Michael Kaminski

Addition requirements for matrix and transposed matrix products

Abstract Let M be an s × t matrix and let M T be the transpose of M . Let x and y be t - and s -dimensional indeterminate column vectors, respectively. We show that any linear algorithm A that computes M x has associated with it a natural dual linear algorithm denoted A T that computes M T y. Furthermore, if M has no zero rows or columns then the number of additions used by A T exceeds the number of additions used by A by exactly s − t . In addition, a strong correspondence is established between linear algorithms that compute the product M x and bilinear algorithms that compute the bilinear form y T Mx .

Context-free languages over infinite alphabets

Abstract. In this paper we introduce context-free grammars and pushdown automata over infinite alphabets. It is shown that a language is generated by a context-free grammar over an infinite alphabet if and only if it is accepted by a pushdown automaton over an infinite alphabet. Also the generated (accepted) languages possess many of the properties of the ordinary context-free languages: decidability, closure properties, etc.. This provides a substantial evidence for considering context-free grammars and pushdown automata over infinite alphabets as a natural extension of the classical ones.

Nonmonotonic default modal logics

Conclusions by failure to prove the opposite are frequently used in reasoning about an incompletely specified world. This naturally leads to logics for default reasoning which, in general, are nonmonotonic, i.e., introducing new facts can invalidate previously made conclusions. Accordingly, a nonmonotonic theory is called (nonmonotonically) degenerate, if adding new axioms does not invalidate already proved theorems. We study nonmonotonic logics based on various sets of defaults and present a necessary and sufficient condition for a nonmonotonic modal theory to be degenerate. In particular, this condition provides several alternative descriptions of degenerate theories. Also we establish some closure properties of sets of defaults defining a nonmonotonic modal logic.

Finite-memory automata

A model of computation dealing with infinite alphabets is proposed. The model is based on replacing the equality test by unification. It appears to be a natural generalization of the classical Rabin-Scott finite-state automata and possesses many of their properties.<<ETX>>

Tree automata over infinite alphabets

A number of models of computation on trees labeled with symbols from an infinite alphabet is considered. We study closure and decision properties of each of the models and compare their computation power.

Regular Expressions for Languages over Infinite Alphabets

In this paper we introduce a notion of a regular expression over infinite alphabets and show that a language is definable by an infinite alphabet regular expression if and only if it is accepted by finite-state unification based automaton - a model of computation that is tightly related to other models of automata over infinite alphabets.

LR(0) conjunctive grammars and deterministic synchronized alternating pushdown automata

In this paper we introduce a sub-family of synchronized alternating pushdown automata, Deterministic Synchronized Alternating Pushdown Automata, and a sub-family of conjunctive grammars, LR(0) Conjunctive Grammars. We prove that deterministic SAPDA and LR(0) conjunctive grammars have the same recognition/generation power, analogously to the classical equivalence between acceptance by empty stack of deterministic PDA and LR(0) grammars. These models form the theoretical basis for efficient, linear, parsing of a rich sub-family of conjunctive languages, which properly includes all the boolean combinations of context-free LR(0) languages.

An algebraic characterization of deterministic regular languages over infinite alphabets

We state and prove an infinite alphabet counterpart of the classical Myhill?Nerode theorem.

Commutation-augmented pregroup grammars and mildly context-sensitive languages

The paper presents a generalization of pregroup, by which a freely-generated pregroup is augmented with a finite set of commuting inequations, allowing limited commutativity and cancelability. It is shown that grammars based on the commutation-augmented pregroups generate mildly context-sensitive languages. A version of Lambek’s switching lemma is established for these pregroups. Polynomial parsability and semilinearity are shown for languages generated by these grammars.

A note on the stable model semantics for logic programs

Abstract The stable model semantics for logic programs is extended from ground literals onto open literals by augmenting the program language with an infinite set of new constants. This, in turn, leads to a natural translation of logic programs into open default theories.