Prioritised Default Logic as Argumentation with Partial Order Default Priorities

Prioritised Default Logic as Argumentation with Partial Order Default Priorities

Anthony P. Young Department of Informatics, King’s College London, Strand, London, U.K.
{peter.young,sanjay.modgil,odinaldo.rodrigues}@kcl.ac.uk
Sanjay Modgil Department of Informatics, King’s College London, Strand, London, U.K.
{peter.young,sanjay.modgil,odinaldo.rodrigues}@kcl.ac.uk
Odinaldo Rodrigues Department of Informatics, King’s College London, Strand, London, U.K.
{peter.young,sanjay.modgil,odinaldo.rodrigues}@kcl.ac.uk
26th July 2019
Abstract

We express Brewka’s prioritised default logic (PDL) as argumentation using ASPIC. By representing PDL as argumentation and designing an argument preference relation that takes the argument structure into account, we prove that the conclusions of the justified arguments correspond to the PDL extensions. We will first assume that the default priority is total, and then generalise to the case where it is a partial order. This provides a characterisation of non-monotonic inference in PDL as an exchange of argument and counter-argument, providing a basis for distributed non-monotonic reasoning in the form of dialogue.111The results of Section 3 first appeared in the preprint [23] and have been published in the conference proceedings of AAMAS2016 [24]. This paper gives the full proofs of these results.

1 Introduction

Dung’s abstract argumentation theory [11] has become established as a means for unifying various nonmonotonic logics (NMLs) [4, 20, 22], where the inferences of a given NML can be interpreted as conclusions of justified arguments. Abstract argumentation defines “justified arguments” by making use of principles familiar in everyday reasoning and debate. This renders the process of inference in the NML transparent and amenable to human inspection and participation, and serves as a basis for distributed reasoning and dialogue.

More precisely, relating NMLs and argumentation is to endow the NML with argumentation semantics. This has already been done for default logic [11], logic programming [11], defeasible logic [12] and preferred subtheories [16]. This allows the application of argument game proof theories [15] to the process of inference in these NMLs, and the generalisation of these dialectical proof theories to distributed reasoning amongst computational agents, where agents can engage in argumentation-based dialogues[17, 14, 1].

Abstract argumentation has been upgraded to structured argumentation theory [3], one example of which is the ASPIC framework for structured argumentation [16]. In ASPIC, arguments are constructed from premises and deductive or defeasible rules of inference. The conclusions of arguments can contradict each other and hence arguments can attack each other. A preference relation over the arguments can be used to determine which attacks succeed as defeats. The arguments and defeats instantiate an abstract argumentation framework, where the justified arguments are determined using Dung’s method. The conclusions of the justified arguments are then identified with the nonmonotonic inferences from the underlying premises and rules of inference. The advantages of ASPIC are that the framework provides a systematic and general method of endowing non-monotonic logics with argumentation semantics, and identifies sufficient conditions on the underlying logic and preference relations that guarantee the satisfaction of various normatively rational desiderata [10].

This paper endows Brewka’s prioritised default logic (PDL) [7] with argumentation semantics. PDL is an important NML because it upgrades default logic (DL) [19] with an explicit priority relation over defaults, so that, for example, one can account for recent information taking priority over information in the distant past. PDL has also been used to represent the (possibly conflicting) beliefs, obligations, intentions and desires (BOID) of agents, and model how these different categories of mental attitudes override each other in order to generate goals and actions that attain those goals [9].

We prove a correspondence between inferences in PDL and the conclusions of the justified arguments defined by the argumentation semantics. We realise these contributions by appropriately representing PDL in ASPIC. The main challenges involve understanding how priorities over defaults in PDL can be represented as an ASPIC argument preference relation, and then applying the properties of this preference relation to prove that the extensions of PDL correspond to the conclusions of justified arguments.

This paper has five sections. In Section 2, we review ASPIC, abstract argumentation, and PDL. In Section 3 we present an instantiation of ASPIC to PDL when the default priority is total. The key results are the design of an appropriate argument preference relation (Section 3.2), and showing that this argument preference relation guarantees that the conclusions of the justified arguments correspond exactly to the PDL extensions by the representation theorem (Section 3.3). We then investigate some properties and directly prove that the normative rationality postulates of [10] are satisfied (Section 3.4).222But in this case we are not leveraging the properties of ASPIC to achieve this. We will discuss this point in Section 5.

