The Turing Centenary Conference CiE 2012: How the World Computes

• Thomas Woolley, Ruth Baker, Eamonn Gaffney and Philip Maini. Stochastic reaction and diffusion on growing domains: understanding the breakdown of robust pattern formation

• Chaudhary Imran Sarwar. Alan Turing and Conventional, Virtual, Robotic, and Mixed Reality Leadership

• David Holcman and Zeev Schuss. Finding small and hidden targets in cells

• Richard Statman. A New Type Assignment for Strongly Normalizable Terms

• Michał B. Paradowski. Humans' linguistic interactions and the upshots for social cognitive systems

• James D Murray. After Turing - Mathematical modelling in the biomedical and social sciences: from animal coat patterns to brain tumours to saving marriages.

• Fernando Ferreira. A short note on Spector's consistency proof of analysis

• Giuseppe Primiero. A verificationist modal semantics for contextual computations

• Fred Gruau. Self Developing Network 2: Universal Machines

• Henson Graves. Structure Diagrams in Type Theory

• Mark Burgin and Eugene Eberbach. Evolutionary Turing in the Context of Evolutionary Machines and Computations

• George Metcalfe and Christoph Röthlisberger. Unifiability and Admissibility in Finite Algebras

• John Preston. What are Computers (if they're not Thinking Things)?

• Ilkka Törmä and Ville Salo. On Shift Spaces with Algebraic Structure

• Michael Fiske. Turing Incomputable Computation

• Steven Ericsson-Zenith. Toward the unification of logic and geometry as the basis of a new computational paradigm

• Ruth Millikan. Natural Information

• Mark Finlayson. Sets of Signals, Information Flow, and Folktales

• Armin Hemmerling. On the Tape-Number Problem for Deterministic Time Classes

• Emmanuel Jeandel. On Immortal Configurations in Turing Machines

• Josef Berger. When is the denial inequality an apartness relation?

• Douglas Bridges and Robin Havea. Square roots and powers in constructive Banach algebra theory

• Samuel Moelius. Characteristics of Minimal Effective Programming Systems

• Anahit Chubaryan and Armine Chubaryan. A Hierarchy of Resolution Systems with Restricted Substitution Rules

• Richard Booth. On Turing Scores

• Isabel Cafezeiro and Ivan Da Costa Marques. Interweavings of Alan Turing's Mathematics and Sociology of Knowledge. Sequent calculus with algorithmic properties for logic of correlated knowledge

• Hartmut Ehrig, Claudia Ermel, Falk Hüffner, Rolf Niedermeier and Olga Runge. Confluence in Data Reduction: Bridging Graph Transformation and Kernelization

• Luís Tarrataca and Andreas Wichert. Iterative Quantum Tree Search

• Fotini Markopoulou. The Computing Spacetime

• Antoine Taveneaux, Bjørn Kjos-Hanssen and Neil Thapen. How much randomness is needed for statistics?

• İlayda Ateş and Çiğdem Gencer. On Models of the Modal Logic GL

• Nikolay Kosovskiy. Polynomial-Time Modifications of Church-Turing Thesis for Three Programming Languages

• Jiri Wiedermann. On the Road to Thinking Machines: Insights and Ideas

• Haroldas Giedra and Jûratė Sakalauskaitė. Sequent calculus with algorithmic properties for logic of correlated knowledge

• Xin-She Yang. Optimality for Exploration and Exploitation in Metaheuristic Optimization Algorithms

• Edgar Daylight. Book Presentation: `The Dawn of Software Engineering: from Turing to Dijkstra'

• Katerina Senkevich and Boris Solon. Relationships between almost partial m-reducibility and other enumeration reducibilities

• Arturo Carsetti. Eigenforms, natural computing and morphogenesis

• Yijia Chen, Jörg Flum and Moritz Mueller. Hard Instances of Algorithms and Proof Systems

• Rod Downey. Randomness, Computation and Mathematics

• William Calhoun. The Legacy of Turing Reducibility

• Stefano Mazzanti. Bases for AC0 and other Complexity Classes

• Vincent Ardourel and Julie Jebeile. Cellular Automata Simulations: Are They Really of a Special Kind?

