https://zh.wikipedia.org/wiki/%E4%BE%9D%E8%B5%96%E7%B1%BB%E5%9E%8B
https://lean-lang.org/documentation/
Who knows and who c who caresareeessss
https://www.youtube.com/watch?v=gIJXcGQrpoM&t=1450s
In formal mathematics, ∀ (forall) and ∃ (there exists) are quantifiers used to express universal and existential properties of mathematical statements. When implemented in Coq, a proof assistant based on dependent types, these quantifiers are naturally represented using dependent types. Coq provides constructs like forall for universal quantification and exists for existential quantification. Here’s an overview of how to use dependent types in Coq to formalize such statements: --- ## 1. Universal Quantification (∀) Universal quantification (forall x, P(x)) asserts that a property ( P(x) ) holds for all ( x ) in a given domain. ### Example: Formalizing "For all ( n ), ( n + 0 = n )" coq Theorem add_zero : forall n : nat, n + 0 = n. Proof. intros n. simpl. reflexivity. Qed. ### Explanation: - forall n : nat, n + 0 = n: States that for every natural number ( n ), ( n + 0 = n ). - intros n: Introduces ( n ) into the proof context. - simpl: Simplifies the left-hand side of the equation. - reflexivity: Proves the equality ( n = n ). --- ## 2. Existential Quantification (∃) Existential quantification (exists x, P(x)) asserts that there exists an ( x ) in the domain such that ( P(x) ) holds. ### Example: Formalizing "There exists an ( n ) such that ( n + 2 = 5 )" coq Theorem exists_example : exists n : nat, n + 2 = 5. Proof. exists 3. (* Providing a witness ) simpl. reflexivity. Qed. ### Explanation: - exists n : nat, n + 2 = 5: Asserts the existence of a natural number ( n ) satisfying ( n + 2 = 5 ). - exists 3: Provides a specific witness, ( n = 3 ). - simpl and reflexivity: Complete the proof by verifying the property for the witness. --- ## 3. Using Dependent Types in Coq Dependent types allow constructing types that depend on values. This is crucial for combining quantifiers and defining more complex properties. ### Example: Dependent Pair (Σ-Type) The dependent pair type, denoted by sig in Coq, is used to represent constructs like exists x, P(x). #### Defining and Using Dependent Pairs coq Require Import Coq.Init.Nat. Definition even (n : nat) : Prop := exists k : nat, n = 2 * k. Definition witness_of_even : {n : nat | even n}. Proof. exists 4. ( Witness ) unfold even. ( Expand definition of even ) exists 2. ( Prove that 4 = 2 * 2 ) reflexivity. Qed. ### Explanation: - {n : nat | even n}: Represents a dependent pair where ( n ) is a natural number, and ( even(n) ) holds. - exists 4: Provides a witness ( n = 4 ). - unfold even: Expands the definition of even. - exists 2: Provides a proof that ( 4 = 2 \times 2 ). --- ## 4. Combining Universal and Existential Quantifiers Sometimes, you need to formalize statements involving both quantifiers, such as: [ \forall x \in \mathbb{N}, \exists y \in \mathbb{N}, x + y = 10. ] ### Coq Formalization coq Theorem forall_exists_example : forall x : nat, exists y : nat, x + y = 10. Proof. intros x. exists (10 - x). ( Choose y = 10 - x ) lia. ( Simplify using linear arithmetic ) Qed. ### Explanation: - forall x : nat, exists y : nat, x + y = 10: Asserts that for any natural number ( x ), there exists a ( y ) such that ( x + y = 10 ). - exists (10 - x): Provides ( y = 10 - x ) as a witness. - lia: Solves the arithmetic. --- ## 5. Dependent Types for Constructive Proofs Coq supports constructive mathematics, so proving ( \exists x, P(x) ) means explicitly constructing ( x ). ### Example: Proving Existence with Constructive Witness coq Definition constructive_witness : exists x : nat, x * x = 16. Proof. exists 4. ( Provide the witness ) reflexivity. Qed. ### Explanation: This proof not only asserts existence but also constructs ( x = 4 ) explicitly. --- ## 6. Advanced Use: Functions Over Dependent Types Dependent types allow constructing functions whose outputs depend on their inputs. ### Example: Function Returning Proofs coq Definition add_zero_proof (n : nat) : n + 0 = n := match n with | 0 => eq_refl | S m => eq_refl end. This function provides a proof that ( n + 0 = n ) for any ( n ). --- ## 7. Interactive Theorem Proving Coq allows interactive proof development using tactics like intros, apply, and exists. ### Example: Proving a Complex Statement [ \forall n \in \mathbb{N}, \exists m \in \mathbb{N}, n \leq m. ] coq Theorem forall_exists_complex : forall n : nat, exists m : nat, n <= m. Proof. intros n. exists n. ( Choose m = n ) apply le_n. ( n <= n *) Qed. --- ## Conclusion Using dependent types in Coq enables precise formalization of mathematical statements involving universal (( \forall )) and existential (( \exists )) quantifiers. Coq's type system ensures constructive proofs by requiring explicit witnesses for existential statements. This approach provides a powerful framework for developing, verifying, and reasoning about mathematical theorems interactively. Let me know if you'd like further elaboration on any part!
