Infinite Games - International Mathematical Union
... Det (il^)=Det (CPCA) implies that all 2?J sets of real numbers are Lebesgue measurable and yields a complete structural theory for levels three and four of the projective hierarchy. J. Mycielski observed that a result of [1] implies that Det (n\) is not provable in the usual set theory ZFC. If one a ...
... Det (il^)=Det (CPCA) implies that all 2?J sets of real numbers are Lebesgue measurable and yields a complete structural theory for levels three and four of the projective hierarchy. J. Mycielski observed that a result of [1] implies that Det (n\) is not provable in the usual set theory ZFC. If one a ...
The Surprise Examination Paradox and the Second Incompleteness
... Berry’s paradox: consider the expression “the smallest positive integer not definable in under eleven words”. This expression defines that integer in under eleven words. To formalize Berry’s paradox, Chaitin uses the notion of Kolmogorov complexity. The Kolmogorov complexity K(x) of an integer x is ...
... Berry’s paradox: consider the expression “the smallest positive integer not definable in under eleven words”. This expression defines that integer in under eleven words. To formalize Berry’s paradox, Chaitin uses the notion of Kolmogorov complexity. The Kolmogorov complexity K(x) of an integer x is ...
03_Artificial_Intelligence-PredicateLogic
... • Intuitionistic first-order logic uses intuitionistic rather than classical propositional calculus; for example, ¬¬φ need not be equivalent to φ • Infinitary logic allows infinitely long sentences; for example, one may allow a conjunction or disjunction of infinitely many formulas, or quantificatio ...
... • Intuitionistic first-order logic uses intuitionistic rather than classical propositional calculus; for example, ¬¬φ need not be equivalent to φ • Infinitary logic allows infinitely long sentences; for example, one may allow a conjunction or disjunction of infinitely many formulas, or quantificatio ...
A Logic of Explicit Knowledge - Lehman College
... having a single state, Γ, accessible to itself, and with an evidence function such that E(Γ, t) is the entire set of formulas. In this model, t serves as ‘universal’ evidence. Also, use a valuation such that V(P ) = {Γ} and V(Q) = ∅. Then we have M, Γ t:P but M, Γ 6 t:(P ∧ Q) because, even though ...
... having a single state, Γ, accessible to itself, and with an evidence function such that E(Γ, t) is the entire set of formulas. In this model, t serves as ‘universal’ evidence. Also, use a valuation such that V(P ) = {Γ} and V(Q) = ∅. Then we have M, Γ t:P but M, Γ 6 t:(P ∧ Q) because, even though ...
We can only see a short distance ahead, but we can see plenty
... (Slaman [personal communication]), but establishing a natural definition of the jump still seems to be the most likely route to a definition of recursive enumerability. (See Slaman [59].) In the area of using classical computability type complexity properties to classify mathematical structures, I w ...
... (Slaman [personal communication]), but establishing a natural definition of the jump still seems to be the most likely route to a definition of recursive enumerability. (See Slaman [59].) In the area of using classical computability type complexity properties to classify mathematical structures, I w ...
A. Formal systems, Proof calculi
... logically valid or no? If there is such an algorithm, then the calculus is decidable. If the calculus is complete, then it proves all the logically valid formulas, and the proofs can be described in an algorithmic way. However, in case the input formula is not logically valid, the algorithm does n ...
... logically valid or no? If there is such an algorithm, then the calculus is decidable. If the calculus is complete, then it proves all the logically valid formulas, and the proofs can be described in an algorithmic way. However, in case the input formula is not logically valid, the algorithm does n ...
SPECTRA OF THEORIES AND STRUCTURES 1. Introduction The
... Hirschfeldt, Khoussainov, Shore, and Slinko [11] proved that the class of graphs is universal for structure spectra in the sense that every spectrum of a structure is the spectrum of a graph. We prove the analogous result for theory spectra. Proposition 2.2. We can translate any theory to the langua ...
... Hirschfeldt, Khoussainov, Shore, and Slinko [11] proved that the class of graphs is universal for structure spectra in the sense that every spectrum of a structure is the spectrum of a graph. We prove the analogous result for theory spectra. Proposition 2.2. We can translate any theory to the langua ...
slides
... BindPatt(φ) ∩ BindPatt(ψ) and φ ⊨ χ ⊨ ψ. Moreover the formula χ in question can effectively constructed from a proof of φ ⊨ ψ (in a suitable proof system). ...
