Foundations for Knowledge
... An important result of this work is that reasoning about knowledge after the execution of actions can be reduced to classical reasoning from an initial first-order theory. However, it is limited in that tests can only refer to what is known about the current state, knowledge about knowledge is not c ...
... An important result of this work is that reasoning about knowledge after the execution of actions can be reduced to classical reasoning from an initial first-order theory. However, it is limited in that tests can only refer to what is known about the current state, knowledge about knowledge is not c ...
Refinement Modal Logic
... equipped with may-transitions and must-transitions. A must-transition is available in every component that implements the modal specification, while a may-transition need not be. This is close to our definition of refinement, as it also is some kind of submodel quantifier, but the two notions are in ...
... equipped with may-transitions and must-transitions. A must-transition is available in every component that implements the modal specification, while a may-transition need not be. This is close to our definition of refinement, as it also is some kind of submodel quantifier, but the two notions are in ...
Part I: Understanding String Theory
... well as adequately frame the question it attempts to answer: What is the fundamental nature of the universe, and how can we describe it? The first concept to analyze is that of unification in Chapter 3, since string theory is a theory of everything, it must somehow combine all the existing framework ...
... well as adequately frame the question it attempts to answer: What is the fundamental nature of the universe, and how can we describe it? The first concept to analyze is that of unification in Chapter 3, since string theory is a theory of everything, it must somehow combine all the existing framework ...
LHC Theory Lecture 1: Calculation of Scattering Cross Sections
... Proton-proton collider: bunches of protons are brought to collision at a center of mass (cms) energy of several TeV /c 2 (Tera ≡ T = 1012 ). About 115 billion particles per bunch of ∼ 7cm length and the diameter of a hair! As protons are very small objects (6 o of about 1fm=10−15 m), nearly all part ...
... Proton-proton collider: bunches of protons are brought to collision at a center of mass (cms) energy of several TeV /c 2 (Tera ≡ T = 1012 ). About 115 billion particles per bunch of ∼ 7cm length and the diameter of a hair! As protons are very small objects (6 o of about 1fm=10−15 m), nearly all part ...
Horn formula minimization - RIT Scholar Works
... and as an inference rule for refutation, complete. We will explain the soundness and completeness of resolution as it applies to Boolean formulas as we illustrate resolution with an example. Resolution is based on the basic inference rule: ((p ∨ F1 ) ∧ (p ∨ F2 )) → (F1 ∨ F2 ) Essentially, resolution ...
... and as an inference rule for refutation, complete. We will explain the soundness and completeness of resolution as it applies to Boolean formulas as we illustrate resolution with an example. Resolution is based on the basic inference rule: ((p ∨ F1 ) ∧ (p ∨ F2 )) → (F1 ∨ F2 ) Essentially, resolution ...
Set theory and logic
... to give self-contained characterizations in turn of the system of integers, of rational numbers, and, finally, of real numbers. This is clone in the last three sections of the chapter. Finally, there is Chapter 9, which is an introductory account of relatively recent investigations of the foundation ...
... to give self-contained characterizations in turn of the system of integers, of rational numbers, and, finally, of real numbers. This is clone in the last three sections of the chapter. Finally, there is Chapter 9, which is an introductory account of relatively recent investigations of the foundation ...
An Introduction to Proof Theory - UCSD Mathematics
... Proof Theory is concerned almost exclusively with the study of formal proofs: this is justified, in part, by the close connection between social and formal proofs, and it is necessitated by the fact that only formal proofs are subject to mathematical analysis. The principal tasks of Proof Theory can ...
... Proof Theory is concerned almost exclusively with the study of formal proofs: this is justified, in part, by the close connection between social and formal proofs, and it is necessitated by the fact that only formal proofs are subject to mathematical analysis. The principal tasks of Proof Theory can ...
Notes on Mathematical Logic David W. Kueker
... P , Q) without any standard or intuitive meanings to mislead one. Thus the fundamental building blocks of our model are the following: (1) a formal language L, (2) sentences of L: σ, θ, . . ., (3) interpretations for L: A, B, . . ., (4) a relation |= between interpretations for L and sentences of L, ...
... P , Q) without any standard or intuitive meanings to mislead one. Thus the fundamental building blocks of our model are the following: (1) a formal language L, (2) sentences of L: σ, θ, . . ., (3) interpretations for L: A, B, . . ., (4) a relation |= between interpretations for L and sentences of L, ...