Class hierarchies in computational complexity and logic theory are frameworks for categorizing problems, functions, or sets according to their complexity or structural properties. Each hierarchy you've listed—Polynomial Hierarchy, Exponential Hierarchy, Grzegorczyk Hierarchy, Arithmetical Hierarchy, and Boolean Hierarchy—has a distinct purpose and applies different criteria to organize classes within the hierarchy. Here’s an overview of each:
The Polynomial Hierarchy is a hierarchy of complexity classes that generalizes P (problems solvable in polynomial time) and NP (nondeterministic polynomial time) by introducing alternating quantifiers.
-
Levels of the Polynomial Hierarchy: The polynomial hierarchy is structured in levels (often denoted by
$\Sigma_k^P$ and$\Pi_k^P$ ), where each level represents a class of problems that can be solved by alternating sequences of existential and universal quantifiers over polynomial-time computations.-
$\Sigma_1^P$ = NP: Problems that can be solved in nondeterministic polynomial time. -
$\Pi_1^P$ = co-NP: Problems whose complements are in NP. -
$\Sigma_2^P$ : Exists-Forall problems, solvable by a nondeterministic polynomial algorithm with an existential quantifier followed by a universal quantifier. - Higher levels (
$\Sigma_k^P$ and$\Pi_k^P$ ) continue by alternating quantifiers.
-
-
Relationship with Complexity Classes: If the Polynomial Hierarchy collapses to some level
$k$ , it means that all levels above$k$ are equivalent, which would imply significant simplifications in complexity theory. -
Importance: The Polynomial Hierarchy captures a wide range of natural computational problems and is essential in understanding the structure of NP and co-NP and the implications for related classes.
The Exponential Hierarchy is a hierarchy of complexity classes similar to the Polynomial Hierarchy but based on exponential time bounds.
-
Definition: Each level in the Exponential Hierarchy corresponds to the classes
$\Sigma_k^{\text{EXP}}$ and$\Pi_k^{\text{EXP}}$ , where:-
$\Sigma_1^{\text{EXP}} = \text{NEXP}$ : Nondeterministic exponential time. -
$\Pi_1^{\text{EXP}} = \text{co-NEXP}$ : Complement of NEXP problems. -
$\Sigma_2^{\text{EXP}}$ : Exists-Forall problems solvable in exponential time.
-
-
Hierarchy Structure: The hierarchy allows alternating quantifiers, with each level increasing the bounds of computation resources exponentially rather than polynomially.
-
Importance: The Exponential Hierarchy is crucial for understanding problems that require exponentially more resources than those in the Polynomial Hierarchy. It’s particularly relevant in areas where problems grow exponentially with input size, like certain types of logical reasoning or combinatorial problems.
The Grzegorczyk Hierarchy is a classification of functions based on computational complexity with respect to primitive recursion and growth rate.
-
Levels of the Grzegorczyk Hierarchy: The hierarchy has levels
$E_n$ , where:-
$E_0$ is the class of constant functions. -
$E_1$ contains addition functions. -
$E_2$ includes multiplication and functions with polynomial growth. - Higher levels
$E_n$ contain functions that grow at increasingly faster rates.
-
-
Relation to Computability: Each level of the Grzegorczyk Hierarchy corresponds to a class of computable functions with restricted growth rates, and
$E_3$ includes all primitive recursive functions. Beyond a certain level, the hierarchy includes functions that are not primitive recursive but still computable. -
Importance: This hierarchy is fundamental in computability theory and recursion theory. It provides insights into the growth rates of functions that are still computable and highlights the boundaries between primitive recursive functions and more complex forms of computation.
The Arithmetical Hierarchy is used in mathematical logic and recursion theory to classify sets of natural numbers (or statements) based on the quantifier complexity of their logical definitions.
