# Fundamenty i rozumowanie

EN: **Foundations and reasoning** · id `foundations`

Path: [AI Atlas](https://aiatlasgraph.com/llms.txt) › Fundamenty i rozumowanie

## Description

- PL: Matematyczny i algorytmiczny język AI, także metod bez sieci neuronowych.
- EN: The mathematical and algorithmic language of AI, including methods without neural networks.

## In one sentence

- PL: To matematyka i sposoby rozumowania, dzięki którym programy liczą, wyciągają wnioski i szukają rozwiązań.
- EN: These are the mathematics and reasoning methods that let programs calculate, draw conclusions and search for solutions.

## Problem it solves

- PL: Trudno rozwiązywać złożone zadania, gdy informacje są niepełne, możliwości liczne, a reguły trzeba stosować konsekwentnie.
- EN: Complex tasks are hard when information is incomplete, possibilities are numerous and rules must be applied consistently.

## What it does

- PL: Zapisuje wiedzę, niepewność i cele w postaci, którą program może przeliczać, porównując możliwe odpowiedzi i kolejne działania.
- EN: It represents knowledge, uncertainty and goals in a form software can calculate with, comparing possible answers and sequences of actions.

## Everyday example

- PL: Planer wybiera kolejność wizyt, uwzględnia dostępne godziny i porównuje możliwe drogi między miejscami.
- EN: A planner chooses the order of visits, accounts for available times and compares possible routes between places.

## This concerns you if you…

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

## Keywords

linear algebra · probability theory · Bayesian inference · gradient descent · information theory · first-order logic · knowledge graphs · constraint satisfaction · heuristic search · automated planning · causal inference · formal theorem proving

## How it works

- PL: Formalne reprezentacje opisują dane, wiedzę i przestrzenie decyzji, a rachunek prawdopodobieństwa modeluje niepewność. Wnioskowanie, przeszukiwanie i optymalizacja wyprowadzają odpowiedzi lub rozwiązania z tych reprezentacji, z założeniami określającymi poprawność i koszt.
- EN: Formal representations describe data, knowledge and decision spaces, while probability models uncertainty. Inference, search and optimisation derive answers or solutions from these representations, under assumptions determining correctness and cost.

## Limits

- PL: Poprawny rachunek nie naprawia niewłaściwej reprezentacji problemu. Dokładne rozwiązania mogą być obliczeniowo niedostępne, a przybliżenia wymagają kontroli błędu i spełnienia założeń.
- EN: Correct computation cannot fix a poor problem representation. Exact solutions can be computationally infeasible, while approximations need error checks and valid assumptions.

## Key terms

- PL: **reprezentacja**: Formalny zapis informacji udostępniony algorytmowi do przetwarzania.
- PL: **wnioskowanie**: Wyprowadzanie odpowiedzi z danych, reguł i założeń modelu.
- PL: **złożoność obliczeniowa**: Zależność potrzebnych zasobów od rozmiaru problemu.
- EN: **representation**: A formal encoding of information supplied to an algorithm.
- EN: **inference**: Deriving answers from data, rules and model assumptions.
- EN: **computational complexity**: How required resources depend on problem size.

