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 Commons@Wayne State University

... he standard logic today is the logic of Frege’s Begriffschrift (1879) and subsequently of Russell and Whitehead’s great book, Principia Mathematica (1913) of Quine’s Mathematical Logic (1940) and Methods of Logic (4th ed.,1982) and of hundreds of other textbooks and treatises which have the same set ...

... he standard logic today is the logic of Frege’s Begriffschrift (1879) and subsequently of Russell and Whitehead’s great book, Principia Mathematica (1913) of Quine’s Mathematical Logic (1940) and Methods of Logic (4th ed.,1982) and of hundreds of other textbooks and treatises which have the same set ...

Predicate Logic

... Binding variables and scope When a quantifier is used on the variable x we say that this occurrence of x is bound. When the occurrence of a variable is not bound by a quantifier or set to a particular value, the variable is said to be free. The part of a logical expression to which a quantifier is a ...

... Binding variables and scope When a quantifier is used on the variable x we say that this occurrence of x is bound. When the occurrence of a variable is not bound by a quantifier or set to a particular value, the variable is said to be free. The part of a logical expression to which a quantifier is a ...

Ribbon Proofs - A Proof System for the Logic of Bunched Implications

... the level of propositions. It generalizes box proofs (as in Fitch[10]), which are essentially onedimensional, into two dimensions. The horizontal structure of the proof is used to model the resource-sensitive part of the logic. We will develop this system informally as an attractive graphical notati ...

... the level of propositions. It generalizes box proofs (as in Fitch[10]), which are essentially onedimensional, into two dimensions. The horizontal structure of the proof is used to model the resource-sensitive part of the logic. We will develop this system informally as an attractive graphical notati ...

Structural Proof Theory

... Since 1970, a branch of proof theory known as constructive type theory has been developed. A theorem typically states that a certain claim holds under given assumptions. The basic idea of type theory is that proofs are functions that convert any proofs of the assumptions of a theorem into a proof of ...

... Since 1970, a branch of proof theory known as constructive type theory has been developed. A theorem typically states that a certain claim holds under given assumptions. The basic idea of type theory is that proofs are functions that convert any proofs of the assumptions of a theorem into a proof of ...

Graphical Representation of Canonical Proof: Two case studies

... An interesting problem in proof theory is to find representations of proof that do 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 ...

... An interesting problem in proof theory is to find representations of proof that do 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 ...

Color - Alex Kocurek

... showing that such models are L-equivalent easier, we can appeal to the notion of a bisimulation.16 The notion of a bisimulation for first-order modal logic has not been discussed much until recently.17 Below, we extend the notion of bisimulation in order to ensure modal equivalence for formulas invo ...

... showing that such models are L-equivalent easier, we can appeal to the notion of a bisimulation.16 The notion of a bisimulation for first-order modal logic has not been discussed much until recently.17 Below, we extend the notion of bisimulation in order to ensure modal equivalence for formulas invo ...

Functional Dependencies in a Relational Database and

... few in the database area that is both intuitively simple and yet complex enough that an advanced development is possible (see, for example, [2-61). Functional dependencies are important tools for database design: in fact, in one approach to database design [6], they are essentially the only input. B ...

... few in the database area that is both intuitively simple and yet complex enough that an advanced development is possible (see, for example, [2-61). Functional dependencies are important tools for database design: in fact, in one approach to database design [6], they are essentially the only input. B ...

Announcement as effort on topological spaces

... (7.3) On the left-hand-side, we have S([ϕ]ψ ∧ [ϕ]χ) = 1 + 4(S(ϕ) + 4)(S(ψ) + S(χ)). However, S([ϕ](ψ ∧ χ)) = 4(S(ϕ) + 4)(1 + S(ψ) + S(χ)) = 4(S(ϕ) + 4) + 4(S(ϕ) + 4)(S(ψ) + S(χ)). Thus, S([ϕ]ψ ∧ [ϕ]χ) < S([ϕ](ψ ∧ χ)). Moreover, d([ϕ]ψ ∧ [ϕ]χ) = max{d(ϕ), d(ψ), d(χ)} = d([ϕ](ψ ∧ χ)) (This is similar ...

