54, Argumentation Based on Classical Logic

Really well written paper ('Argumentation Based on Classical Logic', 2009, Philippe Besnard, Anthony Hunter). Loads of examples throughout. I like the concept of an argument being 'more conservative' than another (i.e. it is "less demanding on the support and less specific about the consequent") and that of a 'maximally conservative undercut'. The argument trees considered are "merely a representation of the argumentation" and (differently to 'abstract argument systems') do not display cases where the argumentation is infinite and unresolved as being so.

'Sentence'

"a sentence is a 'formula' in which every occurrence of a variabl (if any) is within the scope of a quantifier for that variable."

(Introduction to Logic Programming (page 11), by Chrstopher John Hogger)

26.4-6, Argument-based Negotiation among BDI Agents

Notes taken from 'Argument-based Negotiation among BDI Agents' (2002), by Sonia V. Rueda, Alejandro J. Garcia, Guillermo R. Simari

4, Collaborative Agents

Collaborative MAS: A collaborative Multi-Agent System will be a pair of a set of argumentative BDI agents and a set of shared beliefs.

(Negotiating Beliefs; Proposals and Counterproposals; Side-effects; Failure in the Negotiation)

5, Communication Languages

(Interaction Protocol; Interaction Language; Negotiation Primitives)

6, Conclusions and Future Work...

26.3, Argument-based Negotiation among BDI Agents

Notes taken from 'Argument-based Negotiation among BDI Agents' (2002), by Sonia V. Rueda, Alejandro J. Garcia, Guillermo R. Simari

3, Planning and Argumentation

Argumentative BDI Agent: The agents desires D will be represented by a set of literals that will also be called goals. A subset of D will represent a set of committed goals and will be referred to as the agent intentions... The agent's beliefs will be represented by a restricted Defeasible Logic Program... Besides its beliefs, desires and intentions, an agent will have a set of actions that it may use to change its world.

Action: An action A is an ordered triple (P, X, C), where P is a set of literals representing preconditions for A, X is a consistent set of literals representing consequences of executing A, and C is a set of constraints of the form not L, where L is a literal.

Applicable Action...

Action Effect...

26.1-2, Argument-based Negotiation among BDI Agents

Notes taken from 'Argument-based Negotiation among BDI Agents' (2002), by Sonia V. Rueda, Alejandro J. Garcia, Guillermo R. Simari

"... Here we propose a deliberative mechanism for negotiation among BDI agents based in Argumentation."

1, Introduction

In a BDI agent, mental attitudes are used to model its cognitive capabilities. These mental attitudes include Beliefs, Desires and Intentions among others such as preferences, obligations, commitments, etc. These attitudes represent motivations of the agent and its informational and deliberative states which are used to determine its behaviour.

Agents will use a formalism based in argumentation in order to obtain plans for their goals represented by literals. They will begin by trying to construct a warrant for the goal. That might not be possible because some need literals are not available. The agent will try to obtain those missing literals, regarded as subgoals, by executing the actions it has available. When no action can achieve the subgoals the agent will request collaboration...

2, The Construction of a BDI Agent's Plan

Practical reasoning involves two fundamental processes: decide what goals are going to be pursued, and choose a plan on how to achieve them... The selected options will make up the agent's intentions; they will also have an influence on its actions, restrict future practical reasoning, and persist (in some way) in time...

... Abilities are associated with actions that have preconditions and consequences...

25.6-9, Reasoning About Rational Agents

Notes taken from 'Reasoning About Rational Agents' (2000), by Michael Wooldridge

6, Collective Mental States

(Mutual Beliefs, Desires, and Intentions; Mutual Mental States and Teamwork)

7, Communication

(Speech Acts; Attempts; Informing; Requesting; Composite Speech Acts)

8, Cooperation

(What Is Cooperative Problem Solving?; Recognition; Team Formation; Plan Formation)

9, Logic and Agent Theory

(Specification; Implementation; Verification)

25.4-5, Reasoning About Rational Agents

Notes taken from 'Reasoning About Rational Agents' (2000), by Michael Wooldridge

4, LORA Defined

(Syntax; Semantics; Derived Connectives; Some Properties of LORA)

5, Properties of Rational Agents

BDI Correspondence Theory

Pairwise Interactions between Beliefs, Desires and Intentions

(Int i X) => (Des i X): If an agent intends something, then it desires it. Intuitively, this schema makes sense for rational agents...

