Quantitative Logic Reasoning

In this paper we show several similarities among logic systems that deal simultaneously with deductive and quantitative inference. We claim it is appropriate to call the tasks those systems perform as Quantitative Logic Reasoning. Analogous properties hold throughout that class, for whose members there exists a set of linear algebraic techniques applicable in the study of satisfiability decision problems. In this presentation, we consider as Quantitative Logic Reasoning the tasks performed by propositional Probabilistic Logic; first-order logic with counting quantifiers over a fragment containing unary and limited binary predicates; and propositional Łukasiewicz Infinitely-valued Probabilistic Logic.

[1]  Franz Baader,et al.  Cardinality Restrictions on Concepts , 1994, KI.

[2]  Christos H. Papadimitriou,et al.  A linear programming approach to reasoning about probabilities , 1990, Annals of Mathematics and Artificial Intelligence.

[3]  Armin Biere.  Lingeling Essentials, A Tutorial on Design and Implementation Aspects of the the SAT Solver Lingeling , 2014, POS@SAT.

[4]  Joost P. Warners,et al.  A Linear-Time Transformation of Linear Inequalities into Conjunctive Normal Form , 1998, Inf. Process. Lett..

[5]  A. Mostowski.  On a generalization of quantifiers , 1957 .

[6]  P. Walley,et al.  Direct algorithms for checking consistency and making inferences from conditional probability assessments , 2004 .

[7]  Marcelo Finger,et al.  Probabilistic Satisfiability: Logic-Based Algorithms and Phase Transition , 2011, IJCAI.

[8]  Diego Calvanese,et al.  DL-Lite: Tractable Description Logics for Ontologies , 2005, AAAI.

[9]  Franz Baader,et al.  Pushing the EL Envelope , 2005, IJCAI.

[10]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[11]  Martin Otto,et al.  On Logics with Two Variables , 1999, Theor. Comput. Sci..

[12]  Marcelo Finger,et al.  Probably Half True: Probabilistic Satisfiability over Łukasiewicz Infinitely-Valued Logic , 2018, IJCAR.

[13]  B. D. Finetti.  La prévision : ses lois logiques, ses sources subjectives , 1937 .

[14]  Tommaso Flaminio,et al.  The coherence of Lukasiewicz assessments is NP-complete , 2010, Int. J. Approx. Reason..

[15]  J. Eckhoff.  Helly, Radon, and Carathéodory Type Theorems , 1993 .

[16]  Gai CarSO.  A Logic for Reasoning about Probabilities * , 2004 .

[17]  Marcelo Finger,et al.  Algorithms for Deciding Counting Quantifiers over Unary Predicates , 2017, AAAI.

[18]  D. Mundici,et al.  Algebraic Foundations of Many-Valued Reasoning , 1999 .

[19]  Phokion G. Kolaitis,et al.  On the Decision Problem for Two-Variable First-Order Logic , 1997, Bulletin of Symbolic Logic.

[20]  B. D. Finetti,et al.  Theory of Probability: A Critical Introductory Treatment , 2017 .

[21]  George Boole,et al.  An Investigation of the Laws of Thought: Frontmatter , 2009 .

[22]  Ian Pratt-Hartmann.  On the Computational Complexity of the Numerically Definite Syllogistic and Related Logics , 2008, Bull. Symb. Log..

[23]  Marcelo Finger,et al.  Probabilistic satisfiability: algorithms with the presence and absence of a phase transition , 2015, Annals of Mathematics and Artificial Intelligence.

[24]  Marcelo Finger,et al.  Measuring inconsistency in probabilistic logic: rationality postulates and Dutch book interpretation , 2015, Artif. Intell..

[25]  Fabio Gagliardi Cozman,et al.  Towards classifying propositional probabilistic logics , 2014, J. Appl. Log..

[26]  D. Mundici,et al.  A Rényi Conditional in Łukasiewicz Logic , 2011 .

[27]  Barnaby Martin,et al.  Constraint Satisfaction with Counting Quantifiers , 2012, CSR.

[28]  Alfred Tarski,et al.  Measures in Boolean algebras , 1948 .

[29]  Niklas Sörensson,et al.  An Extensible SAT-solver , 2003, SAT.

[30]  Daniele Mundici,et al.  Bookmaking over infinite-valued events , 2006, Int. J. Approx. Reason..

[31]  Daniele Pretolani,et al.  Easy Cases of Probabilistic Satisfiability , 2001, Annals of Mathematics and Artificial Intelligence.

[32]  Ian Pratt-Hartmann.  Complexity of the Two-Variable Fragment with Counting Quantifiers , 2005, J. Log. Lang. Inf..

[33]  D. Mundici.  Advanced Łukasiewicz calculus and MV-algebras , 2011 .

[34]  Pierre Hansen,et al.  Models and Algorithms for Probabilistic and Bayesian Logic , 1995, IJCAI.

[35]  Perlindström.  First Order Predicate Logic with Generalized Quantifiers , 1966 .

[36]  Pierre Hansen,et al.  Column Generation Methods for Probabilistic Logic , 1989, INFORMS J. Comput..

[37]  Ndrei,et al.  Galois correspondence for counting quantifiers , 2013 .

[38]  David K. Smith.  Theory of Linear and Integer Programming , 1987 .

[39]  John N. Tsitsiklis,et al.  Introduction to linear optimization , 1997, Athena scientific optimization and computation series.

[40]  Theodore Hailperin,et al.  Boole's logic and probability , 1976 .

[41]  B. Finetti.  Sul significato soggettivo della probabilità , 1931 .

[42]  Nils J. Nilsson,et al.  Probabilistic Logic * , 2022 .

[43]  Niklas Sörensson,et al.  Translating Pseudo-Boolean Constraints into SAT , 2006, J. Satisf. Boolean Model. Comput..

[44]  Christos H. Papadimitriou,et al.  Probabilistic satisfiability , 1988, J. Complex..

[45]  B. D. Finetti,et al.  Foresight: Its Logical Laws, Its Subjective Sources , 1992 .