Key Points

Karl Lieberherr

Challenge:old high-level description

• Price• Set of problems

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Challenge:new high-level description

• Challenge(X):– Price– A constructive belief involving

• algorithms in domain X and • a set of problems in X

• A constructive belief (X)– A claim about algorithms in domain X

• A constructive belief defines a protocol to demonstrate that the claim is not supported

• Bryan: A challenge represents a constructive belief

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Algorithms in a domain

• The algorithms are about– Solve problems in the domain– Hide secrets in problems of the domain– Find hard problems

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Market Design

• The market is about constructive beliefs and constructive support for those beliefs.

• Constructive means that the objects that are claimed to exist must be constructed by an algorithm (constructive mathematics). Reference to Specker’s Theorem.

• Constructive support involves an interactive protocol (reference to interactive proofs).

• If the constructive support of a belief fails, the belief is called unsupported.

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Constructive support

• Demonstrations for unsupported claims must be efficiently checkable.

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Unsupported beliefs and counterexamples

• An unsupported belief is not a counter example. When a belief is unsupported, it might still be a true fact. Alice might have failed to find the right f in F.

• When Alice is perfect, she will always be able to support her belief if it is a true fact.

• When Bob is perfect, he will always be able to show that the belief is unsupported if it is a wrong fact.

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Why beliefs?

• Beliefs are easier to work with than theorems.

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Project Summary to Slides• Traditional benchmarks set standards for the evaluation of

software innovation in many disciplines within computer science in both industry and academia. Although necessary, developing benchmarks is challenging and once developed, benchmarks are inflexible, representing only a sample of the application space, and are difficult to maintain as applications typically evolve quickly. The objective of this proposal is to address benchmark problems by creating a dynamic strategy based on {\it artificial markets} of computational challenges. Dynamic market evaluation fosters more innovation through effective feedback, aiding in the discovery of better algorithms and the production of high quality software --- ultimately increasing the overall algorithmic knowledge of a given domain.

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• Given a concise description of a computational problem domain, $X$, it will be possible to create an {\it artificial market} around $X$, \market{X}. Software responsible for the creation of the market will generate a starting {\it agent} and a trust-worthy {\it market administrator}. Teams and individuals competing within the problem domain will improve their agent by participating in regular, administrator run competitions. Throughout the course of a competition, each agent offers and solves computational {\it challenges}, and is rewarded for both offering hard problems in $X$, and solving other agent's challenges effectively. Agents are ranked based on their performance within competitions and the corresponding teams can use the log of a competition to discover hard problems for their agent to offer, and gather ideas to improve their agent's algorithms.

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Additional• Fundamental to the \market(X) design is the

concept of a constructive belief and the concept of supported or unsupported constructive belief. We build on ideas from interactive proofs and program checking (a subarea of interactive proofs). Each challenge represents a constructive belief. Demonstrating that a constructive belief is unsupported does not imply that the belief is wrong. It only implies that the current holder of the belief cannot constructively support the belief. Another holder might be able to support it.

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Additional 2

• A constructive belief is specific to a holder. It is like “I, Alice, believe, theorem z holds” and then Alice must interact with a verifier (Bob, who does not believe theorem z holds) to support her belief.

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Additional 3

• Outcomes– Z holds

• Supported: Alice understands Z and she makes it impossible for Bob to attack the belief.

• Unsupported: Alice does not understand Z and she makes it possible for Bob to attack her belief. Alice is buggy.

– Z does not hold• Supported: Bob does not understand why Z does not hold

and cannot find an attack. Bob is buggy.• Unsupported: Bob understands why Z does not hold and he

can find an attack.

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Additional 4

• True statements may be shown to not be supported. The holder of the true statement (Alice) is buggy. The holder is likely to offer the challenge at a price that is too low.

• Wrong statements may be shown to be supported. The challenger of the wrong statement (Bob) is buggy. The challenger is likely to accept the challenge at a price that is too high.

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Supporting is weaker than proving

• Supporting a belief against a perfect challenger is like proving it.

• To show that a belief is unsupported against a perfect holder is like finding a counterexample.

• A perfect holder or challenger has unlimited resources and the perfect algorithm.

• In reality, holders and challengers are not perfect.

• They will become better by playing the game for a while.

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Witnesses

• Supporting a belief is like finding a positive witness. Replace “for all” by an object and show that prop holds.

• To show that a belief is unsupported is like finding a negative witness.