In Section 4 we lift the assumption that the priority order on the defaults is total. Following the pattern of the previous section we generalise the argument preference (Section 4.1) to accommodate for partial order default priorities. We then prove a generalised representation theorem (Sections 4.2) and prove a partial result concerning the satisfaction of the rationality postulates of [10] (Section 4.3). We conclude in Section 5 with suggestions for future work.

2 Background

2.1 Notation Used in this Paper

In this paper: “” is read “is defined as”. WLOG stands for “without loss of generality”. denotes the set of natural numbers. We denote set difference with . For two sets , denotes their symmetric difference. If is a function and , is the image set of in under . For a set its power set is and its finite power set (set of all finite subsets) is . iff is a finite subset of , therefore . Undefined quantities are denoted by , for example in the real numbers. Order isomorphism is denoted by .

If is a preordered set then the strict version of the preorder is , which is also a strict partial order. If is a strict partial order on and , then we define the set , i.e. the set of all -maximal elements of . We define the set analogously. For a set we define the set of possible strict partial orders on to be . Similarly, the set of all possible strict total orders on is . We will use the terms “total (order)” and “linear (order)” interchangeably. We will also call totally ordered sets either “tosets” or “chains”.

2.2 The ASPIC Framework

Abstract argumentation abstracts from the internal logical structure of arguments, the nature of defeats and how they are determined by preferences, and consideration of the conclusions of the arguments [11]. However, these features are referenced when studying whether any given logical instantiation of a framework yields complete extensions that satisfy the rationality postulates of [10]. ASPIC [16] provides a structured account of abstract argumentation, allowing one to reference the above features, while at the same time accommodating a wide range of instantiating logics and preference relations in a principled manner. ASPIC then identifies conditions under which complete extensions defined by the arguments, attacks and preferences, satisfy the rationality postulates of [10].

In ASPIC, the tuple is an argumentation system, where is a logical language and is the contrary function where is the set of wffs that are inconsistent with . Let be wffs for , is the set of strict inference rules of the form , denoting that if are true then is also true, and is the set of defeasible inference rules of the form , denoting that if are true then is tentatively true. Note . For a strict or defeasible rule , we define ,333Note it is possible to have and hence . and . Finally, is a partial function that assigns a name to some of the defeasible rules. For any we define the set to be the smallest superset of that also contains for all such that . We call the closure under strict rules operator.

In ASPIC, a knowledge base is a set where is the set of axioms and is the set of ordinary premises. Note that . Given an argumentation system and , arguments are defined inductively:

  1. (Base) is a singleton argument with , conclusion , premise set , top rule and set of subarguments .

  2. (Inductive) Let be arguments with respective conclusions and premise sets . If there is a rule , then is also an argument with , premises , and set of subarguments .

Let be the (unique) set of all arguments freely constructed following the above rules. It is clear that arguments are finite objects in that each argument has finitely many premises, and take finitely many rules to reach its conclusion. We define the conclusion map . We can generalise this to arbitrary sets of arguments (abuse of notation):

(2.1)

Two strict or defeasible rules are equal iff they have the same antecedent sets, consequents and name syntactically in the underlying . Two arguments are equal iff they are constructed identically as described above. More precisely, we can define equality of arguments inductively. The base case would be two singleton arguments are equal iff and are syntactically the same formulae. Given arguments and two equal rules and (either both strict or both defeasible) with antecedent , such that is the rule appended to the ’s, and is the rule appended to the ’s, then and are equal arguments.

We say is a subargument of iff and we write . We say is a proper subargument of iff and we write . It can be shown that is a preorder on . A set of arguments is subargument closed iff it is -down closed. Clearly, for every defeasible rule in an argument , there is a subargument of with as its top rule, by the inductive construction of arguments.

An argument is firm iff . Further, is the set of strict rules applied in constructing , and is the set of defeasible rules applied in constructing . We also define and . An argument is strict iff , else is defeasible. We can generalise to sets as well just like Equation 2.1 for .

Given , we introduce the set of all arguments freely constructed with defeasible rules restricted to those in as the set , which are all arguments with premises in , strict rules in and defeasible rules in . Formally, is defined inductively just as how arguments are constructed, but with the choice of defeasible rules restricted to those in . It is easy to show that this definition is equivalent to

(2.2)

Clearly, . Given , exists and is unique.

Let . The set of all strict extensions of is the set where

A set is closed under strict extensions iff for all , .

