Fundamenty i rozumowanie

Foundations and reasoning · id foundations

Matematyczny i algorytmiczny język AI, także metod bez sieci neuronowych.

The mathematical and algorithmic language of AI, including methods without neural networks.

In one sentence

To matematyka i sposoby rozumowania, dzięki którym programy liczą, wyciągają wnioski i szukają rozwiązań.

These are the mathematics and reasoning methods that let programs calculate, draw conclusions and search for solutions.

Problem it solves

Trudno rozwiązywać złożone zadania, gdy informacje są niepełne, możliwości liczne, a reguły trzeba stosować konsekwentnie.

Complex tasks are hard when information is incomplete, possibilities are numerous and rules must be applied consistently.

What it does

Zapisuje wiedzę, niepewność i cele w postaci, którą program może przeliczać, porównując możliwe odpowiedzi i kolejne działania.

It represents knowledge, uncertainty and goals in a form software can calculate with, comparing possible answers and sequences of actions.

Everyday example

Planer wybiera kolejność wizyt, uwzględnia dostępne godziny i porównuje możliwe drogi między miejscami.

A planner chooses the order of visits, accounts for available times and compares possible routes between places.

This concerns you if you…

• chcesz zrozumieć, jak program dochodzi do wyniku
• rozwiązujesz zadania z regułami, wyborami lub niepewnością
• want to understand how software reaches a result
• solve tasks involving rules, choices or uncertainty