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Algorithms

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From teaching to Research

• Undergraduates need to deal with programming errors and conceptual problems.

• Researchers need to deal with finding state-of-the-art techniques.

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Beliefs

• Mathematical – Must be of the form: EAEA…

• Secret (time t)– You cannot find my secret– You cannot approximate my secret

• Duel– I solve your problems better than you solve my

problems

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Beliefs

• With beliefs we can express claims and show that they are supported or unsupported even if we don’t have enough information or knowledge to formally prove the claim.

• A belief, when challenged, is either supported or unsupported. When it is supported against a strong challenger the belief is more likely to hold. When it is unsupported against a strong holder, the belief is more likely to be wrong.

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• {\bf Intellectual Merit:} The main objectives of this project are (1) to study the design of artificial markets of computational challenges with specific objectives; (2) show that artificial markets of computational challenges are a useful dynamic benchmark tool for software; and (3) show that they drive innovation, helping to both find better algorithms and improve understanding of their development. The overall goal is to help program officers, managers, and professors with the evaluation of research prototypes, competing software packages, and student programs, respectively, and to help designers improve their algorithms using feedback from the artificial market. The design of particular artificial markets will drive innovation in specific objective areas, ultimately providing targeted feedback. The development of complex agents, administrators, and game generators will be supported by a number of generic programming tools, libraries, and design patterns that push the state-of-the-art. The resulting languages and methodologies are independent of the problem domain and contribute to programmer productivity and software quality.

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• {\bf Broader Impacts:} The impact of this project will be through advancing discovery and understanding of how to solve computational problems and providing an exciting platform for interdisciplinary teaching and learning. A generic artificial market supports the improvement of algorithmic techniques for a wide variety of problems through an experiential teaching curriculum. Students become engrossed in learning the art of software development, algorithmic analysis, and even modeling, through the refinement of their agent and frequent competitions. Distributed competitions will be run over the Internet allowing agents from different schools, companies, and even countries to compete within a common domain. The resulting algorithms, improved software, and useful tools, libraries and techniques will benefit software developers, scientific communities, and the general public as they make their way into various products and services.

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Compare belief/theorem

Belief B• Supporting B is short.• Supporting !B is short.

Theorem T• Proof of T is long.• Proof for !T is short.

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Definition: Proof System

• Proof system for a language L– P-proof x for y: P(x,y) for y in L.– Soundness: if y has a P-proof, y is in L.– Completeness: If y in L, y has a P-proof.– Effectiveness: P(x,y) is in P.

• http://citeseer.ist.psu.edu/cache/papers/cs/33367/http:zSzzSzwww.math.cas.czzSz~krajicekzSzdehn.pdf/dehn-function-and-length.pdf

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Terminology

• Adversary (instead of verifier)• Supporter (instead of prover)• BSS: questioner / supporter• Adversary Disprover Questioner Challenger• AM: agents challenges• Challenging to disprove• Challenging beliefs that the challenge represents• Questioning challenge

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Assumptions

• All beliefs are constructive beliefs. The challenging of a belief is constructive.

• The beliefs are about properties of algorithms but could be about other topics.

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Definition: Constructive Belief Support System

• For statements y of the form (EA)+ in first-order predicate calculus.– Derive interactive protocol from y. x is the outcome of the

protocol for y.– P-support x for y: P(x,y) . x is the support for y.– P-unsupport x for y: Pun(x,y). x shows that y has no support.– Soundness related

• Weak soundness: if y has P-support with challenger of strength q, y is more likely to hold with confidence q.

– Completeness related• Contra-Weak completeness: If y has P-unsupport with supporter of

strength q, y is more likely to be false with confidence q. • Completeness: if y holds, y has P-support x, P(x,y), for some x.

– Effectiveness: P(x,y) and Pun(x,y) are in P.

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Properties: Constructive Belief Support System

• For statements y of the form (EA)+ in first-order predicate calculus.– Derive interactive protocol from y. x is the outcome of the protocol for y.– P-support x for y: P(x,y) . x is the support for y.– P-unsupport x for y: Pun(x,y). x shows that y has no support.– For all y there is an x so that either P(x,y) or Pun(x,y) and such an x can be

found efficiently by following the protocol.– Soundness related

• Weak soundness: if y has P-support with challenger of strength q, y is more likely to hold with confidence q.

• Perfect soundness: If y has P-support with perfect challenger, y holds.– Completeness related

• Contra-Weak completeness: If y has P-unsupport with supporter of strength q, y is more likely to be false with confidence q.

