FOL Towards an architecture for building autonomous agents from building blocks of first order logic Carolyn Talcott SRI International (joint work with Richard Weyhrauch, IBUKI)

Abstract The FOL system was designed to test some ideas about how to build thinking individuals by mechanizing the activity of reasoning and providing data structures powerful enough to represent both general information and information about reasoning itself. In this talk we will spell out the challenges implicit in the goal of FOL in some detail. Then we will describe and illustrate the basic FOL concepts: contexts, partiality, restartable computations, systems, inference rules as operations on contexts, and behaviors. Finally we will discuss how these concepts have been used to address the building of mind.

The FOL Goal To build (minds of) autonomous agents that can 0 come to ``know'' things 0 interact with their environment An adequate theory of mind must necessarily provide an explanation of how the mind works, but also must provide a blueprint for 0 the building blocks of mind 0 its architecture

Autonomy 0 An autonomous agent is a finite object that evolves over time. 0 An autonomous agent can't be reprogrammed. 0 There must be a finite upper bound on – the complexity of the building blocks – the number of kinds – the parameters of each kind 0 A lower bound is needed for sensible building blocks –atoms vs neurons

Paradigm Shift 0 We claim that it is possible to design/build an autonomous agent that can – reason -- not just explain reasoning – refer -- not just be a theory of reference 0 The paradigm shift is to use logic like builders use mechanical or electrical engineering. –the theory tells us how to put the blocks together. – the physics makes it work. 0 Our job is to describe the components necessary to build an autonomous thinker, NOT just to describe how it works. 0 Satisfaction is the important notion, concentrating on validity is a mistake.

Requirements What are the requirements of an adequate theory of mind? At least it must explain: 0 [1] how perceptual information gets to be congnitive information 0 [2] how we can comprehend infinite objects such as the set of numbers 0 [3] how we can reason about –others beliefs without knowing what they are –time without using indexicals 0 [4] how we can ask `what have I been doing lately?'

Requirements cntd. 0 [5] how minds can forget -- non-monotonic reasoning (forgetting, leaping here and there) is the default. 0 [6] how we can be interrupted 0 [7] how problem solving happens 0 [8] how asynchronous sensing and effecting can happen 0 [9] how can we “know” in an ever changing environment

FOL solutions 0 Building blocks with upper bound on complexity – contexts 0 New foundational understanding of First Order axiomatization 0 Data structures (contexts) as axiomatizations, rather than sets of formulas 0 An architecture for “reasoning on the side” (how many theories of arithmetic do we need?)

FOL solutions cntd. 0 Computational and finitist semantics of First Order Logic that is compatible with the classical semantics 0 Computation Systems –resource limited, restartable computations –supporting the finitist semantics 0 A characterization of admissible change

New Notions in FOL 0 Finite realization of standard notions – language, model, proof 0 partial (incomplete) information 0 mixed syntactic and semantic reasoning 0 FOL Contexts –finite data structures that are the building blocks of mind 0 FOL Systems –collections of contexts constrained by a Physics 0 Inference rules – maps on FOL contexts or systems –moving through time

Data Structures vs Sets Finite Data Structures 0 FOL Language –signature 0 Simulation structure –computations –resource limited, restartable 0 FOL facts –contain justification –finite set Infinite Sets 0 Language –set of wffs 0 Model –set theoretic functions 0 Theory –deductively closed –infinite set of wffs

Computation Systems 0 Universe of values – u, v in U 0 Computational descriptions –d, d' in D 0 A step/reduction relation –d ~> d' 0 A predicate, Done, on descriptions 0 A function, d2u – to extract values from Done computations (think Kleene T predicate)

FOL Contexts 0 An FOL context C (over CSys) is a structure C = 0 L is the language of C 0 SS is the simulation structure of C – –SS determines maps: Exp(L) D 0 F is the set of facts of C { }

FOL Contexts -- example 0 A wristwatch context -- impressionistically ww[n] = { } > | [n] As a data structure: C_ww = L_ww =,,, > T_ww =,, 1 > SS_ww =, >,,, >

Models and Consequences 0 The simulation structure of a context determines a “partial model” 0 The models of an FOL context are the usual classical models of a FO theory, constrained to be compatible with its simulation structure. 0 (Semantic) consequence is then defined as usual. 0 An FOL context is consistent if it has a model.

Inference Rules 0 An FOL inference rule is a map on contexts C -ir-> C ‘ 0 ir is satisfiablity/consistency preserving if it maps consistent contexts to consistent contexts 0. 0 ir is conservative if it is consistency preserving, non-decreasing w.r.t. language, and preserves consequences of the initial language 0 ir is validity preserving if it preserves language and the set of consequences

Wristwatch continued  ww[n] = { } > | [n] 0 ww[n] - tick -> ww[n+1] Three ways to look at tick 0 1. as a function 0 2. as a before/after relation (at the meta-level) 0 3. as and updating operation

Inference Rules: Examples 0 Adding a new symbol – conservative, but not validity preserving 0 Attaching a semantic object to a symbol – preserves satisfaction if there are no facts –could introduce inconsistencies if there are facts 0 Forgetting a fact –satisfaction preserving 0 Adding a computational consequence to the facts –validity preserving 0 Conservative extension results of logic restated as rules are conservative

