Constructive Mathematics, in Theory and Programming Practice
... intervention of the Goldbach Conjecture here is not essential: were that conjecture to be resolved today, we could replace it in our example by any one of a host of open problems of mathematics, including the twin prime conjecture, the conjecture that there are no odd perfect numbers, and the Rieman ...
... intervention of the Goldbach Conjecture here is not essential: were that conjecture to be resolved today, we could replace it in our example by any one of a host of open problems of mathematics, including the twin prime conjecture, the conjecture that there are no odd perfect numbers, and the Rieman ...
Sets
... Standard Symbols which denote sets of numbers N : The set of all natural numbers (i.e.,all positive integers) Z : The set of all integers Z+ : The set of all positive integers Z* : The set of all nonzero integers E : The set of all even integers Q : The set of all rational numbers Q* ...
... Standard Symbols which denote sets of numbers N : The set of all natural numbers (i.e.,all positive integers) Z : The set of all integers Z+ : The set of all positive integers Z* : The set of all nonzero integers E : The set of all even integers Q : The set of all rational numbers Q* ...
minimum models: reasoning and automation
... The basic proof method of Isabelle is resolution with higher-order unification. Resolution can be applied by using one-step commands in the proof or by combining tactics. Isabelle has also automated tools for solving goals. The language for proof strategies (tactics) in Isabelle is ML, which is a co ...
... The basic proof method of Isabelle is resolution with higher-order unification. Resolution can be applied by using one-step commands in the proof or by combining tactics. Isabelle has also automated tools for solving goals. The language for proof strategies (tactics) in Isabelle is ML, which is a co ...
Partial Grounded Fixpoints
... years. They showed that grounded fixpoints are an intuitive concept, closely related to exact (two-valued) stable fixpoints. In the context of logic programming, grounded fixpoints can be characterised using a generalised notion of unfounded set. Grounded fixpoints are lattice elements; in this work ...
... years. They showed that grounded fixpoints are an intuitive concept, closely related to exact (two-valued) stable fixpoints. In the context of logic programming, grounded fixpoints can be characterised using a generalised notion of unfounded set. Grounded fixpoints are lattice elements; in this work ...
![arXiv:1410.5037v2 [cs.LO] 18 Jun 2016](http://s1.studyres.com/store/data/007883898_2-29c425582568c0e0d15c3d815896c9cf-300x300.png)
arXiv:1410.5037v2 [cs.LO] 18 Jun 2016
... In this section we first give the well known definition of generalized quantifiers (Lindström quantifiers [20]). We then show how each generalized quantifier naturally gives rise to a generalized atom. Finally, we discuss on some fundamental properties of first-order logic extended with generalized ...
... In this section we first give the well known definition of generalized quantifiers (Lindström quantifiers [20]). We then show how each generalized quantifier naturally gives rise to a generalized atom. Finally, we discuss on some fundamental properties of first-order logic extended with generalized ...
MODAL LANGUAGES AND BOUNDED FRAGMENTS OF
... Modal Logic is traditionally concerned with the intensional operators "possibly" and "necessary", whose intuitive correspondence with the standard quantifiers "there exists" and "for all" comes out clearly in the usual Kripke semantics. This observation underlies the well-known translation from prop ...
... Modal Logic is traditionally concerned with the intensional operators "possibly" and "necessary", whose intuitive correspondence with the standard quantifiers "there exists" and "for all" comes out clearly in the usual Kripke semantics. This observation underlies the well-known translation from prop ...
3.6 First-Order Tableau
... ψ 0 . The case ψ = ¬∀xS .χ can be shown analogously. Completeness of a first-order calculus is more complicated than in propositional logic, because it requires the consideration of infinite derivations. For the tableau calculus I prove that if a possibly infinite saturated derivation contains an op ...
... ψ 0 . The case ψ = ¬∀xS .χ can be shown analogously. Completeness of a first-order calculus is more complicated than in propositional logic, because it requires the consideration of infinite derivations. For the tableau calculus I prove that if a possibly infinite saturated derivation contains an op ...
