A BOUND FOR DICKSON`S LEMMA 1. Introduction Consider the
... In its usual formulation, Dickson’s lemma (for fixed functions) is a Σ01 formula. In contrast, we shall prove a quantifier-free statement which implies Dickson’s lemma in its usual form, but not vice versa. Our proof can be carried out in the formal system of Elementary Analysis [16, p.144], a conse ...
... In its usual formulation, Dickson’s lemma (for fixed functions) is a Σ01 formula. In contrast, we shall prove a quantifier-free statement which implies Dickson’s lemma in its usual form, but not vice versa. Our proof can be carried out in the formal system of Elementary Analysis [16, p.144], a conse ...
Graphical Representation of Canonical Proof: Two case studies
... not distinguish between proofs that are ‘morally’ the same. For many logics, the presentation of proofs in a traditional formalism, such as Gentzen’s sequent calculus, introduces artificial syntactic structure called ‘bureaucracy’; e.g., an arbitrary ordering of freely permutable inferences. A proof ...
... not distinguish between proofs that are ‘morally’ the same. For many logics, the presentation of proofs in a traditional formalism, such as Gentzen’s sequent calculus, introduces artificial syntactic structure called ‘bureaucracy’; e.g., an arbitrary ordering of freely permutable inferences. A proof ...
THE PARADOXES OF STRICT IMPLICATION John L
... implication. According to it, any two analytic statements must have the same meaning. For example, it must mean the same thing to say that 2 + 2 — 4 as to say that all bachelors are unmarried. But that is absurd. These statements do not mean the same thing. Thus if we took the above view of implicat ...
... implication. According to it, any two analytic statements must have the same meaning. For example, it must mean the same thing to say that 2 + 2 — 4 as to say that all bachelors are unmarried. But that is absurd. These statements do not mean the same thing. Thus if we took the above view of implicat ...
A causal approach to nonmonotonic reasoning
... epistemic understanding of default rules it presupposes. A distinctive feature of both default and modal nonmonotonic logics is that they are inherently epistemic formalisms. Namely, they are essentially based on such notions as belief and knowledge, unlike the extensional classical logic used for a ...
... epistemic understanding of default rules it presupposes. A distinctive feature of both default and modal nonmonotonic logics is that they are inherently epistemic formalisms. Namely, they are essentially based on such notions as belief and knowledge, unlike the extensional classical logic used for a ...
Strong Normalisation for a Gentzen-like Cut
... allow cuts to pass over other cuts). To prove this property, we shall make use of a technique developed in [1]. This technique appeals to the recursive path ordering theorem by Dershowitz. Our proof is more difficult than the one given in [4], which also appeals to the recursive path ordering theore ...
... allow cuts to pass over other cuts). To prove this property, we shall make use of a technique developed in [1]. This technique appeals to the recursive path ordering theorem by Dershowitz. Our proof is more difficult than the one given in [4], which also appeals to the recursive path ordering theore ...
A Qualitative Theory of Dynamic Interactive Belief Revision
... as just one of the many possible options for developing a belief-revisionfriendly notion of update. As already mentioned, it is a generalization of the “maximal-Spohn” revision, already explored in [24] and [2], among many other possible formulas for combining the “degrees of belief” of actions and ...
... as just one of the many possible options for developing a belief-revisionfriendly notion of update. As already mentioned, it is a generalization of the “maximal-Spohn” revision, already explored in [24] and [2], among many other possible formulas for combining the “degrees of belief” of actions and ...
SITUATIONS, TRUTH AND KNOWABILITY — A
... is quite implausible that it should be possible to establish Fitch's strong conclusion: For each agent who is not omniscient, there is a true proposition which the agent cannot know. ...
... is quite implausible that it should be possible to establish Fitch's strong conclusion: For each agent who is not omniscient, there is a true proposition which the agent cannot know. ...
Simply Logical: Intelligent Reasoning by Example
... What can and cannot be found in this book The book consists of three parts. Part I presents the necessary material on Logic and Logic Programming. In an introductory chapter, the main concepts in Logic Programming are introduced, such as program clauses, query answering, proof trees, and recursive d ...
... What can and cannot be found in this book The book consists of three parts. Part I presents the necessary material on Logic and Logic Programming. In an introductory chapter, the main concepts in Logic Programming are introduced, such as program clauses, query answering, proof trees, and recursive d ...
Notes on the ACL2 Logic
... time in physics is that the second law of thermodynamics precludes it. The second law of thermodynamics implies that entropy increases over time. There is an even more fundamental reason why time is not reversible. This second reason has to do with the fundamental laws of physics at the quantum leve ...
... time in physics is that the second law of thermodynamics precludes it. The second law of thermodynamics implies that entropy increases over time. There is an even more fundamental reason why time is not reversible. This second reason has to do with the fundamental laws of physics at the quantum leve ...
Structural Proof Theory
... for that, but some of the basic ideas and their connection to natural deduction and normalization procedures are explained in Appendix B. At present, there are many projects in the territory between logic, mathematics, and computer science that aim at fully formalizing mathematical proofs. These pro ...
... for that, but some of the basic ideas and their connection to natural deduction and normalization procedures are explained in Appendix B. At present, there are many projects in the territory between logic, mathematics, and computer science that aim at fully formalizing mathematical proofs. These pro ...
A Logical Foundation for Session
... the years many research attempts were made in this regard, a Curry-Howard correspondence between linear logic and concurrent computation was only found recently, bridging the proof theory of linear logic and session-typed process calculus. Building upon this work, we have developed a theory of intui ...