• Contra-Perfect completeness: If y has P-unsupport with perfect supporter, y is false. • Completeness: if y holds, y has P-support x, P(x,y), for some x.

– Effectiveness: P(x,y) and Pun(x,y) are in P.

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Supporting a belief against a perfect challenger is like proving it.

Why?

• To show that a belief is unsupported against a perfect supporter is like finding a counterexample.– Because the perfect supporter will find the hardest object.

If the challenger can find an object that negates the predicate, we must have a counterexample.

• Supporting a belief against a perfect challenger is like proving it.– Because the perfect challenger will find the best object

and if this object does not negate the predicate, the supporter must have found the object that the statement claims to exist.

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Definition: ConstructiveBelief Support System, secret form

• For statements y of the secret form: You cannot recover my secret solution for objects satisfying predicate pred.– Derive interactive protocol from y. x is the outcome of the

protocol for y.– P-support x for y: P(x,y) . x is the support for y.– P-unsupport x for y: Pun(x,y). x shows that y has no support.– Soundness related

• Weak soundness: if y has P-support with challenger of strength q, y is more likely to hold with confidence q.

– Completeness related• Contra-Weak completeness: If y has P-unsupport with supporter of

strength q, y is more likely to be false with confidence q. • Completeness: if y holds, y has P-support x, P(x,y), for some x.

– Effectiveness: P(x,y) and Pun(x,y) are in P.

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Definition: Constructive Belief Support System, duel form

• For statements y of the duel form: If you give me k problems (satisfying pred) and I give you k problems satisfying pred, I will solve your problems better than you solve mine. – Derive interactive protocol from y. x is the outcome of the protocol

for y.– P-support x for y: P(x,y) . x is the support for y.– P-unsupport x for y: Pun(x,y). x shows that y has no support.– Soundness related

• Weak soundness: if y has P-support with challenger of strength q, y is more likely to hold with confidence q.

– Completeness related• Contra-Weak completeness: If y has P-unsupport with supporter of strength

q, y is more likely to be false with confidence q. • Completeness: if y holds, y has P-support x, P(x,y), for some x.

– Effectiveness: P(x,y) and Pun(x,y) are in P.

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Terminology

• Challenger supports y.– Challenger cannot find an object negating claim y.

• Challenger unsupports y.– Challenger finds an object negating claim y.

• Finding unsupport. Find an object which negates the claim.

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Properties: Constructive Belief Support System, testing perspective

• For statements y of the form (EA)+ in first-order predicate calculus.– Derive interactive protocol from y. x is the outcome of the protocol for y.– P-support x for y: P(x,y) . x is the support for y.– P-unsupport x for y: Pun(x,y). x shows that y has no support.– For all y there is an x so that either P(x,y) or Pun(x,y) and such an x can be found efficiently by

following the protocol.– Soundness related

• Weak soundness: if y has P-support with challenger of strength q, y is more likely to hold with confidence q.

• Perfect soundness: If y has P-support with perfect challenger, y holds.– Completeness related

• Contra-Weak completeness: If y has P-unsupport with supporter of strength q, y is more likely to be false with confidence q.

• Contra-Perfect completeness: If y has P-unsupport with perfect supporter, y is false. • Completeness: if y holds, y has P-support x, P(x,y), for some x.

– Testing related• If y holds, and challenger unsupports y, supporter is buggy.• If y does not hold, and challenger supports y, challenger is buggy.

– Effectiveness: P(x,y) and Pun(x,y) are in P.

04/21/23 Summary 33Supporting a belief against a perfect challenger is like proving it.

Advantages of Belief Support System

• Given a claim, we can quickly find support or unsupport.• The Belief Support system feeds the game:

– Beliefs which cannot be supported by the supporter, cost money.

– Beliefs must be made and put on the market.– Beliefs which are challenged must be challenged successfully,

otherwise they cost money.– The stronger the agents become, the better the beliefs become:

unchallenged beliefs become truths and challenged beliefs turn into counterexamples.

– The game acts as a truth or belief maintenance system for constructive beliefs about algorithms. Bad beliefs are eliminated because agents having bad beliefs lose money and drop out from the game.

