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Sglavo",authors:[{id:"17426",title:"Prof.",name:"Vincenzo Maria",middleName:null,surname:"Sglavo",fullName:"Vincenzo Maria Sglavo",slug:"vincenzo-maria-sglavo"}]},{id:"40707",title:"Unheated and Heated Batch Methods in Ion Exchange of Clinoptilolite",slug:"unheated-and-heated-batch-methods-in-ion-exchange-of-clinoptilolite",signatures:"Tevfik Ünaldı and Selahattin Kadir",authors:[{id:"143090",title:"Dr",name:"TEVFİK",middleName:null,surname:"Ünaldı",fullName:"TEVFİK Ünaldı",slug:"tevfik-unaldi"},{id:"164885",title:"Dr.",name:"Selahattin",middleName:null,surname:"Kadir",fullName:"Selahattin Kadir",slug:"selahattin-kadir"}]},{id:"40709",title:"The Role of Ion Exchange Chromatography in Purification and Characterization of Molecules",slug:"the-role-of-ion-exchange-chromatography-in-purification-and-characterization-of-molecules",signatures:"Hidayat Ullah Khan",authors:[{id:"140538",title:"Dr.",name:"Hidayat",middleName:null,surname:"Khan",fullName:"Hidayat Khan",slug:"hidayat-khan"}]},{id:"40745",title:"Nitrogen Isotope Separation by Ion Exchange Chromatography",slug:"nitrogen-isotope-separation-by-ion-exchange-chromatography",signatures:"Xingcheng Ding and Xunyue Liu",authors:[{id:"139864",title:"Prof.",name:"Xingcheng",middleName:null,surname:"Ding",fullName:"Xingcheng Ding",slug:"xingcheng-ding"},{id:"139865",title:"Dr.",name:"Xunyue",middleName:null,surname:"Liu",fullName:"Xunyue Liu",slug:"xunyue-liu"}]}]}]},onlineFirst:{chapter:{type:"chapter",id:"75035",title:"Towards a Fuzzy Context Logic",doi:"10.5772/intechopen.95624",slug:"towards-a-fuzzy-context-logic",body:'Fuzzy logic has been employed successfully in intelligent systems, sensor-actuator systems, expert systems, and machine learning techniques for more than 50 years [1]. Being a tool for inference at both the logical and the sensor-actuator systems level its use for reliable and explainable autonomous systems has become a focus of recent research [2, 3, 4, 5]. One key building block for this has been a growing understanding of fuzzy logic semantics over the past 20 years [6] and the position this family of logics assumes within the field of logics in general. In particular, the connection to residuated lattices plays an important role for novel perspectives [7, 8]. One such new perspective is the connection to context logic, which is developed in this chapter.
Context logic was introduced in [9, 10, 11] as a logic for representing context-dependency and context phenomena in pervasive computing systems. Recent developments in context logic focus on a logical actuator control mechanism [12, 13, 14]. This chapter presents the logic with a fuzzy logic lattice semantics highlighting the close relation between the two formalisms and the close relation between context logic and the sensory and machine learning components of intelligent sensor actuator systems (ISAS), such as robotics and autonomous vehicles. We show that context logic can be understood as a fuzzy logic since it can be given an algebraic semantics like that of fuzzy logic as based upon lattice structures.
We briefly review the basics of how fuzzy logic handles quantitative information and contrast this with the approach chosen in context logic. Here, it may appear we go into basic aspects at a greater depth than what may seem necessary. However, to bring the two logics together, establishing the common ground conceptually is a critical first step.
