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This looks at the Semantic Web design in the light a little reading on formal logic, of the Access Limited Logic system, in particular, and in the light of logical languages in general. A problem here is a that I am no logician, and so I am am having to step like a fascinated reporter into this world of which I do not possess intimate experience.
The Semantic Web Toolbox discusses the step from the web as being a repository of flat data without logic to a level at which it is possible to express logic. This is something which knowledge representation systems have been wary of.
The Semantic Web has a different set of goals from most systems of logic. As Crawford and Kuipers put it in [Crawf90],
[...]a knowledge representation system must have the following properties:
- It must have a reasonably compact syntax.
- It must have a well defined semantics so that one can say precisely what is being represented.
- It must have sufficient expressive power to represent human knowledge.
- It must have an efficient, powerful, and understandable reasoning mechanism
- It must be usable to build large knowledge bases.
It has proved difficult, however, to achieve the third and fourth properties simultaneously.
The semantic web goal is to be a unifying system which will (like the web for human communication) be as un-restraining as possible so that the complexity of reality can be described. Therefore item 3 becomes essential. This can be achieved by dropping 4 - or the parts of item 4 which conflict with 3, notably a single, efficient reasoning system. The idea is that, within the global semantic web, there will be a subset of the system which will be constrained in specific ways so as to achieve the tractability and efficiency which is no necessary in real applications. However, the semantic web itself will not define a reasoning engine. It will define valid operations, and will require consistency for them. On the semantic web in general, a party must be able to follow a proof of a theorem but is not expected to generate one.
(This fundamental change goals from KR systems to the semantic web is loosely analogous with the goal change from conventional hypertext systems to the original Web design dropping link consistency in favor of expressive flexibility and scalability.The latter did not prevent individual web sites from having a strict hierarchical order or matrix structure, but it did not require it of the web as a whole.)
If there is a semantic web machine, then it is a proof validator, not a theorem prover. It can't find answers, it can't even check that an answer is right, but it can follow a simple explanation that an answer is right. The Semantic Web as a source of data should be fodder for automated reasoning systems of many kinds, but it as such not a reasoning system.
Most knowledge representation systems distinguish between inference "rules" and other believed information. In some cases, this is because the rules (such as substitution in a formula) cannot be written in the language - they are defined outside the language. In fact the set of rules used by the system is often not only formally quite redundant but arbitrary. However, a universal design such as the Semantic Web must be minimalist. We will ask all logical data on the web to be expressed directly or indirectly in terms of the semantic web - a strong demand - so we cannot constrain applications any further. Different machines which use data from the web will use different algorithms, different sets of inference rules. In some cases these will be powerful AI systems and in others they will be simple document conversion systems. The essential this is that the results of either must be provably correct against the same basic minimalist rules. In fact for interchange of proof, the set of rules is an engineering choice.
There are many related ways in which subsystems can be created
For example, Access Limited Logic is restricted (as I understand it) to relations r(a,b) available when r is accessed, and uses inference rules which only chain forward along such links. There is also a "partitioning" of the Web by making partitioning the rules in order to limit complexity.
For the semantic web as a whole, then, we do require tractable
Philosophically, the semantic web produces more than a set of rules for manipulation of formulae. It defines documents on the Web has having a socially significant meaning. Therefore it is not simply sufficient to demonstrate that one can constrain the semantic web so as to make it isomorphic to a particular algebra of a given system, but one must ensure that a particular mapping is defined so that the web representation of that particular system conveys is semantics in a way that it can meaningfully be combined with the rest of the semantic web. Electronic commerce needs a solid foundation in this way, and the development of the semantic web is (in 1999) essential to provide a rigid framework in which to define electronic commerce terms, before electronic commerce expands as a mass of vaguely defined semantics and ad hoc syntax which leaves no room for automatic treatment, and in which the court of law rather than a logical derivation settles arguments.
Practically, the meaning of semantic web data is grounded in non-semantic-web applications which are interfaced to the semantic web. For example, currency transfer or ecommerce applications, which accept semantic web input, define for practical purposes what the terms in the currency transfer instrument mean.
@@I [DanC] think this section is outdated by recent thoughts  on paradox and the excluded middle
To the level of first order logic, we don't really need to pick one set of axioms in that there are equivalent choices which lead to demonstrably the same results.
(A cute one at the propositional logic level seems [Burris, p126] to be Nicod's set in which nand (in XML toolbox <not>..</not> and below [xy]) is the Sheffer (sole) connective and the only rules of inference are substitution and the modus ponens equivelent that from F and [F[G H]] one can deduce H, and the single axiom [[P[QR]][[S[SS]][[UQ][[PU][PU]]]].)
