First-Order Theorem Proving and Vampire

... Which of the following statements are true? 1. First-order logic is an extension of propositional logic; 2. First-order logic is NP-complete. 3. First-order logic is PSPACE-complete. 4. First-order logic is decidable. 5. In first-order logic you can use quantifiers over sets. 6. One can axiomatise i ...

... Which of the following statements are true? 1. First-order logic is an extension of propositional logic; 2. First-order logic is NP-complete. 3. First-order logic is PSPACE-complete. 4. First-order logic is decidable. 5. In first-order logic you can use quantifiers over sets. 6. One can axiomatise i ...

a-logic - Digital [email protected] State University

... set of theorems, the same semantical foundations, and use the same concepts of validity and logical truth though they differ in notation, choices of primitives and axioms, diagramatic devices, modes of introduction and explication, etc. This standard logic is an enormous advance over any preceding s ...

... set of theorems, the same semantical foundations, and use the same concepts of validity and logical truth though they differ in notation, choices of primitives and axioms, diagramatic devices, modes of introduction and explication, etc. This standard logic is an enormous advance over any preceding s ...

Model Theory of Modal Logic, Chapter in: Handbook of Modal Logic

... between the (ﬁrst-order) Kripke structure semantics and the (second-order) frame semantics, give rise to very distinct model theoretic ﬂavours, each with their own tradition in the model theory of modal logic. Still, these two semantics meet through the notion of a general frame (closely related to ...

... between the (ﬁrst-order) Kripke structure semantics and the (second-order) frame semantics, give rise to very distinct model theoretic ﬂavours, each with their own tradition in the model theory of modal logic. Still, these two semantics meet through the notion of a general frame (closely related to ...

5 model theory of modal logic

... between the (first-order) Kripke structure semantics and the (second-order) frame semantics, give rise to very distinct model theoretic flavours, each with their own tradition in the model theory of modal logic. Still, these two semantics meet through the notion of a general frame (closely related t ...

... between the (first-order) Kripke structure semantics and the (second-order) frame semantics, give rise to very distinct model theoretic flavours, each with their own tradition in the model theory of modal logic. Still, these two semantics meet through the notion of a general frame (closely related t ...

Introduction to Computational Logic

... An important part of the course is the theory of classical and intuitionistic propositional logic. We study various proof systems (Hilbert, ND, sequent, tableaux), decidability of proof systems, and the semantic analysis of proof systems based on models. The study of propositional logic is carried o ...

... An important part of the course is the theory of classical and intuitionistic propositional logic. We study various proof systems (Hilbert, ND, sequent, tableaux), decidability of proof systems, and the semantic analysis of proof systems based on models. The study of propositional logic is carried o ...

a thesis submitted in partial fulfillment of the requirements for the

... from inference rules can be frustrating in proof development. The proof checker described here avoids this problem because the user provides the expected result of each step. The use of a functional meta-language as the user interface to the proof checker makes this approach practical: the user does ...

... from inference rules can be frustrating in proof development. The proof checker described here avoids this problem because the user provides the expected result of each step. The use of a functional meta-language as the user interface to the proof checker makes this approach practical: the user does ...

Predicate Logic

... Proof: Suppose ∀x(P (x) ∧ Q(x)) is true. Then if a is in the domain, P (a) ∧ Q(a) is true, and so P (a) is true and Q(a) is true. So, if a in in the domain P (a) is true, which is the same as ∀xP (x) is true; and similarly, we get that ∀xQ(x) is true. This means that ∀xP (x) ∧ ∀xQ(x) is true. If ∀xP ...

... Proof: Suppose ∀x(P (x) ∧ Q(x)) is true. Then if a is in the domain, P (a) ∧ Q(a) is true, and so P (a) is true and Q(a) is true. So, if a in in the domain P (a) is true, which is the same as ∀xP (x) is true; and similarly, we get that ∀xQ(x) is true. This means that ∀xP (x) ∧ ∀xQ(x) is true. If ∀xP ...

Interpretability formalized

... shall use interpretations to compare theories. Furthermore, we shall also study interpretations as meta-mathematical entities. Roughly, an interpretation j of a theory T into a theory S (we write j : S¤T ) is a structure-preserving map, mapping axioms of T to theorems of S. Structurepreserving means ...

... shall use interpretations to compare theories. Furthermore, we shall also study interpretations as meta-mathematical entities. Roughly, an interpretation j of a theory T into a theory S (we write j : S¤T ) is a structure-preserving map, mapping axioms of T to theorems of S. Structurepreserving means ...