Lemma 2.1.

The set , for any , is closed under strict extensions and subarguments.

Proof.

If and , then so and hence , therefore is subargument closed. Now let , so for all , , therefore . Let , then and hence . Therefore, , therefore is closed under strict extensions. ∎

An argument attacks another argument , denoted as , iff at least one of the following hold, where:

  1. is said to undermine attack on the (singleton) subargument = iff there is some such that .

  2. is said to rebut attack on the subargument iff there is some such that , and .

  3. is said to undercut attack on the subargument iff there is some such that and .

See [16, Section 2] for a further discussion of why attacks are distinguished in this way. We abuse notation to define the attack relation as such that . Notice that by the transitivity of , if and , then .

A preference relation over arguments is then used to determine which attacks succeed as defeats. We denote the preference (not necessarily a preorder for now) such that is at least as preferred as . Strict preference and equivalence are, respectively, and . We define a defeat as

(2.3)

That is to say, defeats (on ) iff attacks on the subargument , and is not strictly preferred to . Notice the comparison is made at the subargument instead of the whole argument . We abuse notation to define the defeat relation as such that . A set of arguments is conflict-free (cf) iff .444Note that [16] studies two different notions of cf sets: one where no two arguments attack each other, and the other where no two arguments defeat each other. We choose the latter notion of cf as this is more commonplace in argumentation formalisms that distinguish between attacks and defeats, e.g. in [18]. Notice that by the transitivity of and that the preference comparison is made at the defeated subargument, if and , then . As relations, .

Preferences between arguments are calculated from the argument structure by comparing arguments at their fallible components, i.e. the ordinary premises and defeasible rules. This is achieved by endowing and with preorders and respectively, where (e.g.) iff is just as preferred or more preferred than (and analogously for ). These preorders are then aggregated to a set-comparison relation between the sets of premises and / or defeasible rules of the arguments, and then finally to , following the method in [16, Section 5].555Note there are many other ways to lift a preference on a set of objects to compare subsets of in various ways that are “compatible” with [2]. We will use a modified version of this lifting, which will be explained in Section 3.2.

Given the preference relation between arguments, we call the structure an ASPIC SAF (structured argumentation framework), or attack graph. Its corresponding defeat graph is , where is defined in terms of and as in Equation 2.3.

Given one can then evaluate the extensions under Dung’s abstract argumentation semantics, and thus identify the inferences defined by argumentation as the conclusions of the justified arguments. We now recap the key definitions of [11]. An argumentation framework is a directed graph , where is the set of arguments and is the defeat relation, such that means is a (successful) counterargument against . The argumentation frameworks we consider are defeat graphs, but this is a general definition.

Let and . defeats iff . is conflict-free (cf) iff . defends iff defeats . The characteristic function is , such that . is an admissible extension iff is cf and . An admissible extension is: a complete extension iff ; a preferred extension iff is a -maximal complete extension; the grounded extension iff is the -least complete extension; a stable extension iff is complete and defeats all arguments .

Let be the set of Dung semantics. An argument is sceptically (credulously) justified under the semantics iff belongs to all (at least one) of the -extensions of .

Instantiations of ASPIC should satisfy some properties to ensure they are rational [10]. Given an instantiation let be its ASPIC attack graph with corresponding defeat graph . Let be any complete extension. The Caminada-Amgoud rationality postulates state:

  1. (Subargument closure) is subargument closed.

  2. (Closure under strict rules) satisfies , where is defined in Equation 2.1.

  3. (Consistency) is consistent.666Notice by properties 2 and 3 above is consistent. ASPIC distinguishes this into direct and indirect consistency given that is in general arbitrary and do not have to be the rules of inference of classical logic. We will not make this distinction because our underlying logic will be first order logic (FOL) (Section 3.1). Further, consistency in the abstract logic of ASPIC is expressed in terms of the contrary function, but since our contrary function will just be classical negation, we can take the usual meaning of consistency in FOL.

An ASPIC instantiation is normatively rational iff it satisfies these rationality postulates. These postulates may be proved directly given an instantiation. ASPIC also identifies sufficient conditions for an instantiation to satisfy these postulates [16, Section 4], which we will discuss in Section 5.