Quantitative Temporal Logics: PSPACE and below - FB3
... PS PACE again by further restricting the values of n and m, e.g., by enforcing that n = 0 [1]. However, in contrast to the E XP S PACE-completeness and undecidability results above, ...
... PS PACE again by further restricting the values of n and m, e.g., by enforcing that n = 0 [1]. However, in contrast to the E XP S PACE-completeness and undecidability results above, ...
A game semantics for proof search: Preliminary results - LIX
... The proviso † requires that t and s are unifiable and θ is their most general unifier (∆θ is the multiset resulting from applying θ to all formulas in ∆). The proviso ‡ requires that t and s are not unifiable. The free variables of a sequent are also called eigenvariables. Notice that the equality r ...
... The proviso † requires that t and s are unifiable and θ is their most general unifier (∆θ is the multiset resulting from applying θ to all formulas in ∆). The proviso ‡ requires that t and s are not unifiable. The free variables of a sequent are also called eigenvariables. Notice that the equality r ...
vmcai - of Philipp Ruemmer
... finite and all leaves are justified by an instance of a rule without premises. To construct a proof for an interpolation problem, we build a proof tree starting from the root Γ ` ∆ I I with unknown interpolant I, i.e., I acts as a place holder. For example, to solve an interpolation problem A ∧ B, w ...
... finite and all leaves are justified by an instance of a rule without premises. To construct a proof for an interpolation problem, we build a proof tree starting from the root Γ ` ∆ I I with unknown interpolant I, i.e., I acts as a place holder. For example, to solve an interpolation problem A ∧ B, w ...
Logical nihilism - University of Notre Dame
... Gentzen described the fact that lem prescribes uses of logical particles other than those given by their introduction and elimination rules as “troublesome.” The way that in the sequent calculus the logical rules are quarantined from the distinction between classical and intuitionistic logic he call ...
... Gentzen described the fact that lem prescribes uses of logical particles other than those given by their introduction and elimination rules as “troublesome.” The way that in the sequent calculus the logical rules are quarantined from the distinction between classical and intuitionistic logic he call ...
The Gödelian inferences - University of Notre Dame
... commitments in no way justified by the proof that follows. First, in what sense does the formula under consideration ‘state’ that the formal system of which it is a formula is consistent? Second, in whatever sense it does, how do we know that there aren’t other formulas that also do, so that it make ...
... commitments in no way justified by the proof that follows. First, in what sense does the formula under consideration ‘state’ that the formal system of which it is a formula is consistent? Second, in whatever sense it does, how do we know that there aren’t other formulas that also do, so that it make ...
The Complete Proof Theory of Hybrid Systems
... locally per proof rule. More intriguingly, however, our logical setting also enables us to ask the converse: is the proof calculus complete, i.e., can it prove all that is true? A corollary to Gödel’s incompleteness theorem shows, however, that hybrid systems do not have a sound and complete calcul ...
... locally per proof rule. More intriguingly, however, our logical setting also enables us to ask the converse: is the proof calculus complete, i.e., can it prove all that is true? A corollary to Gödel’s incompleteness theorem shows, however, that hybrid systems do not have a sound and complete calcul ...
Class Notes
... correct writing, those sentences are written following all the standard conventions of the language in which they are written. Because of the precision of the thoughts that a proof must convey, it is especially important that the prose be clear and correct. In order to help the reader follow the arg ...
... correct writing, those sentences are written following all the standard conventions of the language in which they are written. Because of the precision of the thoughts that a proof must convey, it is especially important that the prose be clear and correct. In order to help the reader follow the arg ...
ANNALS OF PURE AND APPLIED LOGIC I W
... (programs). The set of formulas is defined as the least set containing ASF and ATF, and such that if f and g are formulas, then so are (-,f ), (,f*), ...
... (programs). The set of formulas is defined as the least set containing ASF and ATF, and such that if f and g are formulas, then so are (-,f ), (,f*), ...
PDF
... † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. ...
... † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. ...
The Natural Order-Generic Collapse for ω
... ω-representable databases turn out to be exactly those databases where every relation is defined by an infinitary Boolean combination of order-constraints ...
