Ontologies Reasoning Components Agents Simulations Logic Programming Jacques Robin

Roteiro  A unifying metaphor for computer science  Syntax:  Pure Prolog abstract syntax  Full Prolog abstract syntax  Concrete syntax  Declarative semantics:  Open x closed world hypothesis  Unique name hypothesis  Clark’s completion  Least Herbrand model  Operational semantics  Lifted SLD resolution and backward chaining  Limitations  Of ISO Prolog  Of SLD resolution and pure backward chaining  Tabled logic programming  SLG resolution for pure Prolog  High order logic programming  HiLog abstract syntax  HiLog first-order semantics  Constraint logic programming  Limitations of CSP  CLP engine architeture  CLP abstract syntax  Operational semantics  Other extensions

Formal Logic Theory Intelligent Databases Formal Software Specification Automated Reasoning Declarative Programming Logic Programming: a Versatile Metaphor Logic Programming

Logic Programming: Key Ideas Logical Theory Software Specification Declarative Programming Automated Reasoning Intelligent Databases Logical Formula Abstract Specification Declarative Program / Data Structure Knowledge Base Database Theorem Prover Specification Verifier Interpreter / Compiler Inference Engine Query Processor Theorem Proof Specification Verification Program ExecutionInferenceQuery Execution Theorem Proving Strategy Verification Algorithm Single, Fixed, Problem-Independent Control Structure Inference Search Algorithm Query Processing Algorithm

Logic programming vision logic-based declarative semantics  Single language with logic-based declarative semantics that is: programming  A Turing-complete, general purpose programming language knowledge representation  A versatile, expressive knowledge representation language to support deduction and other reasoning services useful for intelligent agents (abduction, induction, constraint solving, belief revision) Data definition, query and update  Data definition, query and update language for databases with built-in inference capabilities  One tool solves all philosophy:  Any computation = resolution + unification + search  Programming = declaring logical axioms (Horn clauses)  Running the program = query about theorems provable from declared axioms  Algorithmic design entirely avoided (single, built-in control structure)  Same language for formal specification (modeling) and implementation  Same language for model checking and code testing

Prolog  First and still most widely used logic programming language  Falls short to fulfill the vision in many respect:  Many imperative constructs with no logical declarative semantics  No commercial deductive database available  Most other logic programming languages both:  Extend Prolog  Fulfills better one aspect of the logic programming vision  Basis for aborted attempt to Japanese to fully revolutionize all computing from ground up in the 80s

Pure Prolog: abstract syntax Pure Prolog Atom arg Definite Query +connective =  Definite Clause +connective =  headbody 0..1 Numerical Symbol Pure Prolog Term Function-Free TermFunctional Term Variable Symbol functor arg predicate * * * * Definite Logic Program +connective =  c 1 1 (...,X k 1,...) :- p 1 1 (...,X i 1,...),..., p m 1 (...,X j 1,...).... c 1 n (...,X k n,...) :- p 1 n (...,X i n,...),..., p m n (...,X j n,...). clauses parent(al,jim)  parent(jim,joe  anc(A,D)  parent(A,D)  anc(A,D)  parent(A,P)  anc(P,D)

Full Prolog Full Prolog Atom Prolog Literal arg Full Prolog Query +connective =  Full Prologl Clause +connective =  head body 0..1 User-Defined Symbol Full Prolog Term Function-Free Term Functional Term Variable Symbol functor arg predicate * * Prolog Literal +connective = naf Full Prolog Program +connective =  * Built-in Symbol Built-in Logical Symbol Built-in Imperative Symbol Meta-Programming Predicate Symbol Numerical Symbol I/O Predicate Symbol Program Update Predicate Symbol Search Customization Predicate Symbol Semantics of full Prolog: Cannot be purely logic-based Must integrate algorithmic ones

Pure Prolog program declarative formal semantics  Declarative, denotational, intentional:  Clark’s transformation to CFOL formula  First-order formula semantically equivalent to program  Declarative, denotational, extentional, model-theoretic:  Least Herbrand Model  Intersection of all Herbrand Models  Conjunction of all ground formulas that are deductive consequences of program

CFOL x Prolog Semantic Assumptions Classical First-Order Predicate Logic:  No Unique Name Assumption (UNA): two distinct symbols can potentially be paraphrases to denote the same domain entity  Open-World Assumption (OWA): If KB |≠ Q but KB |≠  Q, the truth value of Q is considered unknown  Ex: KB: true  core(ai)  true  core(se)  core(C)  offered(C,T)  true  offered(mda,fall) Q: true  offered(mda,spring)  OWA necessary for monotonic, deductively sound reasoning: If KB |= T then  A, KB  A |= T Logic Programming:  UNA: two distinct symbols necessarily denote two distinct domain entities  Closed-World Assumption (CWA): If KB |≠ Q but KB |≠  Q, the truth value of Q is considered false  Ex: ?- offered(mda,spring) fail  CWA is a logically unsound form of negatively abductive reasoning  CWA makes reasoning inherently non-monotonic as one can have: KB |= T but KB  A |≠ T if the proof KB |= T included steps assuming A false by CWA  Motivation: intuitive, representational economy, consistent w/ databases

Clark’s completion semantics  Transform Pure Prolog Program P into semantically equivalent CFOL formula comp(P)  Same answer query set derived from P (abductively) than from F P (deductively)  Example P: core(se). core(ai). offered(mda,fall). offered(C,T) :- core(C). ?- offered(mda,spring) no ?-  Naive semantics naive(P):  C,T true  core(ai)  true  core(se)  true  offered(mda,fall)  core(C)  offered(C,T) naive(P) |≠  offered(mda,spring)  Clark’s completion semantics comp(P):  C,T,C 1,T 1 (core(C 1 )  (C 1 =ai  C 1 =se))  (offered(C 1,T 1 )  (C 1 =mda  T 1 =fall)  (  C,T (C 1 =C  T 1 =T  core(C))))   (ai=se)   (ai=mda)   (ai=fall)   (se=fall)   (se=mda)   (mda=fall) F p |=  offered(mda,spring)