(Des i X) => (Int i X): If an agent desires something, then it intends it. In other words, an agent intends all its options... This formula does not appear to capture any interesting properties of agents.

(Bel i X) => (Des i X): This is a well-known, if not widely-admired property of agents known as realism ("accepting the inevitable"). For example, suppose I believe that the sun will definitely rise tomorrow. Then, one could argue, it makes no sense for me to desire that the sun will not rise... As a property of rational agents, realism seems too strong...

(Des i X) => (Bel i X): If an agent desires something, then it believes it. To give a concrete example, suppose I desire I am rich: should I then believe I am rich? Clearly not.

(Int i X) => (Bel i x): If an agent intends something, then it believes... Suppose I have an intention to write a book; does this imply I believe I will write it? One could argue that, in general, it is too strong a requirement for a rational agent... While I certainly believe it is possible that I will succeed in my attention to write the book, I do not believe it is inevitable that I will do so...

(Bel i X) => (Int i X): If an agent believes something, then it intends it. Again, this is a kind of realism property... Suppose that I believe that X is true: should I then adopt X as an intention? Clearly not. This would imply that I would choose and commit to everything that I believed was true. Intending something implies selecting it and committing resources to achieving it. It makes no sense to suggest committing resources to achieving something that is already true.

These formulae are a useful starting point for our analysis of the possible relationships that exist among the three components of an agent's mental state. However, it is clear that a finer-grained analysis of the relationships is likely to yield more intuitively reasonable results.

Varieties of Realism

(Int i X) => ¬(Des i ¬X)
(Des i X) => ¬(Int i ¬X)
These properties say that an agent's intentions are consistent with its desires, and conversely, its desires are consistent with its intentions... These schemas, which capture intention-desire consistency, appear to be reasonable properties to demand of rational agents in some, but not all circumstances... Under certain circumstances, it makes sense for an agent to reconsider its intentions - to deliberate over them, and possibly change focus. This implies entertaining options (desires) that are not necessarily consistent with its current intentions...

(Bel i X) => ¬(Des i ¬X)
(Des i X) => ¬(Bel i ¬X)
These schemas capture belief-desire consistency. As an example of the first, if I believe it is raining, there is no point in desiring it is not raining, since I will not be able to change what is already the case. As for the second, on first consideration, this schema seeems unreasonable. For example, I may desire to be rich while believing that I am not currently rich. But when we distinguish between present-directed and future-directed desires and beliefs, the property makes sense for rational agents...

Systems of BDI Logic

The Side-Effect Problem

The side-effect problem is illustrated by the following scenario: "Janine intends to visit the dentist in order to have a tooth pulled. She is aware that as a consequence of having a tooth pulled, she will suffer pain. Does Janine intend to suffer pain?"

... It is generally agreed that rational agents do not have to intend the consequences of their intentions. In other words, Janine can intend to have a tooth pulled, believing that this will cause pain, without intending to suffer pain.

25.1-3, Reasoning About Rational Agents

Notes taken from 'Reasoning About Rational Agents' (2000), by Michael Wooldridge

1, Rational Agents

(Properties of Rational Agents, A Software Engineering Perspective, Belief-Desire-Intention Agents, Reasoning About Belief-Desire-Intention Agents, FAQ)

2, The Belief-Desire-Intention Model

(Practical Reasoning, Intentions in Practical Reasoning, Implementing Rational Agents, The Deliberation Process, Commitment Strategies, Intention Reconsideration, Mental States and Computer Programs)

3, Introduction to LORA

This logic (LORA: "Logic of Rational Agents") allows us to represent the properties of rational agents and reason about them in an unambiguous, well-defined way.