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Challenge / Belief

• Mathematical challenge:– Constructive mathematical belief• Implies protocol for support/unsupport

– Price

• Etc. for other challenges

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About beliefs

• http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.40.2822

• http://en.wikipedia.org/wiki/Truth_maintenance_system

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Focus of proposal

• algorithmic competitions for dynamic benchmarks• The dynamic benchmarks are created by mathematical

beliefs, secret beliefs and duel beliefs.– Mathematical beliefs, optimization problems, EA

• Specific example: CSP– Secret beliefs, optimization problems

• Reproduce• Approximate within bound• Specific example: CSP

– Duel beliefs, optimization problems• Specific example: CSP

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Artificial Market and Constructive Belief Support

• A constructive belief support system is the foundation for the artificial market.– Money is made when a belief is supported: encourages

agents to offer only beliefs they can support. Pushes them towards offering “correct” beliefs.

– Money is made when a belief is successfully challenged: encourages agents to only challenge beliefs they can successfully challenge. Pushes them towards challenging only “wrong” beliefs.

– Efficiency (administrator must be efficient)• It must be easy to express successful support and it must be easy

to check successful support.• It must be easy to express successful challenge and must be easy

to check successful support.

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Difficult Problems for Dynamic Benchmarks

• The beliefs are about what an algorithm can do.

• To support the belief means to have a good algorithm.

• To challenge the belief means to give the algorithm difficult problems.

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Explaining SCG(X)

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ConstructiveBeliefSupportSystem(X)

ArtificialMarket(X)

No pricesNo game rulesSupported andUnsupportedBeliefs

SCG math (X) SCG secret (X) SCG duel (X)

Beliefs aboutoptimization problems

Add market rules

ArtificialMarket optimization (X)

Market Design

• Belief Support Systems• Artificial Markets

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• Bryan– ? \not\exists x . !sup(x,y) => valid(y)– ? \forall x . sup(x,y) => valid(y)– \exists x . !sup(x,y) => !valid(y) if confidence = 1.0

• Karl– \exists x.sup(x,y) = there is support for y

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For interesting y

• Sup(x,y) = x supports claim y• Even if y is true \exists x: !sup(x,y)• Even if y is false \exists x: sup(x,y)

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• There exists a y so that Even if y is true \exists x: !sup(x,y)

• There exists a y so that Even if y is false \exists x: sup(x,y)

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New terminology

• Constructive Belief Support System– Belief: • supporter• questioner = adversary

• Artificial Market– Challenge: • supporter = challenger• questioner = adversary = acceptor

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Research on Constructive Belief Support Systems

• Belief support systems for other claims than mathematical (EA)+, secret and duel.

• Formally defining confidence.• Deriving the protocol from the claim form.• Further formal properties of belief support

systems.

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Research on Constructive Belief Support Systems and Artificial Markets• Different ways of embedding belief support

systems into artificial markets.

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Research on Applicability Range of SCG

What is needed for winning?Programming Skills• Currently: high, although a

baby agent with basic communication skills is generated. Still solid programming skills needed.

• Future: would like to make it easier to transfer knowledge to agent.

Algorithmic Knowledge • Good knowledge about

successfully supporting and questioning beliefs.

• For appropriate beliefs turns into: good knowledge about how pose hard problems and how to solve hard problems

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Special case: Belief form: EA q

Beliefs• Alice supports belief • Bob questions belief

Problems• Alice poses hard problem• Bob solves hard problem

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Interactive Proof SystemConstructive Belief Support System

Interactive Proof System• Prover, all-powerful• Verifier, limited resources• True statement• False statement• Completeness

– If true, verifier will be convinced

• Soundness– If false, verifier will not be

convinced (small prob.)

Constructive Belief Support• Supporter, limited resources• Questioner, limited

resources• Supported statement• Unsupported statement• Completeness

– If true, questioning will not succeed

• Soundness– If false, questioner will succeed

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How to influence the hearts ofcomputer science students

• When the students see their agent suffer when playing with the agents of their peers, they want to help their agent. While doing so they learn a lot of computer science skills while transferring their knowledge of the subject domain into their agent.

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Relevant reference

• http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.4593

• Reasoning about Beliefs and Actions under Computational Resource Constraints (1987) by Eric J. Horvitz

• In Proceedings of the 1987 Workshop on Uncertainty in Artificial Intelligence

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Questions based on Reasoning about Beliefs and Actions under

Computational Resource Constraints• What is the comprehensive value of

computation?– Does more computation help me to create a

belief that I can support?– Does more computation help me to

successfully question this belief.– Cost/benefit tradeoffs.

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Value of partial solution

• The parameter p in belief B(p) must be above p_low and below p_high.

