A Unified Algebraic Framework for Non-Monotonicity

Tremendous research effort has been dedicated over the years to thoroughly investigate non-monotonic reasoning. With the abundance of non-monotonic logical formalisms, a unified theory that enables comparing the different approaches is much called for. In this paper, we present an algebraic graded logic we refer to as LogAG capable of encompassing a wide variety of non-monotonic formalisms. We build on Lin and Shoham's argument systems first developed to formalize non-monotonic commonsense reasoning. We show how to encode argument systems as LogAG theories, and prove that LogAG captures the notion of belief spaces in argument systems. Since argument systems capture default logic, autoepistemic logic, the principle of negation as failure, and circumscription, our results show that LogAG captures the before-mentioned non-monotonic logical formalisms as well. Previous results show that LogAG subsumes possibilistic logic and any non-monotonic inference relation satisfying Makinson's rationality postulates. In this way, LogAG provides a powerful unified framework for non-monotonicity.

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1 Introduction

Non-monotonic logics are attempts to model commonsense defeasible reasoning that allows making plausible, albeit possibly fallible, assumptions about the world in the absence of complete knowledge. The term “non-monotonic” refers to the fact that new evidence can retract previous contradicting assumptions. This contrasts with classical logics where new evidence never invalidates previous assumptions about the world. Modelling non-monotonicity has been the focus of extensive studies in the knowledge representation and reasoning community for many years giving rise to a vast family of non-monotonic formalisms. The currently existing approaches to representing non-monotonicity can be classified into two orthogonal families: fixed point logics and model preference logics

[5]. Fixed point logics define a fixed point operator by which possibly multiple sets of consistent beliefs can be constructed. Typical non-monotonic logics taking the fixed point approach are Reiter’s default logic [28] and Moore’s autoepistemic logic [24, 21]. Model preference logics, on the other hand, define non-monotonic logical inference relations with respect to selected preferred models of the world. Typical model preference logics are probabilistic logic [2, 27], McCarthy’s circumscription [22], system P proposed by Kraus, Lehmann and Magidor [19], and Pearl’s system Z [26]. The wide diversity of all of these logics in addition to their non-standard semantics has rendered the task of gaining a good understanding of them quite hard. For this reason, a unified theory that enables comparing the different approaches is much called for. The purpose of this paper is to present an algebraic graded logic we refer to as [18, 12, 13] capable of encompassing the previously-mentioned non-monotonic logics providing a general framework for non-monotonicity.

Another non-standard attempt at formalizing commonsense non-monotonic reasoning is Lin and Shoham’s argument systems [20]. Argument systems are considered a radical departure from the classical sentence-based approaches as they are based entirely on inference rules. In [20], Lin and Shoham prove that classical non-monotonic approaches such as default logic [28], autoepistemic logic [24], circumscription [22], and the principle of negation as failure [8] are all special cases of argument systems. In this paper, we show that argument systems can be embedded in proving that captures the same non-monotonic logical approaches that argument systems capture. In [14], we proved that subsumes possibilistic logic [10] and any non-monotonic inference relation satisfying Makinson’s rationality postulates. While other unifying frameworks for non-monotonic formalisms such as [4, 6] exist in the literature, non of these frameworks can capture weighted approaches such as possibilistic logic while encompassing the classical previously-mentioned logical approaches like does. In this way, can be considered a powerful algebraic unified framework for non-monotonicity providing a unified understanding of a vast diversity of non-monotonic logical formalisms.

The rest of this paper is organized as follows. In Section 2, will be thoroughly reviewed describing its syntax and semantics. Section 3 will briefly review argument systems. In Section 4, the main results of this paper, proving that subsumes argument systems, will be presented. Finally, some concluding remarks are outlined in Section 5.

2 LogAG

While most of the graded logics we are aware of employ non-classical modal logic semantics by assigning grades to possible worlds [11], is a non-modal logic with classical notions of worlds and truth values. This is not to say that is a classical logic, but it is closer in spirit to classical non-monotonic logics such as default logic and circumscription. Following these formalisms, assumes a classical notion of logical consequence on top of which a more restrictive, non-classical relation is defined selecting only a subset of the classical models. In defining this relation we take the algebraic, rather than the modal, route. The remaining of this section is dedicated to reviewing the syntax and semantics of . A sound and complete proof theory for is presented in [18, 12]. In [18], it was proven that is a stable and well-behaved logic observing Makinson’s properties of reflexivity, cut, and cautious monotony.