Predicate logic definitions
... A derivation in PDE is a series of sentences of PLE, each of which is either an assumption or is obtained from previous sentences by one of the rules of PDE. A sentence P of PLE is derivable in PDE from a set Γ of sentences of PLE, written S ` P, iff there exists a derivation in PDE in which all the ...
... A derivation in PDE is a series of sentences of PLE, each of which is either an assumption or is obtained from previous sentences by one of the rules of PDE. A sentence P of PLE is derivable in PDE from a set Γ of sentences of PLE, written S ` P, iff there exists a derivation in PDE in which all the ...
Euclidian Roles in Description Logics
... Figure 1: The (a) Euclidean closure, (b) grid, (c) role hierarchy and (d) coincidence enforcing As it happens with many role axioms of SRIQ [2], it comes to no surprise that Euclidian roles are a syntactic sugar for SRIQ. More precisely, an axiom Eucl(R) can be represented by the RIA R− R v R. This ...
... Figure 1: The (a) Euclidean closure, (b) grid, (c) role hierarchy and (d) coincidence enforcing As it happens with many role axioms of SRIQ [2], it comes to no surprise that Euclidian roles are a syntactic sugar for SRIQ. More precisely, an axiom Eucl(R) can be represented by the RIA R− R v R. This ...
logic for the mathematical
... also touch briefly on the use in software engineering/programme verification of languages (if not theorems) from logic. There is at least one other major reason for wanting to know this subject: instead of using elementary mathematical techniques to study logic, as we do here, people have in recent ...
... also touch briefly on the use in software engineering/programme verification of languages (if not theorems) from logic. There is at least one other major reason for wanting to know this subject: instead of using elementary mathematical techniques to study logic, as we do here, people have in recent ...
dialogues between abelard and eloise
... Karttunen 1984, Hyttinen 1990) and Vaught formulae (Vaught 1973, Makkai 1977), which all extend First-order logic. They all are very naturally defined using games. In this chapter we will be concerned with one family of alternative ways of defining semantically important notions in logics: game-theo ...
... Karttunen 1984, Hyttinen 1990) and Vaught formulae (Vaught 1973, Makkai 1977), which all extend First-order logic. They all are very naturally defined using games. In this chapter we will be concerned with one family of alternative ways of defining semantically important notions in logics: game-theo ...
Modal fixpoint logic: some model theoretic questions
... Richard Ladner [FL79]. One disadvantage of Dynamic Logic is that unlike temporal logics, it is not adequate for modeling non-terminating programs. Extensions of PDL that can capture some specific infinite behaviors (see for instance [Har84]) have been studied by Robet S. Streett [Str81, Str82] (Delt ...
... Richard Ladner [FL79]. One disadvantage of Dynamic Logic is that unlike temporal logics, it is not adequate for modeling non-terminating programs. Extensions of PDL that can capture some specific infinite behaviors (see for instance [Har84]) have been studied by Robet S. Streett [Str81, Str82] (Delt ...
Mathematical Logic
... Any of these assumptions carries a marker . As markers we use assumption variables ¤0 , ¤1 , . . . , denoted by u, v, w, u0 , u1 , . . . . The (previous) variables will now often be called object variables, to distinguish them from assumption variables. If at a later stage (i.e. at a node below an a ...
... Any of these assumptions carries a marker . As markers we use assumption variables ¤0 , ¤1 , . . . , denoted by u, v, w, u0 , u1 , . . . . The (previous) variables will now often be called object variables, to distinguish them from assumption variables. If at a later stage (i.e. at a node below an a ...
The Metaphysics of Quantities and Their Dimensions˚
... structure of the quantities they are scales for. I won’t go into what that intrinsic structure is in a great deal of detail; it will be enough for our purposes to note that the property of having 4 kg mass is, in some sense, “double” the property of having 2 kg mass, and that it is in virtue of this ...
... structure of the quantities they are scales for. I won’t go into what that intrinsic structure is in a great deal of detail; it will be enough for our purposes to note that the property of having 4 kg mass is, in some sense, “double” the property of having 2 kg mass, and that it is in virtue of this ...
Completeness or Incompleteness of Basic Mathematical Concepts
... concept of set”; of the possibility that new axioms will be found via “more profound understanding of the concepts underlying logic and mathematics.”11 There is nothing to suggest that perception of sets could help in finding new axioms or played a role in finding the old ones. A second relevant-loo ...
... concept of set”; of the possibility that new axioms will be found via “more profound understanding of the concepts underlying logic and mathematics.”11 There is nothing to suggest that perception of sets could help in finding new axioms or played a role in finding the old ones. A second relevant-loo ...
Document
... which do not have any model, meaning that there is no interpretation such that each constraint is satisfied. In this case, an interesting question is to determine an interpretation which maximizes the number of satisfied constraints: this problem is called Max-SAT [10]. We can generalize this proble ...
... which do not have any model, meaning that there is no interpretation such that each constraint is satisfied. In this case, an interesting question is to determine an interpretation which maximizes the number of satisfied constraints: this problem is called Max-SAT [10]. We can generalize this proble ...
DIPLOMAMUNKA
... It is well known that the class of primitive relations is closed under substitution by primitive recursive functions, conjunction, disjunction, negation, bounded quantification and bounded minimization. In other words, if R, R0 are n-ary relations, S is an (n + 1)-ary relation, f0 , f1 , . . . , fn− ...
... It is well known that the class of primitive relations is closed under substitution by primitive recursive functions, conjunction, disjunction, negation, bounded quantification and bounded minimization. In other words, if R, R0 are n-ary relations, S is an (n + 1)-ary relation, f0 , f1 , . . . , fn− ...
institute of theoretical physics - Faculty of Physics University of Warsaw
... are not assigned to particular lecturers or Divisions. The rule is that every lecturer can deliver by turns different lectures. It is otherwise with specialized and monographic lectures on various topics in theoretical and mathematical physics for graduate and postgraduate students. These are delive ...
... are not assigned to particular lecturers or Divisions. The rule is that every lecturer can deliver by turns different lectures. It is otherwise with specialized and monographic lectures on various topics in theoretical and mathematical physics for graduate and postgraduate students. These are delive ...
Belief Revision in non
... Syntax Let LB be the Belnap language composed of a countable set of propositional letters P B = fp; q; r; : : :g and the connectives :, ^ and _. The set of wffs is obtained by the standard construction mechanism of formulae. For the finite case, a Belnap theory can be seen as a single formula given ...
... Syntax Let LB be the Belnap language composed of a countable set of propositional letters P B = fp; q; r; : : :g and the connectives :, ^ and _. The set of wffs is obtained by the standard construction mechanism of formulae. For the finite case, a Belnap theory can be seen as a single formula given ...
Chapter X: Computational Complexity of Propositional Fuzzy Logics
... There is a pattern in results presented in this chapter: for those decision problems whose complexity has been settled (the problems have been proved complete in some complexity class), the situation is analogous to the classical case: satisfiability is NP-complete, while tautologousness and consequ ...
... There is a pattern in results presented in this chapter: for those decision problems whose complexity has been settled (the problems have been proved complete in some complexity class), the situation is analogous to the classical case: satisfiability is NP-complete, while tautologousness and consequ ...
Natural Numbers and Natural Cardinals as Abstract Objects
... Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege’s Grundgesetze in Object Theory∗ Edward N. Zalta† Center for the Study of Language and Information and ...
... Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege’s Grundgesetze in Object Theory∗ Edward N. Zalta† Center for the Study of Language and Information and ...
The substitutional theory of logical consequence
... interested in logical consequence in natural language may take first-order languages as a test case. If the substitutional theory is successful for them, there is at least hope it can be extended to natural language. terminological remark. I apply the terms ‘logical validity’ and ‘formal validity’ t ...
... interested in logical consequence in natural language may take first-order languages as a test case. If the substitutional theory is successful for them, there is at least hope it can be extended to natural language. terminological remark. I apply the terms ‘logical validity’ and ‘formal validity’ t ...
Deep Sequent Systems for Modal Logic
... Labelled systems are formulated in a hybrid language which not only contains modalities but also variables and an accessibility relation. There are some concerns about incorporating the semantics into the syntax of a proof system in this way. Avron discusses them in [1], for example. However, even w ...
... Labelled systems are formulated in a hybrid language which not only contains modalities but also variables and an accessibility relation. There are some concerns about incorporating the semantics into the syntax of a proof system in this way. Avron discusses them in [1], for example. However, even w ...
The Facets of Relativistic Quantum Field Theory1
... particles. The experiments are bound to register particles, whereas the basic concept of the theory is the quantum field, technically seen an operator-valued distribution. Such a mathematical object requires in a quantum field theory showing interaction necessarily a process of renormalization, in c ...
... particles. The experiments are bound to register particles, whereas the basic concept of the theory is the quantum field, technically seen an operator-valued distribution. Such a mathematical object requires in a quantum field theory showing interaction necessarily a process of renormalization, in c ...