McAllester, D. and Givan, R.: Natural Language Syntax and First-Order Inference. Artificial Intelligence. 56 (1992)1--20  Home/Search   Document Details and Download   Summary   Related Articles   Check

....unrestricted natural language, typically do not use first order logic but richer alternatives, and try to perform one step inferences by introducing specialised inference rules that should closely mimic informal human reasoning in natural language. The Montagovian Syntax of McAllester Givan [McAllester Givan 1992] is the system that Iwanska Shapiro [Iwanska Shapiro 2000, p. 3] consider closest to 2 natural language as knowledge representation and reasoning language. In the Montagovian Syntax the sentence Every dog ate a bone. is represented as the quantifier free formula (every dog (ate (some ....

D. McAllester & R. Givan, Natural Language Syntax and First Order Inference, Artificial Intelligence vol. 56, pp. 1-20, 1992; reprinted in

.... specification, following a line of work described in Macias and Pulman 1995 [3] and Pulman 1996 [5] The present work differs from its antecedents in several ways, the most obvious being that the target representation language is a natural language oriented logic developed by McAnester [2], 4] which we call McLogic) We describe extensions to McLogic designed to allow for greater expressivity in the dialect of CPE used; an inference system for McLogic; and two experimental applications: the first is a well known ex ample from the Z programme specification literature ( Wing s ....

....mistake) in a reasonable time, by using a representation language that is expressive and at the same time computationally ecient. The subset U is a dialect of Computer Processable English (CPE) 5] The representation language, which we call McLogic, is an extension of the one defined in [2], 4] This is a system for first order logic (with one or two higher order extensions) which for our purposes has two possible advantages: firsfly, the syntax is relatively English like, making for transparency in the process of formulating and reasoning with specifications; and secondly, it has ....

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McAllester D. and Civan R. Natural language syntax and first-order inference. Arti- ficial Intelligence, 56:1 20, 1992.

....connectives and V. The rule set B expresses some, but not all, of the inferential properties of these connectives. The rule set B can be viewed as a (somewhat obscure) characterization of unit resolution, or as a specification of the Boolean constraint propagation mechanism described in [McAllester, 1989]. The inference relation generated by these rules is linear time decidable. Yet, if the above inference rules are augmented by a simple case analysis sequent rule then the rules become complete for Boolean inference. As another example of a set of inference rules, consider the following rules for ....

....involve quantification. Even without quantifiers, a set of rules can still generate an undecidable or intractable inference relation. On the other hand, the presence of quantifiers does not necessarily prevent tractability. Tractable inference relations involving quan tiffcation are discussed in [McAllester, 1989] and [McAllester et al. 1989] A more general notion of locality will be needed to construct a procedure for automatically recognizing tractability in rule sets that involve quantification. Definition: A well formed expression of kind formula will be called a formula. Definition: An inference ....

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D. McAllester and R. Givan. Natural language syntax and first order inference. Memo 1176, MIT Artificial Intelligence Laboratory, October 1989. 33

....basic definitions. Propositional logic is another example of a language in which reasoning is very efficient. In propositional logic, there is a decision procedure. But, in propositional logic, it is impossible even to express the notion of every or some in a general way. David McAllester [58, 59, 86, 87] has developed languages that are more expressive than first order logic and for which the subset of the language for which there is a decision procedure properly contains the subset of first order logic for which there is a decision procedure. This is a great advantage. The disadvantages include ....

David McAllester and Robert Givan. Natural language syntax and first order inference. Artificial Intelligence, 56:1--20, 1992.

....require large ontologies, as is typically the case in natural language technology. Description Logics (DLs) are logics that are built to deal with huge ontologies. Many DLs can be embedded in GF [16, 28] As a consequence, DLs are less expressive than full first order logics. McAllester and Givan [22] consider a fragment of Montague Semantics that can be expressed in a DL. Formulas belonging to this fragment have to be quantifier free, meaning that they do not contain any lambda abstractions. For instance, 4.b) which is the semantic representation of (4.a) belongs to the fragment, but (5.b) ....

....of (4.a) belongs to the fragment, but (5.b) representing (5.a) does not. 4) a. Mary read a book. b. Mary read ( some book ) 5) a. Mary read a book that John bought. b. Mary read ( some ( #x ( x book # ( John ( bought x ) For this quantifier free fragment, the authors of [22] provide an inference procedure which decides satisfiability of a set of formulas in polynomial time. Of course, the examples in (4) and (5) also illustrate that, despite their inferential advantages, the expressive power of DLs is much too weak to be used for an exhaustive representation of ....

D. McAllester and R. Givan. Natural language syntax and first-order inference. Artificial Intelligence, 56(1):1--20, 1992.

....in most English grammars, then Purdy s system would require translation procedures that are not simpler than those needed into the notation of the standard predicate calculus. Consequently, it is hard to see how Purdy s system can deal with common phenomena like NP coordination in an elegant way. McAllester and Givan (1992) propose an inference system that is sound and complete, and moreover decidable in polynomial time. The system is based on so called class expressions expressions that denote sets. A determiner like every can combine with two class expressions to form an atomic formula. Alternatively, a ....

McAllester, D. A. and Givan, R. (1992). Natural language syntax and first-order inference. Artificial Intelligence, 56:1--20.

....hundred msecs. In this case at least, using a higherorder approach does not automatically lead to a penalty in efficiency. 9 More generally, it is well known in the automated reasoning community that higher order specifications often allow more efficient forms of inference than first order ones [14]. In particular, Kimba is able to quickly compute solutions for combinatorial problems whose standard first order formalizations are not solvable within a reasonable time limit by many state of the art first order theorem provers [12] ....

D. McAllester and R. Givan. Natural language syntax and first order inference. Artificial Intelligence, 56:1--20, 1993.

....This class is of particular interest because (linear time) unit propagation (BCP) is complete for it [Henschen and Wos, 1974; Lewis, 1978] In additional to the clausal target languages we have concentrated on in this paper, one could consider tractable non clausal target languages. For example, McAllester and Given [1992] discuss a logical language based on the structure of natural language, and identify a fragment in which inference can be performed in polynomial time. It would be interesting to see if ordinary first order theories could be (approximately) compiled into this tractable fragment, using our ....

McAllester, D.A. and Givan, R. Natural language syntax and first-order inference. Artificial Intelligence, 56, 1--20. 46

....for the generation process. We have been able to identify some of the additional knowledge that is not necessary for the reasoning process, and hence not included in the formal proof, but that is critical for communication. We have shown how this information can be extracted from the proof. As McAllester and Givan (1992) noted, this information a#ects the ease of understanding mathematical arguments. We intend to explore the question of what additional information is needed for understanding further. We identified that the advantages obtained by using a tactic style theorem prover are balanced by some ....

....form for expressions when logically equivalent options are available; for example, chosing between the statement that A implies that B implies C and A and B implies C . This will require the ability to determine which form is most useful to the reader given subsequent uses of the expression (McAllester Givan 1992). It will also require the ability to create alternate forms of expressions in general cases, something we would be able to do by querying the theorem prover. Additionally, we want to verify that our approach a#ords easy extension to generation from proofs in other domains of mathematics. ....

McAllester, D. A., and Givan, R. 1992. Natural language syntax and first-order inference. Artificial Intelligence 56(1):1--20.

....is highly sensitive to the expressiveness of the description logic, and complete polynomial time algorithms are rare. McAllester and others have defined languages in the context of automated reasoning which are not description logics, but which have some overlap with them (as remarked in [MG92]) These languages are interesting in the context of description logics since they have polynomial time algorithms for solving a subsumption like problem. Two of McAllester s languages are examined, one with a firstorder taxonomic syntax [MGF89, MG93] and the other with a Montagovian syntax ....

....[MG92] These languages are interesting in the context of description logics since they have polynomial time algorithms for solving a subsumption like problem. Two of McAllester s languages are examined, one with a firstorder taxonomic syntax [MGF89, MG93] and the other with a Montagovian syntax [GMS91, MG92]. Both languages are equivalent to first order logic. Taxonomic syntax consists of taxonomic terms and formulas, where taxonomic terms denote sets of objects (just as concepts do) and formulas state relationships between terms (just as terminological axioms and A box assertions do) For example, ....

D. McAllester and R. Givan. Natural language syntax and first-order inference. Artificial Intelligence, 56:1-- 20, 1992.

....disambiguation that semantic processing would make possible would be a valuable aid in eliminating unintended interpretations. For some applications, it is also possible that knowledge representation and reasoning carried out directly in a restricted natural language would be possible: see e.g. McAllester and Givan, 1992 ] Natural language generation is an area of practical application that is currently under exploited. There are many domains where a natural language rendering of information represented in a non linguistic form would be of practical value. Some examples are: verbal summaries of tables of ....

McAllester, D. and Givan, R. 1992. Natural language syntax and first order inference. Artificial Intelligence 56:1--20.

....where the conclusion is the main inclusion and the rest of inclusions in the diagram (memberships) are the conditions. As we show below, these conditional inclusions are slightly more expressive than conventional Horn clauses. We could cover most of FOL by an unrestricted use of inclusions [22]. However, in this paper our interest is not to show the full expressive power of inclusions but to capture with them the core of computational logic. john mary peter parent (a) parent john eric diana (b) parent mary ann bob (c) ancestor parent (d) parent ancestor ancestor (e) descendant ....

....innovative ideas on visual environments are introduced. As shown in Section 2, the intuitive interpretation of diagrams is based on sets and set inclusion. Set terms and set term inclusions provide an alternative syntax for FOL that is very near to the taxonomic and Montagovian syntax for FOL in [22] and [23] respectively. This non standard syntax for FOL possesses benefits other than facilitating visualization as shown here. As McAllester claims in [22, 23] it has a closer connection with natural language than conventional FOL syntax and it may make first order inference more efficient. 5 ....

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David McAllester and Robert Givan. Natural language syntax and first-order inference. Artificial Intelligence, 56:1--20, 1992.

....of complex type) In principle, one could represent natural language in Ontic; however, the type system would have to be enriched, and it would still suffer from the disadvantages outlined for KRYPTON. In later work, McAllester has addressed natural language issues explicitly [ Givan et al. 1991; McAllester and Givan, 1992 ] They argue that the syntactic form of language may be a source of inferentially powerful syntax, corresponding to our notion of the natural form constraint. In this later work, McAllester and Givan argue for the utility of nonstandard logical syntax as an aid to automated inference. They make ....

David A. McAllester and Robert Givan. Natural Language Syntax and First-order Inference. Artificial Intelligence, 56(1):1--20, 1992.

....algorithm takes O(n 3 ) time on a sequential RAM. The negative implications of these results for taxonomic and congruence closure apply to the more powerful taxonomic inference system of McAllester, et al., and to the later extension of this system to Montague literals by McAllester and Givan [9]. I will discuss the significance of these results for engineering applications and for McAllester and Givan s speculation that the decision procedure for Montague literals might explain some aspects of natural language. A Taxonomic Inference System The system for taxonomic inference that I will ....

....closure decision problem, in which only function applications with a single argument are permitted. It follows that congruence closure, taxonomic closure, the taxonomic inference system of McAllester, et al. [10] and the problem of inference with Montague literals of McAllester and Givan [9] are all P complete, even if only monadic function application is permitted (as is always the case with Montague literals) 2 It was seen by him as the word problem for a finitely presented algebra Theorem 5 The monadic congruence closure decision problem is complete for P under log space ....

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McAllester, D. and Givan, B. (1989) Natural language syntax and first order inference, Massachusetts Institute of Technology AI memo no. 1176.

....terms (which may be variables of complex type) In principle, one could represent natural language in Ontic; however, the type system would have to be enriched, and it would still suffer from the disadvantages outlined for KRYPTON. In later work, McAllester has addressed natural language issues [ 19, 30 ] similar this work, particularly the natural form issue. 5 The Knowledge Representation Formalism 5.1 Syntax and Semantics of the Formalism In this section, we provide a syntax and semantics of a logic whose variables are not atomic and have structure. We call these variables structured ....

David A. McAllester and Robert Givan. Natural language syntax and first-order inference. Artificial Intelligence, 56(1):1 -- 20, 1992.

....infers a strict superset of the theorems the other infers. we are especially interested in inferring properties that require expressive representation to state. The answer to this question proves very sensitive to the representation system used in inference. We draw on our previous work with McAllester(1993; 1992) to select a representation for program properties amenable to rapid inference. This representation derives from viewing properties as sets of program values a program has a property if the program returns a value in the corresponding set of values. Taking this view, program properties are ....

.... nil (if x: car l) cdr l) cons (car l) delete x (cdr l) Figure 1: Example program definitions. We omit simple definitions which have no effect on the example inferences claimed later (length, neighbors) Property Expressions Our property language is derived from our previous work with McAllester(1992) on natural language syntax and its relationship to tractable inference. The representational features introduced in that work have never before been exploited in an automated reasoning system. Our property language is essentially the programming language extended by some new constructs that allow ....

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McAllester, D., and Givan, R. 1992. Natural language syntax and first order inference. Artificial Intelligence 56:1--20. ftp.ai.mit.edu:/pub/users/dam/aij1.ps.

....assigned. The relationship between inference and constraint propagation can be made explicit by characterizing constraint propagation processes in terms of inference rules. Boolean constraint propagation can be defined in terms of a certain (incomplete) set of inference rules for Boolean logic [ McAllester, 1989 ] Van Hentenryck also defines the various constraint propagation techniques used in Chip in terms of rules of inference. In fact, virtually any form of constraint propagation can be defined in terms of rules of inference. Constraint propagation inference rules are unusual, as rules of ....