Fuzzy logic [15] was developed as a linguistically motivated logic that was to be more akin to how human beings reason with uncertain information and how experts analyze alternatives and act upon them [16]. Its main cognitive motivation was that human beings are able to relay, for instance, control information without the use of numerical values. In fact, human language outside scientific and technical contexts rarely employs quantities to express relations regarding a scale, amounts, or probabilities. We prefer to say, e.g., rarely rather than giving an estimate about a concrete percentage, or give a color term, such as yellow, instead of providing RGB values and we reason with such information. We “compute with words” [17]. One reason for this is the inherent uncertainty of perceptual or sensory information and the presence of intersubjective differences. Rules we receive or provide verbally benefit from this vagueness, as they have a wide applicability, allow a concise formulation, and allow for intersubjective differences: two people may disagree whether a certain fruit is yellow or rather a light orange, but they will agree that to at least some degree, something that has a light orange color is yellow. A rule given by an expert to a novice, such as “if a fruit is yellow, then it is ripe,” is easy to understand for a human being, and accordingly fuzzy expert systems, fuzzy sensor-actuator systems, and the output of some fuzzy learning systems, can be understood and verified by human beings better than purely numerical systems that operate with numerical equations.
In natural language, human beings convey information about continuous sensory domains, such as color or height, by use of adjectives. The phenomena of vagueness, uncertainty, and context-dependency are the main challenges for formalization from a linguistic point of view [18]. Adjectives can be used in several different ways. The main categories are:
Positive: Anne is tall (for her age).
Comparative: Anne is taller than Betty.
Equative: Ann is as tall as Betty.
Superlative: Ann is the tallest (girl on the team).
While the comparative and equative use are most easily mapped to a corresponding ordering and equivalence relation for the dimension in question (here: height), the positive and superlative can change their applicability depending on context. If we talk about children, 1.50 m (5 ft) may be tall. If we talk about the average European female adult, this is comparatively small. Likewise, the superlative changes with the context: Ann may be the smallest person in the room and still be called the tallest while the current topic is her team. Context logic is interesting from a cognitive science perspective as it enables the modeling of such influence of the context.
From a cognitive science point of view, fuzzy logic is an interesting formalism as it addresses issues of vagueness and uncertainty that appear especially in the semantics of adjectives. But it is also one of only few approaches bridging logical reasoning and machine learning [19].
Fuzzy logic goes beyond multi-valued logics [20] by proposing semantics for approximate reasoning. In particular, [15, p.424] proposes to “[view] the process of inference […] as the solution of a system of relational assignment equations.” This emphasizes the connection to both sensor-actuator systems and classical methods of system modeling and evaluation with recent advances reaching from explainable machine learning [5] to advanced uncertainty mechanisms for ontology design [21]. Combining the two languages promises to make the full expressiveness of natural language adjectives available for modeling, reasoning, and explanation in ISAS design.
While the linguistic background facilitates usability of fuzzy logic, it is easier to see logical connections with respect to a more restrictive and conventional logic syntax. We therefore use a simple propositional logical language as a classical background language in this chapter. We adopt the following syntax for the set of all formulae
Using this syntax, we can formalize a proposition similar to the above example as:
We can use the usual semantics for predicate logics to interpret this sentence based on a structure
A fundamental point where fuzzy logic differs from classical predicate logic is in the interpretation of the predicates and predication: classical logic considers
To evaluate complex formulae, fuzzy logic requires extended semantics for the propositional connectives that can handle arbitrary values in
If the semantics for ∧ are based on a t-norm, this guarantees that important semantic properties of the classical conjunction are retained. A t-norm
Examples are the minimum t-norm (5), used in Gödel logics, and the product t-norm (6), used in probability theory:
The corresponding t-conorms, denoted by the symbol
The corresponding s-norms for the above example t-norms are then
There are several ways to interpret the implication and different approaches are suitable for different purposes (cf. [22], for a detailed overlook and comparison). As with other operators, fuzzy implication should be conservative for values in
The relation between the residual and the t-norm/s-norm are covered by two additional axioms, continuity (11) and pre-linearity (12):
For the above two t-norms
The most widely used examples of functions
The intuition behind this is to map elements
Partial orders and corresponding lattice structures are at the heart of the semantics for context logic, and the two languages can on this basis be combined in a natural manner.
We now specify the context logic language and describe a semantics similarly in terms of a predicate logical language, which in turn can be related to lattice structures and thus fuzzy logical semantics.
