 73 EE Vincenzo Marra , Daniele Mundici: Consequence and Complexity in Infinite-Valued Logic: A Survey. ISMVL 2002 EE Ferdinando Cicalese , Daniele Mundici, Ugo Vaccaro : Least adaptive optimal search with unreliable tests. Theor. Comput. Sci. 270 Stefano Aguzzoli , Daniele Mundici: Weierstrass Approximations by Lukasiewicz Formulas with One Quantified Variable. ISMVL 2001 Daniele Mundici, Giovanni Panti : Decidable and undecidable prime theories in infinite-valued logic. Ann. Pure Appl. Logic 108

22. M_ValuedLETS
Definitions of scope are broad and shall include mvalued logic (e.g., fuzzy logic, Lukasiewicz logic); theory of monetary instruments; related quantum
M-LOGICALLY-VALUED (e.g., fuzzy logic, Lukasiewicz logic)
This site is devoted to all and everything associated with the notion of m-logically-valued monetary units and their applications to LETS, local exchange trading systems. Definitions of scope are broad and shall include: m-valued logic (e.g., fuzzy logic, Lukasiewicz logic); theory of monetary instruments; related quantum theoretical issues; applications technologies (hardware and software); research and development; the involved strategic planning issues; real politik of insinuating m-logically-valued exchange systems into the prevailing Newtonian institutionalization; quantum accounts of self-organization as they apply to questions of monetary theory; autopoiesis and its graphical representation systems; metaphors in theoretical biology, biometeorology, oceanography, and related sciences of multiscale dynamical systems; applicability of complexity theory to monetary systematics; history of any and all related subjects. Definitions of exclusion are narrow and shall be determined only by the propensity of any given contribution to elicit ennui. Content of the site is largely generated by e-mail exchanges. The site editor responds to all website, paper, and book referrals in due course, as time permits and incorporates content of messages and commentary thereupon into postings. Names of e-mail correspondents are not employed unless explicitly requested. Say what you want without risk! This is a conceptual clearinghouse filtered through the editor's literary imagination. Any e-mail correspondent dissatisfied by content of this website is free to create hisher own site. Absent e-mail during a given period, the editor will post his own musings in such manner as to suggest an e-mail exchange. Perhaps the whole site is accomplished in this manner!

23. Shunsuke Yatabe
Distinguishing nonstandard natural numbers in a set theory within Lukasiewicz logic Workshop on Algebra and logic, March 12, 2007, at JAIST.
Shunsuke Yatabe
I moved to:
Research Staff,
Research Center for Verification and Semantics (CVS),
National Institute of Advanced Industrial Science and Technology (AIST).
[My new web site Address: 5th floor, Mitsui Sumitomo Kaijo Senri Bldg, 1-2-14 Shin-Senri Nishi, Toyonaka, Osaka, 560-0083 Japan Tel:+81-6-4863-5031 Fax:+81-6-4863-5052 Email: shunsuke.yatabe(at)
Research interest:
  • functorial semantics of abstraction
  • set theory and truth theory within many-valued logic
  • vagueness
Publications (until September 2007, in English/with referee):
On the combinatorial structure of the real numbers
  • Forcing indestructibility of MAD families
    Jorg Brendle and Shunsuke Yatabe. Annals of Pure and Applied Logic, pp.271-312, vol.132(2-3), March 2005. (Abstract)(pdf)
Set theories within many-valued logic
  • On a Set Theory with Uncertain Membership Relations
    Shunsuke Yatabe, Yuzuru Kakuda and Makoto Kikuchi. Design and application of Hybrid Intelligent Systems, pp.458-467, IOS press, 2003. (Abstract)
  • A note on Hajek, Paris and Shepherdson's theorem

24. Logic Colloquium 2006
However, the Hilbertstyle calculi and algebraic semantics of the resulting logics (MTL, BL, Lukasiewicz logic, Goedel logic, etc.) make no reference to the
main invited contributed registration how to get there
Contributed Talks
This page contains the schedule and abstracts of the contributed talks at the Logic Colloquium 2006
The talks are 15 minutes + 5 minutes for questions. The following facilities will be available in each room:
  • Beamer Laptop (for people who did not bring their own laptop: possible file formats .pdf .ps .ppt) Overhead projector Black board or white board
At your convenience you can send a .pdf file with your slides to Jasper Stein ( ) who will have it pre-installed on the presentation computer.
Room 2
Room 2
Room 2
  • Vadim Puzarenko