Clark’s transformation  Axiomatizes closed-world and unique name assumptions in CFOL  Start from naive Horn formula semantics of pure Prolog program  Partition program in clause sets, each one defining one predicate (i.e., group together clauses with same predicate c(t 1,..., t n ) as conclusion)  Replace each such set by a logical equivalence  One side of this equivalence contains c(X 1,..., X n ) where X 1,..., X n are fresh universally quantified variables  The other side contains a disjunction of conjunctions, one for each original clause  Each conjunction is either of the form:  X i = c i ...  X j = c i, if the original clause is a ground fact   Y i... Y j X i = Y i ...  X j = Y j  p 1 (...) ...  p n (... ) if the original clause is a rule with body p 1 (...) ...  p n (... ) containing variables Y i... Y j  Join all resulting equivalences in a conjunction  Add conjunction of the form  (c i = c j ) for all possible pairs (c i,c j ) of constant symbols in pure Prolog program

SLD Resolution  SLD resolution (Linear resolution with Selection function for Definite logic programs):  Joint use of input and goal-driven set of support heuristics  Always pick last proven theorem clause with next untried axiom clause  Always questions last pick even if unrelated to failure that triggered backtracking  A form of goal-driven backward chaining of Horn clauses seen as deductive rules  Prolog uses special case of SLD resolution where:  Axiom clauses tried in top to bottom program writing order  Atoms to unify in the two picked clauses:  Conclusion of selected axiom clause  Next untried premise of last proven theorem clause in left to right program writing order (goal)  Unification without occur-check  Generates proof tree in top-down, depth-first manner Generates proof tree in top-down, depth-first manner  Failure triggers systematic, chronological backtracking: Failure triggers systematic, chronological backtracking  Try next alternative for last selection even if clearly unrelated to failure  Reprocess from scratch new occurrences of sub-goals previously proven true or false  Simple, intuitive, space-efficient, time-inefficient, potentially non-terminating (incomplete)

Refutation Resolution Proof Example Refutation proof principle:  To prove KB |= F  Prove logically equivalent: (KB  F) |= True  In turn logically equivalent to: (KB   F) |= False (american(P)  weapon(W)  nation(N)  hostile(N)  sells(P,N,W)  criminal(P)) //1  (T  owns(nono,m1)) //2a  (T  missile(m1)) //2b  (owns(nono,W)  missile(W)  sells(west,nono,W)) //3  (T  american(west)) //4  (T  nation(nono)) //5  (T  enemy(nono,america)) //6  (missile(W)  weapon(W)) //7  (enemy(N,america)  hostile(N)) //8  (T  nation(america)) //9  (criminal(west)  F) //0 1. Solve 0 w/ 1 unifying P/west: american(west)  weapon(W)  nation(N)  hostile(N)  sells(west,N,W)  F //10 2. Solve 10 w/ 4: weapon(W)  nation(N)  hostile(N)  sells(west,N,W)  F //11 3. Solve 11 w/ 7: missile(W)  nation(N)  hostile(N)  sells(west,N,W)  F //12 4. Solve 12 w/ 2b unifying W/m1: nation(N)  hostile(N)  sells(west,N,m1)  F //13 5. Solve 13 w/ 5 unifying N/nono: hostile(nono)  sells(west,nono,m1)  F //14 6. Solve 14 w/ 8 unifying N/nono: enemy(nono,america)  sells(west,nono,m1)  F //15 7. Solve 15 w/ 6: sells(west,nono,m1)  F //16 8. Solve 16 w/ 3 unifying W/m1: owns(nono,m1)  missile(m1)  F //17 9. Solve 17 with 2a: missile(m1)  F // Solve 18 with 2b: F

SLD resolution example criminal(P) american(P)weapon(W)nation(N)hostile(N)sells(P,N,W) criminal(west)? missile(W)enemy(N,america)owns(nono,W) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example criminal(west) american(P)weapon(W)nation(N)hostile(N)sells(P,N,W) criminal(west)? missile(W)enemy(N,america)owns(nono,W) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example criminal(west) american(west)weapon(W)nation(N)hostile(N)sells(west,N,W) criminal(west)? missile(W)enemy(N,america)owns(nono,W) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example criminal(west) american(west)weapon(W)nation(N)hostile(N)sells(west,N,W) criminal(west)? missile(W)enemy(N,america)owns(nono,W) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example criminal(west) american(west)weapon(W)nation(N)hostile(N)sells(west,N,W) criminal(west)? missile(W)enemy(N,america)owns(nono,W) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example criminal(west) american(west)weapon(W)nation(N)hostile(N)sells(west,N,W) criminal(west)? missile(W)enemy(N,america)owns(nono,W) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example criminal(west) american(west)weapon(W)nation(N)hostile(N)sells(west,N,W) criminal(west)? missile(W)enemy(N,america)owns(nono,W) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example criminal(west) american(west)weapon(W)nation(N)hostile(N)sells(west,N,W) criminal(west)? missile(m1)enemy(N,america)owns(nono,W) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example criminal(west) american(west)weapon(m1)nation(N)hostile(N)sells(west,N,W) criminal(west)? missile(m1)enemy(N,america)owns(nono,W) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example criminal(west) american(west)weapon(m1)nation(N)hostile(N)sells(west,N,m1) criminal(west)? missile(m1)enemy(N,america)owns(nono,W) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example criminal(west) american(west)weapon(m1)nation(N)hostile(N)sells(west,N,m1) criminal(west)? missile(m1)enemy(N,america)owns(nono,W) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example criminal(west) american(west)weapon(m1)nation(N)hostile(N)sells(west,N,m1) criminal(west)? missile(m1)enemy(N,america)owns(nono,W) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example criminal(west) american(west)weapon(m1)nation(nono)hostile(N)sells(west,N,m1) criminal(west)? missile(m1)enemy(N,america)owns(nono,W) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example criminal(west) american(west)weapon(m1)nation(nono)hostile(nono)sells(west,nono,m1) criminal(west)? missile(m1)enemy(N,america)owns(nono,W) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example criminal(west) american(west)weapon(m1)nation(nono)hostile(nono)sells(west,nono,m1) criminal(west)? missile(m1)enemy(N,america)owns(nono,W) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example criminal(west) american(west)weapon(m1)nation(nono)hostile(nono)sells(west,nono,m1) criminal(west)? missile(m1)enemy(nono,america)owns(nono,W) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example criminal(west) american(west)weapon(m1)nation(nono)hostile(nono)sells(west,nono,m1) criminal(west)? missile(m1)enemy(nono,america)owns(nono,W) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example criminal(west) american(west)weapon(m1)nation(nono)hostile(nono)sells(west,nono,m1) criminal(west)? missile(m1)enemy(nono,america)owns(nono,W) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example criminal(west) american(west)weapon(m1)nation(nono)hostile(nono)sells(west,nono,m1) criminal(west)? missile(m1)enemy(nono,america)owns(nono,W) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example criminal(west) american(west)weapon(m1)nation(nono)hostile(nono)sells(west,nono,m1) criminal(west)? missile(m1)enemy(nono,america)owns(nono,W) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,m1)