.pdf

... This paper explores the design of programming logics in which assumptions about the environment can be given explicitly. Such logics allow us to prove that all feasible behaviors of a program satisfy a property, where the characterization of what is feasible is now explicit and subject to change. We ...

... This paper explores the design of programming logics in which assumptions about the environment can be given explicitly. Such logics allow us to prove that all feasible behaviors of a program satisfy a property, where the characterization of what is feasible is now explicit and subject to change. We ...

How to Go Nonmonotonic Contents David Makinson

... acting on sets A of formulae to give larger sets Cn(A). In effect, the operation gathers together all the formulae that are consequences of given premises. The two representations of classical consequence are trivially interchangeable. Given a relation |-, we may define the operation Cn by setting C ...

... acting on sets A of formulae to give larger sets Cn(A). In effect, the operation gathers together all the formulae that are consequences of given premises. The two representations of classical consequence are trivially interchangeable. Given a relation |-, we may define the operation Cn by setting C ...

MATHEMATICAL LOGIC FOR APPLICATIONS

... experts believe this theory to be a more natural model for differential and integral calculus than the traditional model, the more traditional ε − δ method (besides analysis Robinson’s idea was applied to other areas of Mathematics too, and this is called non-standard mathematics). This connection i ...

... experts believe this theory to be a more natural model for differential and integral calculus than the traditional model, the more traditional ε − δ method (besides analysis Robinson’s idea was applied to other areas of Mathematics too, and this is called non-standard mathematics). This connection i ...

Proofs in theories

... These course notes are organized in four parts. In Chapters 1, 2, and 3, we shall present the basic notions of proof, theory and model used in these course notes. When presenting the notion of proof we emphasize the notion of constructivity and that of cut. When we present the notion of theory, we e ...

... These course notes are organized in four parts. In Chapters 1, 2, and 3, we shall present the basic notions of proof, theory and model used in these course notes. When presenting the notion of proof we emphasize the notion of constructivity and that of cut. When we present the notion of theory, we e ...

Sample pages 2 PDF

... Therefore, ¬A ⇒ B ∧ C ⇔ D effectively means ((¬A) ⇒ (B ∧ C)) ⇔ D. Although we can reduce brackets to a minimum, we usually use brackets to distinguish between ∧ and ∨, and between ⇒ and ⇔. Therefore, we would usually write A ∨ (B ∧ C) even if A ∨ B ∧ C would do. Similarly, we write A ⇔ (B ⇒ C) when ...

... Therefore, ¬A ⇒ B ∧ C ⇔ D effectively means ((¬A) ⇒ (B ∧ C)) ⇔ D. Although we can reduce brackets to a minimum, we usually use brackets to distinguish between ∧ and ∨, and between ⇒ and ⇔. Therefore, we would usually write A ∨ (B ∧ C) even if A ∨ B ∧ C would do. Similarly, we write A ⇔ (B ⇒ C) when ...

The Z/EVES 2.0 User`s Guide - Department of Computer Science

... of order; if such a specification is imported into Z/EVES it is necessary to move the paragraphs into a suitable checking order. Order is also important for theorems, as the proof of a given theorem can only use other theorems that precede the given theorem in the specification. Thus, lemmas must ap ...

... of order; if such a specification is imported into Z/EVES it is necessary to move the paragraphs into a suitable checking order. Order is also important for theorems, as the proof of a given theorem can only use other theorems that precede the given theorem in the specification. Thus, lemmas must ap ...

Termination of Higher-order Rewrite Systems

... Implementing abstract data types; Automated theorem proving, especially for equational logic; Proving completeness of axiomatizations for algebras; Proving consistency of proof calculi for logics. These tasks can be divided into practical and theoretical applications. In theoretical applicat ...

... Implementing abstract data types; Automated theorem proving, especially for equational logic; Proving completeness of axiomatizations for algebras; Proving consistency of proof calculi for logics. These tasks can be divided into practical and theoretical applications. In theoretical applicat ...

A Course in Modal Logic - Sun Yat

... Theorem, Deduction Theorem, Reduction Theorem, PostCompleteness Theorem and Modal Conjunction Normal Form Existence Theorem. Most of them are the natural generalizations of the related theorems for the classical propositional calculus. ...

... Theorem, Deduction Theorem, Reduction Theorem, PostCompleteness Theorem and Modal Conjunction Normal Form Existence Theorem. Most of them are the natural generalizations of the related theorems for the classical propositional calculus. ...

KURT GÖDEL - National Academy of Sciences

... numbers, so that one subset is paired with 0, another with 1, still another with 2, and so on, with every natural number used exactly once. Sets have the same cardinal number if they can be thus paired with each other, or put into a "one-to-one correspondence". Denoting the cardinal number of the na ...

