NONSTANDARD MODELS IN RECURSION THEORY
... of PA and recursion theory in §3. In the final section, we discuss subsystems of second order arithmetic and Ramsey type combinatorial principles. We avoid detailed proofs of any theorem, but provide sketches of the key ideas where appropriate. 2. Fragments of Peano Arithmetic and their models The l ...
... of PA and recursion theory in §3. In the final section, we discuss subsystems of second order arithmetic and Ramsey type combinatorial principles. We avoid detailed proofs of any theorem, but provide sketches of the key ideas where appropriate. 2. Fragments of Peano Arithmetic and their models The l ...
Fluctuations of kinematic quantities in p+p interactions at the CERN
... at 158 GeV/c both for NA61 data and for the VENUS model. The analysis showed several structures. They depend on the multiplicity selection and may be related to quantum effects, resonance decays and conservation laws. Dividing into narrow multiplicity bins allows to better see correlation structures ...
... at 158 GeV/c both for NA61 data and for the VENUS model. The analysis showed several structures. They depend on the multiplicity selection and may be related to quantum effects, resonance decays and conservation laws. Dividing into narrow multiplicity bins allows to better see correlation structures ...
CHAPTER 8 Hilbert Proof Systems, Formal Proofs, Deduction
... only way to construct it is from such and such formulas by the means of one of the inference rules, and that formula can be found automatically. We will see now, that one can’t apply the above argument to the proof search in Hilbert proof systems, which contain Modus Ponens as an inference rule. A g ...
... only way to construct it is from such and such formulas by the means of one of the inference rules, and that formula can be found automatically. We will see now, that one can’t apply the above argument to the proof search in Hilbert proof systems, which contain Modus Ponens as an inference rule. A g ...
One-dimensional Fragment of First-order Logic
... a canonical generalisation of (equality-free) FO2 into contexts with arbitrary relational vocabularies. The fragment UF1 can be regarded as a vectorisation of FO2 that offers new possibilities for extending research efforts concerning two-variable logics. It is worth noting that for example in datab ...
... a canonical generalisation of (equality-free) FO2 into contexts with arbitrary relational vocabularies. The fragment UF1 can be regarded as a vectorisation of FO2 that offers new possibilities for extending research efforts concerning two-variable logics. It is worth noting that for example in datab ...
The Foundations
... xn + yn = zn ---- Fermat’s last theorem 6. “Every even number > 2 is the sum of two prime numbers.” ---Goldbach’s conjecture (1742) ...
... xn + yn = zn ---- Fermat’s last theorem 6. “Every even number > 2 is the sum of two prime numbers.” ---Goldbach’s conjecture (1742) ...
Default reasoning using classical logic
... rest of the paper is organized as follows: After introducing some preliminary de nitions in Section 2, we provide in Section 3 the concept of a model for a default theory and explain the theory behind our translation. In Sections 4 and 5 we discuss how the models presented in Section 3 can be treate ...
... rest of the paper is organized as follows: After introducing some preliminary de nitions in Section 2, we provide in Section 3 the concept of a model for a default theory and explain the theory behind our translation. In Sections 4 and 5 we discuss how the models presented in Section 3 can be treate ...
connections to higher type Recursion Theory, Proof-Theory
... example the constructive domain (PR,PRo,ϕo,≤) of the partial recursive functions: in this case the compact elements, PR o, are given by the functions with a finite graph, enumerated in some canonical way, ϕo, say. Then ϕ:ω→PR is just an (acceptable) gödel-numbering. The same applies to the domain o ...
... example the constructive domain (PR,PRo,ϕo,≤) of the partial recursive functions: in this case the compact elements, PR o, are given by the functions with a finite graph, enumerated in some canonical way, ϕo, say. Then ϕ:ω→PR is just an (acceptable) gödel-numbering. The same applies to the domain o ...
Part3
... A corollary is a result which follows directly from a theorem. Less important theorems are sometimes called propositions. A conjecture is a statement that is being proposed to be true. Once a proof of a conjecture is found, it becomes a theorem. It may turn out to be false. ...
... A corollary is a result which follows directly from a theorem. Less important theorems are sometimes called propositions. A conjecture is a statement that is being proposed to be true. Once a proof of a conjecture is found, it becomes a theorem. It may turn out to be false. ...
Asymptotic analysis and quantum integrable models
... Babylonian civilisation with the notion of angle being, however, mainly due to Greeks. A formula expresses that several quantities are inter-related. Think, for instance, of the area law of a triangle relating the area to a base length and to its height relative to this base. As such, a formula offer ...
... Babylonian civilisation with the notion of angle being, however, mainly due to Greeks. A formula expresses that several quantities are inter-related. Think, for instance, of the area law of a triangle relating the area to a base length and to its height relative to this base. As such, a formula offer ...
1Propositional Logic - Princeton University Press
... sentences and their associated wffs. The truth values are limited to true (T) and false (F) in classical logic. All interpreted wffs have truth values. To determine the truth values of interpreted wffs we need to define the meaning of the logical connectives. This is done by a truth table, which def ...
... sentences and their associated wffs. The truth values are limited to true (T) and false (F) in classical logic. All interpreted wffs have truth values. To determine the truth values of interpreted wffs we need to define the meaning of the logical connectives. This is done by a truth table, which def ...
Remarks on Second-Order Consequence
... sentences coincide with the sentences deducible from it). Thus, when dealing with axiomatic theories couched in a first-order language, we don't have to worry about the distinction between theorems and logical consequences of the axioms. The existence of a complete calculus for first-order logical c ...