• Jürgen Dassow, Florin Manea and Robert Mercas. Connecting Partial Words and Regular Languages

• Thomas Streicher. Computability in Quantum Mechanics

• Sepp Hartung and André Nichterlein. NP-Hardness and Fixed-Parameter Tractability of Realizing Degree Sequences with Directed Acyclic Graphs

• Yasushi Okuno and Shinnosuke Seki. On the behavior of tile assembly system at high temperatures

• Marcel Celaya and Frank Ruskey. An Undecidable Nested Recurrence Relation

• Kenshi Miyabe. An integral test for Schnorr randomness and its applications

• Franck Delaplace, Hanna Klaudel, Tarek Melliti and Sylvain Sené. Analysis of modular organisation of interaction networks based on asymptotic dynamics

• John Case, Sanjay Jain, Samuel Seah and Frank Stephan. Automatic functions, linear time and learning

• David FernC!ndez and Joost Joosten. Turing Progressions and their well-orders

• Sergio Rajsbaum, Maurice Herlihy and Dmitry Koslov. Computability in Distributed Computing

• Paul Cockshott and Greg Michaelson. Tangled Tapes: Infinity, Interaction and Turing Machines

• Alexander Kartzow, Jiamou Liu and Markus Lohrey. Tree-Automatic Well-Founded Trees

• Anat Lerner and Rica Gonen. A Difficulty in the Concept of Constrained Combinatorial Auctions

• Jonathan Sherratt. Turing Patterns in Deserts

• Jaap Kaandorp, Daniel Botman, Carlos Tamulonis and Roland Dries. Multi-scale modeling of gene regulation of morphogenesis

• Anton Konovalov and Victor Selivanov. Boolean Algebras of Regular $\omega$-Languages

• Kenneth Johnson and John V. Tucker. A Model of Computing as a Service for Cost Analysis

• Amin Farjudian. A Semantic Framework for Real Computation with Polynomial Enclosures

• Victor Ocasio. Turing Computable Embeddings and Coding Families of Sets

• Loizos Michael. Computability of Narrative

• Darren Abramson. Alan Turing's Defence of Multiple Realizability

• Peter Koepke and Benjamin Seyfferth. Towards a theory of infinite time Blum-Shub-Smale machines

• Michael Gabbay. A simple and complete model theory for intensional and extensional untyped lambda-equality

• Gordana Dodig Crnkovic. Alan Turing's Legacy: Info-Computational Philosophy of Nature

• Stephane Massonet. Turing and Wittgenstein : from the Foundations of Mathematics to Intelligent machines

• Ken-Ichi Hiraga, Kota Ikeda and Takashi Miura. Slow and steady wins the race: relationship between accuracy and speed of Turing pattern formation

• Martin Huschenbett. Word Automaticity of Tree Automatic Scattered Linear Orderings Is Decidable

• Nir Fresco. Computation as Instructional Information Processing

• Thomas Fernique and Mathieu Sablik. Local Rules for Computable Planar Tilings

• Yuri Gurevich. Foundational Analyses of Computation

• Anders Johansson and James Zou. A Slime Mold Solver for Linear Programming Problems

• Jean Lasségue and Giuseppe Longo. What is Turing's comparison between Mechanism and Writing worth?

• Arturo Berrones. Phase transitions and long range dependence in connectionist models for approximately Bayesian learning

• Aziz F. Zambak. Mind as IRIS versus Mind as Machine

• Shigeru Kondo. Turing pattern formation without diffusion

• Kerstin Andersson. Numerical evaluation of the average number of successive guesses

• Akitoshi Kawamura, Norbert Müller, Carsten Rösnick and Martin Ziegler. Uniformly Polynomial-time Computable Operators on Analytic Functions

• Jeffrey Watumull. The Linguistic Turing Machine

• Tommaso Flaminio and Hykel Hosni. De Finettib SakalauskaitDs bets on partially evaluated frames

• Alexander Kulikov, Olga Melanich and Ivan Mihajlin. A $5n-o(n)$ Lower Bound on the Circuit Size over $U_2$ of a Linear Boolean Function

• Ugo Dal Lago, Tobias Heindel, Damiano Mazza and Daniele Varacca. Computational Complexity of Interactive Behaviors

• Bastien Le Gloannec and Nicolas Ollinger. Substitutions and Strongly Deterministic Tilesets

• Julian-Steffen Müller. Complexity of Model Checking in Modal Team Logic

• Bahareh Afshari and Michael Rathjen. An Ordinal Analysis of the Infinite Ramsey Theorem

