Semi-constr. theories - Stanford Mathematics
... function may be generated in this way. Kleene (1959) introduced restricted recursors R^ satisfying the recursion equations R^xy0z = xz and R^xyn′z = y(R^xynz), where z is a sequence of variables such that xz is of type 0. He showed that the 1-section of the functionals generated from 0, Sc, the K, S ...
... function may be generated in this way. Kleene (1959) introduced restricted recursors R^ satisfying the recursion equations R^xy0z = xz and R^xyn′z = y(R^xynz), where z is a sequence of variables such that xz is of type 0. He showed that the 1-section of the functionals generated from 0, Sc, the K, S ...
Chapter 4. Logical Notions This chapter introduces various logical
... at least two cats, so there is at least one cat is formally valid the logician may paraphrase there are at least two cats as there is an x such that there is a y such that x is a cat and y is cat and it is not the case that x is identical to y. The numerical sentence and its "identity" paraphrase a ...
... at least two cats, so there is at least one cat is formally valid the logician may paraphrase there are at least two cats as there is an x such that there is a y such that x is a cat and y is cat and it is not the case that x is identical to y. The numerical sentence and its "identity" paraphrase a ...
Completeness and Decidability of a Fragment of Duration Calculus
... Duration Calculus (DC) was introduced by Zhou, Hoare and Ravn in 1991 as a logic to specify the requirements for real-time systems. DC has been used successfully in many case studies, see e.g. [ZZ94,YWZP94,HZ94,DW94,BHCZ94,XH95], [Dan98,ED99]. In [DW94], we have developed a method for designing a re ...
... Duration Calculus (DC) was introduced by Zhou, Hoare and Ravn in 1991 as a logic to specify the requirements for real-time systems. DC has been used successfully in many case studies, see e.g. [ZZ94,YWZP94,HZ94,DW94,BHCZ94,XH95], [Dan98,ED99]. In [DW94], we have developed a method for designing a re ...
Document
... A predicate P, or propositional function, is a function that maps objects in the universe of discourse to propositions Predicates can be quantified using the universal quantifier (“for all”) or the existential quantifier (“there exists”) Quantified predicates can be negated as follows x P(x) ...
... A predicate P, or propositional function, is a function that maps objects in the universe of discourse to propositions Predicates can be quantified using the universal quantifier (“for all”) or the existential quantifier (“there exists”) Quantified predicates can be negated as follows x P(x) ...
Logic is a discipline that studies the principles and methods used in
... A predicate P, or propositional function, is a function that maps objects in the universe of discourse to propositions Predicates can be quantified using the universal quantifier (“for all”) ∀ or the existential quantifier (“there exists”) ∃ Quantified predicates can be negated as follows ¬∀x P(x) ...
... A predicate P, or propositional function, is a function that maps objects in the universe of discourse to propositions Predicates can be quantified using the universal quantifier (“for all”) ∀ or the existential quantifier (“there exists”) ∃ Quantified predicates can be negated as follows ¬∀x P(x) ...
Proof theory of witnessed G¨odel logic: a
... (First-order) Gödel logic is a prominent example of both a many-valued and a superintuitionistic logic. The importance of Gödel logic is emphasized by the fact that it turns up naturally in a number of different contexts; among them relevance logics, fuzzy logic, and logic programming. Witnessed G ...
... (First-order) Gödel logic is a prominent example of both a many-valued and a superintuitionistic logic. The importance of Gödel logic is emphasized by the fact that it turns up naturally in a number of different contexts; among them relevance logics, fuzzy logic, and logic programming. Witnessed G ...
pdf
... completeness only says that it must be possible to prove every valid formula correct with the tableau method but it doesn’t require that any attempt will succeed. And the fact that we weren’t able to find a proof with a not so bright approach doesn’t mean that there is none at all. Fortunately, we c ...
... completeness only says that it must be possible to prove every valid formula correct with the tableau method but it doesn’t require that any attempt will succeed. And the fact that we weren’t able to find a proof with a not so bright approach doesn’t mean that there is none at all. Fortunately, we c ...
Probabilistic Theorem Proving - The University of Texas at Dallas
... function of a PKB K is given by Z(K) = x i φi i . The conditional probability P (Q|K) is simply a ratio of two partition functions: P (Q|K) = Z(K ∪ {Q, 0})/Z(K), where Z(K ∪ {Q, 0}) is the partition function of K with Q added as a hard formula. The main idea in PTP is to compute the partition functi ...
... function of a PKB K is given by Z(K) = x i φi i . The conditional probability P (Q|K) is simply a ratio of two partition functions: P (Q|K) = Z(K ∪ {Q, 0})/Z(K), where Z(K ∪ {Q, 0}) is the partition function of K with Q added as a hard formula. The main idea in PTP is to compute the partition functi ...
![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
... The satisfiability problem of two-variable logic FO2 was shown to be NEXPTIME-complete in [9]. The extension of two-variable logic with counting quantifiers, FOC2 , was proved decidable in [10,21], and it was subsequently shown to be NEXPTIME-complete in [22]. Research on extensions and variants of ...
... The satisfiability problem of two-variable logic FO2 was shown to be NEXPTIME-complete in [9]. The extension of two-variable logic with counting quantifiers, FOC2 , was proved decidable in [10,21], and it was subsequently shown to be NEXPTIME-complete in [22]. Research on extensions and variants of ...
Section.8.3
... The order of a predicate is 1 if its arguments are terms. Otherwise the order is n + 1 where n is the maximum order of the arguments that are not terms. The order of a function is always 1 since it’s arguments are always terms. Examples. In the wff p(x) q(x, p) the order of p is one and the order ...
... The order of a predicate is 1 if its arguments are terms. Otherwise the order is n + 1 where n is the maximum order of the arguments that are not terms. The order of a function is always 1 since it’s arguments are always terms. Examples. In the wff p(x) q(x, p) the order of p is one and the order ...
The modal logic of equilibrium models
... We relate the language of equilibrium logic to our bimodal language by means of a translation tr. The main clause of the translation is: tr(ϕ ⇒ ψ) = (tr(ϕ) → tr(ψ)) ∧ [T](tr(ϕ) → tr(ψ)) We prove that ϕ has a HT model if and only if its translation tr(ϕ) is satisfiable in MEM. This paves the way to t ...
... We relate the language of equilibrium logic to our bimodal language by means of a translation tr. The main clause of the translation is: tr(ϕ ⇒ ψ) = (tr(ϕ) → tr(ψ)) ∧ [T](tr(ϕ) → tr(ψ)) We prove that ϕ has a HT model if and only if its translation tr(ϕ) is satisfiable in MEM. This paves the way to t ...
Cut-Free Sequent Systems for Temporal Logic
... which clearly violates the subformula property since ψ is an arbitrary formula. Systems of the first kind can be found for example in Kawai [9]. Gudzhinskas [6] and Paech [14] give systems of the second kind. An exception is Pliuškevičius [15], who gives a finitary and truly cut-free system for a ...
... which clearly violates the subformula property since ψ is an arbitrary formula. Systems of the first kind can be found for example in Kawai [9]. Gudzhinskas [6] and Paech [14] give systems of the second kind. An exception is Pliuškevičius [15], who gives a finitary and truly cut-free system for a ...
Computing Default Extensions by Reductions on OR
... the authors state a modal reduction theorem to the effect that a formula O Rϕ is logically equivalent to a disjunction Oϕ1 ∨ · · · ∨ Oϕn , where each ϕk is a propositional formula. Because each such disjunct Oϕ k has a unique model, it is possible, within the logic itself, to break down a formula O ...
... the authors state a modal reduction theorem to the effect that a formula O Rϕ is logically equivalent to a disjunction Oϕ1 ∨ · · · ∨ Oϕn , where each ϕk is a propositional formula. Because each such disjunct Oϕ k has a unique model, it is possible, within the logic itself, to break down a formula O ...
Part 1 - Logic Summer School
... First-order logic is a maximal logic possessing both the Compactness Theorem and the Löwenheim-Skolem Theorem, i.e., no logic that is compact and satisfies the Löwenheim-Skolem property can properly extend FO. ...
... First-order logic is a maximal logic possessing both the Compactness Theorem and the Löwenheim-Skolem Theorem, i.e., no logic that is compact and satisfies the Löwenheim-Skolem property can properly extend FO. ...
Basic Logic and Fregean Set Theory - MSCS
... Arithmetic may be considered a proof interpretation in which there is a universe of ‘proofs’ encoded in numbers. However, a formula of first-order arithmetic is realized by a number if and only if that formula is derivable from HA plus a formalized version of Church’s Thesis [20, page 196]. It appe ...
... Arithmetic may be considered a proof interpretation in which there is a universe of ‘proofs’ encoded in numbers. However, a formula of first-order arithmetic is realized by a number if and only if that formula is derivable from HA plus a formalized version of Church’s Thesis [20, page 196]. It appe ...
Logic
... statements are claimed to follow from others, this may in fact not be the case. • Example: “If I win the lottery, then I’m happy. However, I did not win the lottery. Therefore, I am not happy.” • A piece of reasoning is valid if the statements that are claimed to follow from previous ones do indeed ...
