Logical Formalization of intelligent agent systems

Logical Formalization of intelligent agent systems Cheng-Chia Chen Department of Computer Science, National Cheng-Chi University

Contents • Introduction • Review of basic nonclassical logics used in agent theory • formalize knowledge, belief and actions of agents • Formalize motivational attitudes of agents • Formalize communication between agents • Conclusion

I. Introduction • What is an agent ? • Applications • Why an agent theory?

I. Introduction What is an agent ?

Some definitions Collected from: http://www.msci.memohis.edu/~franklin/AgentProg.html Is it an Agent, or just a Program?: A Taxonomy for Autonomous Agents Stan Franklin and Art Graesser Institute for Intelligent Systems University of Memphis

The AIMA Agent [Russell and Norvig] • Anything that can be viewed as • perceiving its environment through sensors and • acting upon that environment through effectors. • too general such that by the definition, every program is an agent.

The Maes Agent [MIT] • Autonomous agents are • computational systems that • inhabit some complex dynamic environment, • sense and act autonomously in this environment, and by doing so • realize a set of goals or tasks for which they are designed.

The Hayes-Roth Agent [KSL,1995] • Intelligent agents continuously perform three functions: • perception of dynamic conditions in the environment; • action to affect conditions in the environment; and • reasoning to interpret perceptions, solve problems, draw inferences, and determine actions.

The IBM Agent • Intelligent agents are software entities that • carry out some set of operations • on behalf of a user or another program • with some degree of independence or autonomy, and in so doing, • employ some knowledge or representation of the user's goals or desires.

So, what is an agent ? • An autonomous agent is a system • situated within and is a part of an environment that • senses that environment and • acts on it, over time, • in pursuit of its own agenda and • so as to effect what it senses in the future.

Agent Properties • Property Other Names Meaning • autonomous exercises control over its own actions • goal-oriented pro-active does not simply act in response purposeful to the environment • communicative social ability communicates with other agents, perhaps including people • reactive (sensing and acting) responds in a timely fashion to changes in the environment

Agent Properties (cont’d) • Property Other Names Meaning • temporally is a continuously running continuous process • learning adaptive changes its behavior based on its previous experience • mobile able to transport itself from one machine to another • flexible actions are not scripted • character believable "personality" and emotional state.

Applications • Systems and Network Management • Mail and Messaging • filtering of email, news etc. • Information Access and Management: • Information filtering and discovery • Collaboration • Electronic Commerce • Adaptive user Interfaces • Entertainment. • and more...

Why an agent theory ? • For specifications and verification of agent systems • For reasoning about agent systems • For knowledge representation formalisms • for formalizing concepts of interest in philosophy, cognitive science. • Basis for analyzing micro and macro aspects of agent society.

II. Review of basic logics • Classical propositional logic (CPL) • Basic modal logic • logic of knowledge and belief • deontic logic • logic of actions and programs(PDL)

Elements of a Logic • Language • syntax (formal language) • semantics (model theory) • axiomatics (proof theory) • decidability & complexity (computation theory) • automated deduction (Theorem proving)

Classical Propositional Logic(CPL) • The language L: • a set of proposition symbols (PV) : • p,q, r ... means it-is-raining, it-is-cloudy, ... • logical connectives: /\ (and), ~ (negation) • (well-formed) formulas (abstract syntax): P ::= p | P /\ Q | ~P • Definitions: P \/ Q abbreviates ~(~P /\ ~Q) P => Q abbreviates ~(P /\ ~Q)

The semantics for CPL • Goals: • 1. define the contexts in which formulas can be given truth values. • 2. define the truth conditions for formulas. • interpretation (world, state): any assignment of truth value {1,0} to propositional symbols • Truth conditions (or satisfaction relation) |= : • I |= p iff I(p)=T; • I |= P /\ Q iff I |= P and I |= Q • I |= ~P iff not I |= P • If I |= A, then say I is a model of A.

Some logical notions • A formula is satisfiable iff it is true in some world. • A formula is valid (a tautology) (|= A) if it is true in all worlds. • A is a logical consequence of a set of formulas S (S |= A) iff A is true in all models of S. • Problems : How to characterize the set {A | A is a tautology} ?

Calculus and provability • A calculus C over a language L is a finite set of rules, each of the form: • (A1,A2, ..., An, B) • A1,A2,...,An : Premises • B: conclusion • if n = 0 => axioms • Example: (A, B, A /\B), (A, A=>B, B), (A=>B, B, A),...

Provability • Given a calculus C, • The set C = {A | A is C-provable(denoted |-C A)} is defined recursively as follows: • Basis:If (A) is a rule, then A in C ---axioms • Ind: If (A1,..,An,B) is a rule & • A1,...,An in C, then B in C.

