Active logic semantics for a single agent in a static world

Abstract

For some time we have been developing, and have had significant practical success with, a time-sensitive, contradiction-tolerant logical reasoning engine called the active logic machine (ALMA). The current paper details a semantics for a general version of the underlying logical formalism, active logic. Central to active logic are special rules controlling the inheritance of beliefs in general (and of beliefs about the current time in particular), very tight controls on what can be derived from direct contradictions (P&¬P), and mechanisms allowing an agent to represent and reason about its own beliefs and past reasoning. Furthermore, inspired by the notion that until an agent notices that a set of beliefs is contradictory, that set seems consistent (and the agent therefore reasons with it as if it were consistent), we introduce an “apperception function” that represents an agent's limited awareness of its own beliefs, and serves to modify inconsistent belief sets so as to yield consistent sets. Using these ideas, we introduce a new definition of logical consequence in the context of active logic, as well as a new definition of soundness such that, when reasoning with consistent premises, all classically sound rules remain sound in our new sense. However, not everything that is classically sound remains sound in our sense, for by classical definitions, all rules with contradictory premises are vacuously sound, whereas in active logic not everything follows from a contradiction.

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Active logic semantics for a single agent in a static world
Michael L. Andersona,d,∗, Walid Gomaab,e, John Grantb,c, Don Perlisa,b
aInstitute for Advanced Computer Studies, University of Maryland, College Park, MD 20742, USA
bDepartment of Computer Science, University of Maryland, College Park, MD 20742, USA
cDepartment of Mathematics, Towson University, Towson, MD 21252, USA
dDepartment of Psychology, Franklin & Marshall College, Lancaster, PA 17604, USA
eDepartment of Computer and Systems Engineering, Alexandria University, Alexandria, Egypt
Received 7 December 2006; received in revised form 14 November 2007; accepted 16 November 2007
Abstract
For some time we have been developing, and have had significant practical success with, a time-sensitive, contradiction-tolerant
logical reasoning engine called the active logic machine (ALMA). The current paper details a semantics for a general version of the
underlying logical formalism, active logic. Central to active logic are special rules controlling the inheritance of beliefs in general
(and of beliefs about the current time in particular), very tight controls on what can be derived from direct contradictions (P&¬P),
and mechanisms allowing an agent to represent and reason about its own beliefs and past reasoning. Furthermore, inspired by the
notion that until an agent notices that a set of beliefs is contradictory, that set seems consistent (and the agent therefore reasons
with it as if it were consistent), we introduce an “apperception function” that represents an agent’s limited awareness of its own
beliefs, and serves to modify inconsistent belief sets so as to yield consistent sets. Using these ideas, we introduce a new definition
of logical consequence in the context of active logic, as well as a new definition of soundness such that, when reasoning with
consistent premises, all classically sound rules remain sound in our new sense. However, not everything that is classically sound
remains sound in our sense, for by classical definitions, all rules with contradictory premises are vacuously sound, whereas in active
logic not everything follows from a contradiction.
© 2007 Elsevier B.V. All rights reserved.
Keywords: Logic; Active logic; Nonmonotonic logic; Paraconsistent logic; Semantics; Soundness; Brittleness; Autonomous agents; Time
1. Introduction
Real agents have some important characteristics that we need to take into account when thinking about how they
might actually reason logically: (a) their reasoning takes time, meaning that agents always have only a limited, evolv-
ing awareness of the consequences of their own beliefs,1and (b) their knowledge is imperfect, meaning that some
of their beliefs will need to be modified or retracted, and they will inevitably face direct contradictions and other in-
*Corresponding author at: Department of Psychology, Franklin & Marshall College, P.O. Box 3003, Lancaster, PA 17604-3003, USA.
E-mail addresses: michael.anderson@fandm.edu (M.L. Anderson), wgomaa@alex.edu.eg (W. Gomaa), jgrant@towson.edu (J. Grant),
perlis@cs.umd.edu (D. Perlis).
1Levesque’s distinction between explicit and implicit beliefs [29] points to this same issue; however, our approach is precisely to model the
evolving awareness itself, rather than trying to model the full set of (implicit) consequences of a given belief set.
0004-3702/$ – see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.artint.2007.11.005