Context logic has only one type of basic entity, context variables, and a single partial order relation
The language provides three term operators
Since any pre-order can be expressed as a sub-relation of a partial order relation, and be extended to a partial order relation over its equivalence classes, the single sub-context relation together with the
This and the following examples feature one simple spatial sub-context and one temporal sub-context relation. We can in the same manner however express, for instance, directional relations [25], temporal ordering relations (bi-directionally branching), and class hierarchies [9]. Ordering relations between thematic values, such as expressed by the comparative use of adjectives (Section 2) can also be added in the same way. The main purpose of the language is to facilitate expressing the common partial order core of all these theories, including the tractable fragments of these theories in a unified syntax.
A syntactic shorthand reflects – linguistically speaking – a topicalized adverbial position:
Spatially, London is a sub-context of England. Temporally, March 2017 is a sub-context of the year 2017. For entities such as cities or months, this may seem redundant. But contexts, such as a birthday party, which have both temporal and spatial extent can thus be located temporally within one context and spatially within another:
We can also reflect that speakers may choose to topicalize the other way around [26], as the last two sentences are logically equivalent to the following:
or, leveraging the propositional second layer,
where, for any propositional junctor
Regarding John’s birthday party: the location is in London, the time is in March 2017. Moreover, we can allow contexts to be stacked or combined, in order to express more complex contextualization:
Similarly to how we would express conflicting opinions in natural language, we can equivalently state:
Regarding John’s birthday party and the time, Mary says in March 2017 and Tom says in August 2017. Context logic thus allows to reflect colloquial contextualizations well, but also to represent conflicting information.
Context logic thus employs two syntactic layers: the term layer with the term operators
Context formulae
We further define:
Different variant semantics have been proposed [10, 11, 26]. The different approaches slightly differ in the resulting semantics, but all three employ a lattice structure for specifying the meanings of context terms, assigning a partial order to give a semantics to
We note that although we introduce new variables
Consequently, the fragment of predicate logic required in application of
If x, y are tuples of variables, α(x, y) is an atomic formula, ψ(x,y) is in GF, and free (ψ ⊆ free (α) = {x,y}, where free (ϕ) is the set of the free variables of ϕ, then the formulae
In order to obtain the reasoning capabilities, however, we would need to add pre-order axioms for
and we see that transitivity (13) cannot be axiomatized in the two-variable fragment, as it requires three variables. Fortunately, [28, 29] have shown that for GF2 + PG – the guarded fragment limited to two variables and a single binary pre-order that can only appear in the guard – is in 2-EXPTIME. Moreover, this result is a loose upper bound, since the language under inspection here can be expressed using the transitive binary relation
In addition to the pre-order axioms, we can also add a localized guarded variant of the so-called weak supplementation principle [30, Ch. 3] for
The principle says that, if for any
In order to do this, however, we have to employ
Intuitively, the meaning of
The complement
A crucial consequence of adopting weak supplementation (35) is (2). It says that if all parts
Proof (⫤): this holds immediately with the reflexivity (34) and transitivity (33) of
Proof (
It can be shown (Section 4.4) that the definition of
This section shows context logic as specified above is a two-layered language with a generalized
To see that the context terms
Proof (3): if every
The generalized De Morgan law connects t-norms with s-norms (7). It follows for the translations of
The residual can then be derived from its characterization:
The operation
has the required property
We prove that for any
and the term
Proof: assume the antecedent is true, then because of transitivity of
This result indicates that, at least with respect to the supplementation property expressed through (35),
We are thus justified to say that context logic terms have a generalized
We obtain: a t-norm-based classical semantics for context logic is a structure
With
It only remains to show that the context term operators indeed support the lattice requirements.
For the purpose of completeness, the proofs are listed here in detail. However, the results are part of basic, fundamental lattice theory and no novelty is claimed.