• Les Hatton. Equilibriation of information in software systems

• Makoto Fujiwara, Kojiro Higuchi and Takayuki Kihara. Recursive marriage theorems and reverse mathematics

• Armando Matos and Andreia Teixeira. Lossless compressors: Degree of compression and optimality

• Lorenzo Carlucci and Konrad Zdanowski. A note on Ramsey Theorems and Turing Jumps

• Andris Ambainis, ArtE+rs BaD kurs, Kaspars Balodis, Agnis E kuE!kovniks, Juris Smotrovs and Madars Virza. Worst case analysis of non-local games

• Filip Mazowiecki. Continuous reductions of regular languages

• Sam Sanders. Nonstandard Analysis: A New Way to Compute

• Matthew Szudzik. Is Turing's Thesis the Consequence of a More General Physical Principle?

• Pierre Clairambault, Julian Gutierrez and Glynn Winskel. Imperfect Information in Logic and Concurrent Games

• Andreas Weiermann and Alan Woods. Some natural zero one laws for ordinals below $\varepsilon_0$

• Liesbeth De Mol and Maarten Bullynck. A short history of small machines

• Szilard Zsolt Fazekas, Robert Mercas and Kayoko Shikishima-Tsuji. Palindromic completion, a basis for hairpin completion

• A.C. Cem Say and Abuzer Yakaryilmaz. Finite state verifiers with constant randomness

• Madalina Erascu and Tudor Jebelean. Automated Certification of a Logic-Based Verification Method for Imperative Loops

• Ray Dougherty. Turing Machines Expressed as Chomsky Grammars in the Computational Universe: A Linguistic Complexity Metric

• F. Félix Lara-Martín and Andrés Cordón-Franco. Local Induction and Provably Total Computable Functions: a Case Study

• Vladimir Podolskii. Lower Bound on Weights of Large Degree Threshold Functions

• Ludwig Staiger. A Correspondence Priciple for Exact Constructive Dimension

• Simon Perdrix. A Lower Bound on the Minimal Resources for Measurement-based Quantum Computation

• Rusins Freivalds. Ultrametric versus Archimedean automata

• Takayuki Kihara. A hierarchy of immunity and density for sets of reals

• Lars Kristiansen. Degrees of Total Algorithms versus Degrees of Honest Functions

• Jose Espirito Santo. Intuitionistic asymmetry: how the CBN/CBV duality breaks

• Juris Hartmanis. Turing Machine-Inspired Computer Science Results

• Oznur Yasar Diner, Dariusz Dereniowski and Danny Dyer. Complexity of Fast Searching

• Péter Németi and Gergely Székely. Existence of Faster Than Light Signals Implies Hypercomputation Already in Special Relativity

• Loizos Michael. Simultaneous Learning and Prediction

• Vladimir Aristov and Andrey Stroganov. Computing without a computer: the analytical calculus by means of formalizing classical computer operations

• Eric Allender. Curiouser and Curiouser: The Link between Incompressibility and Complexity

• Alexander Golovnev. Approximating Asymmetric Metric TSP in Exponential Time

• David Edward Bruschi, Ivette Fuentes, Jorma Louko, Antony Lee, Andrzej Dragan and Nicolai Friis. Cavity motion affects entanglement

• Paul Smolensky. Subsymbolic Computation Theory for the Human Intuitive Processor

• Roussanka Loukanova. Computational Syntax-Semantics Interface of Passive Forms with The Language of Acyclic Recursion

• Anderson De Araújo. The world as a quantum system with degrees of incomputability

• Francis George Cabarle and Henry Adorna. Notes on Spiking Neural P systems and Petri nets

• Yoshihiro Horihata. Weak theories of concatenation and their mutual interpretabilities

• Arno Pauly. Multi-valued functions in Computability Theory

• Kojiro Higuchi and Takayuki Kihara. Effective strong nullness and effectively closed sets

• Marc Groz. Systems and Methods for Data and Algorithm Protection

• James Royer and Norman Danner. Ramified Structural Recursion and Corecursion

• Vladimir Kanovei. On effective sigma-boundedness and sigma-compactness

• Patrick Allo. Information and Logical Discrimination

• Petar Iliev and Wiebe Van Der Hoek. On the relative succinctness of "everybody knows" and "somebody knows"

