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Algorithms of Artificial Intelligence

Lecture 8: Synthesis of algorithms and plans

E. Tyugu

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Where• Behavior of robots

• Simulation of complex systems

• CAD

• HW/SW synthesis and codesign

• Reconfigurable computing

• Flexible OO interfaces

• Active networks (metaprotocols)

• Context-aware computing

• Just-in-time-synthesis of services

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How

• Transformational synthesis• Inductive Synthesis• Deductive synthesis• Nonmonotonic planning

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Transformational synthesis

Specification --------------> program rewriting

R. M. Burstall, J. Darlington. A transformation system for developing recursive programs. J. ACM, 24(1), 1977, 44-67.

+ many theoretical works continuously

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Inductive synthesis

Examples of input-output pairs are used as specifications:

1. (A B C D) gives (D C B A), construct a Lisp program for lists.

2. 1 gives 1, 2 gives 4, 3 gives 9, …, construct an arithmetic program.

A.W. Biermann, R. Kirshnaswany. Constructing programs from example computations. IEEE Trans. On Software Engineering, vol. 2, Sept.1976, 141 -153

More recently -- inductive logic programming (S. Muggleton)

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Deductive synthesis 1.Prove the theorem: “a solution of the given problem

exists”

2. Extract a program from the proof.

Z. Manna, R. Waldinger. Towards automatic program synthesis. Comm.ACM, 14(3), 1971, 151 -164.

(C. Green did it already in 1969 at IJCAI no. 1.)

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Idea of structural synthesis of programs

• Problem: Given programs p1,…,pn with specifications A1,…An, and a

specification G called a goal, find a program p that satisfies G.

• Solution: 1) find a (constructive) proof of G from A1,…,An. 2) extract the required program p from the proof.

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Intuitionistic logic

• Main intuitionistic requirement: a proof of existence of an entity (e.g. of a solution of a

problem) must include construction of such an entity.

• Examples of formulas not valid in intuitionistic logic:

A ~A~ ~A A

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Example

Are there irrational numbers a and b such that ab is rational?

Classical proof:

Take a=b=2, if 2 is rational, or

a = and b = 2, in the other case - then ab=2 is rational.

This is not a proof in intuitionistic logic, because one can not show the required irrational numbers.

2

22

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Some inference rules (natural deduction)

A1 … An A

A (& +) Ai (& -)

A

:

B A A B

A B ( +) B ( -)

A1,…,An, B denote formulas, A is an abbreviation for A1&…&An

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Realizations of formulas In the intuitionistic logic, every formula has a realization

(meaning) in a functional language:

• propositional variables mean objects, or object variables (if they are assumptions)

• if A1, …,An have realizations a1,…,an, then A has the realization (a1,…,an)

• realization of A B is a function f for computing a realization of B from any given realization of A.

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Building new realizations

A1 a1 … An an A a A a Ai ai

A x : B e A a A B f A B x.e B f(a)

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Implicative fragment of IPL

• Propositional variables: A, B, Ai,… are formulas

• Conjunctions: if A1,…,An are formulas, then

A1&…&An is a formula (we denote this also by A, if the formulas differ only by indices).

• Implications: if A, B are formulas, then

A B is a formula.

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Language of the structural synthesis of programs (SSP)

The general form of formulas of SSP is as follows (A, B, C, D denote propositional variables)

(A B) & C D

The implications A B are called subtasks.

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Subtasks explained

(A B) & … & (C D) & X Y f . . . g f…g x . h

realizations of subtasks, and of the whole axiom.

This can be read as follows:

h computes y from f, … ,g, x, where f computes b from a, … , g computes d from c.

NB! f, …, g are functional variables, h is a higher-order function.

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Using preprogrammed control structures in SSP

SSP:

(A B) & … & (C D) (X Y) f ….. g h

Prolog:

P :- Q, …, R

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Two examples

1. h computes maximal value of a function f given as a realization of subtask arg res :

(arg res) max f f.h

n 2. h computes the sum s = ab

b=1

(b a) & n s f f.h

(here we use same notations for propositional variables and respective object variables)

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Proof rules of SSP

Proof rules of SSP are introduction and elimination rules of conjunction and implication, and the following admissible rule:

(A B) & C D A & F B C & F D

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An example(Y A) B(A B) XX & Y A

Goal: B

(Y A) B A A (A B) X A B

X & Y A X Y Y (Y A) B Y A

B

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Simple program extraction

A1 x1 … An xn :

f A B A a B f B f(a) A B x.f(x)

F (A B) & C D A B f C B F(f(a),z)

where z are variables for C.