## Subtopics (7 direct, 47 in total)

- [Logika i reprezentacja wiedzy (Logic and knowledge representation)](https://aiatlasgraph.com/t/logic.md): How to write down facts, rules and dependencies.
  - [AI neurosymboliczna (Neuro-symbolic AI)](https://aiatlasgraph.com/t/neurosymbolic.md): 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))](https://aiatlasgraph.com/t/theoremproving.md): 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)](https://aiatlasgraph.com/t/kgraphs.md): Concepts and relations written as a graph, retrieval included.
  - [Logika zdań i logika pierwszego rzędu (Propositional and first-order logic)](https://aiatlasgraph.com/t/fol.md): 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))](https://aiatlasgraph.com/t/ontologies.md): Standards for describing concepts, classes and relations so machines can check consistency and infer implicit facts.
  - [Programowanie logiczne (Logic programming)](https://aiatlasgraph.com/t/logicprogramming.md): Declarative paradigm: a program is a set of logical clauses and computation is inference (resolution, fixpoint or stable models).
  - [Rozumowanie zdroworozsądkowe (Commonsense reasoning)](https://aiatlasgraph.com/t/commonsense.md): AI subfield on representing and using everyday knowledge of the physical, social and temporal world.
  - [Solwery SAT i SMT (SAT and SMT solvers)](https://aiatlasgraph.com/t/satsmt.md): 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)](https://aiatlasgraph.com/t/math.md): How to describe and optimise models.
  - [Algebra liniowa: wektory i macierze (Linear algebra: vectors and matrices)](https://aiatlasgraph.com/t/linearalgebra.md): The language for data, weights and embeddings: vectors, matrices, products and matrix decompositions.
    - [Rozkłady macierzy: wartości własne i SVD (Matrix decompositions: eigendecomposition and SVD)](https://aiatlasgraph.com/t/matrixdecomp.md): Matrix factorisations (eigen, SVD, QR, Cholesky) revealing structure, rank and dominant directions of data.
  - [Rachunek różniczkowy i gradienty (Calculus and gradients)](https://aiatlasgraph.com/t/calculus.md): 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)](https://aiatlasgraph.com/t/mathoptimization.md): 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)](https://aiatlasgraph.com/t/convexopt.md): 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)](https://aiatlasgraph.com/t/infotheory.md): Measures of information and of difference between distributions; cross-entropy is the standard loss of language models and classifiers.
- [Modele probabilistyczne (Probabilistic models)](https://aiatlasgraph.com/t/probabilistic.md): How to model dependencies and hidden variables.
  - [Algorytm EM i modele mieszanin (Expectation-maximisation and mixture models)](https://aiatlasgraph.com/t/emmixtures.md): 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)](https://aiatlasgraph.com/t/factorgraphs.md): Undirected graphical models and bipartite factor graphs, with inference by the sum-product algorithm (belief propagation).
  - [Procesy gaussowskie (Gaussian processes)](https://aiatlasgraph.com/t/gaussianprocesses.md): Non-parametric Bayesian model: a distribution over functions where any finite set of values is jointly Gaussian.
  - [Programowanie probabilistyczne (Probabilistic programming)](https://aiatlasgraph.com/t/probprogramming.md): Languages and libraries where a generative model is written as code with sampling and conditioning statements, and inference runs automatically.
  - [Sieci bayesowskie (Bayesian networks)](https://aiatlasgraph.com/t/bayesnets.md): 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)](https://aiatlasgraph.com/t/hmm.md): 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)](https://aiatlasgraph.com/t/approxinference.md): When the exact posterior is intractable: approximate it with samples (MCMC) or by optimising a simpler distribution (VI).
  - [Wnioskowanie przyczynowe (Causal inference)](https://aiatlasgraph.com/t/causalinference.md): Telling causation from correlation: interventions, counterfactuals, confounders and discovering causal structure from data.
- [Przeszukiwanie i planowanie (Search and planning)](https://aiatlasgraph.com/t/search.md): 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))](https://aiatlasgraph.com/t/adversarialsearch.md): Choosing moves when an opponent plays against you: game trees, minimax, alpha-beta pruning, Monte Carlo tree search.
  - [Planowanie automatyczne (PDDL) (Automated planning (PDDL))](https://aiatlasgraph.com/t/classicalplanning.md): 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))](https://aiatlasgraph.com/t/csp.md): Assigning values to variables so all constraints hold — schedules, layouts, sudoku.
  - [Przeszukiwanie heurystyczne (A*) (Heuristic search (A*))](https://aiatlasgraph.com/t/astar.md): 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))](https://aiatlasgraph.com/t/localsearch.md): 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))](https://aiatlasgraph.com/t/uninformedsearch.md): Systematically exploring a state space without knowing where the goal is.
- [Statystyka i niepewność (Statistics and uncertainty)](https://aiatlasgraph.com/t/probability.md): How to reason from incomplete data.
  - [Estymacja parametrów (największa wiarygodność) (Parameter estimation (maximum likelihood, MAP))](https://aiatlasgraph.com/t/mle.md): 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)](https://aiatlasgraph.com/t/distributions.md): Random variables, distributions, expectation and variance; conditional probability and independence.
  - [Testy statystyczne i przedziały ufności (Hypothesis tests and confidence intervals)](https://aiatlasgraph.com/t/hypothesistesting.md): 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)](https://aiatlasgraph.com/t/bayesinference.md): Updating beliefs after seeing data: prior, likelihood, posterior.
- [Teoria gier i decyzji (Game theory and decision theory)](https://aiatlasgraph.com/t/gametheory.md): 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)](https://aiatlasgraph.com/t/shapleyvalue.md): 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)](https://aiatlasgraph.com/t/expectedutility.md): 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)](https://aiatlasgraph.com/t/mechanismdesign.md): 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)](https://aiatlasgraph.com/t/nashequilibria.md): 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)](https://aiatlasgraph.com/t/socialchoice.md): Formal analysis of rules aggregating individual preferences into a group decision, with axioms, impossibility theorems and computational complexity.
- [Teoria uczenia statystycznego (Statistical learning theory)](https://aiatlasgraph.com/t/learningtheory.md): 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)](https://aiatlasgraph.com/t/nofreelunch.md): Result stating that, averaged over all possible problems, no learning algorithm beats another; success requires inductive bias.
  - [Podwójne zejście (Double descent)](https://aiatlasgraph.com/t/doubledescent.md): 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)](https://aiatlasgraph.com/t/pacvc.md): 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)](<https://stanford-cs221.github.io/autumn2025/>) `cs221` · reference: Rozumowanie, przeszukiwanie, logika i modele probabilistyczne.
- [Stanford CS229 · Machine Learning (2018, archiwum)](<https://cs229.stanford.edu/syllabus-autumn2018.html>) `cs229` · reference: Klasyczne ML, statystyka i podstawy uczenia. Użyte jako stabilny szkielet, nie przegląd nowości.

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AI Atlas revision 122 (2026-10-30) · HTML: https://aiatlasgraph.com/t/foundations · index: https://aiatlasgraph.com/llms.txt · everything: https://aiatlasgraph.com/llms-full.txt · updated 2026-10-30