2.1 LogAG Syntax

consists of algebraically constructed terms from function symbols. There are no sentences in ; instead, we use terms of a distinguished syntactic type to denote propositions. is a variant of [16] and [17], which are algebraic languages for reasoning about, respectively, beliefs and temporal phenomena. Propositions are included as first-class individuals in the ontology and are structured in a Boolean algebra giving us all standard truth conditions and classical notions of consequence and validity. The inclusion of propositions in the ontology, though non-standard, has been suggested by several authors [7, 3, 25, 30]. We refer the reader to [16, 30]

for arguments in favour of adopting this approach in the representation of propositional attitudes in artificial intelligence. Additionally,

grades are introduced as first-class individuals in the ontology. As a result, propositions about graded propositions can be constructed, which are themselves recursively gradable.

A language is a many-sorted language composed of a set of terms partitioned into three base sorts: is a set of terms (including the term ) denoting propositions, is a set of terms denoting grades, and is a set of terms denoting anything else. A alphabet includes a non-empty, countable set of constant and function symbols each having a syntactic sort from the set and of syntactic sorts. Intuitively, is the syntactic sort of function symbols that take a single argument of sort , , or and produce a functional term of sort . Given the restriction of the first argument of function symbols to base sorts, is, in a sense, a first-order language. In addition, an alphabet includes a countably infinite set of variables of the three base sorts; a set of syncategorematic symbols including the comma, various matching pairs of brackets and parentheses, and the symbol ; and a set of logical symbols defined as the union of the following sets: , , , and . Terms involving 111Through out this paper, we will use to denote material implication. and can always be expressed in terms of the above logical operators and . The terms containing are referred to grading terms, while the terms not including are referred to as non-grading terms.

The following are some examples of well-formed terms permissible by the syntax of .

The first two well-formed terms denote properties of grades: (1) denotes the proposition that the equality relation of grades is symmetric, and (2) denotes the proposition that the ordering of grades is linear. (3) denotes the proposition that the grade of is 2. (4) and (5) illustrate the de re and de dicto grading, respectively. The syntax of allows the nesting of grading terms forming grading chains as shown in (6). One possible use of such nesting is to express information from various knowledge sources with different trust degrees [18].

2.2 From Syntax to Semantics

A key element in the semantics of is the notion of a structure.

Definition 2.1.

A structure is a quintuple , where

• , the domain of discourse, is a set with two disjoint, non-empty, countable subsets: a set of propositions , and a set of grades .

• is a complete, non-degenerate Boolean algebra.

• is an ordering function.

• is an equality function, where for every :
if , and otherwise.

A valuation of a language is a triple , where is a structure, is a function that assigns to each function symbol an appropriate function on , and is a function mapping each variable to a corresponding element of the appropriate block of . An interpretation of terms is given by a function . Figure 1 summarizes the operation of .

Definition 2.2.

Let be a language and let be a valuation of . An interpretation of the terms of is given by a function :

• , for a variable

• , for a constant

• , for an -adic () function symbol

2.3 Beyond Classical Logical Consequence

We define logical consequence using the familiar notion of filters from Boolean algebra [29].

Definition 2.3.

A filter of a boolean algebra is a subset of such that:

1. ;

2. If , then ; and

3. If and , then .

A propositional term is a logical consequence of a set of propositional terms if it is a member of the filter of the interpretation of , denoted .

Definition 2.4.

Let be a language. For every and , is a logical consequence of , denoted , if, for every -valuation , where

In general, the elements of will be referred to as the graded consequences at level . The rest of this section is dedicated to formally defining graded filters together with our graded consequence relation based on graded filters. In the sequel, for every and , will be taken to represent a grading proposition that grades . Moreover, if , then is graded in . The set of graders in is defined to be the set and grades . Throughout, a structure is assumed.

As a building step towards formalizing the notion of a graded filter, the structure of graded propositions should be carefully specified. For this reason, the following notion of an embedded proposition is defined.

Definition 2.5.

Let . A proposition is embedded in if (i) (ii) or if, for some , is embedded in . Henceforth, let is embedded in .

Since a graded proposition might be embedded at any depth , the degree of embedding of a graded proposition is defined as follows.

Definition 2.6.

For , let the degree of embedding of in be a function , where

1. if , then ; and

2. if , then , where .

For notational convenience, we let the set of embedded propositions at depth be , for every .

Definition 2.7.

Let be a structure with a depth- and fan-out-bounded 222 is depth-bounded if every grading chain has at most distinct grading propositions and is fan-out-bounded if every grading proposition grades at most propositions where [12].. A telescoping structure for is a quadruple , where

• , referred to as the top theory;

• is an ultrafilter of the subalgebra induced by (an ultrafilter is a maximal filter with respect to not including ) [29];

• ; and .

