Unifying Logic, Dynamics and Probability: Founda9ons, Algorithms - PowerPoint PPT Presentation

Unifying Logic, Dynamics and Probability: Founda9ons, Algorithms and Challenges Vaishak Belle University of Edinburgh 1

About this tutorial pages.vaishakbelle.com/logprob • On unifying logic and probability • Slides online at end of tutorial 2

Scope • Symbolic analogues for graphical models; * • First-order logics for probability • Overview of formal construc>ons and some algorithmic ideas * Discussion limited to inference (i.e., no learning) 3

Structure of tutorial • Mo$va$on • Logical formula$ons of graphical models • Algorithms • Logics for reasoning about probability • Reduc$on theorems • Outlook and Open Ques$ons 4

Mo#va#on 5

2 disciplines in AI Propositional logic, Bayesian network, First-order Logic Markov network Symbolic AI Statistical AI SAT, Theorem proving Sampling, filtering 6

Complementary views of AI Logic and probability are some3mes seen as complementary ways of represen3ng and reasoning about the world: • Symbolic vs numeric • Qualita4ve vs quan4ta4ve • Rela4ons, objects vs random variables Clearly, we o+en need features of both worlds! 7

The expressivity argument Encoding of rules of chess: • 1 page in FOL pages in PL (~ graphical models) • Encoding of Alice's knowledge: Hand a card each to A ( ) and B in PL • in FOL • 8

The database argument Data is o(en stored in (rela/onal) databases, industry and scien/fic fields alike This data is almost always probabilis1c (NELL, Probbase, Knowledge Vault) • Rela&ons extracted by mining and learning from unstructured texts • Probabili&es on tuples approximate confidence 9

The implicit beliefs argument It is impossible to store the implicit beliefs of an explicit representa3on in a logically complete manner • E.g., from , we get , etc. Verifying behavioural proper2es of the system's design (safety, liveness) 10

Top-down vs bo,om-up Gary Marcus: To get computers to think like humans, we need a new A.I. paradigm, one that places “top down” and “bo<om up” knowledge on equal foo?ng. Bo<om-up knowledge is the kind of raw informa?on we get directly from our senses, like pa<erns of light falling on our re?na. Top- down knowledge comprises cogni?ve models of the world and how it works. 11

So, useful to combine, but how? • Theorists such as Boole and de Fine3 discussed the connec4ons between logic and probability, and considered probability to be part of logic • Heavy development in the last 70 years, star4ng with Gaifman and lots of exci4ng work in AI • Key idea: for , accord weights to 12

A spectrum (roughly speaking) BLOG, Languages for Relational BNs, MLNs, Probabilistic logic programming logic, probability, action WMC Symbolic (propositional) Generalised measures, Some first-order features languages undecidable in general 13

Logical formula-ons of graphical models 14

Bayesian network Construct a proposi-onal theory, e.g: • • • • • 15

Probabilis)c rela)onal languages Finite-domain FOL + graphical models for compact codifica8on, and have natural logical encodings: • Atoms = random variables • Weights on formulas = poten8als on cliques 16

Algorithms 17

Compu&ng probabili&es Logical constraints = zero-probability regions How do we design effec-ve algorithms that embody implicit beliefs and determinis-c reasoning? Weighted model coun/ng 18

SAT, #SAT and WMC Given proposi+onal CNF : • SAT: find a model • #SAT: count models; here 5: (0,1,0), (0,1,1), (1,1,0), (1,1,1), (1,0,0) • WMC: count weights of models 19

An illustra+on and • M1 M2 omi)ed p, q p, !q • • .4 .6 20

WMC for BNs Suppose encodes a BN. Then: Axioms of probability and BN condi5onal independence proper5es carry over to encoding: correctness prese rving. 21

Advanced topics • Caching sub-formula evalua5ons: component caching • Con%nuous distribu%ons: weighted model integra/on • Exploi(ng symmetry over predicate instances: li#ed reasoning • Learning weights and theories • Modeling dynamic BNs 22

Is this enough? Do we need to go beyond symbolic analogues? 23

The John McCarthy cri0que 1. It is not clear how to a/ach probabili2es to statements containing quan2fiers in a way that corresponds to the amount of convic2on people have. 2. The informa2on necessary to assign numerical probabili2es is not ordinarily available. Therefore, a formalism that required numerical probabili2es would be epistemologically inadequate. 24