We prove that
These properties hold, since
We show the translations:
We can see that all translations of properties are tautologies and follow directly from the properties of
The property (47) holds because
We can now prove the semilattice laws for
When we translate idempotency (48):
we see that the translation of
The other laws follow in a similar manner. We show associativity (49):
By proving the following for any
We prove in two steps.
Proof (
Proof (⫤): assume we have for each
Applying this result twice via the associativity and commutativity of
Theorem 14 holds immediately given the definition of the translation for
Proving the neutral element property (51) requires (35).
The proof follows immediately by (2).
In summary, we needed (35) for proving idempotency (48) and the neutral element (51). Associativity (49) and commutativity (50) were proven without using (35).
We have thus shown that
For (52):
we show that for any
Proof (⫤): this holds because of transitivity (33) and reflexivity (34) of
Proof (
We prove (53):
by showing for any
Proof (
The relation between the residual and the t-norm were covered by two additional axioms above: continuity (11) and pre-linearity (12):
We prove continuity (54) by translation using
Proof (
Proof (⫤): assume the antecedent
We prove pre-linearity (55):
by showing for any
Proof: we obtain for the antecedent:
Since this holds for all
We rename the variables to better show the structure:
and by (2):
We now know that
which entails the consequent.
The mereologically interested reader may notice that adding even the weakened variant of the weak supplementation principle is sufficient to collapse context logic term structures to a single level by (2). The reason for this is that the weak supplementation principle considerably strengthens the expressiveness of negation, which given the principle always ensures the existence of a fully negative individual. This is the case, although our system mereologically speaking is an MM system, i.e., supports M1-M4 [30] only, with M4 acting as an axiom schema.
We may note also, that we need not ensure product (M5) or sum (M6) to exists, nor do we need or posit a universal
To make the discussion more concrete, we briefly sketch a set-theoretical interpretation. An example of a suitable model is the set-theoretic lattice, assuming the set of all subsets of a base universe as the universe for the interpretation of the translation
We can show that, if the canonical interpretation
The pre-order axioms for
Proof: assume a set
We prove that
The case of (20) is immediately clear. For (21), we look at the definition of
Proof (
Proof (⫤): assume that for every
For (22), we similarly look at the definition of
Proof (
Proof (⫤): as in the proof for
We have thus seen that the set-thoretical standard model is a concrete example of a structure for interpreting context terms and formulae.
The key to the proposed fuzzy context logic is to additionally provide a fuzzy interpretation for the atomic formulae, via the symbol
The fuzzy semantics is defined by two lattices: a bounded lattice
with the
We will need to characterize a fuzzified variant of
On this basis, the interpretation of formulae can then follow one of the standard models of fuzzy logic in
The key is to provide a function
What is a good choice depends on both
Two regions are adjacent iff they overlap but only in a part smaller than grain-size.
Two regions are spatially indistinguishable iff they differ only in a part smaller than grain-size.
Two regions relevantly overlap iff they overlap in a part larger than grain-size and differ in a part larger than grain-size.
We can generalize this notion using a
Generally, we can employ a granularity function
We obtain a fully specified family of fuzzy context logics. Note that with
For
Among the potential applications, a two-layered fuzzy logic can help to reason about fuzzy logic systems. The base logic being decidable for the classical semantics, we can, at least for the classical case, make absolute guarantees for a given system. We can prove whether a given fuzzy system, e.g., the output of a machine learning mechanism, such as an ANFIS, together with a description of possible situations in the domain and desirable properties yields a tautology, thus proving that the system has the desirable properties under all possible circumstances. If we are interested in gaining an understanding of systems that are not tautological in this sense, so as to obtain, e.g., degrees of possibility of failure under certain circumstances, more advanced fuzzy proof methods are required.
This chapter illustrated that the lwo-layered logic context logic and fuzzy logic can be combined in a meaningful way. We first mapped both logics to a predicate logical background language, so as to highllight their commonalities and differences and to obtain a background compatible with both. In both cases, we discussed a common set-theoretical model that can be used to interpret the background language. We formally proved that the lattice-based generalized
The formula-layer of context logic could then additionally be imbued with a
This work was financially supported by the Hanse-Wissenschaftskolleg, Delmenhorst, Germany; and received infrastructural support from the University of Bremen, Germany.