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consistencies. Indeed, real agents will not only often find their beliefs contradicted by experience, but will sometimes
find that their beliefs have been internally inconsistent for some time, although they are only now in a position to
notice this inconsistency, having derived a certain set of consequences that makes it apparent. The challenge from the
standpoint of classical logical formalisms is that, if an agent’s knowledge base can be inconsistent, then according to
classical logic, it is permissible to derive any formula from it.
This fact about classical logics is commonly known by the Latin phrase ex contradictione quodlibet: from a con-
tradiction everything follows. However, Graham Priest has coined the somewhat more vivid term explosive logics: a
logic is explosive iff for all formulas A and B, (A&¬A) |= B. Priest defines a paraconsistent logic precisely as one
which is not explosive [40–42]. Now, clearly real agents cannot tolerate the promiscuity of belief resulting from ex-
plosive logics, and must somehow maintain control over their reasoning, watching for and dealing with contradictions
as they arise. The reasoning of real agents, that is, must be paraconsistent. But what sort of paraconsistent logic might
agents usefully employ, what methods might agents use to control inference and deal with contradictions, and how
can these logics (and methods) be modeled in terms of truth and consequence in structures?
In the current paper we are primarily interested in the last of these questions. For some time we have been devel-
oping, and have had significant practical success with a time-sensitive, contradiction-tolerant logical reasoning engine
called the active logic machine (ALMA) [46]. Because ALMA was designed with the above challenges in mind,
its underlying formalism, active logic [17,18,33,34], includes special rules controlling the inheritance of beliefs in
general (and of beliefs about the current time in particular), very tight controls on what can be derived from direct
contradictions (P&¬P), and mechanisms allowing an agent to represent and reason about its own beliefs and past
reasoning.
Here we offer a semantics for a general version of active logic. We hope and expect it will be of interest as a specific
model of formal reasoning for real-world agents that have to face both the relentlessness of time, and the inevitability
of contradictions.
In Sections 2–6 we will introduce the formal semantics for active logic, discuss a new definition of the conse-
quence relation, and give examples of sound and unsound active logic inferences. This will be followed by some more
informal discussion of the various properties of active logic (Section 7), a comparison of active logic with related
approaches (Section 8), and a discussion of the practical issues involved with the use of active logic in real-world
agents (Sections 9 and 10).
2. A semantics for real-world reasoning
In this section we propose a semantics for a time-sensitive, contradiction-tolerant reasoning formalism, incorporat-
ing the basic features of active logic.
2.1. Starting assumptions
In order to make the problem tractable for our first specification of the semantics, we will work under the following
assumptions concerning the agent, the world (i.e., everything apart from the agent), and their interactions:
• There is only one agent a.
• The agent starts its life at time t = 0 (t ∈ N) and runs indefinitely.
• The world is stationary for t ? 0. Thus, changes occur only in the beliefs of the agent a.
Given these assumptions, there is one and only one true complete theory of the world; however, given that the
agent’s beliefs evolve over time, there is a different true complete theory of the agent for each time t.
2.2. The language L
In order to express theories about such an agent-and-world, we define a sorted first-order language L. We define
it in two parts: the language Lw, a propositional language in which will be expressed facts about the world, and the
language La, a first-order language used to express facts about the agent, including the agent’s beliefs, for instance that
the agent’s time is now t, that the agent believes P, or that the agent discovered a contradiction in its beliefs at a given