SLD resolution example criminal(west) american(west)weapon(m1)nation(nono)hostile(nono)sells(west,nono,m1) criminal(west)? missile(m1)enemy(nono,america)owns(nono,m1) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,m1)

SLD resolution example criminal(west) american(west)weapon(m1)nation(nono)hostile(nono)sells(west,nono,m1) criminal(west)? missile(m1)enemy(nono,america)owns(nono,m1) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,m1)

SLD resolution example criminal(west) american(west)weapon(m1)nation(nono)hostile(nono)sells(west,nono,m1) criminal(west)? missile(m1)enemy(nono,america)owns(nono,m1) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,m1)

SLD resolution example criminal(west) american(west)weapon(m1)nation(nono)hostile(nono)sells(west,nono,m1) criminal(west)? missile(m1)enemy(nono,america)owns(nono,m1) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,m1)

SLD resolution example criminal(west) american(west)weapon(m1)nation(nono)hostile(nono)sells(west,nono,m1) criminal(west)? missile(m1)enemy(nono,america)owns(nono,m1) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,m1)

SLD resolution example criminal(west) american(west)weapon(m1)nation(nono)hostile(nono)sells(west,nono,m1) criminal(west)? missile(m1)enemy(nono,america)owns(nono,m1) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,m1)

SLD resolution example criminal(west) american(west)weapon(m1)nation(nono)hostile(nono)sells(west,nono,m1) criminal(west)? missile(m1)enemy(nono,america)owns(nono,m1) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,m1)

SLD resolution example criminal(west) american(west)weapon(m1)nation(nono)hostile(nono)sells(west,nono,m1) criminal(west)? yes missile(m1)enemy(nono,america)owns(nono,m1) missile(m1)american(west)nation(nono) nation(america) enermy(nono,america)owns(nono,m1) sells(west,nono,m1)

SLD resolution example with backtracking criminal(west) american(west)weapon(m1)nation(N)hostile(N)sells(west,N,m1) criminal(west)? missile(m1)enemy(N,america)owns(nono,W) missile(m1)american(west) nation(nono) nation(america)enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example with backtracking criminal(west) american(west)weapon(m1)nation(N)hostile(N)sells(west,N,m1) criminal(west)? missile(m1)enemy(N,america)owns(nono,W) missile(m1)american(west) nation(nono) nation(america)enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example with backtracking criminal(west) american(west)weapon(m1)nation(america)hostile(N)sells(west,N,m1) criminal(west)? missile(m1)enemy(N,america)owns(nono,W) missile(m1)american(west) nation(nono) nation(america)enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example with backtracking criminal(west) american(west)weapon(m1)nation(america)hostile(america)sells(west,america,m1) criminal(west)? missile(m1)enemy(N,america)owns(nono,W) missile(m1)american(west) nation(nono) nation(america)enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example with backtracking criminal(west) american(west)weapon(m1)nation(america)hostile(america)sells(west,america,m1) criminal(west)? missile(m1)enemy(N,america)owns(nono,W) missile(m1)american(west) nation(nono) nation(america)enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example with backtracking criminal(west) american(west)weapon(m1)nation(america)hostile(america)sells(west,america,m1) criminal(west)? missile(m1)enemy(N,america)owns(nono,W) missile(m1)american(west) nation(nono) nation(america)enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example with backtracking criminal(west) american(west)weapon(m1)nation(america)hostile(america)sells(west,america,m1) criminal(west)? missile(m1)enemy(N,america)owns(nono,W) missile(m1)american(west) nation(nono) nation(america)enermy(nono,america)owns(nono,m1) sells(west,nono,W) fail

SLD resolution example with backtracking criminal(west) american(west)weapon(m1)nation(america)hostile(america)sells(west,america,m1) criminal(west)? missile(m1)enemy(N,america)owns(nono,W) missile(m1)american(west) nation(nono) nation(america)enermy(nono,america)owns(nono,m1) sells(west,nono,W) fail backtrack

SLD resolution example with backtracking criminal(west) american(west)weapon(m1)nation(N)hostile(N)sells(west,N,m1) criminal(west)? missile(m1)enemy(N,america)owns(nono,W) missile(m1)american(west) nation(nono) nation(america)enermy(nono,america)owns(nono,m1) sells(west,nono,W) backtrack

SLD resolution example with backtracking criminal(west) american(west)weapon(m1)nation(N)hostile(N)sells(west,N,m1) criminal(west)? missile(m1)enemy(N,america)owns(nono,W) missile(m1)american(west) nation(nono) nation(america)enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example with backtracking criminal(west) american(west)weapon(m1)nation(nono)hostile(N)sells(west,N,m1) criminal(west)? missile(m1)enemy(N,america)owns(nono,W) missile(m1)american(west) nation(nono) nation(america)enermy(nono,america)owns(nono,m1) sells(west,nono,W)

SLD resolution example with bactracking criminal(west) american(west)weapon(m1)nation(nono)hostile(nono)sells(west,nono,m1) criminal(west)? missile(m1)enemy(N,america)owns(nono,W) missile(m1)american(west) nation(nono) nation(america)enermy(nono,america)owns(nono,m1) sells(west,nono,W)