2.3 Brewka’s Prioritised Default Logic

In this section we recap Brewka’s prioritised default logic (PDL) [7]. We work in first order logic (FOL) of arbitrary signature where the set of first-order formulae is and the set of closed first order formulae777i.e. first order formulae without free variables a.k.a. sentences is , with the usual quantifiers and connectives. Entailment is denoted by . Logical equivalence of formulae is denoted by . Given , the deductive closure of is , and given , the addition operator is defined as .

A normal default is an expression where and read “if is the case and is consistent with what we know, then jump to the conclusion even if it does not deductively follow”. In this case we call the antecedent and the consequent. A normal default is closed iff . We will assume all defaults are closed and normal unless stated otherwise. Given , a default is active (in ) iff .

A finite prioritised default theory (PDT) is a structure , where the set of facts is not necessarily finite and is a finite strict partially ordered set of defaults that nonmonotonically extend . The priority relation is such that is more888 We have defined the order dually to [7] so as to comply with orderings over the ASPIC defeasible inference rules. This goes against the tradition in NML where the smaller item in is the more preferred one. prioritised than . All PDTs in this paper are finite.

The inferences of a PDT are defined by its extensions. Let be a linearisation of . A prioritised default extension (with respect to ) (PDE) is a set built inductively as:

(2.4)
(2.5)

where “property 1” abbreviates “ is the consequent of the -greatest999See Footnote 8. default active in ”. Intuitively, one first generates all classical consequences from the facts , and then iteratively adds the nonmonotonic consequences from the highest priority default to the lowest. Notice if is inconsistent then . For this paper we will assume is always consistent.

For finite it can be shown that the ascending chain stabilises at some finite and that is consistent provided that is consistent. does not have to be unique because there are many distinct linearisations of . We say sceptically infers iff for all extensions of .

A PDT for which is a strict total order is a linearised PDT (LPDT). If is total then there is only one way to apply the defaults in by Equation 2.5, hence the extension is unique. We will use the notation to emphasise that the priority is total, and the notation to denote an arbitrary LPDT.

For the rest of this paper, if we declare to be a PDT, we mean where each component is defined above, and we make no further assumptions on each component. If we declare to be an LPDT, we mean where is a strict total order on .

3 From ASPIC to PDL

3.1 Representing PDL in ASPIC

We now instantiate ASPIC to PDL. Let be an LPDT.101010We will lift this assumption of a total order priority in Section 4.

  1. Our arguments are expressed in FOL, so our set of wffs is .

  2. The contrary function syntactically defines conflict in terms of classical negation. For all , unless has the syntactic form for some , then . As is singleton, we will abuse notation and write to refer to its element.

  3. The set of strict rules characterises inference in FOL. Notice is closed under transposition, i.e. for all ,

    We leave the proof theory implicit. instantiates to deductive closure.

  4. The set of defeasible rules is defined as:

    with the naming function . Clearly, there is a bijection where

    (3.1)

    and we will define the strict version of the preorder over as111111From Footnote 8, we do not need to define as the order-theoretic dual to , avoiding potential confusion as to which item is more preferred.

    (3.2)

    We can see that the strict toset is order isomorphic to .

  5. The set of axiom premises is , because we take to be the set of facts. Furthermore, .

The set of ASPIC arguments are defined as in Section 2.2.121212As is a countably infinite set, is also a countably infinite set. All arguments are firm because , and so there are no undermining attacks. As is undefined, no attack can be an undercut. Therefore, we only have rebut attacks,

(3.3)

Defeats are defined as in Equation 2.3. In the next section, we will define the argument preference , based on the strict total order over .

3.2 A Suitable Argument Preference Relation

We wish to define a suitable argument preference relation such that the conclusion set of the stable extension defined by corresponds to the extension of the underlying PDT.131313In Section 3.4, we will show that for the resulting defeat graphs there is only one extension in that is stable, grounded and preferred. The first place to look for such a relation is in the existing relations of ASPIC [16, Definition 19]. However, simple counterexamples can be devised to show the inferences of the PDT and its argumentation counterpart do not correspond.

The difference between PDL and ASPIC is in how blocked defaults are treated. In PDL, blocked defaults are simply excluded from the extension. In ASPIC, it is possible to construct arguments with defeasible rules that correspond to blocked defaults. If such that is arbitrary, there is no guarantee that the blocked defaults will be positioned in the chain such that arguments with blocked defaults are always defeated by arguments with only non-blocked defaults.141414We will see this explicitly in Example 1 later. To ensure that arguments with blocked defaults are defeated and hence the conclusions of the justified arguments form the extension of the PDT, we need to rearrange the rules in to take into account the structure of arguments. ASPIC does allow for explicit reference to argument structure, i.e. we can tell which defeasible rules preceed which within an argument.