... statements are claimed to follow from others, this may in fact not be the case. • Example: “If I win the lottery, then I’m happy. However, I did not win the lottery. Therefore, I am not happy.” • A piece of reasoning is valid if the statements that are claimed to follow from previous ones do indeed ...
Basic Metatheory for Propositional, Predicate, and Modal Logic
... whether every truth function is expressed by some formula of L P . The issue here hinges on the connectives of L P . A set of connectives in an interpreted language (i.e., a language together with its semantics) for propositional logic is said to be adequate iff every truth function can be expressed ...
... whether every truth function is expressed by some formula of L P . The issue here hinges on the connectives of L P . A set of connectives in an interpreted language (i.e., a language together with its semantics) for propositional logic is said to be adequate iff every truth function can be expressed ...
Logic - Disclaimer
... statements are claimed to follow from others, this may in fact not be the case. • Example: “If I win the lottery, then I’m happy. However, I did not win the lottery. Therefore, I am not happy.” • A piece of reasoning is valid if the statements that are claimed to follow from previous ones do indeed ...
... statements are claimed to follow from others, this may in fact not be the case. • Example: “If I win the lottery, then I’m happy. However, I did not win the lottery. Therefore, I am not happy.” • A piece of reasoning is valid if the statements that are claimed to follow from previous ones do indeed ...
.pdf
... tableau cannot be extended any further, because all formulas have been decomposed. Since the propositional tableau method terminates after finitely many steps, this was an easy thing to define. In the first-order case, however, we have to be a bit more careful. We know that because of γ-formulas, pr ...
... tableau cannot be extended any further, because all formulas have been decomposed. Since the propositional tableau method terminates after finitely many steps, this was an easy thing to define. In the first-order case, however, we have to be a bit more careful. We know that because of γ-formulas, pr ...
• Above we applied the unit resolution inference rule: ℓ1 ∨ … ∨ ℓ k
... • Term f(t1, …,tn) refers to the object that is the value of function F applied to objects d1, …,dn, where F is the interpretation of f and d1, …,dn are the objects that the argument terms t1, …,tn refer to • Atomic sentence P(t1, …,tn) is true if the relation referred to by the predicate symbol P h ...
... • Term f(t1, …,tn) refers to the object that is the value of function F applied to objects d1, …,dn, where F is the interpretation of f and d1, …,dn are the objects that the argument terms t1, …,tn refer to • Atomic sentence P(t1, …,tn) is true if the relation referred to by the predicate symbol P h ...
Search problems
... Hard to identify “individuals.” E.g., Mary, 3 Can’t directly talk about properties of individuals or relations between individuals. E.g. “Bill is tall” ...
... Hard to identify “individuals.” E.g., Mary, 3 Can’t directly talk about properties of individuals or relations between individuals. E.g. “Bill is tall” ...
slides (modified) - go here for webmail
... Propositional Logic :Syntax Propositional Logic :Semantics Satisfiability and validity Modeling with Propositional logic Normal forms Deductive proofs and resolution ...
... Propositional Logic :Syntax Propositional Logic :Semantics Satisfiability and validity Modeling with Propositional logic Normal forms Deductive proofs and resolution ...
Ambient Logic II.fm
... The π-calculus notion of name restriction [12], initially intended to represent hidden communication channels, has been used also to represent hidden encryption keys [2] and as the basis for definitions of secrecy [2, 4]. In the context of the ambient calculus [6], name restriction can be used to re ...
... The π-calculus notion of name restriction [12], initially intended to represent hidden communication channels, has been used also to represent hidden encryption keys [2] and as the basis for definitions of secrecy [2, 4]. In the context of the ambient calculus [6], name restriction can be used to re ...
com.1 The Compactness Theorem
... Problem com.2. In the standard model of arithmetic N, there is no element k ∈ |N| which satisfies every formula n < x (where n is 0...0 with n 0’s). Use the compactness theorem to show that the set of sentences in the language of arithmetic which are true in the standard model of arithmetic N are a ...
... Problem com.2. In the standard model of arithmetic N, there is no element k ∈ |N| which satisfies every formula n < x (where n is 0...0 with n 0’s). Use the compactness theorem to show that the set of sentences in the language of arithmetic which are true in the standard model of arithmetic N are a ...
Bisimulation and public announcements in logics of
... to extend the language of LP by introducing modals Ki for each i = 1, 2, . . . , n. We call this extended language the language of evidence-based knowledge or, more briefly, the EBK language. Fitting models for the EBK language are obtained from the Fitting models defined above by adding a reflexive ...
... to extend the language of LP by introducing modals Ki for each i = 1, 2, . . . , n. We call this extended language the language of evidence-based knowledge or, more briefly, the EBK language. Fitting models for the EBK language are obtained from the Fitting models defined above by adding a reflexive ...