Let us assume the properties of first order logic here.
If we add anything else we have to be careful that it should either be definable in terms of the first order set or that the resulting language is a subset of a well proven logical system -- or else we have a lot of work to do in establishing a new system!
These are two goals to which we explicitly do not aspire in the Semantic Web in order to get in return expressive power. (We still require consistency!). The world is full of undecidable statements, and intractable problems. The semantic web has to give the power to express such things.
Crawford and Kuipers The same explain in the introduction their Negation in ALL paper,
"Experience with formally specified knowledge representation systems has revealed a trade-off between the expressive power of knowledge representation systems and their computational complexity. If, for example, a knowledge representation system is as expressive as first order predicate calculus, then the problem of deciding what an agent could logically deduce from its knowledge base is unsolvable"
Do we need in practice to decide what an agent could deduce from its logic base? No, not in general. The agent may have various kinds of reasoning engine, and in practice also various amounts of connectivity, storage space, access to indexes, and processing power which will determine what it will actually deduce. Knowing that a certain algorithm may be nondeterministic polynomial in the size of the entire Web may not be at all helpful, as even linear time would be quite impractical. Practical computability may be assured by topological properties of the web, or the existence of know shortcuts such as precompiled indexes and definitive exclusive lists.
Keeping a language less powerful than first order predicate calculus is quite reasonable within an application, but not for the Web.
A dream of logicians in the last century to find languages in which all sentences were either true or false, and provably so. This involved trying to restrict the language so as to avoid the possibility of (for example) self-contradictory statements which can not be categorized as a true or not true.
On the Semantic Web, this looks like a very academic problem, when in fact one anyway operates with a mass of untrustworthy data at any point, and restricts what one uses to a limited subset of the web. Clearly one must not be able to derive a self-contradictory statement, but there is no harm in the language being powerful enough to express it. Indeed, endorsement systems must give us the power to say "that statement is false" and so loops which if believed prove self-contradictory will arise by accident or design. A typical response of a system which finds a self-contradictory statement might be similar to the response to finding a contradiction, for example, to cease to trust information from the same source (or public key).
@@hmm... better section heading? maybe just quoting, or contexts? one place where we really do seem to need more than HOL is induction.
The fact that there is [Burris p___] "no good set of axioms and rules for higher order logic" is frustrating not only in that it stumps the desire to write common sense mathematically, but also because operations which seem natural for electronic commerce seem at first sight to demand higher order logic. There is also a fundamental niceness to having a system powerful enough to describe its own rules, of course, just as one expects to be able to write a compiler for a programming language in the same language (@@need to study references from Hayes, esp "Tarski's results on meta-descriptions (a consistent language can't be the same expressive power as its own metatheory), Montague's paradox (showing that even quite weak languages can't consistently describe their own semantics)". When Frege tried second-order logic, I understand, Russel showed that his logic was inconsistent. But can we make a language in which is consistent (you can't derive a contradiction from its axioms) and yet allows enough to for example:-
The sort of rule it is tempting to write is such as to allow the inference of an RDF triple from a message whose semantic content one can algebraically derive that triple.
forall message,t, r, x, y ( (signed(message,K) & derivable(t, message) & subject(t, x) & predicate(t, r) & object(t, y)) -> r(x,y) )
(where K is a specific constant public key, and t is a triple)
This breaks the boundary between the premises which deal with the mechanics of the language and the conclusion which is about the subject-matter of the language. Do we really need to do this, or can we get by with several independent levels of machinery, letting one machine prepare a "believable" message stream and parse it into a graph, and then a second machine which shares no knowledge space with the first, do the reasoning on the result? To me this seems hopeless, as one will in practice want to direct the front end's search for new documents from the needs of the reasoning by the back end. But this is all hunch.
Peregrin tries to categorize the needs for and problems with higher order logic (HOL) in [Peregrin]. His description of Henkinian Understanding of HOL in which predicates are are subclass of objects ("individuals") seems to describe my current understanding of the mapping of RDF into logic, with RDF predicates, binary relations, being subclass of RDF nodes. Certainly in RDF the "property" type can be deduced from the use of any URI as a predicate:
forall p,x,y p(x,y) -> type(p, property)
and we assume that the "assert" predicate <rdf:property> is equivalent to the predicate itself.
forall p,x,y assert(p,x,y) <--> p(x,y)
so we are moving between second-order formulation and first-order formulation.