... BindPatt(φ) ∩ BindPatt(ψ) and φ ⊨ χ ⊨ ψ. Moreover the formula χ in question can effectively constructed from a proof of φ ⊨ ψ (in a suitable proof system). ...
Herbrands Theorem
... It is infeasible to consider all interpretations over all domains in order to prove unsatisfiability Instead, we try to fix a special domain (called a Herbrand universe) such that the formula, S, is unsatisfiable iff it is false under all the interpretations over this domain ...
... It is infeasible to consider all interpretations over all domains in order to prove unsatisfiability Instead, we try to fix a special domain (called a Herbrand universe) such that the formula, S, is unsatisfiable iff it is false under all the interpretations over this domain ...
Second-Order Logic and Fagin`s Theorem
... must ascertain the symbol ρt̄ that is read by N . ρt̄ is equal to σi where Si (t̄′ ) holds, and t̄′ is the last time before t̄ that the head was in its present location (or it is the corresponding input symbol if this is the first time the head is at this cell). To express ρt̄ , we need to express t ...
... must ascertain the symbol ρt̄ that is read by N . ρt̄ is equal to σi where Si (t̄′ ) holds, and t̄′ is the last time before t̄ that the head was in its present location (or it is the corresponding input symbol if this is the first time the head is at this cell). To express ρt̄ , we need to express t ...
pdf
... If a first-order formula X is valid, then X the there is an atomically closed tableau for F X. ...
... If a first-order formula X is valid, then X the there is an atomically closed tableau for F X. ...
Modal Logic
... for basic modal logic is quite general (although it can be further generalized as we will see later) and can be refined to yield the properties appropriate for the intended application. We will concentrate on three different applications: logic of necessity, temporal logic and logic of knowledge. T ...
... for basic modal logic is quite general (although it can be further generalized as we will see later) and can be refined to yield the properties appropriate for the intended application. We will concentrate on three different applications: logic of necessity, temporal logic and logic of knowledge. T ...
On the error term in a Parseval type formula in the theory of Ramanujan expansions,
... b) For fixed n, cr (n) is a multiplicative function i.e. for r1 , r2 with gcd(r1 , r2 ) = 1 we have cr1 r2 (n) = cr1 (n)cr2 (n). This is essentially due to the fact that, for r1 , r2 with gcd(r1 , r2 ) = 1, the fields Q(ζr1 ) and Q(ζr2 ) are linearly disjoint. c) cr (·) is a periodic function with per ...
... b) For fixed n, cr (n) is a multiplicative function i.e. for r1 , r2 with gcd(r1 , r2 ) = 1 we have cr1 r2 (n) = cr1 (n)cr2 (n). This is essentially due to the fact that, for r1 , r2 with gcd(r1 , r2 ) = 1, the fields Q(ζr1 ) and Q(ζr2 ) are linearly disjoint. c) cr (·) is a periodic function with per ...
Maximal Introspection of Agents
... (ii’) Everything believed by an agent in T is true (in T ). The theory considered in Example 3.1 is a base theory. A base theory describes the environment in which the agents are situated as well as the agents’ firstorder beliefs about this environment. Condition (ii) simply says that all (firstorde ...
... (ii’) Everything believed by an agent in T is true (in T ). The theory considered in Example 3.1 is a base theory. A base theory describes the environment in which the agents are situated as well as the agents’ firstorder beliefs about this environment. Condition (ii) simply says that all (firstorde ...
The Surprise Examination Paradox and the Second Incompleteness
... The second incompleteness theorem follows directly from Gödel’s original proof for the first incompleteness theorem. As described above, Gödel expressed the statement “this statement has no proof” and showed that, if the theory is consistent, this is a true statement (over N) that has no proof. In ...
... The second incompleteness theorem follows directly from Gödel’s original proof for the first incompleteness theorem. As described above, Gödel expressed the statement “this statement has no proof” and showed that, if the theory is consistent, this is a true statement (over N) that has no proof. In ...
Complexity of Existential Positive First-Order Logic
... to a primitive positive sentence (by moving all existential quantifiers to the front). It is clear that T has polynomial running time, and that Φ is true in Γ if and only if there exists a computation of T on Φ that computes a sentence that is true in Γ . We now show that E X P OS(Γ ) is hard for CS ...
... to a primitive positive sentence (by moving all existential quantifiers to the front). It is clear that T has polynomial running time, and that Φ is true in Γ if and only if there exists a computation of T on Φ that computes a sentence that is true in Γ . We now show that E X P OS(Γ ) is hard for CS ...