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\\n\\n3.1. ERRATUM
\\n\\nAn Erratum will be issued by the Academic Editor when it is determined that a mistake in a Chapter originates from the production process handled by the publisher.
\\n\\nA published Erratum will adhere to the Retraction Notice publishing guidelines outlined above.
\\n\\n3.2. CORRIGENDUM
\\n\\nA Corrigendum will be issued by the Academic Editor when it is determined that a mistake in a Chapter is a result of an Author’s miscalculation or oversight. A published Corrigendum will adhere to the Retraction Notice publishing guidelines outlined above.
\\n\\n4. FINAL REMARKS
\\n\\nIntechOpen wishes to emphasize that the final decision on whether a Retraction, Statement of Concern, or a Correction will be issued rests with the Academic Editor. The publisher is obliged to act upon any reports of scientific misconduct in its publications and to make a reasonable effort to facilitate any subsequent investigation of such claims.
\\n\\nIn the case of Retraction or removal of the Work, the publisher will be under no obligation to refund the APC.
\\n\\nThe general principles set out above apply to Retractions and Corrections issued in all IntechOpen publications.
\\n\\nAny suggestions or comments on this Policy are welcome and may be sent to permissions@intechopen.com.
\\n\\nPolicy last updated: 2017-09-11
\\n"}]'},components:[{type:"htmlEditorComponent",content:'IntechOpen’s Retraction and Correction Policy has been developed in accordance with the Committee on Publication Ethics (COPE) publication guidelines relating to scientific misconduct and research ethics:
\n\n1. RETRACTIONS
\n\nA Retraction of a Chapter will be issued by the Academic Editor, either following an Author’s request to do so or when there is a 3rd party report of scientific misconduct. Upon receipt of a report by a 3rd party, the Academic Editor will investigate any allegations of scientific misconduct, working in cooperation with the Author(s) and their institution(s).
\n\nA formal Retraction will be issued when there is clear and conclusive evidence of any of the following:
\n\nPublishing of a Retraction Notice will adhere to the following guidelines:
\n\n1.2. REMOVALS AND CANCELLATIONS
\n\n2. STATEMENTS OF CONCERN
\n\nA Statement of Concern detailing alleged misconduct will be issued by the Academic Editor or publisher following a 3rd party report of scientific misconduct when:
\n\nIntechOpen believes that the number of occasions on which a Statement of Concern is issued will be very few in number. In all cases when such a decision has been taken by the Academic Editor the decision will be reviewed by another editor to whom the author can make representations.
\n\n3. CORRECTIONS
\n\nA Correction will be issued by the Academic Editor when:
\n\n3.1. ERRATUM
\n\nAn Erratum will be issued by the Academic Editor when it is determined that a mistake in a Chapter originates from the production process handled by the publisher.
\n\nA published Erratum will adhere to the Retraction Notice publishing guidelines outlined above.
\n\n3.2. CORRIGENDUM
\n\nA Corrigendum will be issued by the Academic Editor when it is determined that a mistake in a Chapter is a result of an Author’s miscalculation or oversight. A published Corrigendum will adhere to the Retraction Notice publishing guidelines outlined above.
\n\n4. FINAL REMARKS
\n\nIntechOpen wishes to emphasize that the final decision on whether a Retraction, Statement of Concern, or a Correction will be issued rests with the Academic Editor. The publisher is obliged to act upon any reports of scientific misconduct in its publications and to make a reasonable effort to facilitate any subsequent investigation of such claims.
\n\nIn the case of Retraction or removal of the Work, the publisher will be under no obligation to refund the APC.
\n\nThe general principles set out above apply to Retractions and Corrections issued in all IntechOpen publications.
\n\nAny suggestions or comments on this Policy are welcome and may be sent to permissions@intechopen.com.
\n\nPolicy last updated: 2017-09-11
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