-
Levels of the Arithmetical Hierarchy: Levels are defined using
$\Sigma_n$ and$\Pi_n$ notation, where:-
$\Sigma_0 = \Pi_0 = \Delta_0$ : Decidable (computable) sets. -
$\Sigma_1$ : Sets definable with an existential quantifier followed by a computable predicate. -
$\Pi_1$ : Sets definable with a universal quantifier followed by a computable predicate. - Higher levels alternate quantifiers.
-
-
Applications: The hierarchy classifies logical statements based on their complexity, which is useful for understanding provability and decidability in formal systems. For example,
$\Sigma_1$ -statements include existential questions about natural numbers, while$\Pi_1$ -statements include universal questions. -
Importance: The Arithmetical Hierarchy provides a framework for analyzing the complexity of mathematical statements, particularly in fields like model theory and proof theory.
The Boolean Hierarchy builds on NP and is a hierarchy of complexity classes created by combining NP sets with Boolean operations like union, intersection, and complementation.
-
Levels in the Boolean Hierarchy:
- The hierarchy begins with NP (problems solvable by nondeterministic polynomial-time algorithms).
- The next levels are formed by taking Boolean combinations of NP sets, such as:
- NP ∩ co-NP: Problems that belong to both NP and co-NP.
- (NP ∩ co-NP) ∪ NP: The union of NP and problems in both NP and co-NP.
- Higher levels involve increasingly complex combinations.
-
Importance: The Boolean Hierarchy is critical in understanding how NP problems interact with co-NP problems and provides insights into the complexity of Boolean operations on NP problems. It helps clarify relationships between different classes within NP and their interactions with co-NP.
| Hierarchy | Focus Area | Key Features |
|---|---|---|
| Polynomial Hierarchy (PH) | Complexity Theory | Alternating quantifiers over polynomial time; generalizes NP and co-NP. |
| Exponential Hierarchy (EH) | Complexity Theory | Alternating quantifiers over exponential time; captures higher complexity problems. |
| Grzegorczyk Hierarchy | Computability Theory | Classifies functions by growth rate; includes primitive recursive and hyper-primitive classes. |
| Arithmetical Hierarchy | Mathematical Logic | Categorizes sets/statements by quantifier complexity; used in recursion and proof theory. |
| Boolean Hierarchy | Complexity Theory | Layers Boolean combinations of NP problems; studies interactions between NP and co-NP. |
These hierarchies provide structured ways to analyze complexity and logical properties across different domains of theoretical computer science and mathematical logic. Each hierarchy serves a distinct purpose, from understanding polynomial-time computations to exploring logical quantifier depth, making them fundamental tools for formalizing and studying computational and logical problems.
https://en.wikipedia.org/wiki/Dependent_type
https://web.archive.org/web/20201109032232/http://sage.soe.ucsc.edu/
在基於推理規則的證明中，常用的寫法包括以下幾種：
-
直接推理：
- 前提：( A \rightarrow B )
- 前提：( B \rightarrow C )
- 結論：( A \rightarrow C ) （通過 連鎖律 (Hypothetical Syllogism)）
-
反證法：
- 前提：假設 ( \neg P ) 為真
- 推導出矛盾：( Q \land \neg Q )
- 結論：( P ) 必為真 （通過 歸謬法 (Reductio ad Absurdum)）
-
反證法（Contrapositive）：
- 前提：( A \rightarrow B )
- 反證：( \neg B \rightarrow \neg A ) （通過 對換律 (Contraposition)）
- 結論：若反證為真，則原命題 ( A \rightarrow B ) 為真。
-
歸謬法：
- 假設：( P )
- 得出矛盾：( \neg Q )
- 結論：( \neg P ) 必為真（通過 矛盾律 (Contradiction)）
-
循環證明：
- 前提：( A_1 \rightarrow A_2 )
- 前提：( A_2 \rightarrow A_3 )
- ⋮
- 前提：( A_{n-1} \rightarrow A_n )
- 前提：( A_n \rightarrow A_1 )
- 結論：所有命題 ( A_1, A_2, \dots, A_n ) 均為真 （通過 循環推理 (Cyclic Reasoning)）