Dual use of Prolog for AI knowledge representation  As knowledge representation language programming  As Turing-complete programming language Knowledge Base: Prolog Facts and Rules Inference Engine: Prolog Compiler/ Interpreter Ask Tell Retract Knowledge Base: Sentences in formalism F Inference Engine Implementing in Prolog Reasoning R in formalism F Ask Tell Retract Inference Engine: Prolog Compiler/ Interpreter

Logic vs. imperative and functional programming  Single relational definition: append([],L,L). append([H|T1],L,[H|T2]) :- append(T1,L,T2).  Usable as verification function: ?- append([a,b],[c],[a,b,c]). -> yes. ?- append([a,b],[c],[a]). -> no.  Usable as function and its various input/output converse functions: ?- append([a,b],[c],R). -> R = [a,b,c]. ?- append(H,[c],[a,b,c]). -> H = [a,b].  Usable as constraint solver: ?- append(X,Y,[a,b,c]). -> X = [], Y = [a,b,c] ; -> X = [a], Y = [b,c]...  Usable for enumeration: ?- append(X,Y,Z). -> X = [], Y =[], Z = [] ; -> X = [_], Y = [], Z = [_]...  Implements alone 8 different functions or methods!

Limitations of Pure Prolog and ISO Prolog knowledge representation For knowledge representation:  search customization:  No declarative constructs  Limited support procedimental constructs (cuts)  no support for uncertain reasoning  forces unintuitive rule-based encoding of inherently taxonomic and procedural knowledge  knowledge base updates non- backtrackable and without logical semantics  ab :- assert(a), b. if b fails, a remains as true programming For programming:  no fine-grained encapsulation  no code factoring (inheritance)  poor data structures (function symbols as only construct)  mismatch with dominant object- oriented paradigm  not integrated to comprehensive software engineering methodology  IDE not friendly enough  scarce middleware  very scarce reusable libraries or components (ex, web, graphics)  mono-thread declarative logic For declarative logic programming:  imperative numerical computation, I/O and meta-programming without logical semantics

Limitation of SLD resolution engines  Unsound: unification without occur-check  Incomplete: left-recursion  Correct ancestor Prolog program: anc(A,D) :- parent(A,D). anc(A,D) :- parent(P,D), anc(A,P).  Logically equivalent program (since conjunction is a commutative connective) that infinitely loops: anc(A,D) :- anc(A,P), parent(P,D). anc(A,D) :- parent(A,D).  Inefficient:  repeated proofs of same sub-goals  irrelevant chronological backtracking

Prolog’s imperative arithmetics fac(0,1) :- !. fac(I,O) :- I1 is I - 1, fac(I1,O1), O is I * O1. ?- fac(1,X). X = 1 ?- fac(3,X). X = 6 ?- I1 is I -1 error ?- I1 = I -1 I1 = I -1 ?- I = 2, I1 = I -1 I1 = 2 -1 ?- I = 2, I1 is I -1 I1 = 1  Arithmetic functions and predicates generate exception if queried with non-ground terms as arguments  is requires a uninstantiated variable on the left and an arithmetic expression on the right  Relational programming property lost  Syntax more like assembly than functional ! variable assignment  is Prolog’s sole explicit variable assignment predicate (only for arithmetics) query  = is a bi-directional, general- purpose unification query predicate

Prolog’s redundant sub-goal proofs

Extensions of pure Prolog Pure Prolog ISO Prolog High-Order LP Functional LP Constraint LP Abductive LP Transaction LP Frame (OO) LP Preference LP Tabled LP Inductive LP XSB Aleph Flora

Table logic programming: SLG resolution  Substitute goal stack by goal and answer table caching containing for each goal:  Calling goal (parent in top-down proof tree)  Pointer to variant goal modulo:  variable renaming, ex, p(f(X),Y)  p(f(A),B)  or term subsumption, ex, p(X,g(d))  p(A,B)  Instantiated program clause which conclusion it unified with  Position of that clause in the program  All its answers  To prove goal G:  Look for goal G’ variant of G in the table  If there is one, then use answers of G’ to answer G  Else backchain on G to get first answer to G  After finding such answer (or failing):  cache it in the table  backtrack to find all the other answers and cache them as well  Resume proof to next premise of calling goal

SLG resolution for definite programs anc(A,D) :- parent(A,D). %C1 anc(A,D) :- parent(P,D), anc(A,P). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = dan; A = fay; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- parent(A,dan)C1

SLG resolution for definite programs anc(A,D) :- parent(A,D). %C1 anc(A,D) :- parent(P,D), anc(A,P). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = dan; A = fay; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- parent(A,dan)C1 2aparent(A,dan)1aparent(fay,dan)C3A = fay

SLG resolution for definite programs anc(A,D) :- parent(A,D). %C1 anc(A,D) :- parent(P,D), anc(A,P). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = dan; A = fay; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- parent(A,dan)C12a parent(A,dan)1aparent(fay,dan)C3A = fay

SLG resolution for definite programs anc(A,D) :- parent(A,D). %C1 anc(A,D) :- parent(P,D), anc(A,P). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = dan; A = fay; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- parent(A,dan)C12a parent(A,dan)1aparent(fay,dan)C3A = fay 2b1aparent(jim,dan)C4A = jim

SLG resolution for definite programs anc(A,D) :- parent(A,D). %C1 anc(A,D) :- parent(P,D), anc(A,P). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = dan; A = fay; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- parent(A,dan)C12a, 2b 2aparent(A,dan)1aparent(fay,dan)C3A = fay 2b1aparent(jim,dan)C4A = jim

SLG resolution for definite programs anc(A,D) :- parent(A,D). %C1 anc(A,D) :- parent(P,D), anc(A,P). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = dan; A = fay; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- parent(A,dan)C12a, 2b 1banc(A,dan) :- parent(P,dan), anc(A,P)C2 2aparent(A,dan)1aparent(fay,dan)C3A = fay 2b1aparent(jim,dan)C4A = jim