Lecturecise 19 Proofs and Resolution Compactness for
... What the infinite formula D breaks is the second part, which, from the existence of interpretations that agree on an arbitrarily long finite prefix derives an interpretation for infinitely many variables. Indeed, this part explicitly refers to a finite number of variables in the formula. ...
... What the infinite formula D breaks is the second part, which, from the existence of interpretations that agree on an arbitrarily long finite prefix derives an interpretation for infinitely many variables. Indeed, this part explicitly refers to a finite number of variables in the formula. ...
De Jongh`s characterization of intuitionistic propositional calculus
... Moreover, it can be shown that for every point w in T (n), there exists a point v ∈ U (n) such that wRv. In other words, universal models are “upper parts” of Henkin models. As we saw in Corollary 2.2, the n-universal model of IPC carries all the information about the formulas in n-variables. Unfort ...
... Moreover, it can be shown that for every point w in T (n), there exists a point v ∈ U (n) such that wRv. In other words, universal models are “upper parts” of Henkin models. As we saw in Corollary 2.2, the n-universal model of IPC carries all the information about the formulas in n-variables. Unfort ...
Semantics of intuitionistic propositional logic
... Remark 3.2 An intuitive reading of the above is to think of S as the set of possible worlds and the relation p A as A is true in world p. The judgement p ≤ q indicates that q is accessible from p. A further suggestive reading is to think of worlds as states of knowledge, and then p ≤ q indicates t ...
... Remark 3.2 An intuitive reading of the above is to think of S as the set of possible worlds and the relation p A as A is true in world p. The judgement p ≤ q indicates that q is accessible from p. A further suggestive reading is to think of worlds as states of knowledge, and then p ≤ q indicates t ...
Multi-Agent Only
... If Alice believes that all that Bob knows is that birds normally fly and that Tweety is a bird, then Alice believes that Bob believes that Tweety flies. But technically things were surprisingly cumbersome! The problem lies in the complexity in what agents consider ...
... If Alice believes that all that Bob knows is that birds normally fly and that Tweety is a bird, then Alice believes that Bob believes that Tweety flies. But technically things were surprisingly cumbersome! The problem lies in the complexity in what agents consider ...
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 ...
An Introduction to Löb`s Theorem in MIRI Research
... programs, but I won’t assume any background in mathematical logic beyond knowing the usual logical operators, nor that you’ve even heard of Löb’s Theorem before. To motivate the mathematical sections that follow, let’s consider a toy problem. Say that we’ve designed Deep Thought 1.0, an AI that rea ...
... programs, but I won’t assume any background in mathematical logic beyond knowing the usual logical operators, nor that you’ve even heard of Löb’s Theorem before. To motivate the mathematical sections that follow, let’s consider a toy problem. Say that we’ve designed Deep Thought 1.0, an AI that rea ...
Definability properties and the congruence closure
... property is Lo,,~itself, by LindstrSm's argument coding partial isomorphisms. [] Compare this result with Theorem 4.8 in Krinicki [K]. There are many interesting congruence closed logics, which by the first part of the Corollary can not be sublogics of L,o,o(Th). For example L~o~withK > co1 (these a ...
... property is Lo,,~itself, by LindstrSm's argument coding partial isomorphisms. [] Compare this result with Theorem 4.8 in Krinicki [K]. There are many interesting congruence closed logics, which by the first part of the Corollary can not be sublogics of L,o,o(Th). For example L~o~withK > co1 (these a ...
T - RTU
... For a first-order predicate calculus sentence S and an interpretation I: • An interpretation I that makes a sentence (expression) S true is said to satisfy S. • An interpretation I that satisfies every member of a set of expressions is said to satisfy the set. ...
... For a first-order predicate calculus sentence S and an interpretation I: • An interpretation I that makes a sentence (expression) S true is said to satisfy S. • An interpretation I that satisfies every member of a set of expressions is said to satisfy the set. ...
Mathematical Logic
... Different Forms of Reasoning Deduction: Given a set of premises Γ and a conclusion φ show that indeed Γ |= φ (this includes Validity: Γ = ∅) Abduction/Induction: given a theory T and an observation φ, find an explanation Γ such that T ∪ Γ |= φ Satisfiability Checking: given a set of formulae Γ, che ...
... Different Forms of Reasoning Deduction: Given a set of premises Γ and a conclusion φ show that indeed Γ |= φ (this includes Validity: Γ = ∅) Abduction/Induction: given a theory T and an observation φ, find an explanation Γ such that T ∪ Γ |= φ Satisfiability Checking: given a set of formulae Γ, che ...