ICML-Tutorial, Banff, Canada, 2004 Overview 1.Introduction to PLL 2.Foundations of PLL –Logic Programming, Bayesian Networks, Hidden Markov Models, Stochastic Grammars 3.Frameworks of PLL –Independent Choice Logic,Stochastic Logic Programs, PRISM, –Bayesian Logic Programs, Probabilistic Logic Programs,Probabilistic Relational Models –Logical Hidden Markov Models 4.Applications

ICML-Tutorial, Banff, Canada, 2004 (Propositional) LP – Some Notations Clauses: IF burglary and earthquake are true THEN alarm is true Clause burglary. earthquake. alarm :- burglary, earthquake. marycalls :- alarm. johncalls :- alarm. Herbrand Base (HB) = all atoms in the program burglary, earthquake, alarm, marycalls, johncalls Program atom body head

ICML-Tutorial, Banff, Canada, 2004 Logic Programming (LP) Two views on definite clausal theories 1)A model theoretic one 2)A proof theoretic one Both views are connected

ICML-Tutorial, Banff, Canada, 2004 Model Theoretic: Restrictions on Possible Worlds burglary. earthquake. alarm :- burglary, earthquake. marycalls :- alarm. johncalls :- alarm. Herbrand Interpretation –Truth assigments to all elements of HB An interpretation is a model of a clause C  If the body of C holds then the head holds, too. burglary earthquake alarm marycalls johncalls true false true false true false true false true false

ICML-Tutorial, Banff, Canada, 2004 Goal Proof Theoretic (Entailment): Restrictions on Possible Derivations burglary. earthquake. alarm :- burglary, earthquake. marycalls :- alarm. johncalls :- alarm. :- johncalls. :- alarm. :- burglary, earthquake. :- earthquake. {} A set of clauses can be used to prove that atoms are entailed by the set of clauses.

ICML-Tutorial, Banff, Canada, 2004 Bayesian Networks [Pearl 91] Qualitative part: Directed acyclic graph Nodes - random vars. Edges - direct influence Compact representation of joint probability distributions Quantitative part: Set of conditional probability distributions e b e be b b e BE P(A | B,E) Earthquake JohnCalls Alarm MaryCalls Burglary P(E,B,A,M,J) Together: Define a unique distribution in a compact, factored form P(E,B,A,M,J)=P(E) * P(B) * P(A|E,B) * P(M|A) * P(J|A) [illustration inspired by Kevin Murphy]

ICML-Tutorial, Banff, Canada, 2004 Traditional Approaches P(j)= P(j|a) * P(m|a) * P(a|e,b) * P(e) * P(b) + P(j|a) * P(m|a) * P(a|e,b) * P(e) * P(b) e b e be b b e BE P(A | B,E) Earthquake JohnCalls Alarm MaryCalls Burglary Model Theoretic... + P(j|a) * P(m|a) * P(a|e,b) * P(e) * P(b) burglary. earthquake. alarm :- burglary, earthquake. marycalls :- alarm. johncalls :- alarm. Bayesian Networks [Pearl 91]

ICML-Tutorial, Banff, Canada, 2004 (Hidden) Markov Models [Rabiner 89] coin 2 coin : head 0.7 : tail 0.5 : head 0.5 : tail Moore coin 2 coin 1 0.5*0.5 : head 0.5*0.5 : tail 0.5*0.3 : tail 0.5*0.7 : head 0.5*0.5 : head 0.5*0.5 : tail 0.5*0.7 : head 0.5*0.3 : tail Mealy Observations: t, Hidden States: c1, c2, c1,c2,... Statistical models for sequences, i.e. observations over time T=0,1,2,3,... h, t,t,... Not observed

ICML-Tutorial, Banff, Canada, 2004 (Hidden) Markov Models coin 2 coin 1 0.5*0.5 : head 0.5*0.5 : tail 0.5*0.3 : tail 0.5*0.7 : head 0.5*0.5 : head 0.5*0.5 : tail 0.5*0.7 : head 0.5*0.3 : tail coin2. coin1 0.5*0.3 : tail [Rabiner 89] coin 2 coin 1 coin 2 coin 1 tail coin 2 coin 1 tail coin 2 coin 1 head Prior... = P P1 + P2 + P3 P11P12 P13 * * P10 = P20 + P4... Proof Theoretic *