(2000) The experience of the [PCA] work seems to demonstrate that higher order logic is a very realistic way of unifying these systems.
(2001) The treatment of contexts in [CLA] seems consistent with the design we've implemented.
It seems clear that FOL is insufficient in that some sort of induction seems necessary.
I agree with Tait (Finitism, J. of Philosophy, 1981, 78, 524-546) that PRA is THE NECESSARY AND SUFFICIENT logic for talking about logics and proofs
Robert S. Boyer, 18 Apr 93
also: pra.n3, an N3 transcription of Peter Suber, Recursive Function Theory
also: ACL2: A Precise Description of the ACL2 Logic Kaufmann and Moore 22 Apr 1998, rdf scratchpad entry 26Mar
(for another sort of induction, i.e. as opposed to deduction, see: Circumscription by McCarthy, 1980.)
I personally need to understand what the limitations are on higher order logics...and which are hard limitations and which are unresolved areas.
Was Frege's formulation of second order logic (which should be of course a formulation of common sense) buggy in that Russel found it could derive a contradiction, or are we really finding it is not possible to represent a mathematical system to follow common sense reasoning? Yes, Frege seems to have used the classical logic in which any proposition must be true or false.
When we say that there are valid sentences which cannot be derived - what exactly do we mean by "valid"? In predicate logic, validity can be defined by a truth table, ie evaluation for all truth evaluations of the variables [Burris]. In fact this method equates to a resolution by mapping algebraically into disjoint (say) normal form, and so is not an operation "outside" the language. In any logic with unbounded sets this is not practical. Typically we fall back on some logic such as common sense outside the language under discussion. In higher order logic, intuition suggests we should be able to to construct such a proof of validity in the logic itself. Goedel's incompleteness theorem specifically addresses the difference between validity and derivability, so perhaps a good exposition of that [DanC recommends The Unknownable@@]would contain the necessary distinctions.
I wonder whether the answers can be found on the web...
After a hallway chat with Peter SP-S, DMAL meeting 2001/2/14:
The paradox problem lies not simply in being able to express a paradox, but the logic being such that merely considering a paradox leads one to be able to deduce a falsehood. One can take issue then, with either the ability to express the paradox, or with the "p or not p" axiom. Logics which try to get above first order logic can be divided into two groups in this way. There are sets of logics which use sets, but limit them in some way - for example, by requiring sets to be "well formed", meaning they are either empty or contain at least one element which is disjoint with the set itself. These tricks limit the logic so that you can't in fact have the Russel paradox set (of all sets which are not members of themselves) as you exclude such things as not well-formed. As Peter admitted, on the web it seems silly to use this route when people will be expressing paradoxes.
The other route alsob has many followers. Some of them are, people say, complicated. But it seems that the path can only be in that direction. Meanwhile, back at the ranch, I notice that most of the semantic web applications I have been playing with involve rules and axioms which are not at all complete. Each application has its own sets of axioms and can be individually proved consistent (or not). So one way forward for standards would be to instantiate that, allowing each document and message on the semantic web to have a pointer to the vocabulary its uses, including its varieties of logical connectives and their assocaited axioms.
Mapping RDB and SM models - defining URIs; nulls; etc.
Pat Hayes mentioned a logic in which the law of the excluded middle does not exist - which si important as you can always considera paradox which is neither true nor false. See email 2000/09/01
I think that a more suitable 'basic' logic might be generalized horn clause logic. This is similar to the horn-clause subset of FOL which is used in logic programming, but instead of full negation (which is computationally expensive) or negation-by-failure (which is cheap but grubbily nonmonotonic), it uses an elegant intermediate idea called intuitionistic (or constructive) negation (which is like full negation but rules out proofs by contradiction: basically, it insists on a higher standard of provability than classical negation. In intuitionistic reasoning, Holmes' famous advice about eliminating the impossible does not always apply.) It is computationally tractable, completely monotonic, and has a neat, elegant semantic theory behind it. It escapes some of the limitations of Horn logic while retaining a lot of its advantages. Its been implemented and seems to have been applied to some neat applications. It might be just what you need. For details see the papers at http://citeseer.nj.nec.com/hogan98putting.htmlhttp://citeseer.nj.nec.com/hogan98putting.html, especially the 1996 "meta-programming...." and the main citation at the top. For the raw logic follow the McCarty 88 references.