• Natasha Jonoska, Mile Krajcevski and Gregory McColm. Languages Associated with Crystallographic Structures

• Rebecca Steiner. Low_n Boolean Subalgebras

• Roland Omanadze. ON SOME PROPERTIES OF MAXIMAL SETS AND Q-REDUCIBILITY

• Chengling Fang, Shenling Wang and Guohua Wu. Highness and Local Noncappability

• George Barmpalias and Zhenhao Li. Limit computable infinite games and the structure of their degrees of non-determinacy

• Luis Antunes, Andre Souto and Andreia Teixeira. Roboustness of Logical Depth

• Yoshii Keisuke and Kazuyuki Tanaka. Infinite Games and Transfinite Recursion of Multiple Inductive Definitions

• Vasco Brattka, StC)phane Le Roux and Arno Pauly. On the Computational Content of the Brouwer Fixed Point Theorem

• Masaharu Mizumoto. How to Accept Turingb SakalauskaitDs Thesis without Proof or Conceptual Analysis without also Feeling Qualms

• Ningning Peng, Kojiro Higuchi, Takeshi Yamazaki and Kazuyuki Tanaka. Relative Randomness for Martin-Löf random Sets

• Robert Rettinger. Compactness and the Effectivity of Uniformization

• Ranjit Kumar Upadhyay and Sharada Nandan Raw. Emergence of Spatial Patterns in a Model of Wetland Ecosystem: Application to Keoladeo National Park (KNP), India

• Krishna S, Bogdan Aman and Gabriel Ciobanu. On the Computability Power of Membrane Systems with Controlled Mobility

• Gregorio de Miguel Casado and Manuel GonzC!lez Bedia. Revamping the Turing Test by Computing Interaction Coupling

• Daniela Genova and Kalpana Mahalingam. Generating DNA Code Words Using Forbidding and Enforcing Systems

• Peter Schuster and Matthew Hendtlass. A direct proof of Wiener's theorem

• Andrey Verisokin, Darya Verveyko and Eugene Postnikov. Mathematical Model of Chaotic Oscillations and Oscillatory Entrainment in Glycolysis Originated from Periodic Substrate Supply

• Darya Verveyko and Andrey Verisokin. Computer Simulation of Three-Dimensional Reaction-Diffusion Models. Case Study: Glycolytic Reaction in Open Spatial Reactor

• Leon N Cooper and Sorin Istrail. Mental Experience and The Turing Test: This Double Face is the Face of Mathematics

• Mike Stannett and Marian Gheorghe. Using Topological Spaces to Control P Systems

• Di Cai and Lee McCluskey. A simple method for estimating term mutual information

• Sorin Istrail and Solomon Marcus. Turing and von Neumannb SakalauskaitDs Brains and their Computers

• Alexander Shen. Game arguments in computability theory and algorithmic information theory

• Cameron Buckner. Heuristics and Intelligence: Realizing Turing's Vision

• Pierre Guillon and Charalampos Zinoviadis. Densities and entropies in cellular automata

• Grant Passmore and Paul Jackson. Abstract Partial Cylindrical Algebraic Decomposition I: The Lifting Phase

• Dan Brumleve, Joel David Hamkins and Philipp Schlicht. The mate-in-n problem of infinite chess is decidable

• Benedek Nagy, Veronika Halsz, Lszl Heged and Istvn Hornyk. Solving application oriented graph theoretical problems with DNA computing

• Jesse Alama and Reinhard Kahle. A Turing test for mathematical proof

• Marcin Mostowski. On representing real numbers in finite models

• Bruno Bauwens. A simple proof that complexity of complexity can be large, and that some strings have maximal plain Kolmogorov complexity and non-maximal prefix-free complexity.