The role of probabili-es When are probabili-es appropriate? Which events are to be assigned probabili-es, and how to assign them? Some%mes need to say: the probability of E is less than F, and at least twice the probability of G . A basket contains an unknown number of fruits (bananas, oranges). We draw some fruits, observing the type of each and replacing it. We cannot tell fruits of the same type apart. How many fruits are in the basket? 25

Long-lived systems When it comes to reasoners and learners that poten1ally run forever, we need to be able to reason about probabilis1c events in a more general way. For example, we may need to compare the probabili1es of hypothe1cal outcomes, or analyse the behaviour of non-termina1ng probabilis1c programs. A general first-order logical language as a mathema4cal framework, with "reduc4on theorems." 26

Logics for reasoning about probabili2es 27

Seman&cal set up Exis%ng case: A finite set of finite sta$c structures, a distribu2on on them M1 M2 p, q p, !q .4 .6 28

Seman&cal set up contd. The general case: An infinite set of infinite dynamic structures, infinitely many distribu5ons on them a a p, q, … !p, !q, … p, q, … !p, q, … a a !p, !q, … p, q, … b b p, q, … p, !q, … .4 .2 .6 .8 29

Language and seman+cs • Predicates of every arity, connec2ves • Constants made of object and ac2on terms • For every wff • Ground atoms: constants applied to predicates • Worlds: Interps X ActSeq • Belief state: set of distribu3ons on worlds 30

Proper&es (i.e., validi&es) • if , then • • • • 31

Using the language A probabilis+c Situa+on Calculus, as an ac#on theory Example: decremen.ng a counter • • • • 32

Regression Because 33

Proper&es of regression • Subsumes products of Gaussians • Distribu5on transforma5ons • Works for arbitrarily complex discrete-con5nuous noise models Closed-form solu-on! 34

Implementa)on in ac)on + github.com/vaishakbelle/PREGO > (regr (<= c 7) ((see 5) (dec -2))) ’(/ (INTEGRATE (c z) (* (UNIFORM c 2 12) (GAUSSIAN z -2 1.0) (GAUSSIAN 5 c 4.0) (if (<= (max 0 (- c z)) 7) 1.0 0.0))) (INTEGRATE (c w) (* (UNIFORM c 2 12) (GAUSSIAN w -2 1.0) (GAUSSIAN 5 c 4.0)))) > (eval (<= c 7) ((see 5) (dec -2))) 0.47595449413426844 + Con&nuous counter, noisy ac&on, unique distribu&on assump&on 35

Advanced topics • Handling non-unique distribu3ons (e.g., via linear programming as upper and lower measures, credal networks) • Progressing belief states 36

Ac#on-centric probabilis#c programming Primi%ve programs = ac%ons prim (begin prog1 . . . progn ) (if prog1 prog2) (let((var1 term1)...(varn termn)) prog) (until form prog) github.com/vaishakbelle/ALLEGRO Sample worlds and updates them 37

A simple program (until (> (pr (and (>= c 2) (<= c 6))) .8) (until (> (conf c .4) .8) (see)) (let ((diff (- (exp c) 4))) (dec diff))) > (online-do prog) Execute action: (see) Enter sensed value: 4.1 Enter sensed value: 3.4 Execute action: (dec 1.0) ... > (pr (and (>= c 2) (<= c 6))) 0.8094620133032484 0 5 10 15 20 38

Outlook 39

Spectrum revisited Formal languages increasingly becoming popular to declara4vely design ML pipelines But logic provides access to truth-theore3c reasoning, determinis3c constraints Discussed 2 extremes: • Symbolic analogue to graphical models • General framework for reasoning about probabili:es 40

Outlook for symbolic analogues State-of-the-art in some regards 1. How closely can we .e learning techniques to mainstream ML methods (e.g., deep nets), perhaps via a graphical model formula.on? 2. How can we leverage symbolic + logical aspects to address explainability + commonsensical reasoning? 41

Outlook for general frameworks O"en builds on and extends mainstream (logical) KR languages 1. What type of seman0c jus0fica0on is needed to incorporate them in ML pipelines (e.g., open-world assump0on)? 2. Approaches to correctness oDen shy way from rigorous logical formula0on: when are less rigorous formula0ons not acceptable? 3. Tractability vs expressivity 42

Mee#ng midway Begin with proposi.onal formalisms and extend them to handle "first-order" features, e.g.: • BLOG, which allows Skolem constants • Open-world probabilis;c databases • Model coun;ng with infinite domains • Model coun;ng with existen;als and func;on symbols 43