SLG resolution for definite programs anc(A,D) :- parent(A,D). %C1 anc(A,D) :- parent(P,D), anc(A,P). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = dan; A = fay; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- parent(A,dan)C12a, 2b 1banc(A,dan) :- parent(P,dan), anc(A,P)C2 2aparent(A,dan)1aparent(fay,dan)C3A = fay 2b1aparent(jim,dan)C4A = jim 3aparent(P,dan)21b

SLG resolution for definite programs anc(A,D) :- parent(A,D). %C1 anc(A,D) :- parent(P,D), anc(A,P). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = dan; A = fay; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- parent(A,dan)C12a, 2b 1banc(A,dan) :- parent(P,dan), anc(A,P)C2 2aparent(A,dan)1aparent(fay,dan)C3A = fay 2b1aparent(jim,dan)C4A = jim 3aparent(P,dan)21b 4aanc(A,fay)1banc(A,fay) :- parent(A,fay)C1

SLG resolution for definite programs anc(A,D) :- parent(A,D). %C1 anc(A,D) :- parent(P,D), anc(A,P). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = dan; A = fay; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- parent(A,dan)C12a, 2b 1banc(A,dan) :- parent(P,dan), anc(A,P)C2 2aparent(A,dan)1aparent(fay,dan)C3A = fay 2b1aparent(jim,dan)C4A = jim 3aparent(P,dan)21b 4aanc(A,fay)1banc(A,fay) :- parent(A,fay)C1 5aparent(A,fay)4afail

SLG resolution for definite programs anc(A,D) :- parent(A,D). %C1 anc(A,D) :- parent(P,D), anc(A,P). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = dan; A = fay; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- parent(A,dan)C12a, 2b 1banc(A,dan) :- parent(P,dan), anc(A,P)C2 2aparent(A,dan)1aparent(fay,dan)C3A = fay 2b1aparent(jim,dan)C4A = jim 3aparent(P,dan)21b 4aanc(A,fay)1banc(A,fay) :- parent(A,fay)C15a parent(A,fay)4afail

SLG resolution for definite programs anc(A,D) :- parent(A,D). %C1 anc(A,D) :- parent(P,D), anc(A,P). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = dan; A = fay; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- parent(A,dan)C12a, 2b 1banc(A,dan) :- parent(P,dan), anc(A,P)C2 2aparent(A,dan)1aparent(fay,dan)C3A = fay 2b1aparent(jim,dan)C4A = jim 3aparent(P,dan)21b 4aanc(A,fay)1banc(A,fay) :- parent(A,fay)C15a 4b1banc(A,fay) :- parent(P,fay), anc(A,P)C2 5aparent(A,fay)4afail

SLG resolution for definite programs anc(A,D) :- parent(A,D). %C1 anc(A,D) :- parent(P,D), anc(A,P). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = dan; A = fay; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- parent(A,dan)C12a, 2b 1banc(A,dan) :- parent(P,dan), anc(A,P)C2 2aparent(A,dan)1aparent(fay,dan)C3A = fay 2b1aparent(jim,dan)C4A = jim 3aparent(P,dan)21b 4aanc(A,fay)1banc(A,fay) :- parent(A,fay)C15a 4b1banc(A,fay) :- parent(P,fay), anc(A,P)C2 5aparent(A,fay)4afail 6aparent(P,fay)54b

SLG resolution for definite programs anc(A,D) :- parent(A,D). %C1 anc(A,D) :- parent(P,D), anc(A,P). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = dan; A = fay; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- parent(A,dan)C12a, 2b 1banc(A,dan) :- parent(P,dan), anc(A,P)C2 2aparent(A,dan)1aparent(fay,dan)C3A = fay 2b1aparent(jim,dan)C4A = jim 3aparent(P,dan)21b 4aanc(A,fay)1banc(A,fay) :- parent(A,fay)C15a 4b1banc(A,fay) :- parent(P,fay), anc(A,P)C26a 5aparent(A,fay)4afail 6aparent(P,fay)54b

SLG resolution for definite programs anc(A,D) :- parent(A,D). %C1 anc(A,D) :- parent(P,D), anc(A,P). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = dan; A = fay; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- parent(A,dan)C12a, 2b 1banc(A,dan) :- parent(P,dan), anc(A,P)C2 2aparent(A,dan)1aparent(fay,dan)C3A = fay 2b1aparent(jim,dan)C4A = jim 3aparent(P,dan)21b 4aanc(A,fay)1banc(A,fay) :- parent(A,fay)C15a 4b1banc(A,fay) :- parent(P,fay), anc(A,P)C26a 5aparent(A,fay)4afail 6aparent(P,fay)54b 7aanc(A,jim)1banc(A,jim) :- parent(A,jim)C1

SLG resolution for definite programs anc(A,D) :- parent(A,D). %C1 anc(A,D) :- parent(P,D), anc(A,P). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = dan; A = fay; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- parent(A,dan)C12a, 2b 1banc(A,dan) :- parent(P,dan), anc(A,P)C2 2aparent(A,dan)1aparent(fay,dan)C3A = fay 2b1aparent(jim,dan)C4A = jim 3aparent(P,dan)21b 4aanc(A,fay)1banc(A,fay) :- parent(A,fay)C15a 4b1banc(A,fay) :- parent(P,fay), anc(A,P)C26a 5aparent(A,fay)4afail 6aparent(P,fay)54b 7aanc(A,jim)1banc(A,jim) :- parent(A,jim)C18a 7b1banc(A,jim) :- parent(P,jim), anc(A,P)C29a 8aparent(A,jim)4afail 9aparent(P,jim)54b

SLG resolution for definite programs anc(A,D) :- parent(A,D). %C1 anc(A,D) :- parent(P,D), anc(A,P). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = fay; A = jim; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- parent(A,dan)C12a, 2b 1banc(A,dan) :- parent(P,dan), anc(A,P)C2fail 2aparent(A,dan)1aparent(fay,dan)C3A = fay 2b1aparent(jim,dan)C4A = jim 3aparent(P,dan)21b 4aanc(A,fay)1banc(A,fay) :- parent(A,fay)C15a 4b1banc(A,fay) :- parent(P,fay), anc(A,P)C26a 5aparent(A,fay)4afail 6aparent(P,fay)54b 7aanc(A,jim)1banc(A,jim) :- parent(A,jim)C18a 7b1banc(A,jim) :- parent(P,jim), anc(A,P)C29a 8aparent(A,jim)4afail 9aparent(P,jim)54b