An axomatization for CPL • Let CPL be the calculus: (1) Axiom schema: • A => (B => A) • (A=>(B =>C)) => ((A=>B)=>(A=>C)) • (~A => ~B) => (B => A) (2) Inference rule: • from A and A => B infer B (MP) • Theorem: A is valid in CPL iff A is CPL-provable

Basic Modal logic • The logical study of necessity and possibility • The language: • CPL augmented with two modal operators: [] (necessity) and ⃟ (possibility). • P : any proposition , then []P (<>P) means “P is necessarily (possibly) true”. • Meaning of []p: • depends on the context it is used, not only determined by the truth value of p • A family of logics instead of a single logic

Types of necessity • logical necessity: • e.g, p \/ ~p is logically necessarily true. • physical necessity: • F=ma • Epistemic necessity: • e.g., It is believed(known) that ... • Normal necessity: • e.g., It is obligated (permitted, forbidden) that ... • time-related (always, eventual) • Others: • After the programs terminates P must holds,...

Formal Definition • The language: • Alphabet (S): • PV: a set of propositional variables. • logical connectives: ~ (not), /\ (and), [] (necessity) • MF: a set of modal formulas defined inductively: • A ::= p | A /\ B | ~ A | []A • Abbreviations (Macros) • (A \/ B) abbreviates ~(~A /\ ~B); • (A  B) abbreviates ~(A /\ ~B) • ⃟ A abbreviates ~[]~A

Possible-world Semantics for modal logic • Truth conditions for p /\ q, p \/ q, p  q, and ~p . • Let p = “I win the game”, • q = “It is 5 p.m.” • Assume I win the game and • the present time is 3 p.m, • then p/\q: false, p\/q: true and pq: false. • But how about the statement: []p =It must be the case that I win the game. “

Meaning of necessity and possibility: • The game: • Two players A,B, each getting a card from four cards labeled 1,2,3,4 randomly. • rule: • The player who get a card larger than the other’s wins.

Scenario I: A gets “2”. • Then consider the following sentences: • 1. “A may possibly win” • = “It is possibly true that A win” = “⃟A_win” • 2. “A may possibly not win” • 3. “A must win” • 4. “B must not get “2”” • Which is right ? why?

The answer: • Statement 1 is right • since (2,1) may be the real world, in which A wins. • Statement 2 is right • since (2,3), (2,4) are possible, in which A does not win. • statement 3 is false • since there are cases (e.g., (2,3), (2,4)) in which A does not win. • Statement 4 is true since in all possible cases B does not get 2.

(3,4) The Rule: (2,1) A_win ~B_2 (2,4) ~A_win ~B_2 (2,3) ~A_win ~B_2 Impossible worlds ~[]A_win ⃟ A_win ⃟ ~A_win [] ~B_2 (2,?) Possible worlds Real world

The Possible-world Semantics: • Let W = the set of worlds • e.g, {(x,y) | x = 1..4, y =1..4 & x ¹ y} • Let V : W x PV -> {0,1} be a valuation function s.t., V(w,p) =1 iff p is assigned true at world w. • e.g, V((2,1), A-win) = 1 • R be a binary relation (I.e., subset of WxW) s.t. wRw’ iff w’ is a possible world of w. • e.g, (2,x)R(2,1), (2,x)R(2,3), (2,x)R(2,4). • The triple M=<W,R,V> is called a (possible-world) structure.

Truth-conditions for modal formulas M = <W,R,V>: a possible world structure; w: a world ∈ W, • The statement : “A is true at world w in structure M” is defined as follows: • M,w |= p iff V(w,p) = 1 • M,w |= A /\ B iff M,w |= A and M,w |= B • M,w |= ~A iff not M,w |= A. • M,w |= ⃟ A iff • A is true at some possible world of w. • M,w |= [] A iff A is true at all possible worlds of w.

Some definitions • A: modal formula, M: structure, • C: a class of structures • A is valid iff it is true in all worlds of all structures. • A is C-valid iff it is true at all worlds of all structures of C. • Problem: Given a class of structures C, • {A | A is C-valid } = ?

Interesting classes of structures • Class name Property of R • T reflexive: wRw. • D serial: for all w, there is w’ s.t. w R w’. • 4 transitive: wRw’ & w’Rw’’ ⇒ wRw’’. • 5 Eulidean: wRw’ & wRw’’ ⇒ w’ R w’’. • B symmetric: wRw’ ⇒ w’Rw. • r: any string from {T,D,4,5,B} without repetition. • Kr = the class of the structures whose R satisfying all properties mentioned in r. • (I.e., Every theorem of the logic Kr is valid in all Kr-struture, and vice versa.)

Axiomatization of modal logics • Axioms definitions • PC all truth-functional tautologies • K [](PQ) ([]P []Q) • T []P  P • D []P  ~[]~p • 4 []P [][]P • 5 ~[]P []~[]P • B ~P []~[]P. • Inference rule: MP: from P, P  Q infer Q Nec: from P infer []P

Axiomatizations of modal logic • r: any subset {T,D,4,5,B}. • Kr = the axiom system (calculus) including axioms K, PC and all of r and inference rules MP and Nec. • Kr-provable formulas are defined recursively as follows: • 1. Every axioms of Kr is Kr-provable. • 2. If P, P  Q are Kr-provable then so is Q (MP) • 3. If P is Kr-provable, then so is []P (Nec). • Theorem[Chellas80]: • A is Kr-valid iff A is Kr-provable.