Like any logic, LORA has a syntax, a semantics, and a proof theory. The syntax of LORA defines a set of acceptable constructions known as well-formed formulaue (or just formulae). The semantics assign a precise meaning to every formula of LORA. Finally, the proof theory of LORA tells us some basic properties of the logic, and how we can establish properties of the logic.

The language of LORA combines four distinct components:

1. A first-order component, which is in essence classical first-order logic...
2. A belief-desire-intention component, which allows us to express the beliefs, desires, and intentions of agents within a system.
3. A temporal component, which allows us to represent the dynamic aspects of systems - how they vary over time.
4. An action component, which allows us to represent the actions that agents perform, and the effects of these actions.

Deduction, Induction, Abduction

Deduction: An analytic process based on the application of the general rules to particular cases, with the inference of a result.

Induction: Synthetic reasoning which infers the rule from the case and the result.

Abduction: Another form of synthetic inference, but of the case from a rule and a result.

19, Assumption-based argumentation for epistemic and practical reasoning

Notes taken from 'Assumption-based argumentation for epistemic and practical reasoning' (2007), by Francesca Toni

"Assumption-based argumentation can serve as an effective computational tool for argumentation-based epistemic and practical reasoning, as required in a number of applications. In this paper we substantiate this claim by presenting formal mappings from frameworks for epistemic and practical reasoning onto assumption-based argumentation frameworks..."

1, Introduction

... In this paper, we consider two forms of reasoning that rational agents may need to perform, namely reasoning as to which beliefs they should hold (epistemic) and reasoning as to which course of action/decision they should choose (practical)...

2, Abstract and assumption-based argumentation...

3, Epistemic Reasoning...

3.1, Epistemic frameworks without preference rules...

3.2, Epistemic frameworks with preference rules...

4, Practical reasoning...

5, Example...

6, Conclusions

We have proposed concrete instances of assumption-based argumentation for epistemic reasoning... and practical reasoning...

... Within the ARGUGRID project, our approach to (epistemic and) practical reasoning can be used to model decisions concerning the orchestration of services available over the grid, taking into account preferences by the users and/or the service providers...

Predicate Logic, Nonmonotonic Rules and Priorities

Quotes taken from 'A Semantic Web Primer' (2004), by Grigoris Antoniou and Frank van Harmelen

(page 94, An axiomatic semantics for RDF and RDF Schema)

... we formalize the meaning of the modeling primitives of RDF and RDF Schema. Thus we capture the semantics of RDF and RDFS.

The formal language we use is predicate logic , universally accepted as the foundation of all (symbolic) knowledge representation. Formulas used in the formalization are referred to as axioms.

By describing the semantics of RDF and RDFS in a formal language like logic we make the semantics unambiguous and machine accessible. Also, we provide a basis for reasoning support by automated reasoners manipulating logical formulas.

(page 161, Nonmonotonic rules: Motivation and syntax)

... we turn our attention to nonmonotonic rule systems. So far (i.e. with monotonic rules), once the premises of a rule were proved, the rule could be applied and its head could be derived as a conclusion. In nonmonotonic rule systems, a rule may not be applied even if all premises are known because we have to consider contrary reasoning chains. In general, the rules we consider from now are called defeasible, because they can be defeated by other rules. To allow conflicts between rules, negated atomic formulas may occur in the head and the body of rules...

... To distinguish between defeasible rules and standard, monotonic rules, we use a different arrow:

p(X) => q(X)
r(X) => ¬q(X)

In this example, given also the facts

p(a)
r(a)

we conclude neither q(a) nor ¬q(a). It is a typical example of two rules blocking each other. This conflict may be resolved using priorities among rules. Suppose we knew somehow that the first rule is stronger than the second; then we could indeed derive q(a).