... ω-representable databases turn out to be exactly those databases where every relation is defined by an infinitary Boolean combination of order-constraints ...
a Decidable Language Supporting Syntactic Query Difference
... (k 1), we say the query is in outward form. When the number of variables is minimized, but k is maximized, we say that the formula is in inward form. We now arrive at the main result of this section. ...
... (k 1), we say the query is in outward form. When the number of variables is minimized, but k is maximized, we say that the formula is in inward form. We now arrive at the main result of this section. ...
Nelson`s Strong Negation, Safe Beliefs and the - CEUR
... Since it was introduced in [3], strong negation has been well accepted in the answer set programming community2 . However, this connective has not received a fair treatment. While the answer set semantics has been extended to always more flexible classes of logic programs where conjunctions, disjunc ...
... Since it was introduced in [3], strong negation has been well accepted in the answer set programming community2 . However, this connective has not received a fair treatment. While the answer set semantics has been extended to always more flexible classes of logic programs where conjunctions, disjunc ...
Complexity of Recursive Normal Default Logic 1. Introduction
... are several such conditions in the published literature. Some of these will be used below. These include the notion of stratification [ABW88] and its generalization, local stratification [Prz88]. These conditions guarantee (when described in the language of nonmonotonic rule systems) the existence o ...
... are several such conditions in the published literature. Some of these will be used below. These include the notion of stratification [ABW88] and its generalization, local stratification [Prz88]. These conditions guarantee (when described in the language of nonmonotonic rule systems) the existence o ...
Master Thesis - Yoichi Hirai
... assumes global clock within the use of the adjective “current”. In classical epistemic logic, the description of knowledge relies on the notion of possible worlds. An agent can distinguish some pairs of possible worlds while he or she cannot distinguish the other pairs of possible worlds. When the a ...
... assumes global clock within the use of the adjective “current”. In classical epistemic logic, the description of knowledge relies on the notion of possible worlds. An agent can distinguish some pairs of possible worlds while he or she cannot distinguish the other pairs of possible worlds. When the a ...
Decision procedures in Algebra and Logic
... perspicuous, composed solely of identities. Algebraic structures that are not varieties are described in the following section, and differ from varieties in their metamathematical properties. In this section and the following one, structures are listed in approximate order of increasing complexity, ...
... perspicuous, composed solely of identities. Algebraic structures that are not varieties are described in the following section, and differ from varieties in their metamathematical properties. In this section and the following one, structures are listed in approximate order of increasing complexity, ...
Kripke Models of Transfinite Provability Logic
... However, as a modal logic, it is much more ill-behaved than GL. Most notably, over the class of GLP Kripke frames, the formula [1]⊥ is valid! This is clearly undesirable. There are ways to get around this, for example using topological semantics. However, Ignatiev in [7] showed how one can still get ...
... However, as a modal logic, it is much more ill-behaved than GL. Most notably, over the class of GLP Kripke frames, the formula [1]⊥ is valid! This is clearly undesirable. There are ways to get around this, for example using topological semantics. However, Ignatiev in [7] showed how one can still get ...
Subset Types and Partial Functions
... where x is of type ι. This is to be contrasted with constructive type theories like that of [15], where to type an application of car, that function would have to be applied to an inclusion i(x), not just x. This inclusion will only be typable if cons? x is provable, but that is not the case here. ...
... where x is of type ι. This is to be contrasted with constructive type theories like that of [15], where to type an application of car, that function would have to be applied to an inclusion i(x), not just x. This inclusion will only be typable if cons? x is provable, but that is not the case here. ...
Recent progress in additive prime number theory
... Now we turn from random models to another aspect of prime number theory, namely sieve theory. One way to approach the primes is to start with all the integers in a given range (e.g. from N/2 to N) and then sift out all the non-primes, for instance by removing the multiples of 2, then √ the multiples ...
... Now we turn from random models to another aspect of prime number theory, namely sieve theory. One way to approach the primes is to start with all the integers in a given range (e.g. from N/2 to N) and then sift out all the non-primes, for instance by removing the multiples of 2, then √ the multiples ...