....that do not correspond to any standard constraint propagation technique. For example, the inference rules that define BCP can be combined with the standard inference rules for equality, including the substitution of equals for equals, and the resulting rule set is still polynomial time decidable [ McAllester, 1989 ] In [ McAllester et al. 1989 ] it is argued that the power of tractable rule sets for first order inference is sensitive to the syntax in which formulas are expressed an alternative syntax based on taxonomic relationships between classes yields a more powerful tractable rule set. In [ ....

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D. McAllester and R. Givan. Natural language syntax and first order inference. Memo 1176, MIT Artificial Intelligence Laboratory, October 1989.

....language. In particular the set expression R(every C) is taken to be the set fx : 8y 2 C R(x; y)g. This provides a natural meaning for English verb phrases such as contains every prime number . One simple but expressive Montagovian concept language has a polynomial time satisfiability problem [McAllester and Givan, 1992]. A third class of set formalisms consists of modal and temporal logics. These logics involve formulas which are true or false of possible worlds in Kripke structures. However, it is natural to view a Kripke structure as an ordinary first order structure whose domain is the set of possible ....

....operations respectively. Once we allow an arbitrary lattice (rather than require a complemented distributive lattice) and only require that relations denote monotone operations on lattice elements, then the positive entailment problems for L(3; is decidable in polynomial time [Givan and McAllester, 1992]. Theorem: The positive entailment problem for L(2; 3; is EXPTIME hard. Proof: The proof is by reduction of the acceptance problem for linear space bounded alternating Turing machines. In an alternating Turing machine the states are classified into universal and existential states and for ....

D. McAllester and R. Givan. Natural language syntax and first order inference. Artificial Intelligence, 56:1--20, 1992. internet file ftp.ai.mit.edu:/pub/dam/aij1.ps. 42

....and Montague grammar for natural language. In particular the set expression is taken to be the set . This provides a natural meaning for English verb phrases such as contains every prime number. One simple but expressive Montagovian concept language has a polynomial time satisfiability problem (McAllester and Givan, 1992). Tarskian set expressions have been studied by J nnson and Tarski in the framework of Boolean algebra with operations (J nnson and Tarski, 1951) J nnson and Tarski, 1952) In the work of J nnson and Tarski the operation f in the expression actually denotes a relation on arguments. More ....

....their computational properties have not been widely studied. It is shown in (McAllester and Givan, 1993) that satisfiability of nonrecursive Tarskian set constraints not involving Boolean operations is decidable in cubic time (assuming unit time hash table operations) It is shown in (Givan and McAllester, 1992) that satisfiability of constraints on expressions involving meets, joins, and monotone applications in an arbitrary lattice is similarly decidable in cubic time. The results of this paper are summarized in the table below. We categorize Tarskian set constraint satisfiability problems by the ....

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McAllester and Givan, 1992 [26] McAllester, D. and Givan, R. (1992), Natural language syntax and first order inference, Artificial Intelligence, 56, 1--20. Also available as internet file http://www.ece.purdue.edu/~givan/papers/aij1.ps.

....the reasoning process to be handled by efficient type inference mechanisms. The use of taxonomic syntax is part of a larger program of basing inference procedures on nonstandard syntactic constructions. A syntax based on natural language under Montague semantics seems to be particularly effective [5], 4] 2] The Montagovian syntax for first order logic is not discussed here. Ontic is best viewed as an extension of pure Lisp. Nondeterminism is introduced through the use of McCarthy s amb operator [8] which is renamed in Ontic to either. From an operational viewpoint, when evaluating an ....

D. McAllester and R. Givan. Natural language syntax and first order inference. Artificial Intelligence, 56:1--20, 1992.

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McAllester, D. and Givan, R.: Natural Language Syntax and First-Order Inference. Artificial Intelligence. 56 (1992)1--20

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D. A. McAllester and R. Givan. Natural language syntax and first-order inference. Artificial Intelligence, 56:1--20, 1992.

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David A. McAllester and Robert Givan. Natural language syntax and first-order inference. Artificial Intelligence, 56:1--20, 1992.

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D.A. McAllester and R. Givan. Natural language syntax and first-order inference. Artificial Intelligence, 56(1):1--20, 1992.

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McAllester, David and Robert Givan. 1989a. Natural language syntax and first order inference. A.I. Memo No. 1176, Massachusetts Institute of Technology, Cambridge, Massachusetts, October.

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