Priorities arise naturally in practice, and may be based on various principles:
- The source of one rule may be more reliable than the source of the second, or may even have higher priority. For example, in law, federal law preempts state law...
- One rule may be preferred over another because it is more recent.
- One rule may be preferred over another because it is more specific. A typical example is a general rule with some exceptions; in such cases, the exceptions are stronger than the general rule.

Specificity may often be computed based on the given rules, but the other two principles cannot be determined from the logical formalization. Therefore, we abstract from the specific prioritization principle used, and assume the existence of an external priority relation on the set of rules. To express the relation syntactically, we extend the rule syntax to include a unique label, for example,

r1: p(X) => q(X)
r2: r(X) => ¬q(X)

Then we can write

r1 > r2

to specify that r1 is stronger than r2.

We do not impose many conditions on >. It is not even required that the rules form a complete ordering. We only require the priority relation to be acyclic. That is, it is impossible to have cycles of the form

r1 > r2 > ... rn > r1

Note that priorities are meant to resolve conflicts among competing rules. In simple cases two rules are competing only if the head of one rule is the negation of the head of the other. But in applications it is often the case that once a predicate p is derived, some other predicates are excluded from holding...

Negation in Computing Argumentation

I was just looking through your 1999 paper 'Computing Argumentation in Logic Programming' and noticed that there was no mention of classical negation (like '¬p') anywhere. Is there a reason for this? Would propositions of the form '¬p' and hypotheses of the form 'not ¬p' require alterations to the paper and its findings?

In our approach classical negation 'not p' is meant to be represented by means of additional atoms (like) not_p. This means that in some cases, if the programs are not well written, then an inconsistency (p and not_p) may hold in some extension. This can be avoided if assumptions are properly positionsed, e.g.
'p if q' and 'not p if r'
really are written as
'p if q, a' and 'not_p if r, b'
where a, b are assumptions and contrary(a)=not_p and contrary(b)=p.

So, basically, we can ignore classical negation, and the overall approach is still fine.

Just to check a few things:

- In the paper, 'not p' is negation as failure, right? Is it to be interpreted as "p cannot be shown to hold and so we assume it does not hold".

Correct.

- Are rules of the form 'not p' allowed as the conclusions of rules?

No, conclusions 'not p', where 'not' is negation as failure, are not allowed in rules.

- Strictly speaking, would a program of the form
"p if q, a
not_p if r, b
where a,b are assumptions and contrary(a)=not_p and contrary(b)=p"
be allowed given that assumptions must be of the form of negative literals (like 'not p')?

Assumptions are negation as failure literals in this case, sorry I have written them as 'a' and 'b'. You see, we have proven that naf literals are really assumptions in the AIJ97 paper, so I often simply write them as assumptions. In terms of naf, my example could become what you suggest below or alternatively
p if q, not not_p
not_p if r, not p
where not in the premises of the rules is negation as failure.

I suggest you read my paper Abstract Argumentation (AI and Law 1996 or 1997): you can find it on my web page.

So, instead, would we have to say something like:
"p if q, not a
not_p if r, not b
where not a, not b are assumptions and contrary(a)=not_p and contrary(b)=p"?
Is this a correct translation?

Yes, this would be fine too.

10.2, Argumentation, N-Person Games and Stable Marriage Problem

Notes taken from ‘On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games’, by Phan Minh Dung (1995)

2.1, Argumentation in N-Persons Games

Stable extensions do not capture the intuitive semantics of every meaningful argumentation system… As preferred extensions exist for every argumentation framework, we can introduce the preferred solutions to n-person games by defining them as the preferred extensions of the corresponding argumentation system… The new solutions satisfy both conditions of a rational standard behaviour: freeness from inner contradiction and the ability to withstand any attack from outside. This is clearly a contribution to the theory of n-person games.

2.2, Argumentation and the Stable Marriage Problem

Let P be a knowledge base represented either as a logic program, or as a non-monotonic theory or as an argumentation framework. Then there is not necessarily a “bug” in P if P has no stable semantics (as demonstrated by the stable marriage love triangle example).