... the years many research attempts were made in this regard, a Curry-Howard correspondence between linear logic and concurrent computation was only found recently, bridging the proof theory of linear logic and session-typed process calculus. Building upon this work, we have developed a theory of intui ...
Foundations of Databases - Free University of Bozen
... • An extensional relation of P is a relation occurring only in rule bodies of P • An intensional relation of P is a relation occurring in the head of some rule in P • The extensional schema of P , edb(P ), consists of all extensional relations of P • The intensional schema of P , idb(P ), consists o ...
... • An extensional relation of P is a relation occurring only in rule bodies of P • An intensional relation of P is a relation occurring in the head of some rule in P • The extensional schema of P , edb(P ), consists of all extensional relations of P • The intensional schema of P , idb(P ), consists o ...
Duplication of directed graphs and exponential blow up of
... positively. A strong connective is either an ∧ occurring positively or an ∨ occurring negatively. In the following we will frequently use the notion of occurrence of a formula in a proof as compared to the formula itself which may occur many times. 2.1. Cut elimination In 1934 Gentzen ([12]; see als ...
... positively. A strong connective is either an ∧ occurring positively or an ∨ occurring negatively. In the following we will frequently use the notion of occurrence of a formula in a proof as compared to the formula itself which may occur many times. 2.1. Cut elimination In 1934 Gentzen ([12]; see als ...
pdf
... Consider the boolean formula satisfiability problem, SAT. For formulas in SAT, there is always a short proof of satisfiability – a satisfying truth assignment – and therefore SAT is trivially in NP. However, for formulas not in SAT, it is not that clear what a proof of unsatisfiability could be. Som ...
... Consider the boolean formula satisfiability problem, SAT. For formulas in SAT, there is always a short proof of satisfiability – a satisfying truth assignment – and therefore SAT is trivially in NP. However, for formulas not in SAT, it is not that clear what a proof of unsatisfiability could be. Som ...
Language, Proof and Logic
... By modus ponens, we conclude Small(d). But d denotes an arbitrary object in the domain, so our conclusion, ∀x Small(x), follows by universal generalization. Any proof using general conditional proof can be converted into a proof using universal generalization, together with the method of conditional ...
... By modus ponens, we conclude Small(d). But d denotes an arbitrary object in the domain, so our conclusion, ∀x Small(x), follows by universal generalization. Any proof using general conditional proof can be converted into a proof using universal generalization, together with the method of conditional ...
Discrete Mathematics for Computer Science Some Notes
... is much more easily grasped than semantic concepts. In this approach, we follow Peter Andrew’s motto [1]: “To truth through proof”. We present various natural deduction systems due to Prawitz and Gentzen (in more modern notation), both in their intuitionistic and classical version. The adoption of n ...
... is much more easily grasped than semantic concepts. In this approach, we follow Peter Andrew’s motto [1]: “To truth through proof”. We present various natural deduction systems due to Prawitz and Gentzen (in more modern notation), both in their intuitionistic and classical version. The adoption of n ...
Sound and Complete Inference Rules in FOL Example
... atomic formula. An atomic formula is also called a positive literal, and the negation of an atomic formula is called a negative literal. A clause is a disjunction of literals. There is a special clause called empty which is equivalent to false. Definition. A FOL formula is in conjunctive normal form ...
... atomic formula. An atomic formula is also called a positive literal, and the negation of an atomic formula is called a negative literal. A clause is a disjunction of literals. There is a special clause called empty which is equivalent to false. Definition. A FOL formula is in conjunctive normal form ...
Tableau techniques for ALC
... prove that a formula P is a tautology (or valid), we start with ¬P and produce a contradiction. The procedure for doing this involves expanding ¬P so that inessential details of its logical structure are cleared away. In tableau proofs, such an expansion takes the form of a tree, where nodes are lab ...
... prove that a formula P is a tautology (or valid), we start with ¬P and produce a contradiction. The procedure for doing this involves expanding ¬P so that inessential details of its logical structure are cleared away. In tableau proofs, such an expansion takes the form of a tree, where nodes are lab ...
A Logic for Perception and Belief Department of Computer Science
... rule out all values incompatible with a particular reading. For example, P(T = 10) implies 1 UD (T = 10, T = 7); using proposition (4), we obtain Pi(T = 7). It should be noted that type (i) axioms will not always be adequate. For suppose we have an additional sensor, with identical accuracy, for the ...
... rule out all values incompatible with a particular reading. For example, P(T = 10) implies 1 UD (T = 10, T = 7); using proposition (4), we obtain Pi(T = 7). It should be noted that type (i) axioms will not always be adequate. For suppose we have an additional sensor, with identical accuracy, for the ...
Algebraic logic, I. Monadic boolean algebras
... that space, every open set is closed, or, equivalently, every closed set is open. (The proof is an easy application of Lemma 5.) In such a space the relation R, defined by writing x R y whenever x belongs to the closure of the one-point set {y}, is an equivalence whose associated quotient space is d ...
... that space, every open set is closed, or, equivalently, every closed set is open. (The proof is an easy application of Lemma 5.) In such a space the relation R, defined by writing x R y whenever x belongs to the closure of the one-point set {y}, is an equivalence whose associated quotient space is d ...
Incompleteness in the finite domain
... terms. The “finite domain” in the title refers to the fact that the lengths of computations and lengths of proofs of instances of the problems that we consider are at most exponential, hence there is a finite bound on them. Perhaps, a more precise term would be “exponential domain”. In previous prese ...
... terms. The “finite domain” in the title refers to the fact that the lengths of computations and lengths of proofs of instances of the problems that we consider are at most exponential, hence there is a finite bound on them. Perhaps, a more precise term would be “exponential domain”. In previous prese ...