FOL Systems 0 An FOL system is simply a finite collection of labeled contexts. 0 An FOL system is consistent if its each of its component contexts is consistent. No mutual consistency is necessary. 0 A physics is a collection of systems. Given by –a predicate P –a collection of inference rules –a combination of the above – P(C) implies P(f(C))

FOL Systems -- example 0 A theory of time context totC[m,l] = { le(then,now) } > | | | [m] [l] <= 0 totC[m,l] - update-then -> totC[m,m] 0 A theory of time system: totS[n,m,l] = { wwC[n], totC[m,l] } for m <= n 0 tick and update-then lift to the system 0 totS[n,m,l] - look -> totS[n,n,l]

Physics of Time I Wristwatch physics 0 WW(C) iff – Lang[C] has a single individual constant, now – now is attached to a natural number –there are no facts 0 tick preserves WW -- WW(C) implies WW(tick(C)) Theory of Time physics 0 ToT(C) iff –Lang[C] has 2 individual constants (now,then), and a binary relation symbol (le) –now, then are attached to natural number m >= l, le attached to the <= relation –there is a single fact -- an axiom stating le(then,now) 0 update-then preserves ToT

Physics of Time Theory of time system physics 0 TotS(S) iff S has the form { Cww, Ctot } where –WW(Cww), ToT(Ctot), and val[Ctot,now] <= val[Cww,now] 0 tick, update-then, lookup preserve ToTS 0 Stability Theorem (asynchronous updating): any collection of applications of tick, update-then, look preserves ToTS Note (then <= now): a system with ToTS physics reasons about time, over time, asynchronously, without using indexicals, and always contains the fact that “then comes before now”.

MetaSystems / MetaPhysics 0 Meta( {M, C }) means that M is a meta theory of C. 0 Exist M0 such that Meta( {M0, C }) for any context C (including M0) 0 Semantic valuation at the meta level is Meta preserving. { M, C } -SVal(e,tc) -> { SVal(M,e,tc), C } SVal(M,e,tc) adds a fact equating expression, e and its semantic simplification, tc 0 If M is a universal meta context then lifting object level deduction preserves Meta. 0 If M contains a fact such as ``the number of facts is 3'’ then adding to the facts of C is not Meta preserving.

Reflection principles as inference rules IndConst-able(tc) : Fact[M] IC-down Val[M](tc) = c c : IndConst[C'] { M, C } - IC-down(tc,c)-> { M, declareIndConst(C,c) } c : IndConst[C] IC-up Val[M'](tc) = c IndConst(tc) : Fact[M'] { M, C } - IC-up(tc,c)-> { M', C } M' = addFact(attachIndConst(M, tc, c), IndConst(tc)) Note: satisfaction but not validing preserving.

The FOL reflection rule Assume < ai, (forall f0, f1)(Fact(f0) and Fact(f1) implies FactAble(andi(f0,f1))) > is in the facts of M, and F0, F1 are facts of C { M, C } - Reflect(ai, F0, F1) -> { M, C1} C1 = addFact(C, mkand(F0,F1)) } by the following steps { M, C } - Fact-up(tf0,F0) -> { M1, C } - Fact-up(tf1,F1) -> { M2, C } - AE(ANDI,tf0,tf1) -> { M3, C } - SVal[M3](andi(tf0,tf1), tf) -> { M4, C } - Fact-down(tf) -> { M4, C1} - Forget(tf0,tf1,tf) -> { M, C1 }

Problem Solving 0 A problem solving context needs to provide at least a partial explanation of the predicate Solves(pr,s) for problems, pr and solutions, s Example: 0 find a proof of w from C (w in the language of C). ProofProb( ) 0 Solves(, ) if 0 C < C' 0 in Facts(C') 0 Acceptabel proof of w extractable from J in C’

Problem Solving Concepts & Principles Simple Principle: 0 InFacts(f,C) and w = AssertOf(f,C) implies Solves(, ) Tactic: solve a more specific problem (natural dedn) 0 Solves(, ) implies Solves(, )

Problem Concepts and Principles II Is an Appropriate Context 0 IAC(pr,C) – C is appropriate for formalizing, and solving, pr. IAC Principles: 0 IsFactOf(C,false) implies not(IAC(pr,C)) – for any pr, C 0 NdProb( ) implies IAC(, C)

Problem Solving Concepts & Principles III Can Be Believed - formalizing a means of introducing new problem solving information 0 0 CBB(w,pr,C) - w can be believed for the purpose of solving pr in C CBB Principles: 0 IAC(pr,C) and CBB(w,pr,C) implies IAC(pr, addFact(C,w)) 0 Taut(w,C) implies CBB(w,pr,C)

Problem Solving Concepts & Principles IV Ask an expert: 0 Expert(pr,A) and Confirms(A, pr, C, w) implies 0 IAC(pr,C) and CBB(w,pr,C) 0 TrustedProver(tp,pr) and ProofProblem(pr) implies Expert(pr,tp)

Behaviors 0 Boundaries –to interact with an autonomous agent need to know what is inside and what is outside 0 Dynamic contexts / systems –making systems interactive –perception -> cognition 0 Maintaining consistency / physics –successful agent can be interrupted when changes occur –react to restore physics

The End (the beginning?)