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Program extraction(general case)

F (A B)&C D A & G B f C & G D F(f(a,y),z)

where y, z are variables for G and C respectively.

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ExampleAxioms: i, j, val e

(j e) max (i max) minimaxGoal: val minimax

Proof: (j e) max i, j, val e

(i max) minimax i, val max

val min

j

i eval

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Example of program extraction

Axioms: (j e) max f (i max) minimax g i, j, val e i,j,v. selectGoal: val min

Proof: (j e) max i, j, val e (i max) minimax i, val max v,i. g( j.select) val min v. f(i. g( j.

select))

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Proof strategy

• Use the subtasks as late as possible

• Use connection graph for axioms

• Do goal-driven (backward) search for choosing axioms with subtasks

• Do forward search for simple implications

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Extended structural synthesis

It appeared that also disjunction can be handled efficiently in practical proofs. So one has extended the SSP with disjunnctions, and got ESSP with formulas of the form

(A B) & C D F G Programs derived in ESSP include branching, beause a

realization of a disjunction F G is of the form

if left then f else g,

where f and g are realizations of F and G respectively.

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Spec language 1) specification of an objectid : type [binging, ...] Examples: i: num;

I1: Integrator T =0.01;

2) axiomprecondition -> postcondition{implementation}Example: (i -> u)& I1.T -> delta; 3) equationexp1 = exp2Example: I * U = R;

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Translation of specs

• Let a type of Q have the semantics given by a set of formulae FQ. Then

the translation of a statement x : Q is the set of formulae obtained from FQ by prefixing all propositional variables occuring in formulae with

the prefix “x.”

• Semantics of primitive types is empty set of formulae.

• An equation is translated (has semantics) as a set of formulae showing what is computable from what by the equation.

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Spec language continued

Example:

x1: Oscillator1;

x2: Oscillator2;

process: RungeKutta

model1 = x1,

model2 = x2,

from = 0,

to = 115;

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Example of synthesis of schemes

I1 : integral to = 1;I2 : integral res = I1.val;F1:I1.arg & I2.arg -> I1.res {a realization of this axiom};F2:I1.arg -> I2.to {a realization of this axiom};Goal: -> I2.val;

I1

I2

F1

F2

fA

BC

A & B C (U V) & X Y

Y

U V

X

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Planning

Planning is synthesis of an algorithm for achieving a given goal. It can be, in principle, done by means of program synthesis methods. However, there are several special features of planning that prevent the usage of conventional program synthesizers:

• nonmonotous character of achieved subgoals

• frame problem.

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Planning continuedPlans tend to be shorter and with simpler structure than

computational algorithms.

Common formalisms for planning are

• first-order logic and

• situation calculus.

Quite helpful is hierarchical planning.

Quite conventional are linear and branching plans, iterative and recursive plans appear very seldom.

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Planning continued Naive planning (that is the most common planing method) uses knowledge

about a set of possible actions where, for each action, its pre- and postconditions are given.

A common example is planning in a block world where actions are for moving blocks:

PICKUP, PUTDOWN, STACK, UNSTACK.

More precise example with pre- and postconditions:

PICKUP (block) possible when ONTABLE(block) CLEAR(block) gives HOLDING(block) NOT CLEAR(block) NOT ONTABLE(block)

We can see the similarity to a rule-based knowledge representation. The planning algorithm is similar to application algorithm of rules.

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Planning continued

Example of a planning problem:

A B

A

B

C

C

Initial state: Goal state

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Situation calculus

Situation calculus uses predicate logic extended with situation constants. Each literal, e.g.

Above(x,y)

is extended with an argument showing the situation, e.g. (time moment usually) when this literal is valid:

Above(x,y,s1)

This adds much information and creates the frame problem:

when proceeding from one situation to another, all literals must be recomputed, or there must be some rules for deciding which literals remain unchanged.

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Hierarchical plans

To avoid large amount of search, planning can be performed in several stages: first rough planning that produces a plan including macro actions that must be made precise either by more planning or by macro substitution.

Planning with time constraints is scheduling. Impressive examples of automatic scheduling by means of program synthesis technique exist, e.g. PLANWARE developed in the Kestrel Institute for scheduling military air transportation in the Pacific area.