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time. We write SnKto mean the set of sentences of any language K. We are using the complete set of connectives
{¬,→} from whichotherconnectives,suchas ∧, and ∨,can bederived.We assumethat doublenegationsare removed
from formulas. For the sentence symbols the subscripts are used to indicate different propositional sentences and, for
a fixed subscript, the superscripts are used to indicate different apperceptions (see Section 4) of the agent of the same
proposition. The superscript 0 is used for the original sentence symbol (without superscript).
Definition 1. Let Lwbe a propositional language consisting of the following symbols:
• a set S of sentence symbols (propositional or sentential variables) S = {Sj
numbers)
• the propositional connectives ¬ and →
• left and right parentheses ( and )
i: i,j ∈ N} (N is the set of natural
SnLwis the set of sentences of Lwformed in the usual way. These represent the propositional beliefs of the agent
about the world. For instance S0
1might mean “John is happy”. For later use we assume there is a fixed lexicographic
ordering for the sentences in SnLw.
Definition 2. Let σ,θ ∈ SnLw. We say that {σ,θ} is a direct contradiction if one of the following holds: either θ is the
formula σ preceded by a negation, or σ is the formula θ preceded by a negation, that is θ = ¬σ or σ = ¬θ.
Before giving the definition of the language La, we remark the following:
i. In its current version Lais a restricted form of first-order logic that is essentially propositional. In future work we
intend to extend it to the full power of first-order logic.
ii. Lacontains a belief predicate that captures the fact that the agent believed a certain proposition at some time t.
We allow for sentences of the form: at time s the agent believed that she believed that she .... To allow for this
indefinite (however finite) nesting, the definition of Lahas to be inductive where at stage n+1 all sentences from
the previous levels are captured for the belief predicate.
Definition 3. The language Lais a sorted restricted version of first-order logic having three sorts:
• S1is used to represent SnLw.
• S2is used to represent time.
• S3is used to represent SnLat its various stages of construction as shown below.
Lawill be defined as the union of a sequence of languages {Ln}n∈Nwhich are defined as follows:
• n = 0: L0is a restricted first-order sorted language that does not contain variables or quantifiers, and consists of
the following symbols:
– the propositional connective ¬
– a set of constant symbols C = {i: i ∈ N} of sort S2(this represents the time indices)
– a set of constant symbols D = {σ: σ ∈ SnLw}, each is of sort S1(here for simplicity, the constant symbols used
to represent SnLware the sentences themselves)
– a set of constant symbols E0= {θ: θ ∈ SnLw}, each is of sort S3(again for simplicity the sentences themselves
are used as constant symbols)
– the unary predicate symbol Now of sort S2
– the binary predicate symbol Contra of sort (S1×S2).
• n ? 1: Assume that Lmhas already been defined for all 0 ? m < n. Lnis a restricted first-order sorted language
that does not contain variables or quantifiers. In addition to the symbols of Ln−1, it contains a set of constant
symbols En= {θ: θ ∈ SnLn−1} of sort S3. Also, L1(and hence all Li, 1 ? i) contains a binary predicate symbol
Bel of sort (S3×S2).

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Let E =?
The sort S2stands for time. In La, Now is used to express the agent’s time, Contra is used to indicate the existence
of a direct contradiction in its beliefs, and Bel expresses the fact that the agent had a particular belief at a given
time. We use these predicate symbols because they are crucial for active logic. The semantics for these predicates
will be defined formally in Definition 8. Note that La contains only the connective ¬; hence statements such as
Bel(θ,t) → Bel(θ,t +1) are not in the language.
However, we do not specify one specific set of active logic rules. This means that the semantics we will specify is
applicable to a class of active logics with different rules that share a few common features.
i∈NEi. Let the language La=?
n∈NLnso SnLa=?
n∈NSnLn. C,D,E are sets of constant symbols.
All of these constant symbols are terms in the language La.
Definition 4. Let L = La∪Lw, in the sense that SnL= SnLa∪SnLw.
Definition 5. Let the agent’s knowledge base at time t, KBt, be a finite set of sentences from L, that is, KBt⊆ SnL. In
the case of KB0the only formulas of SnLwwe allow are those whose superscripts are all 0.
We can imagine KB0containing any sentences from L with which a system designer would equip an agent. KB0
may or may not be consistent (no system designer is perfect!). After time 0, each KBt can be different from KB0
because of inference, observation, forgetting, and the like. Also, although all the sentences about the world initially
have superscript 0 for the sentence symbols, as we will see later the agent may assign different superscripts to the
sentence symbols thereby changing a possibly inconsistent set to one that is consistent.
2.3. The semantics of Lw
In the following several definitions, we define the semantics of the formalism given above, in the standard way.
Definition 6. An Lw-truth assignment is a function h : S → {T,F} defined over the set S of sentence symbols in Lw.
Definition 7. An Lw interpretation h (we keep the same notation for this induced interpretation) is a function
h:SnLw→ {T,F} over SnLwthat extends an Lw-truth assignment h as follows:
h(¬ϕ) = T ⇐⇒ h(ϕ) = F
h(ϕ → ψ) = F ⇐⇒ (h(ϕ) = T and h(ψ) = F)
We also stipulate a standard definition of consistency for Lw: a set of Lwsentences is consistent iff there is some
interpretation h in which all the sentences are true. Notationally we write the usual h |= Σ, to mean all the sentences
of Σ are assigned T by h.
We call Wt the set of all L-expressible facts about the external world. Thus, Wt is consistent at every time t.
This means that for every t there exists an Lw-truth assignment function ht, such that ht|= Wt. We also call the
induced interpretation ht, keeping the same notation. This result does not depend upon the assumption that the world
is stationary. In a stationary world a single htwill work for all t; in a changing world there may be a different ht, but
there will still be some ht, for each t. For a brief discussion of how we intend to approach the issue of a changing
world in future work, see Section 11.
This does not mean that the agent’s beliefs about the world are all true, consistent and constant (indeed, we expect
they will contain contradictions and change over time), only that there is some set of true and consistent sentences
that describe the world for every t. We’ll detail later how to interpret and model the agent’s world-knowledge (this
being the crux of the issue when dealing with inconsistency). First, however, we turn to a model of the agent’s meta-
knowledge.
3. A model of the agent’s Labeliefs
First of all it is important to note that, even in the case where the agent’s beliefs are incomplete, incorrect, or
inconsistent, there is always a complete and consistent theory of those beliefs at the meta level, and this theory can be