• There is one right answer. • How can we refine this result with increasing

amounts of computation?

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Dempster-Shafer Theory ???

• Epistemic or subjective uncertainty (lack of knowledge about a system)

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Implications between beliefs

• If x supports belief y1 then x supports belief y2– E.g.: Ef AJ < 0.6 implies Ef AJ < 0.7

• A claim that is never successfully questioned is true?

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Measure confidenceAccumulate evidence

• What can we learn from a long game history about the confidence in agents and statements?

• How can we measure the confidence of an agent? Confidence in agents: use something similar to the FIDE rating approach for Chess.

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Difficult Problems for Dynamic Benchmarks

• The beliefs are about what an algorithm can do.

• To support the belief means to have a good algorithm.

• To challenge the belief means to give the algorithm difficult problems.

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THIS IS WRONG: SEE NEXT SLIDE

Difficult Problems for Dynamic Benchmarks

• The beliefs are about what an algorithm can do. Belief is of the form: Exists f in F For all J in fs(f) q(f,J) < c

• To support the belief means to find hard problems f.

• To challenge the belief means to efficiently solve hard problems f.

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extra

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Relation numbers

• AND = 128• NAND = 127• OR = 254• NOR = 1

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Alternative names

• Hypothesis management system• Hypothesis support system

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Artificial Markets of Computational Challenges

• Constructive beliefs: supported or unsupported

• Challenge: (belief(p), price p): I believe I can support belief belief(p) and you cannot successfully question belief(p). If you can, you win 2*p, if you cannot you get 0. In both cases you had to pay p to accept the challenge.

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Interesting Belief(price,predicate,quality)

• Computational domain.• Predicate selects set of problems.

Challenge(Belief(price,predicate,quality), price)• If I give you problem p satisfying predicate, you

cannot find a solution of quality. If you can, you get 2*price, otherwise 0. The cost of accepting the challenge is price.

• Belief form: Exists problem: “your algorithm cannot do a task”

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Belief support system

• Does not need the notion of a computational domain. Just supported and unsupported beliefs.

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Artificial Markets• Based on beliefs. • Creating challenges based on beliefs that you can support,

makes money.• Accepting challenges based on beliefs that you can

successfully question, makes money.• Supporter (prover) and questioner (verifier) have both

limited resources.• If a belief is supported, there is evidence that the belief is a

theorem. The stronger the questioner, the stronger the evidence.

• If a belief is unsupported, there is evidence that the belief is not a theorem. The stronger the supporter, the stronger the evidence.

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Constructive Belief math

• Problem domain. Define feasible solutions fs(f) for a problem f. Define quality function q(f,J).

• Predicate F: defining a subset of problems in domain.

• Belief math(F, q(f,J) < c, c): Alice believes that if she chooses f in F and Bob chooses J in fs(f), q(f,J) < c (belief is supported by (f,J)).

• If !(q(f,J)<c), belief is not supported by (f,J).

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Example

• Domain: MAX CSP, relations of arity 3• Belief math(F,q(f,J)<c,c):– F: relation one in three– Second parameter: fsat(f,J) < c– c = 0.4

• Bob can show that this belief is not supported.• If c = 0.5, Alice can prevent Bob from showing

that belief is unsupported.• 0.444 break-even price04/21/23 Summary 68

Constructive Belief secret

• Problem domain. Define feasible solutions fs(f) for a problem f.

• Predicate F: defining a subset of problems in domain.

• secret(f): maps f to a secret feasible solution.• Belief secret(F,q(f,J,Js)): I believe if Alice chooses f

in F and secret(f) and Bob chooses J in fs(f) in time t, q(f, J,secret(f)) (belief is supported by (f,J)).

• If ! q(f, secret(f), J), belief is not supported.

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Example: secret Belief

• Domain: MAX CSP, relations of arity 3• Belief secret(F,q(f,J)<c,c):– F: relation one in three– Second parameter: fsat(f,J) < c– c = 0.4

• Bob can show that this belief is not supported.• If c = 0.5, Alice can prevent Bob from showing

that belief is unsupported.• 0.444 break-even price04/21/23 Summary 70

Transition to Challenge

• Belief math(F,q(f,J,c),c) is combined with profit: Alice offers challenge math (belief math (F,q(f,J,c),c), price) – If Bob accepts challenge, he must show that belief

is not supported. • If he succeeds, he gains g(price, quality(f,J))• If he fails, he looses price.

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Proposed Research

• 4 Design

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