Recasting the familiar notion of a kernel of a belief base [15] into the context of structures, we say that a -kernel of is a subset-minimal inconsistent set such that is improper () where is the set of all embedded graded propositions in the filter of . Let be the set of kernels that entail . A proposition survives in if is not the weakest proposition (with the least grade) in . In what follows, the fused grade of a proposition in according to a telescoping structure will be referred to as .

Definition 2.8.

For a telescoping structure and a fan-in-bounded 333 is fan-in-bounded if every graded proposition is graded by at most grading propositions where [12]. , if , then survives given if

1. is ungraded in ; or

2. there is some ungraded such that ; or

3. there is some graded such that and .

The set of kernel survivors of given is the set

if then survives given .

The notion of a proposition being supported in is defined as follows.

Definition 2.9.

Let . We say that is supported in given if

1. ; or

2. there is a grading chain of in with where every member of is supported in .

The set of propositions supported in given is denoted by .

Observation 2.1.

, for some set of embedded graded propositions in .

The -induced telescoping of is defined as the set of propositions supported given in the set of kernel survivors of .

Definition 2.10.

Let be a telescoping structure for . If such that is fan-in-bounded, then the -induced telescoping of is given by

Observation 2.2.

If is proper, then is proper.

Proof.

If is not proper, then has at least one kernel . According to Definitions 2.8 and 2.9, this can only happen if . Thus, is proper. ∎

Definition 2.11.

If has finitely-many grading propositions, then is defined, for every telescoping structure . Hence, provided that the right-hand side is defined, let

 τ_Tn(Q)={Qif n=0τ_T(τ_Tn−1(Q))otherwise

A graded filter of a top theory , denoted , is defined as the filter of the -induced telescoping of of degree .

In the following example, we now go back to the example we introduced at the beginning of this section in Figure 2. We show how the formal construction of the graded filters matches the intuitions we pointed out earlier.

Example 2.1.

Consider and where , and . In what follows, let be an abbreviation for .

• [leftmargin=*]

• =

Upon telescoping to degree 1, there are no contradictions in (no kernel )). Hence, everything in survives telescoping and is supported (notice the equality of , the kernel survivors, and the supported propositions in ). At level 1, we believe that the bird we saw is indeed a penguin and accordingly has wings and does not fly.

• Upon telescoping to degree 2, there are two kernels and . survives the first kernel as it is not graded in . survives the first kernel as well as it is the only graded proposition in the kernel with another member . does not survive the second kernel as the kernel contains another graded proposition and the grade of (2) is less than the fused grade of . Accordingly, loses its support and is not supported in the set of kernel survivors. The graded filter of degree 2 does not contain or , but is retained as it has nothing to do with the contradiction. At level 2, we start taking into account the information our brother told us. Since our combined trust in our brother and sister is higher that our trust in what we saw, we end up not believing that the bird we saw is a penguin since we believe that it flies.

• reaching a fixed point.∎

Henceforth, given a theory and a valuation , let the valuation of be denoted as . We use graded filters to define graded consequence as follows. Further, for a structure , an grading canon is a triple where and and are as indicated in Definition 2.7.

Definition 2.12.

Let be a theory. For every , valuation where has a set which is depth- and fan-out-bounded, and grading canon , is a graded consequence of with respect to , denoted , if is defined and for every telescoping structure for where extends 444An ultrafilter extends a filter , if ..

It is worth noting that reduces to if or if does not contain any grading propositions. However, unlike , is non-monotonic in general. In what follows, let . When we are considering a set of canons which only differ in the value of , we write instead of .

The upcoming example showcases the operation of on the classical non-monotonic reasoning example of birds fly, but penguins are special birds that do not fly.

Example 2.2.

Consider the following theory .

We show next the relevant graded consequences of with respect to a series of canons, with .

Upon telescoping to , we believe that Tweety flies and Opus does not fly. The embedded proposition that Opus flies does not survive telescoping since we trust that Opus does not fly, being a penguin, more than we trust that it flies, being a bird. is a fixed point.

Now, consider the theory which is similar to , but with propositions (1) and (2) replaced by “” and “”, respectively. Thus, we trade the “de re” representation of for the “de dicto” representation in . This change results in a change in the fixed point that we reach. In , as in , we end up believing that Opus does not fly. Unlike however, we give up our belief in the proposition that birds fly and, hence, cannot conclude that Tweety flies. Being able to grade only the consequent in , as in , allows us to give up believing that Opus flies while keeping the rule which allows to conclude that Tweety flies. Grading only part of the rule is one of the strengths of which is not the possible in many weighted logics. ∎

3 Argument Systems

Argument systems [20] assume a logical language which is a set of well-formed formulas (wffs) restricted to contain if is itself a wff. The operator has no logical properties in argument systems. The most distinctive feature of argument systems is that they are based entirely on inference rules which are defined as follows.

