Information and evidence in logic systems

Abstract A logic system S describing a world W contains certain semantic information about W. We define the quantity of this information, and the information value of a formula F relative to S. A close relationship is revealed between the information of S and the cardinality of the set of its models. Then the evidence of a formula F given by S is introduced providing basis for reasoning by evidence as a way of non-monotonic reasoning. The latter is presented in the form of plausible world assumption, PWA possessing certain advantages over the known approaches to non-monotonic reasoning: PWA computes a plausible model for any first-order logic system, and provides a high degree of relative monotonicity to its beliefs. The most important feature of PWA is its semantic plausibility demonstrated by a number of examples.

[1]  J. Keynes.  A Treatise on Probability. , 1923 .

[2]  R. Hartley.  Transmission of information , 1928 .

[3]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[4]  Norbert Wiener,et al.  Cybernetics: Control and Communication in the Animal and the Machine. , 1949 .

[5]  E. Jaynes.  Information Theory and Statistical Mechanics , 1957 .

[6]  M. Tribus.  Information Theory as the Basis for Thermostatics and Thermodynamics , 1961 .

[7]  Henry Ely Kyburg,et al.  Probability and Inductive Logic , 1970 .

[8]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.

[9]  Edward H. Shortliffe,et al.  Computer-based medical consultations, MYCIN , 1976 .

[10]  Keith L. Clark,et al.  Negation as Failure , 1987, Logic and Data Bases.

[11]  Raymond Reiter.  On Closed World Data Bases , 1977, Logic and Data Bases.

[12]  I. Good,et al.  The Maximum Entropy Formalism. , 1979 .

[13]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..

[14]  E. T. Jaynes,et al.  Where do we Stand on Maximum Entropy , 1979 .

[15]  John McCarthy,et al.  Circumscription - A Form of Non-Monotonic Reasoning , 1980, Artif. Intell..

[16]  P. M. Williams.  Bayesian Conditionalisation and the Principle of Minimum Information , 1980, The British Journal for the Philosophy of Science.

[17]  Raymond Reiter,et al.  A Logic for Default Reasoning , 1987, Artif. Intell..

[18]  John McCarthy,et al.  SOME PHILOSOPHICAL PROBLEMS FROM THE STANDPOINT OF ARTI CIAL INTELLIGENCE , 1987 .

[19]  S. Ayme,et al.  APPROXIMATE REASONING IN MEDICAL GENETICS , 1981 .

[20]  Jack Minker,et al.  On Indefinite Databases and the Closed World Assumption , 1987, CADE.

[21]  Matthew L. Ginsberg,et al.  Non-Monotonic Reasoning Using Dempster's Rule , 1984, AAAI.

[22]  John McCarthy,et al.  Applications of Circumscription to Formalizing Common Sense Knowledge , 1987, NMR.

[23]  David Heckerman,et al.  Probabilistic Interpretation for MYCIN's Certainty Factors , 1990, UAI.

[24]  Vladimir Lifschitz,et al.  Computing Circumscription , 1985, IJCAI.

[25]  C. R. Smith,et al.  Maximum-Entropy and Bayesian Methods in Inverse Problems , 1985 .

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

[27]  J. Aczél,et al.  Maximum Entropy and Bayesian Methods in Applied Statistics: Generalized Entropies and the Maximum Entropy Principle , 1986 .

[28]  James F. Baldwin,et al.  Evidential support logic programming , 1987 .

[29]  Wlodek Zadrozny,et al.  Intended Models, Circumscription and Commonsense Reasoning , 1987, IJCAI.

[30]  R. Loui.  Response to Hanks and McDermott: Temporal Evolution of Beliefs and Beliefs about Temporal Evolution , 1987, Cogn. Sci..

[31]  V. Lifschitz,et al.  The Stable Model Semantics for Logic Programming , 1988, ICLP/SLP.

[32]  Kenneth A. Ross,et al.  Unfounded sets and well-founded semantics for general logic programs , 1988, PODS.

[33]  Adrian Walker,et al.  Towards a Theory of Declarative Knowledge , 1988, Foundations of Deductive Databases and Logic Programming..

[34]  Kurt Konolige,et al.  Representing defaults with epistemic concepts , 1989, Comput. Intell..

[35]  Joseph Y. Halpern.  An Analysis of First-Order Logics of Probability , 1989, IJCAI.

[36]  Judea Pearl,et al.  Probabilistic Semantics for Nonmonotonic Reasoning: A Survey , 1989, KR.

[37]  Donald Nute,et al.  Defeasible Logic and The Frame Problem , 1990 .

[38]  Noam Nisan,et al.  Approximate Inclusion-Exclusion , 1990, Comb..

[39]  Edward H. Shortliffe,et al.  The Dempster-Shafer theory of evidence , 1990 .

[40]  Michael Luby,et al.  On deterministic approximation of DNF , 1991, STOC '91.

[41]  V. S. Subrahmanian,et al.  Relating Dempster-Shafer Theory to Stable Semantics , 1991, ISLP.

[42]  Eliezer L. Lozinskii.  Counting Propositional Models , 1992, Inf. Process. Lett..

[43]  Moisés Goldszmidt,et al.  A Maximum Entropy Approach to Nonmonotonic Reasoning , 1990, IEEE Trans. Pattern Anal. Mach. Intell..