11.1-11.2, Computing Argumentation in Logic Programming

Notes taken from ‘Computing Argumentation in Logic Programming’, by Antonio C. Kakas and Francesca Toni (1999)

1, Introduction

In the argumentation-based approach to the semantics of LP, a logic program P is seen as a theory in a background monotonic logic (the logic of Horn clauses) that can be extended by suitable subsets, delta, of a given set of 'hypotheses' H consisting of all negative (variable-free) literals of the form 'not p' that can be constructed in the language of P. A set of hypotheses delta can be seen as an 'argument' supporting the consequences of the extended theory 'P U delta' derived in the underlying monotonic logic, where the (negative) hypotheses in delta are understood as (ground) atomic facts. In general, not all subsets of H are suitable to extend P. For example, a subset delta may be 'self-conflicting', in the sense that 'P U delta' entails an atom p, with 'not p' in delta. More interestingly, two different sets of hypotheses (or arguments) delta and delta' can be 'conflicting' with each other with only one of them forming an 'allowed extension' of P.

The various (logic programming) LP semantics can be understood via different criteria for selecting appropriate subsets of (hypotheses) H to provide allowed extensions of a logic program (P). These different criteria can all be defined in terms of a common notion of ‘attack’ between subsets of H... The different selection criteria are then based upon comparing candidate sets of hypotheses with attacking sets of hypotheses...

2, Argumentation semantics for logic programming

2.1, A set of hypotheses A attacks another set delta (on an hypothesis not p) iff ‘P U A’ |= p for some not p in delta.

2.3, A set of hypotheses delta is admissible iff for all sets of hypotheses A, if A attacks delta, then delta attacks ‘A – delta’.

Intuitively, delta is admissible iff delta defends itself against each attack A. Note that a set of hypotheses that attacks itself cannot be admissible, because there is no attack against the empty set.

… every admissible set of hypotheses is ‘contained’ in some preferred extension, and thus, in order to determine whether a given query holds with respect to the preferred extension and partial stable model semantics, it is sufficient to determine whether the query holds with respect to the semantics of admissible sets.

… a program may admit several admissible (and preferred) sets of hypotheses which may be incompatible with each other. Any proof theory for the admissibility/preferred extension semantics would therefore need to allow a non-deterministic choice among such sets.

… the construction of an admissible set delta can be done incrementally, starting from a given set of hypotheses delta0, by adding to delta0 suitable defences for it. The existence of several admissible (and preferred) sets of hypotheses reflects itself on the existence of several suitable defences for a given delta0 and imposes a non-deterministic choice among defences in the proof theory.

2.8, A set of hypotheses delta is weakly stable iff for all sets of hypotheses A, if A attacks delta, then ‘delta U A’ attacks ‘A – delta’.

Intuitively, the attack A against delta can be used in conjunction with delta to defend delta against A. Note that, similarly to the case of admissible sets, a set of hypotheses that attacks itself cannot be weakly stable.

Trivially, every admissible set of hypotheses is weakly stable, but not vice versa…

As for admissibility and preferred extension semantics, it can be proved that every weakly stable set of hypotheses is ‘contained’ in some stable theory…

2.9, A set of hypotheses delta is acceptable to another set of hypotheses delta0 iff for all sets of hypotheses A, if A attacks ‘delta – delta0’, then there exists a set of hypotheses D such that D attacks ‘A – (delta0 U delta)’ and D is acceptable to ‘A U delta0 U delta’.
A set of hypotheses is acceptable iff it is acceptable to the empty set of hypotheses.

The base case for the above recursive definition is given by
(BC1) a set of hypotheses delta is acceptable to another set delta0 if there exists no set of hypotheses A attacking (delta – delta0).
… (BC2) delta is acceptable to delta0 if delta is a subset of delta0.
Note that a set of hypotheses delta that attacks itself cannot be acceptable, because there is no attack against the empty set.