-
數學歸納法：
- 基礎步驟：證明 ( P(1) ) 為真
- 歸納步驟：假設 ( P(k) ) 為真，證明 ( P(k+1) ) 為真 （通過 數學歸納法 (Mathematical Induction)）
- 結論：( P(n) ) 對所有自然數 ( n ) 為真
在基於推理規則的證明中，常用的寫法包括以下幾種：
-
直接推理：
- 前提：( A \rightarrow B )
- 前提：( B \rightarrow C )
- 結論：( A \rightarrow C ) （通過 連鎖律 (Hypothetical Syllogism)）
-
反證法：
- 前提：假設 ( \neg P ) 為真
- 推導出矛盾：( Q \land \neg Q )
- 結論：( P ) 必為真 （通過 歸謬法 (Reductio ad Absurdum)）
-
反證法（Contrapositive）：
- 前提：( A \rightarrow B )
- 反證：( \neg B \rightarrow \neg A ) （通過 對換律 (Contraposition)）
- 結論：若反證為真，則原命題 ( A \rightarrow B ) 為真。
-
歸謬法：
- 假設：( P )
- 得出矛盾：( \neg Q )
- 結論：( \neg P ) 必為真（通過 矛盾律 (Contradiction)）
-
循環證明：
- 前提：( A_1 \rightarrow A_2 )
- 前提：( A_2 \rightarrow A_3 )
- ⋮
- 前提：( A_{n-1} \rightarrow A_n )
- 前提：( A_n \rightarrow A_1 )
- 結論：所有命題 ( A_1, A_2, \dots, A_n ) 均為真 （通過 循環推理 (Cyclic Reasoning)）
-
數學歸納法：
- 基礎步驟：證明 ( P(1) ) 為真
- 歸納步驟：假設 ( P(k) ) 為真，證明 ( P(k+1) ) 為真 （通過 數學歸納法 (Mathematical Induction)）
- 結論：( P(n) ) 對所有自然數 ( n ) 為真
In proofs based on rules of inference, the typical formats include:
-
Direct Reasoning:
- Premise: ( A \rightarrow B )
- Premise: ( B \rightarrow C )
- Conclusion: ( A \rightarrow C ) (using Hypothetical Syllogism)
-
Proof by Contradiction:
- Premise: Assume ( \neg P )
- Derive a contradiction: ( Q \land \neg Q )
- Conclusion: ( P ) must be true (using Reductio ad Absurdum)
-
Proof by Contrapositive:
- Premise: ( A \rightarrow B )
- Contrapositive: ( \neg B \rightarrow \neg A ) (using Contraposition)
- Conclusion: If the contrapositive is true, then the original proposition ( A \rightarrow B ) is true.
-
Proof by Contradiction:
- Assume: ( P )
- Derive a contradiction: ( \neg Q )
- Conclusion: ( \neg P ) must be true (using Contradiction)
-
Cyclic Proof:
- Premise: ( A_1 \rightarrow A_2 )
- Premise: ( A_2 \rightarrow A_3 )
- ⋮
- Premise: ( A_{n-1} \rightarrow A_n )
- Premise: ( A_n \rightarrow A_1 )
- Conclusion: All propositions ( A_1, A_2, \dots, A_n ) are true (using Cyclic Reasoning)
-
Mathematical Induction:
- Base case: Prove ( P(1) ) is true
- Inductive step: Assume ( P(k) ) is true, prove ( P(k+1) ) is true (using Mathematical Induction)
- Conclusion: ( P(n) ) is true for all natural numbers ( n )
En las pruebas basadas en reglas de inferencia, los formatos típicos incluyen:
-
Razonamiento Directo:
- Premisa: ( A \rightarrow B )
- Premisa: ( B \rightarrow C )
- Conclusión: ( A \rightarrow C ) (usando Silogismo Hipotético)
-
Prueba por Contradicción:
- Premisa: Supongamos ( \neg P )
- Derivamos una contradicción: ( Q \land \neg Q )
- Conclusión: ( P ) debe ser verdadero (usando Reducción al Absurdo)
-
Prueba por Contraposición:
- Premisa: ( A \rightarrow B )
- Contraposición: ( \neg B \rightarrow \neg A ) (usando Contraposición)
- Conclusión: Si la contraposición es verdadera, entonces la proposición original ( A \rightarrow B ) es verdadera.
-
Prueba por Contradicción:
- Supuesto: ( P )
- Derivamos una contradicción: ( \neg Q )
- Conclusión: ( \neg P ) debe ser verdadero (usando Contradicción)
-
Prueba Cíclica:
- Premisa: ( A_1 \rightarrow A_2 )
- Premisa: ( A_2 \rightarrow A_3 )
- ⋮
- Premisa: ( A_{n-1} \rightarrow A_n )
- Premisa: ( A_n \rightarrow A_1 )
- Conclusión: Todas las proposiciones ( A_1, A_2, \dots, A_n ) son verdaderas (usando Razonamiento Cíclico)
-
Inducción Matemática:
- Caso base: Probar que ( P(1) ) es verdadero
- Paso inductivo: Suponer que ( P(k) ) es verdadero, probar que ( P(k+1) ) es verdadero (usando Inducción Matemática)
- Conclusión: ( P(n) \
Matters or not. Does it matters? It is a mater? Are they matters? Do they matter? English majors. English majores in majors. English majores.