... sentences coincide with the sentences deducible from it). Thus, when dealing with axiomatic theories couched in a first-order language, we don't have to worry about the distinction between theorems and logical consequences of the axioms. The existence of a complete calculus for first-order logical c ...
Proof analysis beyond geometric theories: from rule systems to
... When sequent calculi are used for the analysis of mathematical theories, a first limitation is encountered: if the theories are formulated as axioms, or equivalently as axiomatic sequents, full elimination of cuts is lost. What one can obtain is a generalized Hauptsatz that reduces all cuts in deriv ...
... When sequent calculi are used for the analysis of mathematical theories, a first limitation is encountered: if the theories are formulated as axioms, or equivalently as axiomatic sequents, full elimination of cuts is lost. What one can obtain is a generalized Hauptsatz that reduces all cuts in deriv ...
Modal Logic for Artificial Intelligence
... These course notes were written for an introduction in modal logic for students in Cognitive Artificial Intelligence at Utrecht University. Earlier notes by Rosalie Iemhoff have been used both as a source and as an inspiration, the chapters on completeness and decidability are based on her course no ...
... These course notes were written for an introduction in modal logic for students in Cognitive Artificial Intelligence at Utrecht University. Earlier notes by Rosalie Iemhoff have been used both as a source and as an inspiration, the chapters on completeness and decidability are based on her course no ...
Document
... Most of the meson spectra can be described assuming that the qq component is not only the dominant one, but also the only one. However, there are some exceptions like mesons with exotic quantum numbers or those with properties which are difficult to explain (like the light scalars, the X(3872) or th ...
... Most of the meson spectra can be described assuming that the qq component is not only the dominant one, but also the only one. However, there are some exceptions like mesons with exotic quantum numbers or those with properties which are difficult to explain (like the light scalars, the X(3872) or th ...
Transfinite progressions: A second look at completeness.
... PA also proves, for any n > 0, a formalization of For any φ, xφ(x) is a true Σn -sentence if and only if φ(k) is a true Πn 1 -sentence for some k and similarly for Πn -sentences. Using such restricted definitions of semantic concepts, PA proves e.g., that Thmφ (u) holds if and only there is a y such ...
... PA also proves, for any n > 0, a formalization of For any φ, xφ(x) is a true Σn -sentence if and only if φ(k) is a true Πn 1 -sentence for some k and similarly for Πn -sentences. Using such restricted definitions of semantic concepts, PA proves e.g., that Thmφ (u) holds if and only there is a y such ...
Experimental investigation of ultracold atom
... for lithium, sodium and potassium atom-diatom collisions is higher than the elastic rate coefficient and does not depend on temperature, again in agreement with the Wigner threshold law. In Refs. [6, 7] the inelastic collision rate coefficient for He-H2 and H-H2 collisions is found to strongly incre ...
... for lithium, sodium and potassium atom-diatom collisions is higher than the elastic rate coefficient and does not depend on temperature, again in agreement with the Wigner threshold law. In Refs. [6, 7] the inelastic collision rate coefficient for He-H2 and H-H2 collisions is found to strongly incre ...
Uniform satisfiability in PSPACE for local temporal logics over
... A dependence alphabet is a pair (Σ, D) where Σ is finite alphabet of actions and D ⊆ Σ2 is a reflexive and symmetric relation on Σ called dependence relation. A trace over (Σ, D) is (an isomorphism class of) a labeled, at most countably infinite partial order t = (V, , λ) such that (V, ) is a part ...
... A dependence alphabet is a pair (Σ, D) where Σ is finite alphabet of actions and D ⊆ Σ2 is a reflexive and symmetric relation on Σ called dependence relation. A trace over (Σ, D) is (an isomorphism class of) a labeled, at most countably infinite partial order t = (V, , λ) such that (V, ) is a part ...
Quantifiers
... Individual Constants • An individual constant is a name for an object. • Examples: john, marie, a, b • Each name is assumed to refer to a unique individual, i.e. we will not have two objects with the same name. • However, each individual object may have more than one name. ...
... Individual Constants • An individual constant is a name for an object. • Examples: john, marie, a, b • Each name is assumed to refer to a unique individual, i.e. we will not have two objects with the same name. • However, each individual object may have more than one name. ...
Tableau-based decision procedure for the full
... procedure for the logic ATEL, which subsumes the basic branching-time logic considered in [7] and this paper, was presented. In our view (to be explained further), however, [11] should be seen as a contribution to the complexity-theoretic analysis of the temporal-epistemic logics rather than to the ...
... procedure for the logic ATEL, which subsumes the basic branching-time logic considered in [7] and this paper, was presented. In our view (to be explained further), however, [11] should be seen as a contribution to the complexity-theoretic analysis of the temporal-epistemic logics rather than to the ...
Chapter 6: The Deductive Characterization of Logic
... But what is an inference rule? What does it to say that something follows by one? We will largely leave this open. Basically, an inference rule is a computable relation R from sets of formulas to formulas. To say that α follows from Γ by R is to say that the pair 〈Γ,α〉 stand in relation R. To say th ...
... But what is an inference rule? What does it to say that something follows by one? We will largely leave this open. Basically, an inference rule is a computable relation R from sets of formulas to formulas. To say that α follows from Γ by R is to say that the pair 〈Γ,α〉 stand in relation R. To say th ...