Subtopics

• Logika i reprezentacja wiedzy (Logic and knowledge representation): How to write down facts, rules and dependencies.
  • AI neurosymboliczna (Neuro-symbolic AI): Combining neural networks (learning from data) with symbolic systems (logic, rules, solvers), e.g. an LLM translating a problem into a formal language for a solver.
  • Automatyczne dowodzenie twierdzeń (Lean) (Automated and formal theorem proving (Lean)): Models generate formal proofs checked by the kernel of a system such as Lean. Verification establishes that an accepted proof is correct relative to the formalized statement, declared axioms, and trusted kernel; the faithfulness of autoformalization remains a separate issue.
  • Grafy wiedzy (Knowledge graphs): Concepts and relations written as a graph, retrieval included.
  • Logika zdań i logika pierwszego rzędu (Propositional and first-order logic): Formal notation for facts and rules, and automatic derivation of conclusions from them.
  • Ontologie i sieć semantyczna (RDF, OWL) (Ontologies and the Semantic Web (RDF, OWL)): Standards for describing concepts, classes and relations so machines can check consistency and infer implicit facts.
  • Programowanie logiczne (Logic programming): Declarative paradigm: a program is a set of logical clauses and computation is inference (resolution, fixpoint or stable models).
  • Rozumowanie zdroworozsądkowe (Commonsense reasoning): AI subfield on representing and using everyday knowledge of the physical, social and temporal world.
  • Solwery SAT i SMT (SAT and SMT solvers): Engines deciding satisfiability of propositional formulas (SAT) and of formulas with background theories such as arithmetic or arrays (SMT).
• Matematyka i optymalizacja (Mathematics and optimization): How to describe and optimise models.
  • Algebra liniowa: wektory i macierze (Linear algebra: vectors and matrices): The language for data, weights and embeddings: vectors, matrices, products and matrix decompositions.
    • Rozkłady macierzy: wartości własne i SVD (Matrix decompositions: eigendecomposition and SVD): Matrix factorisations (eigen, SVD, QR, Cholesky) revealing structure, rank and dominant directions of data.
  • Rachunek różniczkowy i gradienty (Calculus and gradients): Derivatives, gradients and the chain rule — the basis for computing how to change weights to reduce error (backpropagation).
  • Spadek gradientu i optymalizacja ciągła (Gradient descent and continuous optimisation): Mathematical foundations of minimising a loss: gradient descent, convexity, constraints. Practical training tuning lives under “Training optimization”.
    • Optymalizacja wypukła i dualność (Convex optimisation and duality): Minimising a convex function over a convex set, with Lagrangian duality theory and KKT optimality conditions.
  • Teoria informacji: entropia i dywergencja KL (Information theory: entropy, cross-entropy and KL divergence): Measures of information and of difference between distributions; cross-entropy is the standard loss of language models and classifiers.
• Modele probabilistyczne (Probabilistic models): How to model dependencies and hidden variables.
  • Algorytm EM i modele mieszanin (Expectation-maximisation and mixture models): Iterative method for maximum-likelihood estimates in models with latent variables, such as Gaussian mixtures.
  • Pola losowe Markowa i grafy czynników (Markov random fields and factor graphs): Undirected graphical models and bipartite factor graphs, with inference by the sum-product algorithm (belief propagation).
  • Procesy gaussowskie (Gaussian processes): Non-parametric Bayesian model: a distribution over functions where any finite set of values is jointly Gaussian.
  • Programowanie probabilistyczne (Probabilistic programming): Languages and libraries where a generative model is written as code with sampling and conditioning statements, and inference runs automatically.
  • Sieci bayesowskie (Bayesian networks): A directed graph whose nodes are random variables and edges are dependencies; lets you compute probabilities from partial knowledge.
  • Ukryte modele Markowa i filtrowanie (Hidden Markov models and filtering): Models of hidden variables that change over time; tracking state from noisy observations.
  • Wnioskowanie przybliżone: próbkowanie i metody wariacyjne (Approximate inference: MCMC sampling and variational methods): When the exact posterior is intractable: approximate it with samples (MCMC) or by optimising a simpler distribution (VI).
  • Wnioskowanie przyczynowe (Causal inference): Telling causation from correlation: interventions, counterfactuals, confounders and discovering causal structure from data.
• Przeszukiwanie i planowanie (Search and planning): How to find a route to a goal in a space of possibilities.
  • Gry i przeszukiwanie z przeciwnikiem (minimax) (Adversarial search and games (minimax, alpha-beta)): Choosing moves when an opponent plays against you: game trees, minimax, alpha-beta pruning, Monte Carlo tree search.
  • Planowanie automatyczne (PDDL) (Automated planning (PDDL)): Finding a sequence of actions that reaches a goal from a formal world description; today also as verifiable support for LLM agent planning.
  • Problemy z ograniczeniami (CSP) (Constraint satisfaction problems (CSP)): Assigning values to variables so all constraints hold — schedules, layouts, sudoku.
  • Przeszukiwanie heurystyczne (A*) (Heuristic search (A*)): A* orders nodes by the sum of path cost so far and heuristic cost to the goal. An admissible heuristic guarantees optimality in tree search; graph search must additionally handle newly found cheaper paths to previously visited states correctly.
  • Przeszukiwanie lokalne (wspinaczka, wyżarzanie) (Local search (hill climbing, simulated annealing)): Improving a candidate solution step by step by exploring its neighbours instead of the whole space: hill climbing, simulated annealing, tabu search.
  • Przeszukiwanie ślepe (BFS, DFS) (Uninformed search (BFS, DFS, uniform-cost)): Systematically exploring a state space without knowing where the goal is.
• Statystyka i niepewność (Statistics and uncertainty): How to reason from incomplete data.
  • Estymacja parametrów (największa wiarygodność) (Parameter estimation (maximum likelihood, MAP)): Maximum likelihood estimation chooses parameters that maximize the likelihood of the observed data. In typical probabilistic classification and language models, minimizing the corresponding cross-entropy is equivalent to maximizing log-likelihood.
  • Rozkłady prawdopodobieństwa (Probability distributions): Random variables, distributions, expectation and variance; conditional probability and independence.
  • Testy statystyczne i przedziały ufności (Hypothesis tests and confidence intervals): Hypothesis tests and confidence intervals help assess observations relative to sampling variation and describe uncertainty. They do not by themselves determine whether an effect is true or practically important; they are used, among other things, in model comparisons, benchmarks, and A/B tests.
  • Twierdzenie Bayesa i wnioskowanie bayesowskie (Bayes' theorem and Bayesian inference): Updating beliefs after seeing data: prior, likelihood, posterior.
• Teoria gier i decyzji (Game theory and decision theory): Theory of rational choice for a single decision-maker (utility) and many interacting players (equilibria, mechanism design, social choice).
  • Gry kooperacyjne i wartość Shapleya (Cooperative games and the Shapley value): Cooperative-game solution assigning each player their average marginal contribution over all orders of coalition formation.
  • Oczekiwana użyteczność i decyzje w niepewności (Expected utility and decisions under uncertainty): Normative theory of choice: a rational agent picks the action maximising the expected value of a utility function over outcomes.
  • Projektowanie mechanizmów i aukcje (Mechanism design and auctions): Reverse game theory: constructing rules (a mechanism) so players' strategic behaviour achieves a set goal, e.g. efficient allocation.
  • Równowaga Nasha i rozwiązywanie gier (Nash equilibrium and solving games): Game solution concept: a strategy profile in which each player's strategy is a best response to the others'.
  • Wybór społeczny: łączenie preferencji wielu osób (Computational social choice): Formal analysis of rules aggregating individual preferences into a group decision, with axioms, impossibility theorems and computational complexity.
• Teoria uczenia statystycznego (Statistical learning theory): Theory of generalisation: error on unseen data, hypothesis-class complexity and the sample size needed to learn.
  • Brak darmowego obiadu i założenia wstępne modelu (No free lunch theorem and inductive bias): Result stating that, averaged over all possible problems, no learning algorithm beats another; success requires inductive bias.
  • Podwójne zejście (Double descent): Non-monotonic test-risk curve: classical U shape, a peak at the interpolation threshold and a second descent in the overparameterised regime.
  • Uczenie PAC i wymiar VC (PAC learning and VC dimension): Formal learnability model: with probability at least 1−δ the hypothesis error is ≤ ε, with sample size depending on the class's VC dimension.

Sources

• Stanford CS221 · AI: Principles and Techniques (2025) · reference: Rozumowanie, przeszukiwanie, logika i modele probabilistyczne.
• Stanford CS229 · Machine Learning (2018, archiwum) · reference: Klasyczne ML, statystyka i podstawy uczenia. Użyte jako stabilny szkielet, nie przegląd nowości.