Weakly stable and admissible sets of hypotheses are acceptable.

2.13, A set of hypotheses delta is wf-acceptable iff for all sets of hypotheses A, if A attacks delta, then there exists a set of hypotheses D such that D attacks A and D is wf-acceptable.

The base case for wf-acceptability is given by:
(BC) delta is wf-acceptable if no set of hypotheses attacks delta.

(Lemma 2.15) A set of hypotheses delta is wf-acceptable iff each set of hypotheses delta’ (that is a subset of delta) is wf-acceptable.

Complete sets of hypotheses: Sets of hypotheses that are admissible and contain every hypothesis they ‘defend’.

2.16, A set of hypotheses delta is stable iff delta does not attack itself and delta attacks each hypothesis not in delta.

Logics for Representing Knowledge

What different logics are there for representing knowledge? As an example, if told that "every bird flies, but penguins don't", some possible logics for representing this are:

Default Logic
b(X): f(X)/ f(X)
p(X) -> b(X)
p(X): / ~f(X)

Logic Programming (with negation as failure)
f(X) <- b(X) & not abnormal(f(X))
abnormal(f(X)) <- p(X)
~f(X) <- p(X)

Autoepistemic Logic (where 'L' means 'believes')
b(X) & ~L(~f(X)) -> f(X)
p(X) -> ~f(X)
a -> La (given)
~a -> L~a (given)
L(a -> b) -> (La -> Lb) (given)

Language Literals

What exactly is a “literal”? When we speak of a “literal”, do we speak of it as a language construct irrespective of its truth value? For example, is "~a" a literal as well as "a"? If not, would the term "proposition" be more apt when referring to the truth value of "literals"?

It is either an atom or the negation of an atom. An atom is the "smallest" formula in a logic-based language, namely something that can be either true or false. An atom can be equated to a proposition in propositional logic (logic without any variables and quantifiers). Most of the work in the [argumentation] literature is about propositional logic indeed, although some (including assumption-based argumentation) is not.

Rebuttal and Undercutting Attacks

For the descriptions below assume we have a system of only two agents, x1 and x2, with knowledge bases
KB(x1) = a <- b, a <- c, c
KB(x2) = ~a, ~b
where the set of candidate assumptions is {b}. The two agents are engaged in a dialectical argumentation process wherein x1 is attempting to defend an argument for the proposition 'a' and x2 is attempting to attack it.

Undercutting Attack
Assume agent x1 presents the argument
({a <- b, b}, a)
where 'b' is an assumption made by x1. Agent x2 can and will successfully undercut/attack this argument by putting forward the assumption-free argument
({~b}, ~b).

Rebuttal Attack
Assume agent x1 presents the assumption-free argument
({a <- c, c}, a).
Agent x2 can rebut/attack this argument by putting forward its own assumption-free argument
({~a}, ~a).
Note that x1 could potentially counter-attack this attack by putting forward the original argument again, unless there is a restriction specified by the argumentation protocol. Likewise, x2 could attack the counter-attack with the same argument again, and so on indefinitely.

Reducing a rebuttal to an undercutting attack
If we redefine the knowledge bases of agents x1 and x2 as follows:
KB(x1) = a <- b, a <- c ^ alpha, c
KB(x2) = ~a <- beta, ~b <- gamma
where the set of candidate assumptions is {b, alpha, beta, gamma} and the contrary of b is ~b, the contrary of alpha is ~a, the contrary of beta is a and the contrary of gamma is b.

Now if agent x1 presents the argument
({a <- c ^ alpha, c, alpha}, a)
Agent x2 can undercut/attack this argument by putting forward the argument
({~a <- beta, beta}, ~a).
Note that the culprit here is not the literal 'a' but the assumption 'alpha'.