... numbers, so that one subset is paired with 0, another with 1, still another with 2, and so on, with every natural number used exactly once. Sets have the same cardinal number if they can be thus paired with each other, or put into a "one-to-one correspondence". Denoting the cardinal number of the na ...

A Pebble Weighted Automata and Weighted Logics

... This has already been done by Droste and Gastin [Droste and Gastin 2009]: they have introduced weighted logics with syntax close to monadic second-order logic, extending the semantics by using addition and product of a semiring to evaluate disjunctions/existential quantifications, and conjunctions/u ...

... This has already been done by Droste and Gastin [Droste and Gastin 2009]: they have introduced weighted logics with syntax close to monadic second-order logic, extending the semantics by using addition and product of a semiring to evaluate disjunctions/existential quantifications, and conjunctions/u ...

Lecture Notes on the Lambda Calculus

... a vehicle for studying such extensions, in isolation and jointly, to see how they will affect each other, and to prove properties of programming language (such as: a well-formed program will not crash). The lambda calculus is also a tool used in compiler construction, see e.g. [8, 9]. ...

... a vehicle for studying such extensions, in isolation and jointly, to see how they will affect each other, and to prove properties of programming language (such as: a well-formed program will not crash). The lambda calculus is also a tool used in compiler construction, see e.g. [8, 9]. ...

Automated Theorem Proving in a First

... artificial intelligence. Automated methods of proving theorems precede the existence of computers (see e.g. [15, 17, 23] for a historical survey). In order to be represented in a computer, a mathematical problem must be expressed in a language of some formal logic. Among the logics used for this pur ...

... artificial intelligence. Automated methods of proving theorems precede the existence of computers (see e.g. [15, 17, 23] for a historical survey). In order to be represented in a computer, a mathematical problem must be expressed in a language of some formal logic. Among the logics used for this pur ...

A Judgmental Reconstruction of Modal Logic

... reductions and expansions of proofs, respectively. Note that there are other ways to define meaning. For example, we frequently expand our language by notational definition. In intuitionistic logic negation is often given as a derived concept, where ¬A is considered a notation for A ⊃ ⊥. This means ...

... reductions and expansions of proofs, respectively. Note that there are other ways to define meaning. For example, we frequently expand our language by notational definition. In intuitionistic logic negation is often given as a derived concept, where ¬A is considered a notation for A ⊃ ⊥. This means ...

A Unified View of Induction Reasoning for First-Order Logic

... the induction reasoning is argued by an external global induction discharge condition associated to the proof structure. In the same line, Brotherston and Simpson [14] compared two classical first-order sequent calculus proof systems; the local induction is performed using conventional induction tog ...

... the induction reasoning is argued by an external global induction discharge condition associated to the proof structure. In the same line, Brotherston and Simpson [14] compared two classical first-order sequent calculus proof systems; the local induction is performed using conventional induction tog ...

Formalizing Context (Expanded Notes) - John McCarthy

... always compatible with each other. This is an expanded and revised version of [42]. An earlier version of this paper is the Stanford University Technical Note STAN-CS-TN-94-13. The current version contains new sections §7 and §13, as well as updated bibliographical remarks. Some of the results in th ...

... always compatible with each other. This is an expanded and revised version of [42]. An earlier version of this paper is the Stanford University Technical Note STAN-CS-TN-94-13. The current version contains new sections §7 and §13, as well as updated bibliographical remarks. Some of the results in th ...

How to Write a 21st Century Proof

... considerably in the last few centuries. Mathematicians no longer write formulas as prose, but use symbolic notation such as e iπ + 1 = 0 . On the other hand, proofs are still written in prose pretty much the way they were in the 17th century. The proofs in Newton’s Principia seem quite modern. This ...

... considerably in the last few centuries. Mathematicians no longer write formulas as prose, but use symbolic notation such as e iπ + 1 = 0 . On the other hand, proofs are still written in prose pretty much the way they were in the 17th century. The proofs in Newton’s Principia seem quite modern. This ...

Higher Order Logic - Theory and Logic Group

... are increasingly recognized for their foundational importance and practical usefulness, notably in Theoretical Computer Science. In this chapter we try to present a survey of some issues and results, without any pretense of completeness. Our choice of topics is driven by an attempt to cover the foun ...

... are increasingly recognized for their foundational importance and practical usefulness, notably in Theoretical Computer Science. In this chapter we try to present a survey of some issues and results, without any pretense of completeness. Our choice of topics is driven by an attempt to cover the foun ...