25. Home Page Of A. Di Nola
A. Di Nola, B. Gerla, Algebras of Lukasiewicz s logic and their semiring L.P. Belluce, A. Di Nola, The MValgebra of first order Lukasiewicz logic.
Prof. Antonio Di Nola
Pubblicazioni 2001-2005
    A. Di Nola, B. Gerla, Algebras of Lukasiewicz's Logic and their semiring
    reducts, Contemporary Mathematics, vol. 377, (2005), pp. 131-144.
    L.P. Belluce, A. Di Nola, Frames and MV-algebras, Studia Logica, vol.
    81, (2005), pp. 357-385.
    P. Amato, A. Di Nola, B. Gerla, Neural Networks and Rational Mc-
    Naughton Functions J. of Mult.-Valued Logic and Soft Computing, Vol.
    11, (2005), pp. 95-110.
    A. Di Nola, P. Niederkorn, Natural Dualities for varieties of BL-algebras,
    Archive for Mathematical Logic, vol. 44, (2005), pp. 995-1007.
    A. Di Nola, M. Navara, The sigma-complete MV-algebras which have
    enough states, Colloquium Mathematicum, vol. 103, (2005), pp. 121-130. A. Di Nola, A. Dvurecenskij, J. Jakubik, Good and Bad Infinitesimals, and States on Pseudo MV-algebras, Order, 21, (2004), 293-314. A. Di Nola, A. Dvurecenskij, MV-test Spaces versus MV-algebras. Czechoslo- vak Mathematical Journal vol. 54, (2004), pp. B. Bede, A. Di Nola,(2004). Elementary calculus in Riesz MV-algebras, International Journal of Approximate Reasoning, vol. 36, pp. 129-149.

26. Fuzzy DL Syntax
The implication in (all R C) is interpreted as (or (not R) C); Lukasiewicz logic uses Lukasiewicz negation, conjunction and disjunction and implication.
Concept expressions: *top* (top concept) *bottom* (bottom concept) A (atomic concept) (and C1 C2) (concept conjunction) (or C1 C2) (concept disjunction) (not C1) (concept negation) (some R C1) (existential role restriction) (all R C1) (existential role restriction) (CM C) (modifier name CM applied to a concept), modifiers are defined below CFC (concept name CFC defined with explicit fuzzy membership function), see below DR (datatype restriction), see below (L-and C1 C2) (concept conjunction according to Lukasiewicz T-norm) (L-or C1 C2) (concept disjunction according to Lukasiewicz T-conorm) (G-and C1 C2) (concept conjunction according to G¶del T-norm) (G-or C1 C2) (concept disjunction according to G¶del T-conorm) (P-and C1 C2) (concept conjunction according to Product T-norm) (P-or C1 C2) (concept conjunction according to Product T-conorm) (w-sum (n1 C1) ... (nk Ck)) (weighted concept, ni in [0,1], Ci concept, n1+ ..+nk=1, see below) Note
  • Lukasiewicz T-norm: (and x y) = max(0, x+y -1) Lukasiewicz T-conorm: (or x y) = min(1, x+y) G¶del T-norm: (and x y) = min(x, y)

27. Łukasiewicz Logic - Wikipedia, The Free Encyclopedia
In mathematics, ukasiewicz logic is a nonclassical, many valued logic. It was originally defined by Jan ukasiewicz as a three-valued logic,Łukasiewicz_logic
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Łukasiewicz logic
From Wikipedia, the free encyclopedia
Jump to: navigation search In mathematics Łukasiewicz logic is a non-classical many valued logic. It was originally defined by Jan Łukasiewicz as a three-valued logic , known as trivalent logic it was later generalized to n -valued (for all finite n ) as well as infinitely-many-valued variants, both propositional and first-order. It belongs to the classes of t-norm fuzzy logics and substructural logics
edit Real-valued semantics
Infinite-valued Łukasiewicz logic is a real-valued logic in which sentences from sentential calculus are assigned a truth value of arbitrary precision between and 1. Valuations have a recursive definition in the following sense: The values of and are given explicitly by:
edit Properties of valuations
Under this definition, valuations satisfy the following conditions: and satisfy
  • and . Correspondingly: and and are continuous and are strictly increasing in each component. and are associative in the following sense: F a F b c F F a b c for each
Thus and are both continuous t-norms
  • and is continuous.