7, DeLP an Argumentative Approach

Notes take from ‘Defeasible Logic Programming An Argumentative Approach’ (2004), by Alejandro J. Garcia and Guillermo R. Simari

“… The defeasible argumentation basis of DeLP allows building applications that deal with incomplete and contradictory information in dynamic domains, where information may change. Thus, DeLP can be used for representing agent’s knowledge and for providing an inference engine…”

2, the Language

2.1 Fact: a literal, i.e. a ground atom, or a negated ground atom.
2.2 Strict Rule: an ordered pair, denoted “Head <- Body”.
2.3 Defeasible Rule: an ordered pair, denoted “Head -< Body”.

2.4 Defeasible Logic Program: a possibly infinite set of facts, strict rules and defeasible rules. In a program P, denoted as (H, A), we distinguish the subset H of facts and rules, and the subset A of defeasible rules.

2.5 Defeasible Derivation (monotonic)…
2.6 Strict Derivation: all the rules used in the defeasible derivation are strict rules.
2.7 A set of rules is contradictory iff there exists a defeasible derivation for a pair of complementary literals from the set.

3, Defeasible Argumentation

3.1 Argument Structure (non-monotonic): Denoted as [A, h]… or simply an argument A for h, is a minimal non-contradictory set of defeasible rules, obtained from a defeasible derivation for a given literal h… Note that strict rules are not part of an argument structure.
3.2 [B, q] is a sub-argument structure of [A, h] if B is a subset of A.

3.3 Two literals h and h1 disagree iff the set ‘H U {h, h1}’ is contradictory, where H is the set of facts and rules of the program.
3.4 We say that [A1, h1] counter-argues, rebuts, or attacks [A2, h2] at literal h iff there exists a sub-argument [A, h] of [A2, h2] such that h and h1 disagree.

3.5 (Generalised) Specificity: Criterion which allows discriminating between two conflicting arguments. Intuitively, this notion of specificity favours two aspects in an argument: it prefers an argument (1) with greater information content (and thus more precise) or (2) with less use of rules (more direct and thus more concise).

3.6 Equi-Specificity: Two arguments [A1, h1] and [A2, h2] are equi-specific iff A1 = A2, and the literal h2 has a strict derivation from ‘H U {h1}’, and the literal h1 has a strict derivation from ‘H U {h2}’.

3.7 Argument Comparison Using Rule’s Priorities: The argument [A1, h1] will be preferred (denoted “>”) over [A2, h2] iff:
1. there exists at least one rule ra (from A1) and one rule rb (from A2) such that ra > rb.
2. and there is no rb’ (from A2) and ra’ (from A1) such that rb’ > ra’.

4, Defeaters and Argumentation Lines

4.1 [A1, h1] is a proper defeater for [A2, h2] at literal h iff there exists a sub-argument [A, h] of [A2, h2] such that [A1, h1] counter-argues [A2, h2] at h, and [A1, h1] is strictly more specific than [A, h].
4.2 [A1, h1] is a blocking defeater for [A2, h2] at literal h iff there exists a sub-argument [A, h] of [A2, h2] such that [A1, h1] counter-argues [A2, h2] at h, and [A1, h1] is unrelated by the preference order to [A, h], i.e., neither argument structure is more specific than the other.
4.3 [A1, h1] is a defeater for [A2, h2] iff it is either a proper defeater or a blocking defeater.

4.4 Argumentation Line (for [A0, h0]): A sequence of argument structures from P, denoted [[A0, h0], [A1, h1], [A2, h2] …], where each element of the sequence [Ai, hi], i > 0, is a defeater of its predecessor [Ai-1, hi-1].
4.5 Supporting and Interfering argument structures: Let [[A0, h0], [A1, h1], [A2, h2] …] be an argumentation line, we define the set of supporting argument structures {[A0, h0], [A2, h2], [A4, h4] …} and the set of interfering argument structures {[A1, h1], [A3, h3], [A5, h5] …}.

4.6 A set of arguments {[Ai, hi]} (for i = 1 to n) is concordant iff the set ‘H U A1 U A2 U … U An’ is non-contradictory.