... (7.3) On the left-hand-side, we have S([ϕ]ψ ∧ [ϕ]χ) = 1 + 4(S(ϕ) + 4)(S(ψ) + S(χ)). However, S([ϕ](ψ ∧ χ)) = 4(S(ϕ) + 4)(1 + S(ψ) + S(χ)) = 4(S(ϕ) + 4) + 4(S(ϕ) + 4)(S(ψ) + S(χ)). Thus, S([ϕ]ψ ∧ [ϕ]χ) < S([ϕ](ψ ∧ χ)). Moreover, d([ϕ]ψ ∧ [ϕ]χ) = max{d(ϕ), d(ψ), d(χ)} = d([ϕ](ψ ∧ χ)) (This is similar ...

TR-14-06 - Ynot - Harvard University

... monad, HTT already provides the underlying basis of first-class modules with specifications. However, truly reusable components require that their internal invariants are appropriately abstracted. That is, the interfaces of components and objects need to include not only abstract types, but also abs ...

... monad, HTT already provides the underlying basis of first-class modules with specifications. However, truly reusable components require that their internal invariants are appropriately abstracted. That is, the interfaces of components and objects need to include not only abstract types, but also abs ...

Interpretability formalized

... Interpretations have also been used in partial realizations of Hilbert’s programme and other attempts to settle foundational questions ([Sim88], [Fef88], [Nel86]). For another occurrence of interpretations, we can think of translations of classical propositional calculus into intuitionistic proposit ...

... Interpretations have also been used in partial realizations of Hilbert’s programme and other attempts to settle foundational questions ([Sim88], [Fef88], [Nel86]). For another occurrence of interpretations, we can think of translations of classical propositional calculus into intuitionistic proposit ...

numbers and uniform ergodic theorems

... general dependence structures. The lectures begin (Chapter 1) with a review and description of classic laws of large numbers and ergodic theorems, their connection and interplay, and their infinite dimensional extensions towards uniform theorems with applications to dynamical systems. The first appr ...

... general dependence structures. The lectures begin (Chapter 1) with a review and description of classic laws of large numbers and ergodic theorems, their connection and interplay, and their infinite dimensional extensions towards uniform theorems with applications to dynamical systems. The first appr ...

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... environment. Logics to reason about real-time, for example, axiomatize assumptions about how time advances while the program executes. These assumptions abstract the effects of the scheduler and the execution times of various atomic actions. A logic to reason about the consequences of resource const ...

... environment. Logics to reason about real-time, for example, axiomatize assumptions about how time advances while the program executes. These assumptions abstract the effects of the scheduler and the execution times of various atomic actions. A logic to reason about the consequences of resource const ...

A logic-based theory of deductive arguments

... that there exist two distinct subsets of ∆ (namely, Φ and Ψ ) supporting α ∧ β. Whilst equivalent arguments make the same point (that is, the same inference), we do want to distinguish equivalent arguments from each other. What we do not want is to distinguish between arguments that are more conserv ...

... that there exist two distinct subsets of ∆ (namely, Φ and Ψ ) supporting α ∧ β. Whilst equivalent arguments make the same point (that is, the same inference), we do want to distinguish equivalent arguments from each other. What we do not want is to distinguish between arguments that are more conserv ...

PhD Thesis First-Order Logic Investigation of Relativity Theory with

... observers is formulated by means of the world-view relation. We visualize an observer as “sitting” at the origin of the space part of its reference frame, or equivalently, “living” on the time-axis of the reference frame. We distinguish inertial and noninertial observers. For the time being, inertia ...

... observers is formulated by means of the world-view relation. We visualize an observer as “sitting” at the origin of the space part of its reference frame, or equivalently, “living” on the time-axis of the reference frame. We distinguish inertial and noninertial observers. For the time being, inertia ...

Symmetry in Nature

... anticipated in the fourteenth century by Jean Buridan and Nicole Oresme: the laws of nature that we discover do not change their form if we observe nature within a moving laboratory, traveling at constant velocity. The fact that the earth is speeding around the sun, for instance, does not affect the ...

... anticipated in the fourteenth century by Jean Buridan and Nicole Oresme: the laws of nature that we discover do not change their form if we observe nature within a moving laboratory, traveling at constant velocity. The fact that the earth is speeding around the sun, for instance, does not affect the ...

Sample pages 2 PDF

... We have just defined what a formulae in propositional calculus should look like. We have also seen how such formulae correspond to English sentences, which can be either true or false. The next step is to build a series of mathematical tools to reason about these formulae. For instance, some stateme ...

... We have just defined what a formulae in propositional calculus should look like. We have also seen how such formulae correspond to English sentences, which can be either true or false. The next step is to build a series of mathematical tools to reason about these formulae. For instance, some stateme ...