ICML-Tutorial, Banff, Canada, 2004 Stochastic Grammars Weighted Rewrite Rules S NPVP PP i saw VNPP Det N N man with thetelescopethe 1.0 : S  NP, VP 1/3 : NP  i 1/3 : NP  Det, N 1/3 : NP  NP, PP 1.0 : Det  the 0.5 : N  man 0.5 : N  telescope 0.5 : VP  V, NP 0.5 : VP  VP, PP 1.0 : PP  P, NP 1.0 : V  saw 1.0 : P  with 1.0 *1/3 * 0.5 * 1.0 *... = Proof Theoretic [Manning, Schütze 99] Upgrade HMMs (regular languages) to more complex languages such as context-free languages.

ICML-Tutorial, Banff, Canada, 2004 Upgrading to First-Order Logic The maternal information mc/2 depends on the maternal and paternal pc/2 information of the mother mother/2 : mchrom(fred,a). mchrom(fred,b),... or better mc(P,a) :- mother(M,P), pc(M,a), mc(M,a). mc(P,a) :- mother(M,P), pc(M,a), mc(M,b). mc(P,b) :- mother(M,P), pc(M,a), mc(M,b).... father(rex,fred). mother(ann,fred). father(brian,doro). mother(utta, doro). father(fred,henry). mother(doro,henry). pc(rex,a). mc(rex,a). pc(ann,a). mc(ann,b)....

ICML-Tutorial, Banff, Canada, 2004 Upgrading - continued Full Clausal Logic Functors aggregate objects Relational Clausal Logic Constants and variables refer to objects Propositional Clausal Logic Expressions can be true or false alarm :- burglary, earthquake. atom clause head body Substitution: Maps variables to terms: { M / ann }: mc(P,a) :- mother(ann,P),pc(ann,a),mc(ann,a). Herbrand base: set of ground atoms (no variables): {mc(fred,fred),mc(rex,fred),…} atom mc(P,a) :- mother(ann,P),pc(ann,a),mc(ann,a). clause head body variable (placeholder) constant terms nat(0). nat(succ(X)) :- nat(X). atom clause head body variable constant functor term Interpretations can be infinite ! nat(0),nat(succ(0)), nat(succ(succ(0))),...

ICML-Tutorial, Banff, Canada, 2004 Forward Chaining father(rex,fred). mother(ann,fred). father(brian,doro). mother(utta, doro). father(fred,henry). mother(doro,henry). pc(rex,a). mc(rex,a). pc(ann,a). mc(ann,b).... mc(P,a) :- mother(M,P), pc(M,a), mc(M,a). mc(P,a) :- mother(M,P), pc(M,a), mc(M,b). {M/ann, P/fred} mc(P,a):- mother(M,P), pc(M,a), mc(M,b). mc(fred,a)... mother(ann,fred).pc(ann,a)mc(ann,b) father(rex,fred).... Set of derivable ground atoms = least Herbrand model

ICML-Tutorial, Banff, Canada, 2004 Backward Chaining father(rex,fred). mother(ann,fred). father(brian,doro). mother(utta, doro). father(fred,henry). mother(doro,henry). pc(rex,a). mc(rex,a). pc(ann,a). mc(ann,b).... mc(P,a) :- mother(M,P), pc(M,a), mc(M,a). mc(P,a) :- mother(M,P), pc(M,a), mc(M,b). mother(ann,fred). {M/ann} pc(ann,a),mc(ann,a) mother(ann,fred). {M/ann} pc(ann,a),mc(ann,b) pc(ann,a). mc(ann,a) fail pc(ann,a). mc(ann,b) success mc(fred,a) {P/fred} mother(M,fred),pc(M,a),mc(M,a) mc(P,a):- mother(M,P), pc(M,a), mc(M,a). mother(M,fred),pc(M,a),mc(M,b) mc(P,a):- mother(M,P), pc(M,a), mc(M,b). {P/fred}