• Jole Costanza, Claudio Angione, Pietro Lio' and Giuseppe Nicosia. Bacteria as Unconventional Computing Architectures

• Alison Pease and Ursula Martin. Summary of an ethnographic study of the third Mini-Polymath project

• Mikhail Peretyatkin. On the Tarski-Lindenbaum algebra of the class of all strongly constructivizable prime models

• Amos Golan. On the Foundations and Philosophy of Info-Metrics

• Peter Cholak, Damir Dzhafarov and Jeffry Hirst. Computable Mathias genericity

• Joerg Zimmermann and Armin Cremers. Making Solomonoff Induction Effective or You Can Learn What You Can Bound

• Sorin Istrail. Computer Science Through Urn Games: A Unified Framework for a Hierarchy of Solvable and Unsolvable Problems

• Liliana Badillo and Charles Harris. 1-GENERICITY AND THE N _2 ENUMERATION DEGREES

• Anargyros Sarafopoulos. A Shape Language for IFS

• Anargyros Sarafopoulos. Evolutionary Art, Artificial Critics, Aesthetics, and Hilbert's Entscheidungsproblem

• Sori Lee. Trees for Subtoposes of the Effective Topos

• Assel Altayeva and Iman Poernomo. Towards a Type Theory of Predictable Assembly

• Anupam Das. Complexity of Deep Inference via Flow Graphs

• S. G. Sterrett. Bringing Up Turing's Child-Machine

• Michael Beeson. The logic of ruler and compass constructions

• Bruce Edmonds and Carlos Gershenson. Learning, Social Intelligence and the Turing Test - why an 'out-of-the-box' Turing Machine will not pass the Turing Test

• Ralph Wojtowicz, Selmer Bringsjord and John Hummel. Dynamic Semantics of Tau-N Theories

• Selmer Bringsjord. Unrevealably Secret Communication

• Brian Josephson. On Origins

• Francis Rousseaux, Alain Bonardi and Antoine Vincent. Collection-centred Analysis and Design: Because We Should Not Have to Choose between Ludus and Paidia while Building our Interactive AI Systems

• Chung-Chih Li. Is the Trilogy of Gap, Compression, and Honesty Theorems Still Significant at Type-2?

• Aliakbar Safilian and Farzad Didehvar. Two New Reducibilities Based on Enumeration Orders

• Veronica Becher. Turing's normal numbers: towards randomness

• Olivier Finkel. Ambiguity of $\omega$-Languages of Turing Machines

• Merlin Carl. The distribution of the ITRM-recognizable reals

• Stephen Flood.Reverse mathematics and a packed Ramsey's theorem.

• Ivor Grattan-Guinness. On mathematicians who liked logic: the case of Max Newman

• Martin Lackner. Permutation Pattern Matching and its Parameterized Complexity

• Oron Shagrir. Who is the human computer?

• Manuel Mazzara, Antonio Marraffa, Luca Biselli, Simona De Nicola, Luca Chiarabini and Georgios Papageorgiou. Social Networks and Collective Intelligence

• Nurlan Kogabaev. On Uniform Computability in Familiar Classes of Projective Planes

• Margarita Leontyeva. On Existence of Strongly Computable Copies in Class of Boolean Algebras

• Shannon Bohle. Virtual Worlds as Portals for Information Discovery

• Stefan Gerdjikov. On modernisation of historic texts

• Philippe Codognet. Turing, Mechanism and Blind Thought

• Philippe Codognet. Turing, Search and Randomness

• Mariam Thalos and Oliver Richardson. The Morals of the St. Petersburg Game: Why Statistical Distributions Matter in Prospect Appraisal

• Maarten Mckubre-Jordens and Douglas Bridges. Solving the Dirichlet Problem Constructively

• Ekaterina Fokina and Pavel Semukhin. Structures presentable by various types of automata

• Sebastian Eberhard. Applicative theories for logarithmic complexity classes

• Douglas Cenzer. Bounded Primitive Recursive Randomness

• Russell Miller and Keng Meng Ng. Finitary reducibility on equivalence relations

• Eugene Postnikov. Computational model based on meteorological data for forecast of influenza-like activity

• Andrey Sariev. The Omega-Turing degrees

• Paola Zizzi. The modality "Future" for Probably true propositions

• Marcin Miłkowski. Functional account of computational explanation

• Matthew Wright. Degrees of Relations on Ordinals

• Rema Padman, Daniel Neill and Chris Harle. Information Visualization for Cognitively Guided Health Risk Assessment

• Francisco Javier Almaguer, Arturo Berrones, Mónica Alcalá, Roberto Soto, Oscar Chacón, Ricardo Sánchez and Eduardo Soto. Cellular Automaton in Aggregation and Breaking Processes

• Nazif Khisamiev. On positive and constructive groups

• Duilio D'Alfonso. Algorithmic complexity, entropy and the discovery of causal correlations