SLG resolution for definite programs  Avoids redundant repeated proofs of same subgoals  ex, calculates each fibonacci number exactly once  Also avoids left-recursive loops:  The table allows detection of goal G that is a variant of the goal G’ that called it  Looping is avoided by suspending G and moving to the following goals of the clause body and then backpropagating to G the variable bindings necessary to prove these following goals  When G becomes sufficiently instantiated to no longer be a variant of G’, its proof resumes

SLG resolution for definite programs anc(A,D) :- anc(A,P), parent(P,D). %C1 anc(A,D) :- parent(A,D). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = fay; A = dan; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- anc(A.P), parent(P,dan)C1

SLG resolution for definite programs anc(A,D) :- anc(A,P), parent(P,D). %C1 anc(A,D) :- parent(A,D). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = fay; A = dan; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- anc(A.P), parent(P,dan)C1 2aanc(A,P)1aanc(A.P) :- anc(A,P’), parent(P´,P)C1

SLG resolution for definite programs anc(A,D) :- anc(A,P), parent(P,D). %C1 anc(A,D) :- parent(A,D). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = fay; A = dan; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- anc(A.P), parent(P,dan)C1 2aanc(A,P)1aanc(A.P) :- anc(A,P’), parent(P´,P)C1 3aanc(A,P’)2a suspend

SLG resolution for definite programs anc(A,D) :- anc(A,P), parent(P,D). %C1 anc(A,D) :- parent(A,D). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = fay; A = dan; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- anc(A.P), parent(P,dan)C1 2aanc(A,P)1aanc(A.P) :- anc(A,P’), parent(P´,P)C1 3aanc(A,P’)2a suspend 4aparent(P’,P)1aparent(fay,dan)C3P’= fay

SLG resolution for definite programs anc(A,D) :- anc(A,P), parent(P,D). %C1 anc(A,D) :- parent(A,D). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = fay; A = dan; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- anc(A.P), parent(P,dan)C1 2aanc(A,P)1aanc(A.P) :- anc(A,P’), parent(P´,P)C1 3aanc(A,P’)2a suspend 4aparent(P’,P)1aparent(fay,dan)C3P’= fay, P = dan 4b1aparent(jim,dan)C4P’= jim, P = dan

SLG resolution for definite programs anc(A,D) :- anc(A,P), parent(P,D). %C1 anc(A,D) :- parent(A,D). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = fay; A = dan; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- anc(A.P), parent(P,dan)C1 2aanc(A,P)1aanc(A.P) :- anc(A,P’), parent(P´,P)C1 3aanc(A,P’)2a suspend 3banc(A,fay)3aanc(A,fay) :- anc(A,P’’), anc(P’’,fay)C1 4aparent(P’,P)1aparent(fay,dan)C3P’= fay 4b1aparent(jim,dan)C4P’= jim

SLG resolution for definite programs anc(A,D) :- anc(A,P), parent(P,D). %C1 anc(A,D) :- parent(A,D). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = fay; A = dan; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- anc(A.P), parent(P,dan)C1 2aanc(A,P)1aanc(A.P) :- anc(A,P’), parent(P´,P)C1 3aanc(A,P’)2a suspend 3banc(A,fay)3aanc(A,fay) :- anc(A,P’’), parent(P’’,fay)C1 4aparent(P’,P)1aparent(fay,dan)C3P’= fay 4b1aparent(jim,dan)C4P’= jim 5aanc(A,P’’)2a3asuspend

SLG resolution for definite programs anc(A,D) :- anc(A,P), parent(P,D). %C1 anc(A,D) :- parent(A,D). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = fay; A = dan; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- anc(A.P), parent(P,dan)C1 2aanc(A,P)1aanc(A.P) :- anc(A,P’), parent(P´,P)C1 3aanc(A,P’)2a suspend 3banc(A,fay)3aanc(A,fay) :- anc(A,P’’), parent(P’’,fay)C1 4aparent(P’,P)1aparent(fay,dan)C3P’= fay 4b1aparent(jim,dan)C4P’= jim 5aanc(A,P’’)2a3asuspend 6aparent(P’’,fay)3afail

SLG resolution for definite programs anc(A,D) :- anc(A,P), parent(P,D). %C1 anc(A,D) :- parent(A,D). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = fay; A = dan; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- anc(A.P), parent(P,dan)C1 2aanc(A,P)1aanc(A.P) :- anc(A,P’), parent(P´,P)C1 3aanc(A,P’)2a suspend 3banc(A,fay)3aanc(A,fay) :- anc(A,P’’), parent(P’’,fay)C1fail 3b3aanc(A,fay) :- parent(A,fay)C2 4aparent(P’,P)1aparent(fay,dan)C3P’= fay 4b1aparent(jim,dan)C4P’= jim 5aanc(A,P’’)2a3asuspend 6aparent(P’’,fay)3afail

SLG resolution for definite programs anc(A,D) :- anc(A,P), parent(P,D). %C1 anc(A,D) :- parent(A,D). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = fay; A = dan; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- anc(A.P), parent(P,dan)C1 2aanc(A,P)1aanc(A.P) :- anc(A,P’), parent(P´,P)C1 3aanc(A,P’)2a suspend 3banc(A,fay)3aanc(A,fay) :- anc(A,P’’), parent(P’’,fay)C1fail 3b3aanc(A,fay) :- parent(A,fay)C2 4aparent(P’,P)1aparent(fay,dan)C3P’= fay 4b1aparent(jim,dan)C4P’= jim 5aanc(A,P’’)2a3asuspend 6aparent(P’’,fay)3afail 7aparent(A,fay)6a