28. Fuzzy Logic (Stanford Encyclopedia Of Philosophy)
In ukasiewicz logic this is not the case if has the value 0.5 then its 2000), and logics putting ukasiewicz and product logic together (Esteva
Cite this entry Search the SEP Advanced Search Tools ...
Please Read How You Can Help Keep the Encyclopedia Free
Fuzzy Logic
First published Tue Sep 3, 2002; substantive revision Sun Jul 23, 2006 The term "fuzzy logic" emerged in the development of the theory of fuzzy sets by Lotfi Zadeh ( ). A fuzzy subset A of a (crisp) set X is characterized by assigning to each element x of X the degree of membership of x in A (e.g., X is a group of people, A the fuzzy set of old people in X ). Now if X is a set of propositions then its elements may be assigned their degree of truth intermediate connectives truth functions different from probability theory since the latter is not truth-functional (the probability of conjunction of two propositions is not determined by the probabilities of those propositions). Two main directions in fuzzy logic have to be distinguished (cf. Zadeh 1994 Fuzzy logic in the broad sense (older, better known, heavily applied but not asking deep logical questions) serves mainly as apparatus for fuzzy control, analysis of vagueness in natural language and several other application domains. It is one of the techniques of soft-computing , i.e. computational methods tolerant to suboptimality and impreciseness (vagueness) and giving quick, simple and

29. Quantum Genetics In Terms Of Quantum Reversible Automata And Quantum Computation
Further, detailed studies of nonlinear dynamics in genetic networks were carried out in categories of nvalued, ukasiewicz logic Algebras that showed
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    Quantum Genetics in terms of Quantum Reversible Automata and Quantum Computation of Genetic Codes and Reverse Transcription
    Baianu, Professor I.C. Quantum Genetics in terms of Quantum Reversible Automata and Quantum Computation of Genetic Codes and Reverse Transcription. [Preprint] Full text available as: Preview PDF - Requires a PDF viewer such as GSview Xpdf or Adobe Acrobat Reader
    Item Type: Preprint Additional Information: Keywords: Automata Theory/ Sequential Machines, Bioinformatics, Complex Biological Systems, Complex Systems Biology, Computer Simulations and Modeling, Dynamical Systems , Quantum Dynamics, Quantum Field Theory, Quantum Groups,Topological Quantum Field Theory (TQFT), Quantum Automata, Cognitive Systems, Graph Transformations, Logic, Mathematical Modeling; applications of the Theory of Categories, Functors and Natural Transformations; pushouts, pullbacks, presheaves, sheaves, Categories of sheaves, Topos, n-valued Logic, N-categories/ higher dimensional algebra, Homotopy theory;apllications to physical theories, quantum gravity, complex systems biology, bioengineering, informatics, Bioinformatics, Computer simulations, Mathematical Biology of complex systems and phenomena in various types of Dynamical Systems; bioengineering, Computing, Neurosciences, Bioinformatics, biological and/or social networks; quantitative ecology and quantitative biology/

30. Stefano Aguzzoli
Stefano Aguzzoli An Asymptotically Tight Bound on Countermodels for ukasiewicz logic, International Journal of Approximate Reasoning , 43, pp.
Dipartimento di Scienze dell'Informazione
Via Comelico, 39-41
20135 MILANO

tel.: +39-02- 50316-356,313
fax : +39-02- 50316373
room: S204
PhD Dissertation:
, PhD Program in Mathematical Logic and Theoretical Computer Science University of Siena
Journal Papers:
Stefano Aguzzoli, Daniele Mundici An Algorithmic Desingularization of 3-Dimensional Toric Varieties , pp. 557-572, 1994. Stefano Aguzzoli: The Complexity of McNaughton Functions of One Variable, Advances in Applied Mathematics , pp. 58-77, 1998. Stefano Aguzzoli: A Note on the Representation of McNaughton Lines by Basic Literals, Soft Computing , pp. 111-115, 1998. Stefano Aguzzoli, Agata Ciabattoni and Antonio Di Nola Journal of Logic and Computation , pp. 213-222, 2000. Stefano Aguzzoli, Agata Ciabattoni Journal of Logic, Language and Information , pp. 5-29, 2000. Stefano Aguzzoli, Brunella Gerla Finite-Valued Reductions of Infinite-Valued Logics, Archive for Mathematical Logic , pp. 361-399, 2002.

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