Definition 3.1.

An inference rule is a rule of the following formats:

1. , where is a wff. This is called a base fact.

2. . This is called a monotonic rule.

3. . This is called a non-monotonic rule.

By chaining rules together, we get

arguments which are used to establish propositions.

Definition 3.2.

Let be a set of rules. An argument in is a rooted tree with labelled arcs defined as follows:

1. If is a base fact, the tree consisting of only as a root is an argument.

2. If are arguments whose roots are , and such that is not a node in the trees , then the tree with as its root and as its immediate subtrees is an argument. All the arcs from B to its children is labelled by the monotonic (non-monotonic) rule.

An argument is said to be for (or is supported by ) if is the root of . By grouping arguments together, we get an argument structure. An argument structure can be thought of as the set of logically consistent arguments held by the agent.

Definition 3.3.

Let be a set of rules, an argument structure is defined as follows:

1. if is a base fact, then .

2. , if is a subtree of , then (T is closed)

3. if is formed from by a monotonic rule, then (T is monotonically closed).

4. , does not contain arguments for both and (T is consistent).

For argument structures, a notion of completeness is defined with respect to a formula as follows.

Definition 3.4.

An argument structure is complete with respect to if contains an argument for either or .

Finally, the belief space of an agent is defined as the set of formulas supported by an argument structure.

Definition 3.5.

The set of formulas supported by an argument structure , such that is supported by .

The resulting framework made up of the inference rule, arguments, and argument structures is referred to as an argument system.

Example 3.1.

This example is due to [20]. Let be the following set of rules:

r1: true penguin(A) penguin(A)→bird(A) bird(A),¬abnormal(bird(A))→flies(A) penguin(A),¬abnormal(penguin(A))→¬flies(A) penguin(A)→abnormal(bird(A)) true⇒¬abnormal(penguin(A)) true⇒¬abnormal(bird(A))

There are 8 possible arguments in :

p1: true penguin(A) p2r3−→bird(A) p2r6−→abnormal(bird(A)) p1\xRightarrowr7¬abnormal(penguin(A)) p1\xRightarrowr8¬abnormal(bird(A)) p3,p6r4−→flies(A) p2,p5r5−→¬flies(A)

Further, there are two possible argument structures:

T_1: {p1,p2,p3,p4} {p1,p2,p3,p4,p5,p8}

Only is complete with respect to both and .∎

4 Argument Systems in LogAG

In this section, we show how to encode argument systems in theories, and prove that the graded consequence relation can capture the notions of argument structures and supported propositions by an argument structure. We start by presenting a mapping function from the inference rules of argument systems to propositional terms.

Definition 4.1.

Let the mapping from a set of inference rules of an argument system to a set of propositional terms be defined as follows.

1. If is a base fact, .

2. If is a monotonic rule, .

3. If is a non-monotonic rule, .

Whenever is a set of rules, .

While the mapping function maps monotonic and non-monotonic rules to similar propositional terms, the corresponding mappings will be treated differently when the corresponding theory is constructed as will be shown below.

The following definition describes how to use the mapping function to construct theories capable of capturing argument structures and supported propositions in argument structures.

In the sequel, let the function mapping a rule to a term denoting a grading proposition be defined as follows.

 chain(ϕ,d)={G(ϕ,1)if d=1G(chain(ϕ,d−1),1)otherwise.

Further, let be the set of non-empty subsets of a set . An indexing of is a bijection .

Definition 4.2.

Let be a set of rules of an argument system with and being the sets of monotonic and non-monotonic rules therein respectively, and let be an indexing of . The -translation of to a theory, referred to as , is the union of a monotonic subtheory , and a non-monotonic subtheory . is the smallest set satisfying the following:

1. for all base facts , .

2. for all monotonic rules , .

The non-monotonic subtheory is the smallest set satisfying the following:

1. for all and , .

2. for all and , and .

Example 4.1.

Consider the set of rules of an argument system in Example 3.1. The monotonic subtheory of the corresponding translation is made of the following propositional terms:

t1: true penguin(A) penguin(A)⊃bird(A) bird(A)∧¬abnormal(bird(A))⊃flies(A) penguin(A)∧¬abnormal(penguin(A))⊃¬fli