SLG resolution for definite programs anc(A,D) :- anc(A,P), parent(P,D). %C1 anc(A,D) :- parent(A,D). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = fay; A = dan; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- anc(A.P), parent(P,dan)C1 2aanc(A,P)1aanc(A.P) :- anc(A,P’), parent(P´,P)C1 3aanc(A,P’)2a suspend 3banc(A,fay)3aanc(A,fay) :- anc(A,P’’), parent(P’’,fay)C1fail 3b3aanc(A,fay) :- parent(A,fay)C2fail 4aparent(P’,P)1aparent(fay,dan)C3P’= fay 4b1aparent(jim,dan)C4P’= jim 5aanc(A,P’’)2a3asuspend 6aparent(P’’,fay)3afail 7aparent(A,fay)6a

SLG resolution for definite programs anc(A,D) :- anc(A,P), parent(P,D). %C1 anc(A,D) :- parent(A,D). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = fay; A = dan; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- anc(A.P), parent(P,dan)C1 2aanc(A,P)1aanc(A.P) :- anc(A,P’), parent(P´,P)C1 3aanc(A,P’)2a suspendfail 3banc(A,fay)3aanc(A,fay) :- anc(A,P’’), parent(P’’,fay)C1fail 3b3aanc(A,fay) :- parent(A,fay)C2fail 4aparent(P’,P)1aparent(fay,dan)C3P’= fay 4b1aparent(jim,dan)C4P’= jim 5aanc(A,P’’)2a3asuspend 6aparent(P’’,fay)3afail 7aparent(A,fay)6a

SLG resolution for definite programs anc(A,D) :- anc(A,P), parent(P,D). %C1 anc(A,D) :- parent(A,D). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = fay; A = dan; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- anc(A.P), parent(P,dan)C1 2aanc(A,P)1aanc(A.P) :- anc(A,P’), parent(P´,P)C1 2b1aanc(A,P) :- parent(A,P)C2 3aanc(A,P’)2a suspendfail 3banc(A,fay)3aanc(A,fay) :- anc(A,P’’), parent(P’’,fay)C1fail 3b3aanc(A,fay) :- parent(A,fay)C2fail 4aparent(P’,P)1aparent(fay,dan)C3P’= fay 4b1aparent(jim,dan)C4P’= jim 5aanc(A,P’’)2a3asuspend 6aparent(P’’,fay)3afail 7aparent(A,fay)6a

SLG resolution for definite programs anc(A,D) :- anc(A,P), parent(P,D). %C1 anc(A,D) :- parent(A,D). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = fay; A = dan; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- anc(A.P), parent(P,dan)C1 2aanc(A,P)1aanc(A.P) :- anc(A,P’), parent(P´,P)C1 2b1aanc(A,P) :- parent(A,P)C2 3aanc(A,P’)2a suspendfail 3banc(A,fay)3aanc(A,fay) :- anc(A,P’’), parent(P’’,fay)C1fail 3b3aanc(A,fay) :- parent(A,fay)C2fail 4aparent(P’,P)1aparent(fay,dan)C3P’= fay 4b1aparent(jim,dan)C4P’= jim 5aanc(A,P’’)2a3asuspend 6aparent(P’’,fay)3afail 7aparent(A,fay)6a 8aparent(A,P)4a4a,4b

SLG resolution for definite programs anc(A,D) :- anc(A,P), parent(P,D). %C1 anc(A,D) :- parent(A,D). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = fay; A = dan; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- anc(A.P), parent(P,dan)C1 2aanc(A,P)1aanc(A.P) :- anc(A,P’), parent(P´,P)C1 2b1aanc(A,P) :- parent(A,P)C24a, 4b 3aanc(A,P’)2a suspendfail 3banc(A,fay)3aanc(A,fay) :- anc(A,P’’), parent(P’’,fay)C1fail 3b3aanc(A,fay) :- parent(A,fay)C2fail 4aparent(P’,P)1aparent(fay,dan)C3P’= fay, P = dan 4b1aparent(jim,dan)C4P’= jim, P = dan 5aanc(A,P’’)2a3asuspend 6aparent(P’’,fay)3afail 7aparent(A,fay)6a 8aparent(A,P)4a4a,4b

SLG resolution for definite programs anc(A,D) :- anc(A,P), parent(P,D). %C1 anc(A,D) :- parent(A,D). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = fay; A = dan; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- anc(A.P), parent(P,dan)C1 2aanc(A,P)1aanc(A.P) :- anc(A,P’), parent(P´,P)C1 2b1aanc(A,P) :- parent(A,P)C24a, 4b 3aanc(A,P’)2a suspendfail 3banc(A,fay)3aanc(A,fay) :- anc(A,P’’), parent(P’’,fay)C1fail 3b3aanc(A,fay) :- parent(A,fay)C2fail 4aparent(P’,P)1aparent(fay,dan)C3P’= fay, P = dan 4b1aparent(jim,dan)C4P’= jim, P = dan 5aanc(A,P’’)2a3asuspend 6aparent(P’’,fay)3afail 7aparent(A,fay)6a 8aparent(A,P)4a4a,4b 9aparent(fay,dan) yes

SLG resolution for definite programs anc(A,D) :- anc(A,P), parent(P,D). %C1 anc(A,D) :- parent(A,D). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = fay; A = dan; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- anc(A.P), parent(P,dan)C14a 2aanc(A,P)1aanc(A.P) :- anc(A,P’), parent(P´,P)C1 2b1aanc(A,P) :- parent(A,P)C24a, 4b 3aanc(A,P’)2a suspendfail 3banc(A,fay)3aanc(A,fay) :- anc(A,P’’), parent(P’’,fay)C1fail 3b3aanc(A,fay) :- parent(A,fay)C2fail 4aparent(P’,P)1aparent(fay,dan)C3P’= fay, P = dan 4b1aparent(jim,dan)C4P’= jim, P = dan 5aanc(A,P’’)2a3asuspend 6aparent(P’’,fay)3afail 7aparent(A,fay)6a 8aparent(A,P)4a4a,4b 9aparent(fay,dan)1aparent(fay,dan)yes