4.7 An argumentation line is acceptable iff:
1. It is a finite sequence.
2. The set of supporting arguments is concordant, and the set of interfering arguments is concordant.
3. No argument in the argumentation line is a sub-argument of an argument appearing earlier.
4. For all i, such that the argument [Ai, hi] is a blocking defeater for [Ai-1, hi-1], if [Ai+1, hi+1] exists, then [Ai+1, hi+1] is a proper defeater for [Ai, hi].

It is interesting to note that changes in the definition of acceptable argumentation line may produce a different behaviour of the formalism. Thus, the definition could be used as a way of tuning the system to obtain different results.

5, Warrant through Dialectical Analysis

In DeLP a literal h will be warranted if there exists a non-defeated argument structure [A, h]. In order to establish whether [A, h] is non-deafeated, the set of defeaters for A will be considered. Since each defeater D for A is itself an argument structure, defeaters for D will in turn be considered, and so on. Therefore, more than one argumentation line could arise, leading to a tree structure.

5.1 Dialectical Tree… Every node (except the root) represents a defeater (proper or blocking) of its parent, and leaves correspond to non-defeated arguments. Each path from the root to a leaf corresponds to one different acceptable argumentation line.

Marking of a dialectical tree (a bottom-up process through which we are able to determine the marking of the root):
(1) All leaves in the tree are marked as “U”.
(2) An inner node will be marked as “U” iff every child of it is marked as “D”. Otherwise it will be marked as “D”, i.e. iff it has at least one child marked as “U”.

5.2 Warranted Literals: Let [A, h] be an argument structure and T* its associated marked dialectical tree. The literal h is warranted iff the root of T* is marked as “U”. We will say that A is a warrant for h.

5.3 Answer to Queries: The answers of a DeLP interpreter can be defined in terms of a modal operator B. In terms of B, there are four possible answers for a query h:
- YES, if Bh (h is warranted)
- NO, if B~h (the compliment of h is warranted)
- UNDECIDED, if Bh and B~h (neither h nor ~h are warranted)
- UNKNOWN, if h is not in the language of the program.

The Warrant Procedure with pruning…

6, DeLP Extensions

DeLP with Default Negation… In DeLP “absence of sufficient evidence” means “there is no warrant”. Therefore, the default negation ‘not F’ will be assumed when the literal F is not warranted… Default negation will be allowed only preceding literals in the body of defeasible rules, e.g., ‘~cross_railway_tracks -< not ~train_is_coming’…

Extended Defeasible Rules: defeasible rules that use default negation.

… The reason not allowing default negotiation in strict rules is twofold. On one hand, a strict rule ‘p <- not q’ is not completely strict, because the head ‘p’ will be derived assuming ‘not q’. On the other hand, the set of strict rules and facts could become a contradictory set in many cases…

Extended Defeasible Logic Program: A set of Facts, Strict Rules and Extended Defeasible Rules.

6.1 Extended Defeasible Derivation: Since the decision of assuming an extended literal ‘not L’ will be carried out by the dialectical process, the definition of defeasible derivation is modified accordingly in extended DeLP. The change reflects that when an extended literal is found in the body of a rule, the literal will be ignored…

6.2 Extended Argument Structure: The definition of argument structure is also extended in order to avoid the introduction of self-defeating arguments… The definition is as before but with an addition rule:
- if L is a literal in the defeasible derivation (from the union of the supporting argument, and set of facts and rules) of h, then there is no defeasible rule in the argument containing ‘not L’ in its body.

6.3 In extended DeLP, default negated literals (assumptions on which the derivation is based) will be another point of attack in an argument… An argument structure [A1, h1] is a defeater for [A2, h2] iff it is a proper or blocking defeater for [A2, h2], or an attack to an assumption of [A2, h2].

DeLP with presumptions (a defeasible rule with an empty body, e.g. ‘a -<’)…

7, Implementation and Application (visit http://cs.uns.edu.ar/~ajg/DeLP.html)