· w Å · Some useful modal logics • Logical system Property of R usage • S5 (KT45) equivalence logic of knowledge • KD serial deontic logic • KD45 almost equ. logic of belief • S4 (KT4) ref. tran. Intuitionistic logic • S4.3 linear(total) temporal logic {w’ | w R w’} w · · Å Worlds inside are fully connected · • real world must be possible • real world may and may not be possible

Logic of Knowledge and Belief • Modal logic of knowledge : KT45(S5) • Modal logic of belief: KD45( weak S5) • Epsitemic interpretation of knowledge&belief axioms • KA means A is known; BA means A is believed. • T: []A  A (knowledge axioms) • D: []A  ~[]~A (belief axiom) • 4: []A [][] A (positive introspection) • 5:~[]A []~[]A (negative introspection) • K:[]A /\ [](A  B) []B (distribution axiom) • Nec: From p infer []p -- agent knows the logic

Extensions to multimodal logics: • S5 (KD45) can model only one single agent’s knowledge (believes) • Multi-agent cases: n agents: 1,2,3,...,n; • 2n knowledge(and belief) operators K1,B1,...,Kn,Bn: • KiA ( BiA ) means agent i knows(resp. believes) A. • Resulting logic: S5nWS5n • N copies of S5, and N copies of KD45, each for one agent.e.g., Tj: KjAA where j =1,..,n. • semantics: Structure M=<W,{Ki,Bi}i=1..n, V> • Each Ki is an equivalence relation on W and Bi is a serial,trans. and euclidean relation.

Related Issues[Halpern85] • Logical Omniscience Problem: • Agents with S5 (KD45) ability are perfect logical reasoners, but human never be. • Common knowledge, Distributed knowledge • [E]P = [1]P /\ [2]P.../\[n]P • [C]P = [E]P /\[E][E]P /\ [E][E][E]P /\ ... = [E]P /\[E][C]P • [D]P = P can be known by an agent who knows all what others known (the wisest man). • Needed and useful in many fields (Economics,distributing sys,AI ...)

Deontic interpretation of modal logic • Deontic logic (D or KD) • PA means A is permitted; OA means A is obligated; FA means A is forbidden. • A is (strongly) forbidden = • Doing A or bringing about A will result in punishment (dangerous, disastrous) worlds. • A is obligated = not doing A or not bring about A will result in punishment. = ~A is forbidden. • A is (weekly) permitted = A is not forbidden = doing A may not result in punishment. • Another possible pairs: • weekly forbidden/strongly permitted

Semantic analysis of forbidden, obligation and permission ~drive-car murder ~pay-tax ~dead ~drive-car ~pay-tax dead ~murder drive-car ~dead pay-tax ~ murder ~drive-car pay-tax ~murder ~dead current world drive-car murder pay-tax dead sets of worlds which may become the real world commit-crime or dead (undesired world) F murder : since all murder-worlds are red. O pay-tax: since all ~pay-tax world are red. P drive-car: some drive-car-world is white. Permitted worlds

Formalization of Deontic logic • W: The set of all possible worlds • D: A set of undesired, punishment world • V: WXPV -> {0,1} with the constraint that • V(w,v) = 1 iff w ∈ D. • I.e., we use v to denote all sanction or punishment worlds. • R: a binary relation on W, s.t. • wRw’ means w’ is a possible world that the agent may choose to become the real world from w.

Properties of the deontic logic: • By definition: • FA = [] (A v) ; • OA = F~A = [](~A  v); • PA = ~FA = ⃟ (A /\ ~v); • All KD axioms(K, D) • Desirable property: OA => PA: not valid in K but valid in KD (I.e., R must be serial)

possible past now real history real past real future possible future Temporal interpretation of modal logic • Taxonomy of temporal structures: • linear v.s. branch-time, • past time v.s. future time v.s. past&future • continuous v.s. discrete

Linear discrete time temporal logic • Temporal operators: • FA means A is eventually true • GA means A is always true • A U B means A is true until B becomes true • 0A: A is true at the next time.

Meaning of temporal formulas • Linear discrete-time temporal structure: 0 1 2 3 ..... n n+1 m initial world Fp p Gq q q q q .... q..... q 0r r AUB A A A A B

Meaning of temporal formulas • linear discrete temporal logic: • W = N = {0,1,2,3,...} :time point set • V:NXPV -> {0,1} • Truth conditions: • M,n |= 0A iff M,n+1 |= A. • M,n |= FA iff there is m  n s.t., M,m |= A • M,n |= GA iff for all m  n, M,m |= A. • M,n |= A U B iff there is m  n s.t., M,m|= B & for all m > s  n, M,s |= A.

Logic of programs and actions • Modal logic of programs (Dynamic Logic) • PDL: propositional version of DL • The language: • Primitive programs: a,b,c,... • Primitive propositions: p,q,r... • program constructs: “ ;”, “|”,”*”,”?”. • logic connectives: /\,~, [A] for each program A.

Logical Formalization of intelligent agent systems