SLG resolution for definite programs anc(A,D) :- anc(A,P), parent(P,D). %C1 anc(A,D) :- parent(A,D). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = fay; A = dan; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- anc(A.P), parent(P,dan)C14a 2aanc(A,P)1aanc(A.P) :- anc(A,P’), parent(P´,P)C1 2b1aanc(A,P) :- parent(A,P)C24a, 4b 3aanc(A,P’)2a suspendfail 3banc(A,fay)3aanc(A,fay) :- anc(A,P’’), parent(P’’,fay)C1fail 3b3aanc(A,fay) :- parent(A,fay)C2fail 4aparent(P’,P)1aparent(fay,dan)C3P’= fay, P = dan 4b1aparent(jim,dan)C4P’= jim, P = dan 5aanc(A,P’’)2a3asuspend 6aparent(P’’,fay)3afail 7aparent(A,fay)6a 8aparent(A,P)4a4a,4b 9aparent(fay,dan)1aparent(fay,dan)yes 10aparent(jim,dan)1aparent(jim,dan)yes

SLG resolution for definite programs anc(A,D) :- anc(A,P), parent(P,D). %C1 anc(A,D) :- parent(A,D). %C2 parent(fay,dan). %C3 parent(jim,dan). %C4 ?- anc(A,dan). A = fay; A = dan; no ?- gidgoalvgcginstantiated rulernanswers 1aanc(A,dan)anc(A,dan) :- anc(A.P), parent(P,dan)C14a, 4b 2aanc(A,P)1aanc(A.P) :- anc(A,P’), parent(P´,P)C1 2b1aanc(A,P) :- parent(A,P)C24a, 4b 3aanc(A,P’)2a suspendfail 3banc(A,fay)3aanc(A,fay) :- anc(A,P’’), parent(P’’,fay)C1fail 3b3aanc(A,fay) :- parent(A,fay)C2fail 4aparent(P’,P)1aparent(fay,dan)C3P’= fay, P = dan 4b1aparent(jim,dan)C4P’= jim, P = dan 5aanc(A,P’’)2a3asuspend 6aparent(P’’,fay)3afail 7aparent(A,fay)6a 8aparent(A,P)4a4a,4b 9aparent(fay,dan)1aparent(fay,dan)yes 10aparent(jim,dan)1aparent(jim,dan)yes

HiLog: extending Prolog with high-order syntax  Meta-programming facilities of functional languages  Single language for reasoning and meta-reasoning  Schema browsing in databases Prolog atomsHiLog atoms Prolog Atom Prolog Term Function-Free Term Functional Term Variable Symbol functor arg predicate * * arg HiLog Atom HiLog Term Function-Free Term Functional Term Variable Symbol functor arg predicate * * arg Examples of HiLog atoms that are not Prolog atoms: P(a) p(F(a) f(a)(X,Y) p(f(a)(X,Y)) P(F(X))(G(Y))

HiLog: example Prolog program: parent(bob, bill). parent(bill, mary). supervisor(john, mary). supervisor(mary, kathy). anc(A,D) :- parent(A,D). anc(A,D) :- parent(A,C), anc(C,D). boss(B,E) :- supervisor(B,E). boss(B,E) :- supervisor(B,D), boss(D,E). obeys(P,A) :- anc(A,P). obeys(P,A) :- boss(A,P). ?- obeys(mary,X). X=bill; X=bob; X= mary; X=john; no More concise and extensible HiLog program: parent(bob, bill). parent(bill, mary). supervisor(john, mary). supervisor(mary, kathy). relation(parent). relation(supervisor). trans(R)(X, Y) :- R(X, Y) trans(R)(X, Y) :- R(X, Z), trans(R)(Z,Y). obeys(P)(A) :- relation(R), trans(R)(A,P). ?- obeys(mary)(X). X=bill; X=bob; X= mary; X=john; no

HiLog  High-order syntax... ... but first-order semantics (and thus semi-decidable)  Why?  HiLog is merely a very convenient syntactic sugar since:  All HiLog program can be translated into a barely legible, verbose but semantically equivalent Prolog program  How?  Axiomatize predicates and functions with single additional functor apply  Complex terms or variables as functors become first argument of apply  Arguments of these functors become second to last arguments of apply

HiLog to Prolog translation: example parent(bob, bill). parent(bill, kathy). supervisor(john, mary). supervisor(mary, kathy). relation(parent). relation(supervisor). trans(R)(X, Y) :- R(X, Y) trans(R)(X, Y) :- R(X, Z), trans(R)(Z,Y). obeys(P)(A) :- relation(R), trans(R)(A,P) ?- obeys(kathy)(X). X=bill; X=bob; X= mary; X=john; no parent(bob, bill). parent(bill, kathy). supervisor(john, mary). supervisor(mary, kathy). relation(parent). relation(supervisor). apply(trans(R),X,Y) :- apply(R,X,Y). apply(trans(R),X,Y) :- apply(R,X,Y), apply(trans(R),Z,Y). apply(obeys(P,A)) :- relation(R), apply(trans(R),A,P)). ?- apply(obeys(john),X). X=bill; X=bob; X= mary; X=john; no

Constraint Logic Programming (CLP)  CSP libraries’ strengths:  Specification level, declarative programming for predefined sets of constraints over given domains  Efficient  Extensive libraries cover both finite domain combinatorial problems and infinite numerical domain optimization problems  CSP libraries’ weaknesses:  Constraint set extension can only be done externally through API using general-purpose programming language  Same problem for mixed constraint problems (ex, mixing symbolic finite domain with real variables) Prolog’s s trengths:  Allows specification level, declarative programming of arbitrary general purpose problems  A constraint is just a predicate  New constraints easily declaratively specified through user predicate definitions  Mixed-domain constraints straightforwardly defined as predicate with arguments from different domains  Prolog’s weakness:  Numerical programming is imperative, not declarative  Very inefficient at solving finite domain combinatorial problem

CLP  Integrate Prolog with constraint satisfaction and solving libraries in a single inference engine  Get the best of both worlds:  Declarative user definition of arbitrary constraints  Declarative definition of arbitrarily mixed constraints  Declarative numerical and symbolic reasoning, seamlessly integrated  Efficient combinatorial and optimization problem solving  Single language to program constraint satisfaction or solving problems and the rest of the application

CLP engine architecture

CLP engine reasoning services