None of it matters because nobody cares.
~## p => p (Circular Reasoning)
~p => p (Circular Reasoning?)
The logical statement ( p \Rightarrow q ) ("if ( p ), then ( q )") is logically equivalent to the following:
[ \neg q \Rightarrow \neg p ]
- Meaning: "If ( q ) is false, then ( p ) is false."
- This equivalence holds because ( p \Rightarrow q ) is true in all cases where ( q ) is true or ( p ) is false.
[ \neg p \lor q ]
- Meaning: "Either ( p ) is false, or ( q ) is true."
- This is derived from the truth table for ( p \Rightarrow q ), which is true except when ( p ) is true and ( q ) is false.
[ \neg (p \land \neg q) ]
- Meaning: "It is not the case that ( p ) is true and ( q ) is false."
- This highlights that the only condition where ( p \Rightarrow q ) is false is when ( p ) is true and ( q ) is false.
Breaking it down further, ( p \Rightarrow q ) can also be expressed as: [ \neg p \lor q \equiv \neg (p \land \neg q) ] This demonstrates the equivalence between the disjunction and negation forms.
| ( p ) | ( q ) | ( p \Rightarrow q ) | ( \neg p \lor q ) | ( \neg (p \land \neg q) ) | ( \neg q \Rightarrow \neg p ) |
|---|---|---|---|---|---|
| T | T | T | T | T | T |
| T | F | F | F | F | F |
| F | T | T | T | T | T |
| F | F | T | T | T | T |
All these forms are equivalent.
The Dyson equation is a fundamental concept in quantum field theory and many-body physics, describing how the propagator (or Green's function) of a system is modified by interactions. It is used to understand how particles propagate in the presence of interactions and is central to many-body perturbation theory.
-
General Form: [ G = G_0 + G_0 \Sigma G ] where:
- ( G ): Full propagator (interacting system).
- ( G_0 ): Free propagator (non-interacting system).
- ( \Sigma ): Self-energy, encapsulating all interaction effects.
-
Physical Interpretation:
- The Dyson equation relates the full propagator ( G ) of a particle to the free propagator ( G_0 ), modified by the self-energy ( \Sigma ).
- ( G_0 \Sigma G ) represents the effect of interactions between the particle and the system.
-
Free Propagator (( G_0 )):
- Describes how a particle propagates in a non-interacting system.
- In momentum space, for example, ( G_0(k, \omega) = \frac{1}{\omega - \epsilon_k + i\eta} ), where ( \epsilon_k ) is the energy of a free particle with momentum ( k ).
-
Self-Energy (( \Sigma )):
- Represents all interaction effects, including scattering and virtual processes.
- It modifies the energy and lifetime of particles.
-
Full Propagator (( G )):
- Describes the propagation of a particle in the interacting system, accounting for all interaction effects.
For more complex systems, such as those involving multiple particles or spins, the Dyson equation can be written in matrix form: [ \mathbf{G} = \mathbf{G}_0 + \mathbf{G}_0 \mathbf{\Sigma} \mathbf{G} ]
In momentum-frequency space (often used in condensed matter physics): [ G(k, \omega) = G_0(k, \omega) + G_0(k, \omega) \Sigma(k, \omega) G(k, \omega) ]
Solving for ( G(k, \omega) ): [ G(k, \omega) = \frac{1}{G_0(k, \omega)^{-1} - \Sigma(k, \omega)} ]
-
Quantum Field Theory:
- Used in Feynman diagrams to describe how particles propagate and interact.
-
Many-Body Physics:
- Central to studying electronic properties in materials, including phenomena like superconductivity, magnetism, and strongly correlated systems.
-
Perturbation Theory:
- Forms the backbone of perturbative expansions, where the self-energy ( \Sigma ) is computed iteratively or diagrammatically.
-
Condensed Matter Physics:
- Describes electron dynamics in solids, including quasiparticle behavior.
-
Renormalization:
- The self-energy ( \Sigma ) incorporates renormalization effects, such as shifting particle masses or changing energy levels.
-
Quasiparticles:
- The modified propagator ( G ) reveals the existence of quasiparticles with properties different from free particles.
-
Interaction Effects:
- The self-energy captures all complex interactions, including multi-particle effects and correlations.
The Dyson equation provides a framework for systematically including interaction effects in particle propagation, making it a cornerstone of modern theoretical physics. Would you like a deeper dive into any specific representation or proof?