quoted without permission. DanC recommends Non-Contradiction and Excluded Middle in Peter Suber's Logical Systems course notes as an explanation of intuitionistic logic. TimBL noted the same bit that I did on first reading:
In the world of metamathematics, the intuitionists are not at all exotic, despite the centrality of the PEDC (hence the PEM) to the ordinary sense of consistency. Their opponents do not scorn them as irrationalists but, if anything, pity them for the scruples that do not permit them to enjoy some "perfectly good" mathematics.
is socratic completeness, as in [Crawf90], directly relevant? hmmm@@
solutions to paradoxes around wtr in KIF: complex one in KIFv3, simpler, more limited one in later KIF spec. (KIF/RDF stuff from Jul 2000)
[PCA] Proof-Carrying Authentication. Andrew W. Appel and Edward W. Felten, 6th ACM Conference on Computer and Communications Security, November 1999. (background) Mar 2000 discovery. based on [LF]
[Burris] Stanley N. Burris, Logic for Mathematics and Computer Science, Prentice Hall 1998.
[BurrisSup]Supplementary text to the above.
[Cheng] The ERM model paper, available in [Laplante]
[ConnollySoLREK] Dan Connolly's home page for this sort of stuff. "A Study of Linguistics: Representation and Exchange of Knowledge".
ALL: Formalizing Access Limited Reasoning
J. M. Crawford email@example.com
Benjamin Kuipers firstname.lastname@example.org
Department of Computer Sciences
The University of Texas At Austin
Austin, Texas 78712
25 May 1990
Access-Limited Logic (ALL) is a language for knowledge representation which formalizes the access limitations inherent in a network-structured knowledge base. Where a classical deductive method or logic programming language would retrieve all assertions that satisfy a given pattern, an access-limited logic retrieves all assertions reachable by following an available access path. The complexity of inference is thus independent of the size of the knowledge-base and depends only on its local connectivity. Access-Limited Logic, though incomplete, still has a well defined semantics and a weakened form of completeness, Socratic Completeness, which guarantees that for any query which is a logical consequence of the knowledge-base, there exists a series of queries after which the original query will succeed. This chapter presents an overview of ALL, and sketches the proofs of its Socratic Completeness and polynomial time complexity.
[Crawf91] J. M. Crawford and B.J. Kuipers, "Negation and proof by Contradiction in Access Limited Logic", in Proceedings of the Ninth National Conference on Artificial Intelligence (AAA1-91), Cambridge MA, 1991
[Date] An Introduction to Database Systems, 6th Edn, Adison-Wesley, 1995
[Davis] Randall Davis and Howard Schrobe, "6.871: Knowledge Based Application Systems" course supporting information, 1999
[CLA] Contexts: A Formalization and Some Applications, Ramanathan V. Guha, 1991 Stanford PhD thesis. see also: ContextLogic.lsl, May 2001 transcription into larch.
[HOLintro]M. J. C. Gordon and T. F. Melham, "Introduction to HOL A theorem proving environment for higher order logic", Cambridge University Press, 1993 ISBN 0 521 44189 7
[Laplante] Phillip Laplante (Ed.), "Great Papers in Computer Science", West, 1996, ISBN:0-314-06365-X
[Marchiori98] M. Marchiori, Metalog paper at QL98
[Peregrin] Jaroslav Peregrin, "What Does One Need, When She Needs "Higher-Order Logic?" Proceedings of LOGICA'96, FILOSOFIA, Praha, 1997, 75-92
[VLFM] J. P. Bowen, Formal Methods, in WWW Virtual Library
Much of this is following in the wake of Dan Connolly's investigations, and so many of the references were provided directly or indictly by Dan.Other pointers from Dan Connolly
[LF] Harper, Honsell, Plotkin "A
Framework for Defining Logics", Journal of the ACM,
January 1993 appears to be the seminal paper on ELF paper
connolly's (attempted) transcription into larch
Pat Hayes, Research Interests and Selected Papers contains stuff on time modelsImpl
Reiter, R., On Closed World Data Bases, in: H. Gallaire and J Minker (eds.), Logic and Data Bases, Plenum, 1978, pp. 55-76. Defines Cloased World Assumption? Ref from CIL
Perlis, D., Languages with Self-Reference I: Foundations. (or: We Can Have Everything in First-Other Logic!). Artificial Intelligence, 25:301-322, 1985.
"Second-order logic is rather difficult to study, since it lacks a decent set of axioms and rules of inference, so we have little to say about it" -- Footnote to preface to [Burris], p.xii
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