• Ivan Georgiev. Complexity of the Continued Fractions of Some Subrecursive Real Numbers

• Sergey Ospichev. Computable Numberings in Ershov Hierarchy

• Dirk Sudholt. The Analysis of Evolutionary Algorithms: Why Evolution is Faster With Crossover

• Laurentiu Leustean and Ulrich Kohlenbach. Applications of proof mining in nonlinear ergodic theory

• Magnus Gausdal Find and Joan Boyar. Cancellation-free circuits: An approach for proving superlinear lower bounds for linear Boolean operators

• Akitoshi Kawamura, Carsten Rösnick and Martin Ziegler. Second-Order Representations and Complexity of Compact Sets

• Martin Adamcik. Collective Reasoning under Uncertainty and Inconsistency

• Yiorgos Stavrinos and Anastasia Veneti. Linear aspects of Intersection

• Benoît Monin.Von Neumann's Biased Coin Revisited

• Sabine Broda, António Machiavelo, Nelma Moreira and Rogério Reis.Average Case Complexity of epsilon-NFA's

• Peter Wegner. Alan Turing and Maurice Wilkes

• Hristo Ganchev. The enumeration degrees of semi-recursive sets

• Catherine Stinson. Computational Models as Mechanistic Explanations

• Michael Olinick. An Undergraduate Course on the Life and Work of Alan Turing

• Yun Shang, Xian Lu and Ruqian Lu. Closure Properties of Quantum Turing Languages

• Sergei Podzorov. Computable C-classes

• Hideki Tsuiki. Multi-Head Extension of the Turing Machine for Computation over Continuous Spaces

• Sylvain Gravier, Jérôme Javelle, Mehdi Mhalla and Simon Perdrix. Quantum Secret Sharing with Graph States

• Eva Maia, Nelma Moreira and Rogério Reis. Incomplete Transition Complexity of Basic Operations on Regular Languages

• Iosif Petrakis and Helmut Schwichtenberg. On the Formal Constructive Theory of Computable Functionals TCF+

• Mitko Yanchev. Part restrictions: adding new expressiveness in Description Logics

• Joost Joosten. Turing Machines, revisited

• Karel Chvalovský. Linearization of Proofs in Propositional Hilbert Systems

• Loizos Michael. Ant Collectives as Computers

• Kenji Miyamoto and Helmut Schwichtenberg.Program Extraction with Nested Inductive/Coinductive Definitions

• Juan Carlos Garcia Escartin and Pedro Chamorro-Posada. Distributed computation in inertial frames

• Pieter Adriaans. Facticity as the amount of self-descriptive information in a data set

• Marcin J. Schroeder. Structural versus Quantitative Characteristics of Information in Information Processing: Information Integration

• Loizos Michael. WebSense: Making Sense of the Web

• Yoshihiro Maruyama. Categorical Universal Logic: Duality as Semantics

• Benjamin Rin. The computational strengths of alpha-length infinite time Turing machines

• Christopher Porter. On Analogues of the Church-Turing Thesis in Algorithmic Randomness

• Naveen Sundar Govindarajulu. Uncomputable Games: Toward Crowd-sourced Solving of Truly Difficult Problems

• Ranjit Nair. Computablity, physics and logic

• Naohi Eguchi and Andreas Weiermann. A Simplified Characterisation of Provably Computable Functions of the System ID_1

• Lindsay Smith, Vito Veneziano and Paul Wernick. The use of language and logic in computing technology

• Joseph Shipman and John H. Conway. Geometric Models of Computation

• Ashish Umre and Ian Wakeman. Dynamics of Agent Interaction: In search of strategies for optimal information sharing

• Alison Pease, Simon Colton and John Charnley. The Turing Test and Computational Creativity

• Carolina Blasio and João Marcos.Logical Agency: consulting instead of collecting

• Francisco Hernández-Quiroz and Pablo Padilla-Longoria. Some constraints on the physical realizability of a mathematical construction

• Jiamou Liu. An Automata Model for Computations Over an Arbitrary Structure

• Peter Koepke and Andrey Morozov. On the Computational Strength of Infinite Time Blum Shub Smale Machines

• Hajnal Andréka and István Németi. Turing-degree of first-order logic FOL and reducing FOL to a propositional logic