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expressed using the language La. For instance, if the agent believes both that John is happy (S0
happy (¬S0
not happy” (Bel(¬S0
Now we define an interpretation that models the theory about the agent at the meta level. In what follows, Σ is to
be understood formally as any set of sentences from L; typically we will assume it to be some subset of the agent’s
knowledge base, combining beliefs about the world and about the agent, at some time t. Our point of view is that
at any time t the agent may deduce new beliefs from its knowledge base at time t − 1, may add new sentences, for
example from observations, or delete some sentences.
The following definition consists of ten bullet points: the first identifies the domain, the next three provide the
interpretations for the three sorts; the following three provide the interpretations for all the constant symbols; the last
three provide the interpretations for the three predicate symbols. Now keeps track of the time, and indicates the current
time of the agent’s internal clock. Contra indicates the existence of a direct contradiction in Σ at some time s ? t. Bel
has the rough meaning “believes that”, and states that a given sentence from L was in the agent’s KB at some time
s ? t. We define the interpretation HΣ
1) and that John is not
1), the two sentences “the agent believes that John is happy” (Bel(S0
1)) can both be true at the same time.
1)) and “the agent believes that John is
t+1(at time t +1 based on Σ) modeling facts about the agent as follows.
Definition 8. HΣ
t+1is defined as the following interpretation:
• Domain(HΣ
• HΣ
• HΣ
• HΣ
• ∀n ∈ C: HΣ
• ∀σ ∈ D: HΣ
• ∀θ ∈ E: HΣ
• The predicate symbol Now has the following semantics: HΣ
otherwise HΣ
t+1|= ¬Now(s).
• The predicate symbol Contra has the following semantics: HΣ
Contra(σ,s) ∈ Σ or s = t and ∃σ,¬σ ∈ Σ; otherwise HΣ
• The predicate symbol Bel has the following semantics: HΣ
s = t and θ ∈ Σ; otherwise HΣ
t+1) = N ∪SnL.
t+1(S1) = SnLw(the set of propositions about the world).
t+1(S2) = N (all non-negative integers).
t+1(S3) = SnL(the set of sentences representing the agent’s knowledge base).
t+1(n) = n.
t+1(σ) = σ.
t+1(θ) = θ.
t+1|= Now(s) ⇐⇒ s = t + 1 and Now(t) ∈ Σ;
t+1|= Contra(σ,s) ⇐⇒ either s < t and
t+1|= ¬Contra(σ,s).
t+1|= Bel(θ,s) ⇐⇒ either s < t and Bel(θ,s) ∈ Σ or
t+1|= ¬Bel(θ,s).
4. A model of the agent’s Lwbeliefs
Now we turn to the challenging problem of how to model, at the object level, the agent’s beliefs about the world,
given that these beliefs are not just evolving from moment to moment, but that at any given time, they may be
inconsistent. Our scenario is as follows. At time 0 the agent has a finite set of initial beliefs, KB0, about the world. All
the sentence symbols have superscript 0. Then the agent starts to reason about the world using rules of active logic.
This is where the agent may assign, via its apperception function, non-zero superscripts to some sentence symbols to
avoid inconsistency. The agent may also obtain additional information about the world over time through other means,
such as observations.
We will tackle this problem initially in three steps. First, we define a weak notion of consistency allowing for
inconsistency in the agent’s knowledge about the world; second, we will define a class of “apperception functions”
intended to capture the intuition that an inconsistent KB will not necessarily seem inconsistent to the agent; and finally,
we will show that there is some apperception function that, when applied to a given set of sentences, always produces
a consistent set. Having shown this, we will proceed in the following sections to define a viable notion of active
consequence, discuss the relation of this notion of consequence to the classical notion of logical consequence, and
prove the soundness of some of the central inference rules of active logic.
Recalling that Σ need not be consistent concerning facts about the world we define a weak version of consistency.
Definition 9. A set of sentences Σ ⊆ SnLis said to be Laconsistent iff Σ ∩SnLais classically consistent.