Rearranging to take argument structure into account captures the PDL meaning of “active” default, because defaults are added to when its prerequisite is inferred. This rearrangement will mean that every defeasible rule corresponding to a blocked default will be less preferred than the rules which make up arguments that rebut the argument with as its top rule. We now devise a new ASPIC argument preference relation which incorporates the argument structure into the preorder .

More formally, given any strict total order on , we first define a transformation , where the subscript SP stands for structure-preference. This sorts the defeasible rules in a way compatible with both the priority and their logical structure.

The set is finite because we have assumed that is finite (Sections 2.3 and 3.1). Let . We define to be the -greatest element of the following set:

(3.4)

The intuition is: is the most preferred rule whose antecedent is inferred by the conclusions of all strict arguments, is the next most preferred rule, whose antecedent is amongst the conclusions of all arguments having at most as a defeasible rule. Similarly, is the next most preferred rule, whose antecedent is amongst the conclusions of all arguments having at most and as defeasible rules, and so on until all of the rules of are exhausted. Notice that the second union after the set difference in Equation 3.4 ensures that once a rule is applied it cannot be applied again. We then define as (notice the dual order)

(3.5)

We define the non-strict order to be . This makes sense because is bijective between and . Clearly is a strict total order on . We call this the structure preference order on , which exists and is unique given . This means the transformation is functional, where is total on .

Now let be any strict partial order on . We define the strict set comparison relation on corresponding to . For , the relation , called the disjoint elitist order, is defined as follows:

(3.6)

The lifting is functional. We will focus on the following special case of , where instead of we have :

(3.7)

The corresponding strict argument preference is, for ,

(3.8)

We define the corresponding non-strict preference as

(3.9)

We now show that satisfies the following properties.

Lemma 3.1.

For all , .

Proof.

If then , so . If , then , which means is vacuously true from Equation 3.7 so . ∎

The following result shows that larger arguments, which potentially can contain more fallible information (i.e. defeasible rules), cannot be more preferred than its (smaller) subarguments.

Corollary 3.2.

For all , if then .

Proof.

It can be shown from how ASPIC arguments are constructed (Section 2.2) that , and then invoke Lemma 3.1. ∎

Corollary 3.3.

Strict arguments are -maximal.

Proof.

Let be strict and be arbitrary. Assume for contradiction that . As , Equations 3.7 and 3.8 instantiate to: , which is impossible by the first bounded quantifier. Therefore, if is strict, then for all . ∎

Lemma 3.4.

Let be a strict toset, then is also a strict toset, where is defined in Equation 3.6, here with instead of .

Proof.

We prove is irreflexive, transitive and total on , assuming that is a strict total order on .151515 More generally, it can be shown that for any strict partial order , the relation from Equation 3.6 is acyclic, and hence irreflexive and asymmetric, but not necessarily transitive. If is a modular order [13, Lemma 3.7], then is transitive. Further, if total (recalling that total orders are modular), then is trichotomous, and hence a strict total order. To show irreflexivity, assume for contradiction that there is some such that , which by Equation 3.6 is equivalent to a formula whose first bounded quantifier is “”, which is false, so is irreflexive. To show transitivity, let , such that

(3.10)

All of these elements are distinct. If then the corresponding set is empty. Let , where

We can picture these sets with the Venn diagram in Figure 3.1. The solid outer rectangle represents the set . The three finite sets are the three rectangles within. Each overlapping region has exactly the elements indicated and nothing more. This configuration exhausts all possibilities for and .

Figure 3.1: Venn diagram for the proof of Lemma 3.4.

Now suppose permits , we write this out in terms of elements (Equations 3.2 and 3.12). is equivalent to

(3.11)

Note that there are disjuncts in Equation 3.2. Applying the same reasoning as in Equation 3.2, we can see that is equivalent to

(3.12)

There are disjuncts in Equation 3.2. We need to show that . By the same reasoning as Equations 3.2 and 3.12, this is equivalent to

(3.13)

So, to prove Equation 3.13, we need to show one of the disjuncts of Equation 3.13 i.e. for at least one of or , we show either

(3.14)

by establishing all of the conjuncts. Given , we take the conjunction of Equations 3.2 and 3.12, making disjuncts, which is equivalent to the following expression: